Old-Keynesianism in the New Keynesian Model

Abstract

I investigate an ‘old Keynesian’ fiscal policy in which government spending endogenously responds to inflation and the output gap when the nominal interest rate is pegged at the ZLB. I do so within a standard Representative Agent New Keynesian model (RANK), as well as a two-period Overlapping Generations New Keynesian model (OLG-NK). For both model versions, I find that the equilibrium values for inflation and the output gap under a standard Taylor rule regime can be replicated under the ‘old Keynesian’ regime. However, a unique stable countercyclical equilibrium is only feasible within the OLG-NK model and crucially depends on the pension system.

1 Introduction

The regular New Keynesian model is typically closed by formulating a Taylor rule in which the nominal interest rate endogenously responds to inflation and the output gap (Woodford (2003) and subsequent literature). However, such a rule does not provide a realistic description once the economy lands at the Zero Lower Bound (ZLB), in which case unconventional monetary policies and fiscal policy are typically employed for macroeconomic stabilization. Such a situation particularly applies to advanced economies that spent many years at the ZLB after the Great Financial Crisis of 2007–2009, a period during which nominal policy rates were expected to remain there for many years or even decades to come.

Therefore, I explore an alternative ‘old Keynesian’ fiscal policy in which government spending endogenously responds to inflation and the output gap. I do so within the class of otherwise standard New Keynesian models and investigate whether such a government spending rule can stabilize the macroeconomy when the nominal interest rate is permanently pegged at the ZLB.¹ Within such a framework, several more specific questions arise: is it possible to replicate the equilibrium that would arise under an active Taylor rule through an appropriate choice of the government spending rule? What type of spending rules generate a unique stable equilibrium? Does the answer to this question depend on whether I employ a standard representative agent version of the New Keynesian model or an overlapping generations version?

The main contribution of this paper is that it is the first (to the best of my knowledge) in which government spending endogenously responds to inflation and the output gap, in similar fashion to the nominal interest rate endogenously responding to inflation and the output gap when the economy is away from the ZLB (Taylor, 1993). Since such a spending rule is based on variables that are directly observable, such a fiscal policy could actually be implemented by policymakers. Therefore, this policy sharply contrasts with the rest of the literature, in which fiscal policy rules are either expressed in terms of exogenous shocks and deep parameters that are hard to observe in real time (Correia et al., 2013), or where the level of government spending differs between regimes (ZLB vs no ZLB), but is constant within a regime (Christiano et al., 2011; Eggertsson, 2011; Woodford, 2011; Eggertsson and Krugman, 2012).

Specifically, I employ the standard Representative Agent New Keynesian (RANK) model, as well as a two-period Overlapping Generations New Keynesian (OLG-NK) model. Both model versions feature pricing rigidities a la Calvo (1983) and government spending that is financed by issuing one-period bonds and by levying lump sum taxes. The OLG-NK model is closest in spirit to Galí (2014), and features a young generation that consumes, provides labor, pays lump sum taxes, and saves through government bonds, while the old generation uses gross repayment of bonds and a government transfer to pay for consumption and lump sum taxes. For analytical tractability, I set lump sum taxes on the old equal to their gross interest payments on bonds, and thereby eliminate government debt as a state variable. As a result, the OLG-NK model also features Ricardian equivalence. The production side of the economy and the government budget constraint turn out to be equivalent under both models, except for the government transfer to the old generation in the OLG-NK model. Both model versions do not feature physical capital.

I consider two rules for the government transfer to the old. Under the first, the transfer is linear in the difference between output and government spending, reflecting the fact that increases in government spending (to stabilize the economy) might come at the expense of pensioners. Under the second rule, the transfer is linear in output, thereby assuming that pensioners will not be affected directly by the stabilization policy of the government. It turns out that the linearized version of the first OLG-NK model (with the transfer linear in the difference between output and government spending) is exactly equal to the linearized version of the RANK model.

For all models, I distinguish between the familiar ‘monetary’ regime in which government spending is constant and the nominal interest rate follows an active Taylor rule on the one hand (Taylor 1993; Woodford 2003 and subsequent literature), and a ‘fiscal’ or ‘old Keynesian’ regime in which the nominal interest rate is pegged at the ZLB and government spending endogenously responds to inflation and the output gap on the other. The linearized version of both models can be reduced to the familiar New Keynesian Phillips curve and an aggregate demand equation which relates the (expected) output gap to the return difference between the expected real rate and the natural rate of interest, the market clearing interest rate under perfectly flexible prices (Woodford, 2000). The channel through which endogenous government spending affects the equilibrium is by changing the natural rate of interest via its impact on the endogenous component from government spending $\hat{R}_{t}^{g,*}$. Crucially, the natural rate increases within the RANK model with the difference between today’s and tomorrow’s expected government spending, whereas it increases with the (weighted) sum of today’s and tomorrow’s expected government spending in the second OLG-NK model (the model version with a transfer that is linear in output).

My first result is to show that for both the RANK and the OLG-NK model the aggregate demand equation under the fiscal regime can be obtained by replacing the nominal interest rate $\hat{R}_{t}^{n}$ under the monetary regime by the negative of the endogenous natural rate component $\hat{R}_{t}^{g,*}$. Specifically, the difference between the expected real rate and the natural rate is equal to $\hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \hat{R}_{t}^{z,*}$ under the monetary regime, while it is equal to $- \hat{R}_{t}^{g,*} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \hat{R}_{t}^{z,*}$ under the fiscal regime (where $\hat{R}_{t}^{z,*}$ denotes the exogenous component of the natural rate). As a result, the equilibrium values of the output gap and inflation that arise under the monetary regime can be replicated under the fiscal regime through appropriate choice of the feedback coefficients of inflation and the output gap on government spending. The resulting government spending rule turns out to be countercyclical in inflation and the output gap: the government reduces aggregate demand by reducing spending when inflation and the output gap are positive and vice versa, just as the central bank reduces aggregate demand by raising the nominal and real interest rate under the monetary regime. Therefore, this result shows that being permanently stuck at the ZLB (in expectation) does not prevent the government from achieving the equilibrium values of the output gap and inflation that would arise when conventional monetary policy can be employed.

However, whereas the resulting countercyclical government spending rule is consistent with a unique stable equilibrium within the second OLG-NK model (the model with a transfer linear in output), it turns out that this is not the case within the first OLG-NK model and the RANK model. Therefore, the specific form of the pension system (the transfer rule) is crucial for the stability analysis in the OLG-NK model. To explain the intuition behind these results, let us first consider the countercyclical monetary regime: in that case, the impact from expected inflation $E_{t}\left[ \hat{\pi }_{t+1}\right]$ on the expected real rate $\hat{r}_{t}\equiv \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right]$ works in the opposite direction as the impact from the nominal interest rate $\hat{R}_{t}^{n}$ on the expected real rate $\hat{r}_{t}$. For example, a shock that increases (expected) inflation increases the nominal interest $\hat{R}_{t}^{n}$ on the one hand, which increases the (expected) real rate, everything else equal. However, the accompanying increase in expected inflation $E_{t}\left[ \pi _{t+1}\right]$ simultaneously decreases the (expected) real rate on the other hand, everything else equal. Therefore, a unique stable countercyclical equilibrium is only feasible if the nominal interest rate responds sufficiently strong to more than offset the impact that expected inflation has on the expected real interest rate. This is achieved when the monetary policy rule satisfies the Taylor principle (Taylor, 1993). Similarly, the endogenous component of the natural rate $\hat{R}_{t}^{g,*}$ has to respond sufficiently strong to changes in government spending under the fiscal regime in order for a unique stable countercyclical equilibrium to exist. This is only the case in the second OLG-NK model, where the endogenous natural rate increases with the weighted sum of today’s and tomorrow’s expected government spending, while it is impossible in the RANK model (and the first OLG-NK model) as the endogenous natural rate increases with the difference between today’s and tomorrow’s expected government spending.

Although I restrict my attention to the RANK model and the OLG-NK model, my analysis suggests that the intuition behind the (non-) existence of a unique stable countercyclical spending equilibrium can be applied more generally to any linearized model with a New Keynesian Phillips curve and an aggregate demand equation. Specifically, it suggests that a unique stable countercyclical equilibrium will only be feasible under the fiscal regime in models where the endogenous natural rate $\hat{R}_{t}^{g,*}$ increases sufficiently strong with today’s and tomorrow’s expected government spending.

Finally, these results highlight that employing countercyclical endogenous government spending in an economy that is expected to remain at the ZLB for many years to come is not straightforward, as the workhorse model for policy analysis, the RANK model, predicts that such a policy will feature multiple equilibria. That would be the last thing governments need, as the goal of countercyclical fiscal policy is typically to stabilize the economy. Therefore, at the very least more research is needed if governments were to consider such a policy. Alternative policy options are to employ endogenous unconventional monetary policies at the ZLB, see Sims and Wu (2021) for an analysis within the RANK model.

Literature review

First of all my paper is related to the classic IS-LM literature that started with Hicks (1937) after publication of John Maynard Keynes’ General Theory (Keynes, 1936). This framework encompasses the recommendation of Keynes that fiscal policy should be expansionary in recessions to mitigate the drop in GDP, while it should be contractionary in booms (Keynes, 1936). My model also employs fiscal policy for macroeconomic stabilization, but all stabilization is performed through changes in government spending, as Ricardian equivalence prevents government deficits from affecting the equilibrium in both the RANK model and the OLG-NK model.

Although the primary instrument for macroeconomic stabilization within the standard New Keynesian model is the nominal interest rate, this instrument is no longer available when the economy hits the ZLB like in the Great Financial Crisis of 2007–2009. In response, governments around the world resorted to fiscal policy to provide additional macroeconomic stimulus. This has inspired a whole new strand within the New Keynesian literature in which government spending is increased for as long as the economy is at the ZLB (Christiano et al., 2011; Eggertsson, 2011; Woodford, 2011; Eggertsson and Krugman, 2012). These papers differ in two respects from my paper. First, the economy is only temporarily at the ZLB, and eventually returns to a regime in which monetary policy regains full potency. Second, the level of government spending depends on the regime (ZLB vs. no ZLB) but is exogenous within a particular regime.

Eggertsson et al. (2019) explicitly model how an economy can be permanently at the ZLB as a result of secular stagnation, but do not investigate whether fiscal policy can replicate the equilibrium values for inflation and the output gap that would arise under a monetary regime that allows for negative nominal interest rates. This contrasts with Correia et al. (2013), who show within the standard New Keynesian model that fiscal policy at the ZLB can replicate the equilibrium allocation under a monetary regime with negative nominal interest rates. My paper differs in two dimensions. First, I employ government spending rather than distortionary taxes to achieve the equilibrium values for inflation and the output gap that would arise under a monetary regime with negative nominal interest rates. Second, and more importantly, my fiscal rule depends on endogenous variables that are directly observable to policymakers, whereas the fiscal policy rules of Correia et al. (2013) are expressed in terms of exogenous shocks and deep parameters of the model.

A problem with the RANK model is that it features indeterminacy issues at the (temporarily binding) ZLB (Cochrane, 2017). Within heterogeneous agents models such as Hagedorn (2016) and Hagedorn (2018), this problem is eliminated by specifying fiscal policy in nominal sequences for government spending, government debt, and taxes. As a result, the present value government budget constraint is satisfied at all times and for any price level, which in turn adjusts until demand and supply in the goods market and asset market are equalized. Therefore, the indeterminacy of the price level and inflation when monetary policy is implemented through an interest rate target (Sargent and Wallace, 1975) is eliminated.

Leeper (1991) identifies under which monetary and fiscal policies a unique stable equilibrium is feasible within a stochastic representative agent model. He finds that when the nominal interest rate is pegged to its steady state value, fiscal policy must be active in the sense that the feedback from government debt to lump sum taxes does not respond strongly, or not at all, as in the fiscal theory of the price level (Sims, 1994; Woodford, 1995; Cochrane, 1999). Lump sum taxes in my model, however, abide by the Bohn (1998) condition, and are therefore passive in the terminology of Leeper (1991). The reason why unique stable equilibria are still possible is the fact that government spending is endogenous, unlike the constant real spending in Leeper (1991). Therefore, fiscal policy can be considered active, as the fiscal authority does not take the state of government debt into account when determining how much to spend.

There is also a literature which studies the effects of fiscal policy within endogenous growth models (Barro, 1990; Turnovsky, 1996; Turnovsky, 2000; Agénor, 2008; Barseghyan and Battaglini, 2016). Fiscal policy is endogenous in the sense that government spending depends on the amount of taxes levied, which in turn depends on aggregate production (Barro, 1990; Agénor, 2008), or on an explicit modeling of the legislative bargaining process (Barseghyan and Battaglini, 2016). The focus of most of these papers is on optimal fiscal policy, the fiscal policy that maximizes long-run growth (Barro, 1990; Turnovsky, 1996; Turnovsky, 2000; Agénor, 2008). One exception is Chari et al. (1994), who study optimal fiscal policy within a business cycle model, and determine the optimal tax rate on capital and labor by solving the Ramsey problem. Just as Chari et al. (1994), I focus on business cycle dynamics rather than long-run dynamics, but I refrain from looking at optimal fiscal policy.

My paper is also related to the literature with overlapping generation models, which started with Samuelson (1958). My overlapping generations model is closest to Galí (2014), in which there is price stickiness as well. Galí (2014), however, differs in four important dimensions. First, Galí (2014) features a bubbly asset. Second, production firms operate for two periods, while my firms are infinitely lived to keep the model as comparable with the RANK model as possible. Third, labor supply is inelastic, while it is endogenous in my setup. Fourth, there is no government spending.

Finally, my paper is also related to McKay et al. (2017), as my overlapping generations model also features different coefficients in front of the current and the future expected output gap. However, whereas the anticipated future output gap is discounted with respect to the current output gap in McKay et al. (2017), I find that the current output gap is discounted with respect to the future anticipated output gap. These different results arise from different microfoundations, as the linearized model of McKay et al. (2017) arises from a simplified heterogeneous agents New Keynesian model with infinitely-lived agents subject to idiosyncratic income risk, while my linearized model equations arise from an overlapping generations model.

I describe the model in Sect. 2, and establish analytical results in Sect. 3. I present numerical simulations in Sect. 4, and conclude in Sect. 5.

2 Model

As the RANK model and its derivations are by now standard in the literature, I refer the interested reader to Appendix B for the full description of the model, and immediately present the linearized equations below.

The overlapping generations model is inspired by Galí (2014). Specifically, a generation lives for two periods, the size of which has mass one and is constant across time. Each member of a generation has identical preferences, and is referred to as “the young” in the first period of existence, and as “the old” in the second period of existence. The young receive income from providing endogenous labor and ownership of all production firms, which they spend on consumption, lump sum taxes, and government bonds.² The old receive a transfer from the government and the gross repayment of government bonds, which they spend on consumption and lump sum taxes. I consider two rules for the government transfer to the old. Under the first, the transfer is linear in the difference between output and government spending, while it is linear in output alone under the second rule. Lump sum taxes are levied on both the young and the old, with lump sum taxes on the old equal to the gross interest payments on their government bonds (so they are in effect financing their own repayment of bond holdings), while lump sum taxes on the young respond to the stock of previous period government debt, thereby satisfying the Bohn (1998) principle.³ The government budget constraint is the same as in the RANK-model, except that the government makes a transfer to the old. The production sector is identical to that in the RANK model, except that ownership is in the hands of the young and transferred to the next generation when the young turn old. Therefore, production firms discount future profits using a stochastic discount factor that features the marginal utility of future young generations.

Both the RANK and the OLG-NK model do not feature physical capital to keep the models analytically tractable. Unless otherwise stated, the only exogenous shock in the main text is a productivity shock that follows a regular AR(1) process. A full specification of both models can be found in Appendix B and C.

2.1 The Representative Agent New Keynesian (RANK) Model

I start by linearizing the standard representative agent New Keynesian model in Appendix B, which can eventually be described by two familiar equations. These are the New Keynesian Phillips curve and the aggregated Euler equation which I will refer to as the aggregate demand equation:

$$\begin{aligned} \hat{\pi }_{t}= & {} \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t}, \end{aligned}$$

(1)

$$\begin{aligned} \sigma \left( \bar{y}/\bar{c}\right) \tilde{y}_{t}= & {} \sigma \left( \bar{y}/\bar{c}\right) E_{t}\left[ \tilde{y}_{t+1}\right] - \left( \hat{r}_{t} - \hat{R}_{t}^{*}\right) , \end{aligned}$$

(2)

where $\hat{x}_{t}$ denotes the percentage deviation of variable $x_{t}$ from its steady state. $\tilde{y}_{t}\equiv \hat{y}_{t}-\hat{y}_{t}^{n}$ denotes the output gap, which is the difference between output under the New Keynesian model $\hat{y}_{t}$ and output under perfectly flexible prices $\hat{y}_{t}^{n}$. $\sigma$ denotes households’ coefficient of relative risk aversion, and $\bar{y}$ and $\bar{c}$ denote the steady state level of output and consumption, respectively. The factor $\bar{y}/\bar{c}$ arises because output is not only absorbed by consumption, but also by government spending. $\hat{r}_{t}\equiv \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right]$ is the expected real interest rate, where $\hat{R}_{t}^{n}$ denotes the percentage deviation of the gross nominal interest rate from its steady state value, while $\hat{R}_{t}^{*}$ denotes the natural rate of interest (Woodford, 2000), which can be decomposed into two components:

$$\begin{aligned} \hat{R}_{t}^{*} = \hat{R}_{t}^ {z,*} + \hat{R}_{t}^ {g,*}. \end{aligned}$$

(3)

These two components are given by:

$$\begin{aligned} \hat{R}_{t}^ {z,*}= & {} - \sigma \left( \bar{y}/\bar{c}\right) \left( \frac{1+\varphi }{\sigma \left( \bar{y}/\bar{c}\right) +\varphi }\right) \left( 1-\rho _{z}\right) \hat{z}_{t}, \end{aligned}$$

(4)

$$\begin{aligned} \hat{R}_{t}^ {g,*} = \sigma \left( \bar{y}/\bar{c}\right) \left( \frac{\left( \bar{g}/\bar{y}\right) \varphi }{\sigma \left( \bar{y}/\bar{c}\right) +\varphi }\right) \left( \hat{g}_{t}-E_{t}\left[ \hat{g}_{t+1}\right] \right) , \end{aligned}$$

(5)

where $\hat{z}_{t}$ denotes the exogenous productivity shock, $\varphi$ the inverse Frisch elasticity, and $\bar{g}$ the steady state level of government spending. The term $\hat{g}_{t}-E_{t}\left[ \hat{g}_{t+1}\right]$ arises from the fact that the aggregate demand equation is derived from the households’ Euler equation, which features today’s and tomorrow’s expected consumption. Through substitution of the (linearized version of the) aggregate resource constraint $y_{t}=c_{t}+g_{t}$, these terms introduce today’s and tomorrow’s expected government spending on opposite sides of the equality sign.

A key observation is that the natural rate of interest is no longer exogenous when government spending endogenously responds to inflation and the output gap, as will be the case below. In fact, changing the natural rate of interest is the key channel through which government spending affects the equilibrium of the economy (1)–(2), as government spending does not show up at other places in these equations. This marks a key difference with the textbook case, in which macroeconomic stabilization is performed through adjustment of the nominal interest rate (Woodford, 2003; Galí, 2015).

Before I continue, I discuss the intuition behind the above expressions for the natural rate of interest, where we remember that the natural rate is the equilibrium interest rate in a model with perfectly flexible prices. We see from Eq. (4) that a temporary positive productivity shock reduces the natural rate of interest (assuming constant government spending). Given an AR(1) process for productivity, a positive shock implies that productivity will be higher today than tomorrow. As such, households know that today’s income will be higher than tomorrow’s, everything else equal. To smooth consumption over time, households would like to save part of the additional income that the positive productivity shock generates today by buying additional government bonds. However, the supply of bonds does not increase, while the supply of final goods increases as a result of the productivity shock. The only way to clear both the bond market and the goods market is through a fall in the equilibrium interest rate (Walsh, 2010).

Next, we see from Eq. (5) that a positive government spending shock increases the natural rate of interest. Higher government spending increases the demand for final goods as well as the supply of government bonds. To induce households to reduce consumption and increase savings so that equilibrium in goods and bond markets can be achieved, the natural rate of interest must increase. However, an interesting observation is the fact that this natural rate increases with $\hat{g}_{t}-E_{t}\left[ \hat{g}_{t+1}\right]$. As such, expected government spending tomorrow reduces the natural rate of interest today: an increase in future government spending reduces households’ life-time income, everything else equal, and therefore future consumption. In response, households would like to save more today to smooth consumption over time, which increases the demand for government bonds. As today’s supply of bonds is not directly affected by expected spending tomorrow, the natural rate of interest must decrease to achieve equilibrium in the bond market. As such, the fact that the natural rate depends on $\hat{g}_{t}-E_{t}\left[ \hat{g}_{t+1}\right]$ causes a persistent government spending shock to increase the natural rate by less than when the natural rate only depends on $\hat{g}_{t}$ (Walsh, 2010).

2.2 Overlapping Generations New Keynesian Model (OLG-NK)

Next, I discuss the two-period OLG-NK model. The derivations of the nonlinear first order conditions, and the resulting set of linearized equations can be found in Appendix C. I show that the New Keynesian Phillips curve is the same as in the RANK-model (1). To derive the aggregate demand equation, I start from the young’s (linearized) Euler equation which determines how much to consume and how much to save through government bonds:

$$\begin{aligned} -\sigma \hat{c}_{t}^{1} = - E_{t}\left[ \sigma \hat{c}_{t+1}^{2}\right] + \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] , \end{aligned}$$

(6)

where $\hat{c}_{t}^{1}$ and $\hat{c}_{t}^{2}$ denote consumption of the young and old, respectively. Both the young and old’s coefficient of relative risk aversion is $\sigma$.

To arrive at an aggregate demand equation in terms of inflation and the output gap, I substitute a linearized version of the old’s budget constraint $c_{t}^{2}=s_{t}$, where $s_{t}$ denotes the government transfer to the old⁴:

$$\begin{aligned} s_{t} = \left( \frac{\bar{s}}{\bar{y} - \delta \bar{g}}\right) \left( y_{t} - \delta g_{t}\right) . \end{aligned}$$

(7)

Here, $s_{t}$, $y_{t}$, and $g_{t}$ denote the level of the transfer, output, and government spending, while variables with a bar denote their respective steady state values. In addition to a linearized version of $c_{t}^{2}=s_{t}$, I substitute a linearized version of the aggregate resource constraint $c_{t}^{1} = y_{t}-c_{t}^{2}-g_{t}$ into equation (6). In the rest of the paper, I will restrict my attention to the two corner cases $\delta =0$ and $\delta =1$, the last of which is discussed in Proposition 1.

Proposition 1

The linearized version of the OLG-NK model with $\delta =1$ in Eq. (7) is exactly equal to the linearized version of the RANK model of Sect. 2.1.

Proof

See Appendix C.8.

As a result, the analysis of the RANK model will directly carry over to the OLG-NK model with $\delta =1$, which implies that the statements and conclusions for the RANK model automatically apply to the OLG-NK model with $\delta =1$. Therefore, I will use the term ‘OLG-NK model’ to refer to the OLG-NK model with $\delta =0$ going forward, unless explicitly stated otherwise.

After having substituted the above-described expressions for $\hat{c}_{t}^{1}$ and $\hat{c}_{t}^{2}$, I set $\delta =0$ and substitute $\hat{y}_{t}=\tilde{y}_{t} + \hat{y}_{t}^{n}$, where $\hat{y}_{t}^{n}$ denotes the natural level of output in the OLG-NK model with $\delta =0$. As a result, I arrive at the following aggregate demand equation, a detailed mathematical derivation of which can be found in Appendix C:

$$\begin{aligned} \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1-\frac{\bar{c}_{2}}{\bar{y}}\right) \tilde{y}_{t} = \sigma E_{t}\left[ \tilde{y}_{t+1}\right] - \left( \hat{r}_{t} - \hat{R}_{t}^{*}\right) , \end{aligned}$$

(8)

Compared with the representative agent version of the aggregate demand Eq. (2) (and the OLG-NK model with $\delta =1$), we see that the coefficients in front of $\tilde{y}_{t}$ and $E_{t}\left[ \tilde{y}_{t+1}\right]$ are not the same anymore, which is a result of the fact that the young and old have different budget constraints.⁵ As such, the different numerical values of these two coefficients will at least quantitatively affect the young’s savings decision with respect to the savings decision of the representative household in the RANK model (and the OLG-NK model with $\delta =1$).

Although $\hat{R}_{t}^{*}$ can still be decomposed into the two components of expression (3), the resulting expressions change with respect to their counterparts (4) and (5) in the RANK model:

$$\begin{aligned} \hat{R}_{t}^ {z,*}= & {} - \sigma \left( \left( \bar{y}/\bar{c}_{1}\right) \left( 1-\frac{\bar{c}_{2}}{\bar{y}}\right) - \rho _{z}\right) \left( \frac{\left( 1+\varphi \right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1-\frac{\bar{c}_{2}}{\bar{y}}\right) +\varphi }\right) \hat{z}_{t}, \end{aligned}$$

(9)

$$\begin{aligned} \hat{R}_{t}^ {g,*}= & {} \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( \frac{\left( \bar{g}/\bar{y}\right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1-\frac{\bar{c}_{2}}{\bar{y}}\right) +\varphi }\right) \left( \varphi \hat{g}_{t} + \sigma E_{t}\left[ \hat{g}_{t+1}\right] \right) , \end{aligned}$$

(10)

Comparing the new expression for the natural rate of interest arising from productivity shocks with the equivalent expression in the RANK model, we see that the response will qualitatively be the same as in the RANK model. The young understand that an AR(1) productivity process implies that income today increases by more than income tomorrow ceteris paribus, which increases their desire to save. To achieve clearing in bond and goods market, the natural rate of interest must come down.

The key difference with the natural rate of interest within the RANK model, however, is the component that arises from government spending, expression (10). Compared with the equivalent expression in the RANK-model (5), today’s and tomorrow’s expected government spending terms are now additive ($\varphi \hat{g}_{t} + \sigma E_{t}\left[ \hat{g}_{t+1}\right]$), rather than subtractive ($\hat{g}_{t} - E_{t}\left[ \hat{g}_{t+1}\right]$). The reason for this sign switch has to do with the term relating to tomorrow’s expected consumption by the representative agent and the young, respectively. Whereas tomorrow’s expected consumption by the representative agent is substituted by the difference between output and government spending in the RANK model, tomorrow’s expected consumption by today’s young is substituted by the government transfer in the OLG-NK model, which is linear in output alone.

As a result of today’s and tomorrow’s expected government spending being additive, the same persistent government spending shock increases the natural rate of interest by more in the OLG-NK model than in the RANK model (and the OLG-NK model with $\delta =1$). In addition, a more persistent spending shock leads to a larger change in the natural rate, everything else equal. This sharply contrasts with the RANK model, where more persistent shocks lead to a smaller change in the natural rate. Therefore, government spending is more powerful in changing the natural rate of interest in the OLG-NK model, and will therefore likely be a more effective tool in stabilizing the macroeconomy.

2.3 The Different Policy Regimes

In this subsection I specify the two regimes that I study in this paper. These consist of the regular monetary regime that is typically studied in the literature (see Woodford (2003) and Galí (2015), for example), and the fiscal regime that I define below.

Specifically, the monetary regime is defined by government spending being equal to its steady state value, i.e. $\hat{g}_{t}=0$, and an active Taylor rule for the nominal interest rate (Taylor, 1993):

$$\begin{aligned} \hat{R}_{t}^{n} = \kappa _{\pi }\hat{\pi }_{t} + \kappa _{y}\tilde{y}_{t}, \end{aligned}$$

(11)

which satisfies the Taylor principle $\kappa _{\pi }>1$ and $\kappa _{y}\ge 0$. Therefore, macroeconomic stabilization is performed by adjusting the nominal interest, which in turn changes the expected real interest rate. Both within the RANK model, as well as the OLG-NK model, the natural rate only features the exogenous productivity component, since $\hat{g}_{t}=0$ across time.

The fiscal regime is captured by a nominal interest rate that is equal to its steady state value, i.e. $\hat{R}_{t}^{n}=0$, while government spending is given by:

$$\begin{aligned} \hat{g}_{t} = g_{\pi }\hat{\pi }_{t}+g_{y}\tilde{y}_{t}. \end{aligned}$$

(12)

Under this regime, macroeconomic stabilization is performed by adjusting the natural rate of interest. At the same time, the nominal interest rate is no longer employed for stabilization.

3 Analytical Results

In this section I establish several analytical results. First, I show for both the RANK and the OLG-NK model that the aggregate demand equation under the fiscal regime can be obtained by replacing the nominal interest rate under the monetary regime with the negative of the endogenous natural rate component $\hat{R}_{t}^{g,*}$. Following up on this result, I show that the equilibrium values for inflation and the output gap under the monetary regime can be replicated under the fiscal regime through an appropriate mapping from the monetary feedback coefficients $\kappa _{\pi }$ and $\kappa _{y}$ to the government spending feedback coefficients $g_{\pi }$ and $g_{y}$. This shows that endogenous government spending can ensure that the same equilibrium values for inflation and the output gap can be implemented as in the case where the ZLB is not binding, and conventional monetary policy is employed for macroeconomic stabilization.

Third, I show that such a countercyclical equilibrium is not unique in the RANK model (and the OLG-NK model with $\delta =1$) and that multiple other equilibria exist. This result implies that the RANK model might not be the right framework to explore the consequences of an old-fashioned Keynesian fiscal policy. Fourth, I show that it is always possible to find some coefficients $g_{\pi }<0$ and $g_{y}=0$ such that a unique stable countercyclical equilibrium exists for the OLG-NK model. This result also implies that the specific form of the pension system is crucial for the stability analysis in the OLG-NK model. Fifth, I show that a unique stable procyclical equilibrium exists in the RANK model for sufficiently strong positive feedback coefficients from inflation and output gap on government spending. Finally, I provide some intuition behind the (im)possibility of a unique stable countercyclical equilibrium.

3.1 Equivalence Between Monetary and Fiscal Equilibrium

I start this section by showing for both the RANK and the OLG-NK model that the role of the nominal interest rate under the monetary regime is effectively performed by the negative of the endogenous natural rate component $\hat{R}_{t}^{g,*}$ under the fiscal regime.

Proposition 2

For both the RANK and the OLG-NK model, the aggregate demand equation under the fiscal regime can be obtained by replacing the nominal interest rate under the monetary regime by the negative of the endogenous natural rate component $\hat{R}_{t}^{g,*}$.

Proof

To prove the proposition, I will rewrite the aggregate demand equation under the monetary and fiscal regime, respectively, in order to compare the two regimes. I first do so for the aggregate demand Eq. (2) in the RANK model (and the OLG-NK model with $\delta =1$):

$$\begin{aligned} {\textbf {MR:}}\ \ \sigma \left( \bar{y}/\bar{c}\right) \tilde{y}_{t} = \sigma \left( \bar{y}/\bar{c}\right) E_{t}\left[ \tilde{y}_{t+1}\right] - \left( \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \hat{R}_{t}^{z,*}\right) , \end{aligned}$$

(13)

$$\begin{aligned} {\textbf {FR:}}\ \ \sigma \left( \bar{y}/\bar{c}\right) \tilde{y}_{t}= & {} \sigma \left( \bar{y}/\bar{c}\right) E_{t}\left[ \tilde{y}_{t+1}\right] - \left( - \hat{R}_{t}^{g,*} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \hat{R}_{t}^{z,*}\right) , \end{aligned}$$

(14)

where the abbreviation ‘MR’ stands for monetary regime, and ‘FR’ for fiscal regime. Similarly, I find for the aggregate demand equation (8) in the OLG-NK model:

$$\begin{aligned} {\textbf {MR:}}\ \ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1-\frac{\bar{c}_{2}}{\bar{y}}\right) \tilde{y}_{t}= & {} \sigma E_{t}\left[ \tilde{y}_{t+1}\right] - \left( \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \hat{R}_{t}^{z,*}\right) , \end{aligned}$$

(15)

$$\begin{aligned} {\textbf {FR:}}\ \ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1-\frac{\bar{c}_{2}}{\bar{y}}\right) \tilde{y}_{t}= & {} \sigma E_{t}\left[ \tilde{y}_{t+1}\right] - \left( - \hat{R}_{t}^{g,*} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \hat{R}_{t}^{z,*}\right) , \end{aligned}$$

(16)

Comparing the monetary regime in both models on the one hand (Eqs. (13) and (15)), with the fiscal regime on the other (Eqs. (14) and (16)), we see that the role of the nominal interest rate $\hat{R}_{t}^{n}$ under the monetary regime is effectively performed by $-\hat{R}_{t}^{g,*}$ under the fiscal regime. This concludes the proof.

Observe, however, that despite Proposition 2, there are two differences between the monetary and the fiscal regime. First, while the Taylor rule (11) directly depends on inflation and the output gap, the natural rate component $\hat{R}_{t}^{g,*}$ in (5) and (10) only does so indirectly via government spending, see Eq. (12). Second, the natural rate component $\hat{R}_{t}^{g,*}$ does not only depend on current government spending $\hat{g}_{t}$ but also on next period’s expected government spending $E_{t}\left[ \hat{g}_{t+1}\right]$. Therefore, the natural rate component $\hat{R}_{t}^{g,*}$ will indirectly also depend on expected future inflation and output gap. This contrasts with the Taylor rule (11), which only depends on current inflation and output gap.

Next, I analytically calculate the impulse response functions to a productivity shock using the method of undetermined coefficients. I do so for both the monetary and the fiscal regime, see Appendix B.12 for the RANK model and Appendix C.10 for the OLG-NK model. Doing so will allow me to show that the equilibrium values for inflation and the output gap that arise under the monetary regime can be replicated under the fiscal regime through an appropriate choice of the government spending coefficients $g_{\pi }$ and $g_{y}$ in terms of the monetary feedback coefficients $\kappa _{\pi }$ and $\kappa _{y}$.

Proposition 3

An equivalence exists between the equilibrium values for inflation and the output gap under the monetary and fiscal regime through an appropriate choice of $g_{\pi }$ and $g_{y}$ in terms of the monetary feedback coefficients $\kappa _{\pi }$ and $\kappa _{y}$.

Proof

In Appendix B.12, I show for the RANK model (and thus for the OLG-NK model with $\delta =1$) that the following mapping between the monetary policy coefficients $\kappa _{\pi }$ and $\kappa _{y}$ and the government spending coefficients $g_{\pi }$ and $g_{y}$ results in the exact same equilibrium values for inflation and the output gap in response to productivity shocks:

$$\begin{aligned} \kappa _{\pi }= & {} -B\left( 1 - \rho _{z}\right) g_{\pi }, \end{aligned}$$

(17)

$$\begin{aligned} \kappa _{y}= & {} -B\left( 1 - \rho _{z}\right) g_{y}, \end{aligned}$$

(18)

where B comes from rewriting Eq. (5) as $\hat{R}_{t}^ {g,*} = B\left( \hat{g}_{t}-E_{t}\left[ \hat{g}_{t+1}\right] \right)$, and is therefore given by:

$$\begin{aligned} B = \sigma \left( \bar{y}/\bar{c}\right) \left( \frac{\left( \bar{g}/\bar{y}\right) \varphi }{\sigma \left( \bar{y}/\bar{c}\right) +\varphi }\right) > 0, \end{aligned}$$

(19)

In Appendix C.10 I show the equivalent mapping for the OLG-NK model, which is given by:

$$\begin{aligned} \kappa _{\pi }= & {} -B^{*}\left( \sigma \rho _{z}+\varphi \right) g_{\pi }, \end{aligned}$$

(20)

$$\begin{aligned} \kappa _{y}= & {} -B^{*}\left( \sigma \rho _{z}+\varphi \right) g_{y}, \end{aligned}$$

(21)

where $B^{*}$ comes from rewriting Eq. (10) as $\hat{R}_{t}^ {g,*}=B^{*}\left( \varphi \hat{g}_{t} + \sigma E_{t}\left[ \hat{g}_{t+1}\right] \right)$, and is therefore given by:

$$\begin{aligned} B^{*} = \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( \frac{\left( \bar{g}/\bar{y}\right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1-\frac{\bar{c}_{2}}{\bar{y}}\right) +\varphi }\right) . \end{aligned}$$

(22)

The economic intuition behind the proposition is straightforward and can be explained by considering a positive productivity shock that decreases inflation and the output gap. Under the monetary regime, the central bank reduces the nominal and real interest rate to increase aggregate demand. As a result, inflation and the output gap increase with respect to the initial decrease that resulted from the productivity shock. To achieve the same equilibrium under the fiscal regime, the government also has to increase aggregate demand, which is now achieved by raising government spending.

Interestingly, the above proposition implies that countries, which are currently stuck at the ZLB and predicted to remain there for many years to come, are (theoretically) not in any way limited by their inability to employ conventional monetary policy when it comes to achieving the levels of inflation and output gap under an unconstrained monetary policy. To do so, they can simply employ government spending.⁶ Given these results, it is interesting to observe that Japan has employed fiscal policy much more aggressively in recent years. And although it has not been able to bring inflation back to its target of 2%, Blanchard and Tashiro (2019) suggest that it has been able to close the output gap.⁷

3.2 The (Im)possibility of a Unique Stable Countercyclical Equilibrium

Above we saw that the equilibrium values for inflation and the output gap under the monetary regime can be replicated under the fiscal regime through an appropriate choice of the feedback coefficients of inflation and output gap on government spending. However, the fact that an equilibrium is feasible does not guarantee that it is stable and unique. In this section, I will investigate to what extent a unique stable countercyclical equilibrium exists under the fiscal regime. I will do so for both the RANK model and the OLG-NK model. I start with the RANK model in Proposition 4, and show that a unique stable equilibrium does not exist for any countercyclical government spending rule, i.e. $g_{\pi }<0$ and $g_{y}<0$.

Proposition 4

There is no unique stable equilibrium in the RANK model and the OLG-NK model with $\delta =1$ for countercyclical fiscal policy ($g_{\pi }<0$ and $g_{y}<0$).

Proof

See Appendix B.10.

Therefore, I conclude that despite the fact that the equilibrium values for inflation and the output gap in the RANK model (and the OLG-NK model with $\delta =1$) can be replicated under the fiscal regime, see Proposition 3, the resulting equilibrium is not unique.⁸ Before explaining the intuition behind this result, I first show in Proposition 5 that, unlike the RANK-model, a unique stable countercyclical equilibrium exists in the OLG-NK model.

Proposition 5

For the OLG-NK model, there always exists a unique stable countercyclical equilibrium for some values $g_{\pi }<0$ and $g_{y}=0$ of the government spending rule.

Proof

See Appendix C.9.3.

Observe that Proposition 5 has to be understood as a ‘minimum result’ in the sense that I analytically show that at least a unique stable countercyclical equilibrium exists for some values $g_{\pi }<0$ and $g_{y}=0$. However, we will see in the numerical analysis of Sect. 4.4 that there are many additional combinations of $g_{\pi }<0$ and $g_{y}<0$ for which a unique stable countercyclical equilibrium can exist.

A key conclusion from Propositions 4 and 5 is that the specific form of the pension system (the transfer rule) is crucial for whether or not a unique stable countercyclical equilibrium is feasible. There is no such equilibrium when the transfer is linear in the difference between output and government spending, and increases in spending for macroeconomic stabilization come at the expense of pensioners. However, such an equilibrium is feasible when the transfer is linear in output, and increases in government spending are financed by issuing more government debt.

Now in order to understand the intuition behind Propositions 4 and 5, we first observe that under the monetary regime the two components of the (expected) real rate $\hat{r}_{t}\equiv \hat{R}_{t}^{n} - E_{t}\left[ \pi _{t+1}\right]$ have an opposite effect for a given change in (expected) inflation. For example, a shock that increases (expected) inflation increases the nominal interest $\hat{R}_{t}^{n}$ on the one hand, which increases the (expected) real rate, everything else equal. However, the accompanying increase in expected inflation $E_{t}\left[ \pi _{t+1}\right]$ simultaneously decreases the (expected) real rate on the other hand, everything else equal.

To understand why this is important for the question whether a unique stable equilibrium exists, consider a positive productivity shock. Such a shock decreases the exogenous part of the natural rate $\hat{R}_{t}^{z,*}$, see Eqs. (4) and (9). Simultaneously, the resulting (expected) deflation will lead to an increase of the (expected) real rate of interest $\hat{r}_{t}=\hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right]$, everything else equal. Therefore, a unique stable equilibrium is only feasible when the increase in the real rate due to $E_{t}\left[ \hat{\pi }_{t+1}\right]$ is more than offset by a decrease in the nominal interest rate $\hat{R}_{t}^{n}$, which requires the Taylor principle to be operative Taylor (1993). Only then will the (expected) real rate and the natural rate move in the same direction (decrease), which is necessary for a unique stable equilibrium to exist.

Now, under the fiscal regime, the endogenous natural rate component $-\hat{R}_{t}^{g,*}$ performs the role that the nominal interest rate does under the monetary regime: a countercyclical fiscal policy increases government spending when inflation and the output gap decrease, as a result of which $\hat{R}_{t}^{g,*}$ increases, see Eqs. (5) and (10). Therefore, the same logic applies as under the monetary regime: the impact from expected inflation $E_{t}\left[ \hat{\pi }_{t+1}\right]$ on the real rate $\hat{r}_{t}$ must be more than offset by the change in $- \hat{R}_{t}^{g,*}$ in order for a unique stable equilibrium to exist.

However, we can immediately see from Eq. (5) that this will be impossible to achieve in the RANK model (and the OLG-NK model with $\delta =1$), as $\hat{R}_{t}^{g,*}$ increases with the difference between today’s government spending $\hat{g}_{t}$ and tomorrow’s expected government spending $E_{t}\left[ \hat{g}_{t+1}\right]$. Therefore, a unique stable equilibrium will not be possible in the RANK model with countercyclical fiscal policy, see Proposition 4, as $\hat{R}_{t}^{g,*}$ cannot increase sufficiently. This contrasts with the OLG-NK model, where $\hat{R}_{t}^{g,*}$ increases with the (weighted) sum of today’s government spending $\hat{g}_{t}$ and tomorrow’s expected government spending $E_{t}\left[ \hat{g}_{t+1}\right]$, see Eq. (10). As a result, $\hat{R}_{t}^{g,*}$ is able to more than offset the effect from expected inflation $E_{t}\left[ \hat{\pi }_{t+1}\right]$ on the real rate $\hat{r}_{t}$. Therefore, a unique stable equilibrium with countercyclical fiscal policy is feasible in the OLG-NK model, see Proposition 5.

The above observations also allow me to conclude more generally for which types of models a unique stable countercyclical equilibrium might exist in models with a standard New Keynesian Phillips curve (1) and an aggregate demand equation: the natural rate component $\hat{R}_{t}^{g,*}$ from government spending needs to increase sufficiently and more than offset the opposite effect that expected inflation $E_{t}\left[ \hat{\pi }_{t+1}\right]$ has on the real rate $\hat{r}_{t}$ under a countercyclical spending rule.

Finally, I end this section by discussing Proposition 6 and the intuition behind it.

Proposition 6

There is a unique stable equilibrium in the RANK model (and the OLG-NK model with $\delta =1$) for a sufficiently responsive procyclical fiscal policy in the sense that $g_{\pi }>1/B$ and $g_{y}>\sigma \left( \bar{y}/\bar{c}\right) /B$.

Proof

See Appendix B.11.

The key intuition why a unique stable procyclical equilibrium exists is that under a procyclical spending rule the impact from expected inflation $E_{t}\left[ \hat{\pi }_{t+1}\right]$ on the real rate $\hat{r}_{t}$ works in the same direction as the effect from $-\hat{R}_{t}^{g,*}$: as (expected) inflation increases, (expected) government spending increases, which in turn decreases $-\hat{R}_{t}^{g,*}$. As $\hat{R}_{t}^{g,*}$ and $E_{t}\left[ \hat{\pi }_{t+1}\right]$ now move in the same direction, the required change in $\hat{R}_{t}^{g,*}$ to achieve a unique stable equilibrium is smaller than under a countercyclical policy. Therefore, a unique stable procyclical equilibrium exists, despite $\hat{R}_{t}^{g,*}$ only increasing with the difference between today’s and tomorrow’s expected government spending.

4 Numerical Results

In this section, I numerically investigate the RANK and the OLG-NK model. I begin by investigating the stability properties of the RANK model (and hence the OLG-NK model with $\delta =1$), and show the regions for which this model has a unique and stable equilibrium. Then I show the impulse response functions to a productivity shock, which allow me to explain the dynamics that arise under a procyclical government spending rule and compare the resulting equilibrium with that arising under a countercyclical monetary policy. I will then perform the same stability analysis for the OLG-NK model, after which I again investigate the impact of a productivity shock, but now with a countercyclical rather than a procyclical government spending rule. Again, I compare the resulting impulse response functions with those arising from a countercyclical monetary policy.

4.1 Parameter Values

The numerical analysis is meant to illustrate the qualitative properties of the model, rather than perform a quantitative analysis for a particular country. Therefore, no moments are matched and most parameters have numerical values that are typical within the New Keynesian literature.

Different from the literature, I set the steady state gross nominal interest rate equal to one (and therefore the net nominal rate equal to zero) to capture an economy that is at the ZLB in the long run. I also set steady state net inflation equal to zero. To accommodate these choices, I set the subjective discount factor equal to 1. The coefficient of relative risk aversion $\sigma$ and the inverse Frisch elasticity $\varphi$ are also set to 1. The elasticity of substitution is set to 10, implying a steady state markup of 11%, while the Calvo-probability is set to 0.75. I set steady state government spending over GDP equal to 0.2. As both the RANK model and the OLG-NK model feature Ricardian equivalence, steady state debt-GDP does not affect the equilibrium allocation, and therefore does not need to be chosen. Steady state consumption of the old is 40% of steady state output within the OLG-NK model to make sure that the steady state gross nominal interest rate is also one in the OLG-NK model. The AR(1) coefficient for productivity is 0.95, while the standard deviation is equal to 0.01. These parameter values result in $B=1/9$ and $B^{*}=1/5$. An overview of the parameter values can be found in Table 1 in Appendix D.

4.2 Stability in the RANK Model

I start by investigating for which values of $g_{\pi }$ and $g_{y}$ a unique stable equilibrium exists within the RANK model (and the OLG-NK model with $\delta =1$). I do so by calculating the roots for the system that consists of Eqs. (1) and (14). The results can be found in Fig. 1, where the blue plus sign denotes a unique stable equilibrium, while the red cross sign represents a combination for which multiple equilibria exist.⁹

From Fig. 1 we see that a unique stable equilibrium only exists for strictly positive coefficients $g_{y} > 0$ and $g_{\pi } > 0$, which is in line with Proposition 6. Furthermore, these results also confirm Proposition 4, which says that a unique stable equilibrium does not exist when government spending is countercyclical in both inflation and the output gap.

Fig. 1

Stability properties of the RANK model under the fiscal regime. The horizontal axis features the output coefficient $g_{y}$ and the vertical axis the inflation coefficient $g_{\pi }$. A unique stable equilibrium exists for combinations of $g_{\pi }$ and $g_{y}$ that have a blue plus sign, while the combinations with multiple equilibria are denoted by a red cross

4.3 Dynamic Response to a Productivity Shock in the RANK Model

To follow up on the theoretical analysis of Sect. 3 and the stability properties from the previous subsection, I investigate the RANK economy’s impulse response functions to a productivity shock for a combination of $g_{\pi }$ and $g_{y}$ for which a unique stable procyclical equilibrium exists. Specifically, I investigate in Fig. 2 the impact of a positive productivity shock of 1%, and compare the response under the monetary regime (blue, solid) with that under the fiscal regime with $g_{\pi }=10$ and $g_{y}=20$ (red, slotted).

Fig. 2

Figure displaying the impulse response functions to a 1% positive productivity shock in the RANK model and the OLG-NK model with $\delta =1$ (percentage deviation from steady state). The blue solid line denotes the monetary regime with $\kappa _{\pi }=1.5$ and $\kappa _{y}=0.125$, while the red slotted line denotes the fiscal regime with $g_{\pi }=10$ and $g_{y}=20$. “Expected real rate” refers to the variable $\hat{r}_{t}=\hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right]$. The variable “Natural rate” refers to $\hat{R}_{t}^{*}$ in Eq. (3), “Natural rate (z)” corresponds to $\hat{R}_{t}^{z,*}$ in Eq. (4), while “Natural rate (g)” refers to $\hat{R}_{t}^{g,*}$ in Eq. (5)

Strikingly, we see that inflation and the output gap have opposite signs under the monetary and fiscal regime. Meanwhile, the expected real interest rate $\hat{r}_{t}=\hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right]$ and the natural rate of interest (3) have the same sign. They both decrease in response to the productivity shock. Also observe that there is a small quantitative difference in the natural rate of interest between the two regimes, which is caused by endogenous government spending under the fiscal regime and is equal to $\hat{R}_{t}^{g,*}$ in Eq. (5). Therefore, changes in government spending have a relatively minor influence on the natural rate and most of the change in the natural rate is driven by the exogenous productivity component $\hat{R}_{t}^{z,*}$ in Eq. (4). This is the case despite the fact that government spending increases by almost 1% of steady state, which is a similar percentage change as the productivity shock. Of course, the reason for the small change in $\hat{R}_{t}^{g,*}$ is that it is increasing in the difference between today’s and tomorrow’s expected government spending, as discussed in Sect. 3.2.

To better understand the above impulse response functions, I will first revisit the aggregate demand equation under the monetary regime, and subsequently discuss it under the fiscal regime. Despite the fact that the monetary regime is well known from the literature (Galí (2015) for example), revisiting it allows me to compare and contrast it with the fiscal regime, which in turn will help us understand the response of inflation and the output gap under the fiscal regime.

The aggregate demand equation under the standard monetary regime is again given by Eq. (13), which I repeat below for convenience:

$$\begin{aligned} \sigma \left( \bar{y}/\bar{c}\right) \tilde{y}_{t}= & {} \sigma \left( \bar{y}/\bar{c}\right) E_{t}\left[ \tilde{y}_{t+1}\right] - \left( \hat{r}_{t} - \hat{R}_{t}^ {z,*}\right) , \end{aligned}$$

(23)

In this equation, a positive productivity shock reduces the natural rate of interest $\hat{R}_{t}^{z,*}$ which increases the return difference between the expected real rate $\hat{r}_{t}$ and the natural rate $\hat{R}_{t}^ {z,*}$. As a result, households shift from spending to saving with respect to the flexible prices equilibrium, resulting in a negative output gap. A negative output gap today results in (expected) deflation through the New Keynesian Phillips curve (1), which decreases the nominal and real interest rate through the Taylor rule. This, in turn, (partially) offsets the initial increase in the return difference between the expected real rate $\hat{r}_{t}$ and the natural rate $\hat{R}_{t}^ {z,*}$ that resulted from the productivity shock, allowing for a unique stable equilibrium to emerge.

Next, I move to the fiscal regime. To do so, I substitute the government spending rule (12) into Eq. (14), which results in the following aggregate demand equation:

$$\begin{aligned} \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) \tilde{y}_{t}= & {} \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) E_{t}\left[ \tilde{y}_{t+1}\right] - \left( \hat{r}_{t} - \hat{R}_{t}^ {z,*} - \hat{R}_{t}^{\pi ,*}\right) , \end{aligned}$$

where $\hat{R}_{t}^{\pi ,*} = Bg_{\pi }\left( \hat{\pi }_{t} - E_{t}\left[ \hat{\pi }_{t+1}\right] \right)$. The above expression allows me to explain why a shock that initially increases the return difference between the expected rate of interest $\hat{r}_{t}$ and the natural rate $\hat{R}_{t}^{*}$ results in a positive output gap and inflation, see Fig. 2. Previously, the increase in the return difference would generate a shift from spending to saving. This desire to save, however, is now more than offset by a change in the natural rate arising from a change in the output gap (captured by $Bg_{y}$), since $g_{y} > \sigma \left( \bar{y}/\bar{c}\right) /B$. As a result, households shift from saving to spending instead, resulting in a positive output gap in equilibrium. The additional purchases generate inflation through the New Keynesian Phillips curve (1), which drives down the expected real interest rate $\hat{r}_{t}=-E_{t}\left[ \hat{\pi }_{t+1}\right]$. The increase in the endogenous part of the natural rate $\hat{R}_{t}^{g,*}$ and the decrease in the expected real interest rate offset the initial increase in the return difference between the expected real rate and the natural rate, thereby giving rise to a unique stable equilibrium.

Now that I have explained the economic intuition behind the impulse response functions of the procyclical unique equilibrium in the RANK model, I move on to investigate the OLG-NK model.

4.4 The OLG-NK Model

In this section, I turn my attention to the OLG-NK model. While I already proved in Proposition 5 that there always exists a unique stable countercyclical equilibrium for some values $g_{\pi }<0$ and $g_{y}=0$, I will investigate in this section for what other values of $g_{\pi }$ and $g_{y}$ a unique stable equilibrium exists. In addition, I will also investigate the impulse response functions to the same productivity shock as in the previous section under a countercyclical government spending rule.

First, I investigate in Fig. 3 the stability properties of the OLG-NK model. From the figure we see that no unique stable equilibrium exists when the inflation and output gap coefficients are positive. Therefore, a unique stable equilibrium is only feasible when government spending is countercyclical. The figure not only confirms Proposition 5, as we see that a unique stable countercyclical equilibrium exists for $g_{\pi }<0$ and $g_{y}=0$, but it also shows that such an equilibrium exists for any $g_{y}<0$ as long as $g_{\pi }$ is sufficiently negative.

Fig. 3

Stability properties of the OLG-NK model under the fiscal regime. The horizontal axis features the output coefficient $g_{y}$ and the vertical axis the inflation coefficient $g_{\pi }$. A unique, stable equilibrium exists for combinations of $g_{\pi }$ and $g_{y}$ that have a blue plus sign, while the combinations with multiple equilibria are denoted with a red cross

Next, I investigate in Fig. 4 the impulse response functions of the OLG-NK model in response to the same positive productivity shock as in Fig. 2. The blue solid line denotes the monetary regime with $\kappa _{\pi }=1.5$ and $\kappa _{y}=0.125$, whereas the red slotted line denotes the fiscal regime with feedback coefficients $g_{\pi }=-10$ and $g_{y}=-10$. Interestingly, we see that the output gap and inflation now have the same sign as under the monetary regime, a result that strongly contrasts with the RANK model. Instead, we see that the sign of the expected real interest rate $\hat{r}_{t} = - E_{t}\left[ \hat{\pi }_{t+1}\right]$ and the natural rate $\hat{R}_{t}^{*}$ switch with respect to that under the monetary regime. This also differs from the RANK model, where the expected real interest rate and the natural rate had the same sign as under the monetary regime, and the quantitative difference between the natural rate of the two regimes was small.

Fig. 4

Figure displaying the impulse response functions to a 1% positive productivity shock in the OLG-NK model (percentage deviation from steady state). The blue solid line denotes the monetary regime with $\kappa _{\pi }=1.5$ and $\kappa _{y}=0.125$, while the red slotted line denotes the fiscal regime with $g_{\pi }=-10$ and $g_{y}=-10$. “Expected real rate” refers to the variable $\hat{r}_{t}=\hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right]$. The variable “Natural rate” refers to $\hat{R}_{t}^{*}$ in Eq. (3), “Natural rate (z)” corresponds to $\hat{R}_{t}^{z,*}$ in Eq. (9), while “Natural rate (g)” refers to $\hat{R}_{t}^{g,*}$ in Eq. (10)

To enhance our understanding behind these results, I first study the monetary regime. Just as in the previous section, I write down the aggregate demand Eq. (15) below for convenience:

$$\begin{aligned} \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1-\frac{\bar{c}_{2}}{\bar{y}}\right) \tilde{y}_{t} = \sigma E_{t}\left[ \tilde{y}_{t+1}\right] - \left( \hat{r}_{t} - \hat{R}_{t}^ {z,*}\right) , \end{aligned}$$

(24)

A comparison with the aggregate demand equation under the monetary regime in the RANK model (23) shows that the two equations are qualitatively the same, and only differ to the extent that the coefficients in front of today’s output gap $\tilde{y}_{t}$ and tomorrow’s expected output gap $E_{t}\left[ \tilde{y}_{t+1}\right]$ are quantitatively different. As a result, the impulse response functions of the RANK model and the OLG-NK model are qualitatively the same (compare the blue solid lines in Fig. 2 with the blue solid lines in Fig. 4). Therefore the intuition behind the impulse response functions under the monetary regime of the OLG-NK model is similar to that under the monetary regime of the RANK model.

Next, I turn my attention to the fiscal regime under the OLG-NK model. Just as under the monetary regime, the productivity shock decreases the exogenous part of the natural rate $\hat{R}_{t}^{z,*}$, which in turn increases the return difference between the expected real rate $\hat{r}_{t}$ and the natural rate of interest $\hat{R}_{t}^{*}$. Just as under the monetary regime, there is a shift from spending to saving. A negative output gap emerges, which leads to (expected) deflation through the New Keynesian Phillips curve (1). This expected deflation raises the expected real interest rate under the fiscal regime, which further increases the difference between the real rate $\hat{r}_{t}$ and the natural rate of interest $\hat{R}_{t}^{*}$, everything else equal. However, the government increases spending in response to the negative output gap and deflation, which increases the endogenous component of the natural rate $\hat{R}_{t}^{g,*}$ to such an extent that it more than offsets the decrease in the exogenous part of the natural rate $\hat{R}_{t}^{z,*}$. Therefore, we see that the real rate $\hat{r}_{t}$ and the natural rate $\hat{R}_{t}^{*}$ increase in equilibrium, as a result of which a unique stable equilibrium exists. The reason why the increase in the endogenous component of the natural rate is so large is that $\hat{R}_{t}^{g,*}$ changes with the (weighted) sum of today’s and tomorrow’s expected government spending. Therefore, the impulse response functions confirm the intuition behind the existence of a unique stable countercyclical equilibrium in the OLG-NK model as discussed in Sect. 3.2.

5 Conclusion

In this paper I investigate an ‘old Keynesian’ fiscal policy within the New Keynesian framework in which government spending endogenously responds to inflation and the output gap while the nominal interest rate is pegged at the ZLB. I employ two versions of the New Keynesian framework, both of which do not feature physical capital. The first is the standard Representative Agent New Keynesian (RANK) model, while the second is a two-period Overlapping Generations New Keynesian (OLG-NK) model that is similar in spirit to Galí (2014). Within the OLG-NK model, I consider two rules for the government transfer to the old. Under the first, the transfer is linear in the difference between output and government spending, reflecting the fact that increases in government spending (to stabilize the economy) might come at the expense of pensioners. Under the second rule, the transfer is linear in output alone, reflecting a situation in which the government issues more debt to finance the extra spending. It turns out that the linearized version of the first OLG-NK model is exactly equal to the linearized version of the RANK model.

Both under the standard ‘monetary’ regime, as well as under my ‘fiscal’ or ‘old Keynesian’ regime, government policy affects the economy through the aggregate demand equation, which relates the (expected) output gap to the return difference between the expected real rate and the natural rate of interest. While conventional monetary policy affects the equilibrium by changing the expected real interest rate (through adjustment of the nominal rate), government spending affects the equilibrium by changing the natural rate of interest via its impact on the endogenous component from government spending $\hat{R}_{t}^{g,*}$. A crucial difference between the RANK model and the second OLG-NK model is that $\hat{R}_{t}^{g,*}$ increases in the difference between today’s and tomorrow’s expected government spending in the RANK model (and in the first OLG-NK model), whereas it increases in the (weighted) sum of today’s and tomorrow’s expected spending in the second OLG-NK model. As such, a given path of (expected) government spending has a stronger effect on the natural rate in the second OLG-NK model than in the RANK model and the first OLG-NK model.

My first result is to show that for both the RANK and the OLG-NK model the aggregate demand equation under the fiscal regime can be obtained by replacing the nominal interest rate under the monetary regime by the negative of the endogenous natural rate component $\hat{R}_{t}^{g,*}$. As a result, the equilibrium values for inflation and the output gap under the monetary regime can be replicated under the fiscal regime through an appropriate choice for the feedback coefficients of inflation and the output gap on government spending, both in the RANK model and the OLG-NK model. The resulting government spending rule turns out to be countercyclical in inflation and the output gap. This can be understood by considering a positive productivity shock that decreases inflation and the output gap. Under the monetary regime, the central bank will reduce the nominal and real interest rate to increase aggregate demand, thereby increasing inflation and the output gap with respect to the initial decrease. Under the fiscal regime, the government raises aggregate demand by increasing government spending.

My second and main result is that a unique stable equilibrium with countercyclical government spending is only feasible within the second OLG-NK model (the model with a transfer that is linear in output), while such an equilibrium does not exist in the RANK model and the first OLG-NK model. Therefore, the specific form of the pension system (the transfer rule) is crucial for the stability analysis within the OLG-NK model. To explain the intuition behind these results, remember that under a countercyclical monetary regime the impact from expected inflation on the expected real rate works in the opposite direction as the impact from the nominal interest rate on the expected real rate. Therefore, a unique stable equilibrium is only feasible if the nominal interest rate responds sufficiently strong and more than offsets the effect that expected inflation has on the expected real interest rate. Under the monetary regime, this requires the Taylor principle to be operative. Under the fiscal regime, this requires that the endogenous component of the natural rate $\hat{R}_{t}^{g,*}$ responds sufficiently strong to changes in government spending. Therefore, a unique stable countercyclical equilibrium is only feasible in the second OLG-NK model, where the natural rate increases with the weighted sum of today’s and tomorrow’s expected government spending. Simultaneously, it also explains why no unique stable countercyclical equilibrium exists in the RANK model and the first OLG-NK model, where the endogenous natural rate increases with the difference between today’s and tomorrow’s expected government spending.

These results highlight that employing countercyclical endogenous government spending in an economy that is expected to remain at the ZLB for many years to come is not straightforward, as the workhorse model for policy analysis, the RANK model, predicts that such a policy will feature multiple equilibria. As the goal of countercyclical fiscal policy is typically to stabilize the economy, such an outcome would be the last thing that policymakers need. Therefore, governments considering such a policy would at the very least need to do more research into the effects that countercyclical endogenous government spending has as a stabilization tool. An alternative option is to rely on endogenous unconventional monetary policies at the ZLB, see Sims and Wu (2021) for an analysis of such policies.

Appendices

Appendix A. Empirical Evidence

1.1 Appendix A.1 Expectation of Permanent Low Interest Rates

Figure 5 provides a strong indication that in 2020 financial markets expected short-term interest rates in the Eurozone, Japan, and the United Kingdom to remain at or close to the ZLB. Specifically, Fig. 5a shows that yields on government bonds increased by less than 1% when moving from a maturity of less than one year to a maturity of 30 years. At the same time, their level was below 1% even for debt with a maturity of 30 years. Figure 5b shows the instantaneous forward rate that is implied by the yield curves in Fig. 5a. This forward rate can be understood as the future short-term interest rate expected by financial markets at the time. Figure 5b clearly shows that future short-term interest rates in the Eurozone were not expected to increase above 0.2% during the next 30 years, while they were expected to only temporarily increase above 1% in the United Kingdom, and then revert back below 0.5%. This implies that financial markets at the time expected these economies to remain at or close to the ZLB for the next 30 years.

Fig. 5

The left figure displays yield curves in 2020 for the euro area (blue, solid), the United Kingdom (red, slotted), and Japan (black, dashed). The right figure displays the instantaneous forward curve for the euro area and the United Kingdom, which is the short-term (instantaneous) interest rate for future periods that is implied by the yield curve. Maturity is in years. Sources:European Central Bank (2020), Bank of England (2020), and Ministry of Finance Japan (2020)

Appendix B. Representative Agent New Keynesian (RANK) Model

I employ a standard New-Keynesian model without capital such as can be found in standard textbook treatments such as Galí (2015). Households consume, supply labor, and save through one-period nominal government bonds, which pay a nominal interest rate that is set by the central bank. Within my model, the central bank will set the interest rate equal to its steady state value. The fiscal authority raises revenue from issuing one-period government bonds and lump sum taxes, while these revenues are spent on government purchases of the final good and gross interest payments (including the principal) of government bonds issued in the previous period. Lump sum taxes satisfy the Bohn (1998) principle, which results in my model satisfying Ricardian equivalence. As is standard in the New Keynesian literature, the production sector is three-layered. Final goods producers have a production function that has a constant elasticity of substitution between different retail goods. They operate in a perfectly competitive market, and therefore take prices as given while choosing how many goods to purchase from each retail goods producer. Retail goods producers require one intermediate good to produce one retail good, and operate in a market of monopolistic competition. Therefore, they have the capacity to set prices while taking the demand schedule into account, resulting in a markup over the intermediate goods. However, they are subject to Calvo (1983) pricing frictions which prevents some retail goods producers to change prices in a given period. Due to their monopoly power, retail goods producers make a profit in equilibrium, which is transferred to households. Finally, intermediate goods producers operate in a perfectly competitive market in which they produce using a production function that is linear in labor. They hire labor in a perfectly competitive labor market. As a result, intermediate goods producers take prices and wages as given, and only determine how much labor to hire in equilibrium.

1.1 Appendix B.1. Households

There is a continuum $i\in [0,1]$ of identical households. Each household i receives income from supplying labor $W_{t}h_{t}(i)$, where $W_{t}$ is the nominal wage rate, and $h_{t}(i)$ the number of hours worked. In addition, income is received from gross repayment of nominal one-period bonds $R_{t-1}^{n}B_{t-1}(i)$, where $R_{t}^{n}$ is the gross nominal interest rate set by the central bank. Finally, households receive income $\Omega _{t}(i)$ from profits of firms owned by household i. Income is spent on consumption $C_{t}(i)$, purchases of new government bonds $B_{t}(i)$, and lump sum taxes $P_{t}\tau _{t}(i)$, where $P_{t}$ is the price level of the final good. This gives rise to the following nominal budget constraint for household i:

$$\begin{aligned} C_{t}(i) + B_{t}(i) + P_{t}\tau _{t}(i) = W_{t}h_{t}(i) + R_{t-1}^{n}B_{t-1}(i) + \Omega _{t}(i). \end{aligned}$$

(A1)

Division by the price level $P_{t}$ results in the following budget constraint in terms of the final good:

$$\begin{aligned} c_{t}(i) + b_{t}(i) + \tau _{t}(i) = w_{t}h_{t}(i) + \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1}(i) + \omega _{t}(i), \end{aligned}$$

(A2)

where $\pi _{t}\equiv P_{t}/P_{t-1}$ is the gross inflation rate of the price level. Household i maximizes the expected discounted life-time utility, which is separable in consumption and labor:

$$\begin{aligned} \max _{\left\{ c_{t+s}(i),b_{t+s}(i),h_{t+s}(i)\right\} _{s=0}^{\infty }}E_{t}\left\{ \sum _{s=0}^{\infty }\beta ^{s}\xi _{t+s}\left[ \frac{c_{t+s}(i)^{1-\sigma }-1}{1-\sigma } - \chi _{h}\frac{h_{t+s}(i)^{1+\varphi }}{1+\varphi }\right] \right\} , \end{aligned}$$

where $\xi _{t}$ denotes a preference shock. The Lagrangian of household’s i maximization problem is given by:

$$\begin{aligned} \mathcal {L}= & {} E_{t}\left\{ \sum _{s=0}^{\infty }\beta ^{s}\xi _{t+s}\left[ \frac{c_{t+s}(i)^{1-\sigma }-1}{1-\sigma } - \chi _{h}\frac{h_{t+s}(i)^{1+\varphi }}{1+\varphi }\right] \right\} \\{} & {} + E_{t}\left\{ \sum _{s=0}^{\infty }\beta ^{s}\lambda _{t+s}\left[ w_{t+s}h_{t+s}(i) + \left( \frac{R_{t-1+s}^{n}}{\pi _{t+s}}\right) b_{t-1+s}(i) + \omega _{t+s}(i) - c_{t+s}(i) - b_{t+s}(i) - \tau _{t+s}(i)\right] \right\} \end{aligned}$$

This results in the following first order conditions:

$$\begin{aligned} c_{t}(i)&:&\lambda _{t} = \xi _{t}c_{t}(i)^{-\sigma }, \end{aligned}$$

(A3)

$$\begin{aligned} h_{t}(i)&:&\chi _{h}h_{t}(i)^{\varphi } = w_{t}c_{t}(i)^{-\sigma }, \end{aligned}$$

(A4)

$$\begin{aligned} b_{t}(i)&:&E_{t}\left[ \beta \Lambda _{t,t+1}\left( \frac{R_{t}^{n}}{\pi _{t+1}}\right) \right] = 1, \end{aligned}$$

(A5)

where $\beta \Lambda _{t,+s}=\beta \lambda _{t+s}/\lambda _{t}$ denotes households’ stochastic discount factor.

1.2 Appendix B.2. Production Firms

I explained in the main text that the production sector consists of final goods producers, retail goods producers, and intermediate goods producers. Below I show the formal derivations of their first order conditions.

1.2.1 Appendix B.2.1. Final Goods Producers

Final goods producers purchase retail goods $y_{t}^{f}$ at price $P_{t}^{f}$ from retail goods producer $f\in [0,1]$, and combine these into final goods $y_{t}$ using the following constant elasticity of substitution production technology:

$$\begin{aligned} y_{t} = \left[ \int _{0}^{1}\left( y_{t}^{f}\right) ^{\frac{\epsilon -1}{\epsilon }}df\right] ^{\frac{\epsilon }{\epsilon -1}}, \end{aligned}$$

(A6)

where $\epsilon$ denotes the elasticity of substitution between two retail goods producers. There is perfect competition among final goods producers, hence all final goods producers charge the same price $P_{t}$ for their final goods. They take demand $y_{t}$ for final goods as given, and only decide how many retail goods $y_{t}^{f}$ to buy from each retail goods producer. Hence final goods producers’ optimization problem is given by:

$$\begin{aligned} \max _{\lbrace y_{t}^{f}\rbrace }\ \ \ P_{t}y_{t} - \int _{0}^{1}P_{t}^{f}y_{t}^{f}df, \end{aligned}$$

subject to the production technology (A6). Taking the first order condition with respect to $y_{t}^{f}$ results in the following demand schedule for retail good $f\in [0,1]$:

$$\begin{aligned} y_{t}^{f} = \left( \frac{P_{t}^{f}}{P_{t}}\right) ^{-\epsilon }y_{t}, \end{aligned}$$

(A7)

Substitution of (A7) into final goods prodcuers’ production function (A6) allows me to find the general price level $P_{t}$:

$$\begin{aligned} P_{t}^{1-\epsilon } = \int _{0}^{1}\left( P_{t}^{f}\right) ^{1-\epsilon }df, \end{aligned}$$

(A8)

1.2.2 Appendix B.2.2. Retail Goods Producers

Retail goods producer $f\in [0,1]$ purchases goods $y_{t}^{i}$ at a price $\phi _{t}$ (expressed in terms of final goods) from intermediate goods producers. He converts these goods one for one into a unique retail good $y_{t}^{f} = y_{t}^{i}$. The fact that retail good f is unique provides retail goods producer f a monopoly for good f. As mentioned above, however, due to the fact that final goods producers purchase retail goods from all retail goods producers, retail goods producer f effectively operates in a market with monopolistic competition. However, monopolistic competition allows retail goods producer f to set the price $P_{t}^{f}$ while taking the demand schedule (A7) into account, thereby allowing him to charge a markup over the price $\phi _{t}$ of intermediate goods. Retail goods producers, however, are subject to price-stickiness a la Calvo (1983). This implies that there is a probability $\psi$, which is constant across time and cross-section, that retail goods producer f will not be able to change its nominal price $P_{t}^{f}$ in the future. Hence retail goods producers do not only maximize current profits, but also expected future profits when setting a new price $P_{t}^{*}$ today. Future expected profits are discounted using the households’ stochastic discount factor $\beta ^{s}\Lambda _{t,t+s}$, as they are the ultimate owners of the retail goods producers. The optimization problem is given by:

$$\begin{aligned} \max _{P_{t}^{*}} E_{t}\left\{ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\left[ \frac{P_{t}^{*}}{P_{t+s}} - \phi _{t+s}\right] y_{t+s}^{f}\right\} , \end{aligned}$$

subject to the demand curve (A7). Substitution of this demand curve gives the following optimization objective:

$$\begin{aligned} \max _{P_{t}^{*}} E_{t}\left\{ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\left[ \left( \frac{P_{t}^{*}}{P_{t+s}}\right) ^{1-\epsilon } - \phi _{t+s}\left( \frac{P_{t}^{*}}{P_{t+s}}\right) ^{-\epsilon }\right] y_{t+s}\right\} , \end{aligned}$$

Differentiation with respect to $P_{t}^{*}$ gives the following first order condition:

$$\begin{aligned}{} & {} \left( \epsilon -1\right) E_{t}\left[ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\left( \frac{P_{t}^{*}}{P_{t+s}}\right) ^{1-\epsilon }\frac{y_{t+s}}{P_{t}^{*}}\right] \\{} & {} \quad = \epsilon E_{t}\left[ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\phi _{t+s}\left( \frac{P_{t}^{*}}{P_{t+s}}\right) ^{-\epsilon }\frac{y_{t+s}}{P_{t}^{*}}\right] , \end{aligned}$$

Rearranging this expression gives:

$$\begin{aligned} \frac{P_{t}^{*}}{P_{t}}= & {} \left( \frac{\epsilon }{\epsilon -1}\right) \frac{E_{t}\left[ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\phi _{t+s}\left( \frac{P_{t+s}}{P_{t}}\right) ^{\epsilon }y_{t+s}\right] }{E_{t}\left[ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\left( \frac{P_{t+s}}{P_{t}}\right) ^{\epsilon -1}y_{t+s}\right] },\\= & {} \left( \frac{\epsilon }{\epsilon -1}\right) \frac{E_{t}\left[ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\phi _{t+s}\left( \prod _{j=1}^{s}\pi _{t+j}^{\epsilon }\right) y_{t+s}\right] }{E_{t}\left[ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\left( \prod _{j=1}^{s}\pi _{t+j}^{\epsilon -1}\right) y_{t+s}\right] },\\= & {} \left( \frac{\epsilon }{\epsilon -1}\right) \frac{\Xi _{1,t}}{\Xi _{2,t}}, \end{aligned}$$

where $\Xi _{1,t}$ and $\Xi _{2,t}$ are given by:

$$\begin{aligned} \Xi _{1,t}= & {} E_{t}\left[ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\phi _{t+s}\left( \prod _{j=1}^{s}\pi _{t+j}^{\epsilon }\right) y_{t+s}\right] = \phi _{t}y_{t} + E_{t}\left[ \psi \beta \Lambda _{t,t+1}\pi _{t+1}^{\epsilon }\Xi _{1,t+1}\right] , \end{aligned}$$

(A9)

$$\begin{aligned} \Xi _{2,t}= & {} E_{t}\left[ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}\left( \prod _{j=1}^{s}\pi _{t+j}^{\epsilon -1}\right) y_{t+s}\right] = y_{t} + E_{t}\left[ \psi \beta \Lambda _{t,t+1}\pi _{t+1}^{\epsilon -1}\Xi _{2,t+1}\right] . \end{aligned}$$

(A10)

The price level $P_{t}$ evolves according to the following law of motion, see (A8):

$$\begin{aligned} P_{t}^{1-\epsilon } = \left( 1-\psi \right) \left( P_{t}^{*}\right) ^{1-\epsilon } + \psi \left( 1-\psi \right) \left( P_{t-1}^{*}\right) ^{1-\epsilon } + \psi ^2\left( 1-\psi \right) \left( P_{t-2}^{*}\right) ^{1-\epsilon } + .... \end{aligned}$$

Lagging by one period, and multiplying by $\psi$ gives the following expression:

$$\begin{aligned} \psi P_{t-1}^{1-\epsilon } = \psi \left( 1-\psi \right) \left( P_{t-1}^{*}\right) ^{1-\epsilon } + \psi ^2\left( 1-\psi \right) \left( P_{t-2}^{*}\right) ^{1-\epsilon } + \psi ^3\left( 1-\psi \right) \left( P_{t-3}^{*}\right) ^{1-\epsilon } + .... \end{aligned}$$

Hence I can write the general price level $P_{t}$ as:

$$\begin{aligned} P_{t}^{1-\epsilon } = \left( 1-\psi \right) \left( P_{t}^{*}\right) ^{1-\epsilon } + \psi P_{t-1}^{1-\epsilon }. \end{aligned}$$

(A11)

Division by $P_{t}^{1-\epsilon }$ allows me to express everything in terms of the relative new price $\pi _{t}^{*}\equiv P_{t}^{*}/P_{t}$ and the gross inflation rate $\pi _{t}\equiv P_{t}/P_{t-1}$:

$$\begin{aligned} 1 = \left( 1-\psi \right) \left( \pi _{t}^{*}\right) ^{1-\epsilon } + \psi \pi _{t}^{\epsilon -1}. \end{aligned}$$

(A12)

Finally, I calculate price dispersion, which is defined as:

$$\begin{aligned}{} & {} \mathcal {D}_{t}\equiv \int _{0}^{1}\left( \frac{P_{t}^{f}}{P_{t}}\right) ^{-\epsilon }df\\{} & {} \quad = \left( 1-\psi \right) \left( \frac{P_{t}^{*}}{P_{t}}\right) ^{-\epsilon } + \psi \left( 1-\psi \right) \left( \frac{P_{t-1}^{*}}{P_{t}}\right) ^{-\epsilon } + \psi ^2\left( 1-\psi \right) \left( \frac{P_{t-2}^{*}}{P_{t}}\right) ^{-\epsilon } +....\\{} & {} \quad = P_{t}^{\epsilon }\left[ \left( 1-\psi \right) \left( P_{t}^{*}\right) ^{-\epsilon } + \psi \left( 1-\psi \right) \left( P_{t-1}^{*}\right) ^{-\epsilon } + \psi ^2\left( 1-\psi \right) \left( P_{t-2}^{*}\right) ^{-\epsilon } +....\right] \end{aligned}$$

Lagging by one period, and multiplying with $\psi \left( P_{t}/P_{t-1}\right) ^{\epsilon }$ gives:

$$\begin{aligned} \psi \left( \frac{P_{t}}{P_{t-1}}\right) ^{\epsilon }\mathcal {D}_{t-1}= & {} P_{t}^{\epsilon }\left[ \psi \left( 1-\psi \right) \left( P_{t-1}^{*}\right) ^{-\epsilon } + \psi ^2\left( 1-\psi \right) \left( P_{t-2}^{*}\right) ^{-\epsilon } +....\right] \end{aligned}$$

Hence I find for dispersion $\mathcal {D}_{t}$ the following expression:

$$\begin{aligned} \mathcal {D}_{t}= & {} P_{t}^{\epsilon }\left( 1-\psi \right) \left( P_{t}^{*}\right) ^{-\epsilon } + \psi \left( \frac{P_{t}}{P_{t-1}}\right) ^{\epsilon }\mathcal {D}_{t-1}\nonumber \\= & {} \left( 1-\psi \right) \left( \pi _{t}^{*}\right) ^{-\epsilon } + \psi \pi _{t}^{\epsilon }\mathcal {D}_{t-1}. \end{aligned}$$

(A13)

1.2.3 Appendix B.2.3. Intermediate Goods Producers

The production technology of intermediate goods producer $i\in [0,1]$ is given by:

$$\begin{aligned} y_{t}^{i} = z_{t}h_{t}^{i}, \end{aligned}$$

(Appendix A14)

where $y_{t}^{i}$ is the number of intermediate goods produced, $z_{t}$ productivity, and $h_{t}^{i}$ the amount of labor hired by intermediate goods producer i. Both the labor market and the market for intermediate goods are perfectly competitive, and intermediate goods producers therefore take the wage rate $w_{t}$ and the price of intermediate goods $\phi _{t}$ as given. The government provides a wage subsidy by paying a fraction $\tau ^ {w}$ of wage payments, which I introduce to offset the distortion from monopolistic competition in the steady state (Galí, 2015). Intermediate goods producers’ decision problem is static, and mathematically represented in the following way:

$$\begin{aligned} \max _{\lbrace h_{t}^{i}\rbrace }\ \ \ \phi _{t}y_{t}^{i} - \left( 1 - \tau ^ {w}\right) w_{t}h_{t}^{i}, \end{aligned}$$

subject to the production technology (A14). Taking the derivative with respect to $h_{t}^{i}$ results in the following first order condition:

$$\begin{aligned} \left( 1 - \tau ^ {w}\right) w_{t} = \phi _{t}z_{t}, \end{aligned}$$

(A15)

1.3 Appendix B.3. Government

1.3.1 Appendix B.3.1. Fiscal Authority

The fiscal authority raises revenues through lump sum taxes $P_{t}\tau _{t}$ on households, and issuing one period nominal government bonds $B_{t}$. Revenues are used to purchase final goods $P_{t}g_{t}$, where $g_{t}$ denotes the number of final goods purchased, and for payment of the gross interest rate on debt issued in the previous period $R_{t-1}^{n}B_{t-1}$. Finally, it pays a wage subsidy $\tau ^ {w}W_{t}h_{t}$, where $W_{t}$ denotes the nominal wage rate and where $h_{t}$ denotes aggregate labor supply. Hence the nominal government budget constraint is given by:

$$\begin{aligned} P_{t}\tau _{t} + B_{t} = P_{t}g_{t} + \tau ^ {w}W_{t}h_{t} + R_{t-1}^{n}B_{t-1}. \end{aligned}$$

(A16)

Division by the price level $P_{t}$ results in the following government budget constraint in terms of final goods:

$$\begin{aligned} \tau _{t} + b_{t} = g_{t} + \tau ^ {w}w_{t}h_{t} + \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1}, \end{aligned}$$

(A17)

where $b_{t}\equiv B_{t}/P_{t}$ is the stock of government debt in real terms. Government spending is as specified in the main text. I assume that there is a feedback rule from the stock of government debt on the level of lump sum taxes satisfying the Bohn (1998) principle:

$$\begin{aligned} \tau _{t} = \bar{\tau } + \kappa _{b}\left( b_{t-1}-\bar{b}\right) . \end{aligned}$$

(A18)

Therefore, the model satisfies Ricardian equivalence, and as a result the equilibrium allocation for lump sum taxes $\tau _{t}$ and government bonds $b_{t}$ will not affect the equilibrium allocation of the other variables. Finally, government spending will be equal to steady state under the monetary regime, while it will endogenously respond to inflation and the output gap under the fiscal regime.

1.3.2 Appendix B.3.2. Central Bank

The central bank sets the nominal interest rate $R_{t}^{n}$ on government bonds. I assume that the central bank sets the nominal interest rate according to a standard Taylor rule under the monetary regime, while the interest rate will be equal its steady state value under the fiscal regime:

$$\begin{aligned} R_{t}^{n} = \bar{R}_{n}. \end{aligned}$$

(A19)

1.4 Appendix B.4. Market Clearing

Market clearing occurs when the supply of final goods $y_{t}$ equals demand for final goods:

$$\begin{aligned} y_{t} = c_{t} + g_{t}, \end{aligned}$$

(A20)

1.5 Appendix B.5. Aggregation

I start by observing that there is a mass of one of households, each of which makes the same decisions for consumption and labor supply. Therefore, we know that $c_{t}\equiv \int _{0}^{1}c_{t}(i)di=c_{t}(i)\int _{0}^{1}di=c_{t}(i)$ and $h_{t}\equiv \int _{0}^{1}h_{t}(i)di=h_{t}(i)\int _{0}^{1}di=h_{t}(i)$. Therefore, I can simply replace $c_{t}(i)$ by $c_{t}$ and $h_{t}(i)$ by $h_{t}$ in households’ first order conditions for consumption and labor supply.

Next, I integrate Eq. (A7) over all retail goods producers:

$$\begin{aligned} \int _{0}^{1}y_{t}^{f}df = \int _{0}^{1}\left( \frac{P_{t}^{f}}{P_{t}}\right) ^{-\epsilon }y_{t}df = y_{t}\int _{0}^{1}\left( \frac{P_{t}^{f}}{P_{t}}\right) ^{-\epsilon }df = \mathcal {D}_{t}y_{t}, \end{aligned}$$

Integration over the left hand side occurs by remembering that $y_{t}^{f}=y_{t}^{i}=z_{t}h_{t}^{i}$, and then integrating over all intermediate goods producers:

$$\begin{aligned} \int _{0}^{1}y_{t}^{f}df = \int _{0}^{1}y_{t}^{i}di = z_{t}\int _{0}^{1}h_{t}^{i}di = z_{t}h_{t}. \end{aligned}$$

Therefore, the aggregate equivalent of Eq. (A7) is given by:

$$\begin{aligned} \mathcal {D}_{t}y_{t} = z_{t}h_{t}. \end{aligned}$$

(A21)

1.6 Appendix B.6. Overview First Order Conditions (RANK)

A compeititve equilibrium is a series of quantities $\left\{ c_{t},h_{t},y_{t},g_{t},b_{t},\tau _{t}\right\}$, (shadow) prices

$\left\{ \lambda _{t},R_{t}^{n},\phi _{t},w_{t},\pi _{t},\pi _{t}^{*},\mathcal {D}_{t},\Xi _{1,t},\Xi _{2,t}\right\}$, and exogenous processes $\left\{ z_{t},\xi _{t}\right\}$ satisfying the following equations:

$$\begin{aligned} \lambda _{t}= & {} \xi _{t}c_{t}^{-\sigma }, \end{aligned}$$

(A22)

$$\begin{aligned} \chi _{h}h_{t}^{\varphi }= & {} w_{t}c_{t}^{-\sigma }, \end{aligned}$$

(A23)

$$\begin{aligned} E_{t}\left[ \beta \Lambda _{t,t+1}\left( \frac{R_{t}^{n}}{\pi _{t+1}}\right) \right]= & {} 1, \end{aligned}$$

(A24)

$$\begin{aligned} \left( 1 - \tau ^ {w}\right) w_{t}= & {} \phi _{t}z_{t}, \end{aligned}$$

(A25)

$$\begin{aligned} \mathcal {D}_{t}y_{t}= & {} z_{t}h_{t}, \end{aligned}$$

(A26)

$$\begin{aligned} \pi _{t}^{*}= & {} \left( \frac{\epsilon }{\epsilon -1}\right) \frac{\Xi _{1,t}}{\Xi _{2,t}}, \end{aligned}$$

(A27)

$$\begin{aligned} \Xi _{1,t}= & {} \phi _{t}y_{t} + E_{t}\left[ \psi \beta \Lambda _{t,t+1}\pi _{t+1}^{\epsilon }\Xi _{1,t+1}\right] , \end{aligned}$$

(A28)

$$\begin{aligned} \Xi _{2,t}= & {} y_{t} + E_{t}\left[ \psi \beta \Lambda _{t,t+1}\pi _{t+1}^{\epsilon -1}\Xi _{2,t+1}\right] , \end{aligned}$$

(A29)

$$\begin{aligned} 1= & {} \left( 1-\psi \right) \left( \pi _{t}^{*}\right) ^{1-\epsilon } + \psi \pi _{t}^{\epsilon -1}. \end{aligned}$$

(A30)

$$\begin{aligned} \mathcal {D}_{t}= & {} \left( 1-\psi \right) \left( \pi _{t}^{*}\right) ^{-\epsilon } + \psi \pi _{t}^{\epsilon }\mathcal {D}_{t-1}. \end{aligned}$$

(A31)

$$\begin{aligned} \tau _{t} + b_{t}= & {} g_{t} + \tau ^ {w}w_{t}h_{t} + \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1}, \end{aligned}$$

(A32)

$$\begin{aligned} \tau _{t}= & {} \bar{\tau } + \kappa _{b}\left( b_{t-1}-\bar{b}\right) , \end{aligned}$$

(A33)

$$\begin{aligned} R_{t}^{n}= & {} ...., \end{aligned}$$

(A34)

$$\begin{aligned} g_{t}= & {} ...., \end{aligned}$$

(A35)

$$\begin{aligned} y_{t}= & {} c_{t} + g_{t}, \end{aligned}$$

(A36)

$$\begin{aligned} \log {\left( z_{t}\right) }= & {} \rho _{z}\log {\left( z_{t-1}\right) } + \varepsilon _{z,t}, \end{aligned}$$

(A37)

$$\begin{aligned} \log {\left( \xi _{t}\right) }= & {} \rho _{\xi }\log {\left( \xi _{t-1}\right) } + \varepsilon _{\xi ,t}, \end{aligned}$$

(A38)

where $\beta \Lambda _{t,t+s}\equiv \beta \lambda _{t+s}/\lambda _{t}$ denotes the representative households’ stochastic discount factor. In addition, there is a transversality condition for government bonds, and the process for government purchases $g_{t}$ is as specified in the main text.

1.7 Appendix B.7. Linearized FOCs

$$\begin{aligned} \varphi \hat{h}_{t}= & {} -\sigma \hat{c}_{t} +\hat{w}_{t}, \end{aligned}$$

(A39)

$$\begin{aligned} \hat{\xi }_{t} - \sigma \hat{c}_{t}= & {} E_{t}\left[ \hat{\xi }_{t+1} - \sigma \hat{c}_{t+1}\right] +\hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] ,\end{aligned}$$

(A40)

$$\begin{aligned} \hat{w}_{t}= & {} \hat{\phi }_{t} + \hat{z}_{t},\end{aligned}$$

(A41)

$$\begin{aligned} \hat{\mathcal {D}}_{t} + \hat{y}_{t}= & {} \hat{z}_{t} + \hat{h}_{t},\end{aligned}$$

(A42)

$$\begin{aligned} \hat{\pi }_{t}^{*}= & {} \hat{\Xi }_{1,t} - \hat{\Xi }_{2,t},\end{aligned}$$

(A43)

$$\begin{aligned} \hat{\Xi }_{1,t}= & {} \left( 1-\psi \beta \bar{\pi }^{\epsilon }\right) \left( \hat{\phi }_{t}+\hat{y}_{t}\right) \nonumber \\+ & {} \psi \beta \bar{\pi }^{\epsilon }\left( -\sigma E_{t}\left[ \hat{c}_{t+1}\right] + \sigma \hat{c}_{t} + \epsilon E_{t}\left[ \hat{\pi }_{t+1}\right] +E_{t}\left[ \hat{\Xi }_{1,t+1}\right] \right) ,\end{aligned}$$

(A44)

$$\begin{aligned} \hat{\Xi }_{2,t}= & {} \left( 1-\psi \beta \bar{\pi }^{\epsilon -1}\right) \hat{y}_{t}\nonumber \\+ & {} \psi \beta \bar{\pi }^{\epsilon -1}\left( -\sigma E_{t}\left[ \hat{c}_{t+1}\right] + \sigma \hat{c}_{t} + \left( \epsilon -1\right) E_{t}\left[ \hat{\pi }_{t+1}\right] +E_{t}\left[ \hat{\Xi }_{2,t+1}\right] \right) ,\end{aligned}$$

(A45)

$$\begin{aligned} \left( 1-\psi \right) \left( \bar{\pi }^{*}\right) ^{1-\epsilon }\hat{\pi }_{t}^{*}= & {} \psi \left( \bar{\pi }\right) ^{\epsilon -1}\hat{\pi }_{t},\end{aligned}$$

(A46)

$$\begin{aligned} \hat{\mathcal {D}}_{t}= & {} \left( 1-\psi \left( \bar{\pi }\right) ^{\epsilon }\right) \left( -\epsilon \hat{\pi }_{t}^{*}\right) +\psi \left( \bar{\pi }\right) ^{\epsilon }\left( \epsilon \hat{\pi }_{t}+\hat{\mathcal {D}}_{t-1}\right) ,\end{aligned}$$

(A47)

$$\begin{aligned} \hat{y}_{t}= & {} \left( \bar{c}/\bar{y}\right) \hat{c}_{t} + \left( \bar{g}/\bar{y}\right) \hat{g}_{t},\end{aligned}$$

(A48)

$$\begin{aligned} \bar{\tau }\hat{\tau }_{t} + \bar{b}\hat{b}_{t}= & {} \bar{g}\hat{g}_{t} + \tau ^{w}\bar{w}\bar{h}\left( \hat{w}_{t} + \hat{h}_{t}\right) + \left( \frac{\bar{R}_{n}}{\bar{\pi }}\right) \bar{b}\left( \hat{R}_{t-1}^{n} - \hat{\pi }_{t} + \hat{b}_{t-1}\right) ,\end{aligned}$$

(A49)

$$\begin{aligned} \bar{\tau }\hat{\tau }_{t}= & {} \kappa \bar{b}\hat{b}_{t-1},\end{aligned}$$

(A50)

$$\begin{aligned} \hat{R}_{t}^{n}= & {} ....,\end{aligned}$$

(A51)

$$\begin{aligned} \hat{g}_{t}= & {} ....,\end{aligned}$$

(A52)

$$\begin{aligned} \hat{z}_{t}= & {} \rho _{z}\hat{z}_{t-1} + \varepsilon _{z,t},\end{aligned}$$

(A53)

$$\begin{aligned} \hat{\xi }_{t}= & {} \rho _{\xi }\hat{\xi }_{t-1} + \varepsilon _{\xi ,t}, \end{aligned}$$

(A54)

1.8 Appendix B.8. Further Derivations

Going forward, I assume that gross steady state inflation is equal to one: $\bar{\pi }=1$. Next, I substitute Eqs. (A44) and (A45) into (A43) to obtain:

$$\begin{aligned} \hat{\pi }_{t}^{*} = \left( 1-\psi \beta \right) \hat{\phi }_{t} + \psi \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \psi \beta E_{t}\left[ \hat{\pi }_{t+1}^{*}\right] , \end{aligned}$$

(A55)

Substitution of $\hat{\pi }_{t}^{*}=\left( \psi /\left( 1-\psi \right) \right) \hat{\pi }_{t}$ (A46) delivers the traditional New Keynesian Phillips-curve:

$$\begin{aligned} \hat{\pi }_{t} = \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \zeta \hat{\phi }_{t}, \end{aligned}$$

(A56)

where $\zeta \equiv \left( 1-\psi \beta \right) \left( 1-\psi \right) /\psi$.

I can rewrite the aggregate resource constraint (A48) to obtain an expression for consumption $\hat{c}$:

$$\begin{aligned} \hat{c}_{t} = \frac{\bar{y}}{\bar{c}}\left[ \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] , \end{aligned}$$

(A57)

1.8.1 Appendix B.8.1. The Flexible Prices Equilibrium

Now I aim to derive the flexible prices equilibrium. To do so, I set $\psi =0$ in Eq. (A55)

$$\begin{aligned} \hat{\pi }_{t}^{*} = \hat{\phi }_{t}, \end{aligned}$$

Since I know from Eq. (A46) that $\hat{\pi }_{t}^{*}=0$ when $\psi =0$, I find that $\hat{\phi }_{t}=0$. Next, I substitute expression (A46) into Eq. (A47), and find that $\hat{\mathcal {D}}_{t}=0$, irrespective of whether $\psi =0$ or not.

Now I consider Eq. (A39), and substitute Eq. (A41) for $\hat{w}_{t}$, expression (A42) for $\hat{h}_{t}$, and Eq. (A57) to obtain:

$$\begin{aligned} \varphi \left( \hat{y}_{t} - \hat{z}_{t}\right) = -\sigma \left( \bar{y}/\bar{c}\right) \left[ \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] +\hat{z}_{t}, \end{aligned}$$

Solving for output delivers the flexible prices level of output, or the natural level of output:

$$\begin{aligned} \hat{y}_{t}^{n} = \left( \frac{1+\varphi }{\sigma \left( \bar{y}/\bar{c}\right) + \varphi }\right) \hat{z}_{t} + \sigma \left( \bar{y}/\bar{c}\right) \left( \frac{\left( \bar{g}/\bar{y}\right) }{\sigma \left( \bar{y}/\bar{c}\right) + \varphi }\right) \hat{g}_{t}, \end{aligned}$$

(A58)

1.8.2 Appendix B.8.2. The Sticky Prices Equilibrium

Agiain I consider Eq. (A39), and substitute Eq. (A41) for $\hat{w}_{t}$, expression (A42) for $\hat{h}_{t}$, and Eq. (A57). However, the difference with respect to the flexible prices equilibrium is that $\hat{\phi }_{t}$ is no longer zero in Eq. (A41). I thus obtain:

$$\begin{aligned} \varphi \left( \hat{y}_{t} - \hat{z}_{t}\right) = -\sigma \left( \bar{y}/\bar{c}\right) \left[ \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] +\hat{z}_{t} + \hat{\phi }_{t}, \end{aligned}$$

Rearranging gives:

$$\begin{aligned} \hat{\phi }_{t}= & {} \left( \sigma \left( \bar{y}/\bar{c}\right) + \varphi \right) \hat{y}_{t} - \left( 1+\varphi \right) \hat{z}_{t} - \sigma \left( \bar{y}/\bar{c}\right) \left( \bar{g}/\bar{y}\right) \hat{g}_{t}\nonumber \\= & {} \left( \sigma \left( \bar{y}/\bar{c}\right) + \varphi \right) \left( \hat{y}_{t} - \hat{y}_{t}^{n}\right) = \left( \sigma \left( \bar{y}/\bar{c}\right) + \varphi \right) \tilde{y}_{t}, \end{aligned}$$

(A59)

where $\tilde{y}_{t}\equiv \hat{y}_{t} - \hat{y}_{t}^{n}$ denotes the output gap, the difference between the level of output under the sticky prices equilibrium and the flexible prices equilibrium.

Substitution of expression (A59) into Eq. (A56) delivers the familiar New Keynesian Phillips curve in the output gap $\tilde{y}_{t}$:

$$\begin{aligned} \hat{\pi }_{t} = \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t}, \end{aligned}$$

(A60)

where $\kappa$ is given by:

$$\begin{aligned} \kappa = \left( \sigma \left( \bar{y}/\bar{c}\right) + \varphi \right) \zeta , \end{aligned}$$

(A61)

Finally, I substitute expressions (A57) into the Euler Eq. (A40):

$$\begin{aligned} \hat{\xi }_{t} - \sigma \left( \bar{y}/\bar{c}\right) \left[ \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] = E_{t}\left[ \hat{\xi }_{t+1}\right] - \sigma \left( \bar{y}/\bar{c}\right) E_{t}\left[ \hat{y}_{t+1} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t+1}\right] + \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] , \end{aligned}$$

(A62)

Now I substitute $\hat{y}_{t}=\hat{y}_{t}^{n} + \tilde{y}_{t}$, and substitute expression (A58) to get the aggregate demand equation:

$$\begin{aligned} \sigma \left( \bar{y}/\bar{c}\right) \tilde{y}_{t} = \sigma \left( \bar{y}/\bar{c}\right) E_{t}\left[ \tilde{y}_{t+1}\right] - \left( \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \hat{R}_{t}^{*}\right) , \end{aligned}$$

(A63)

where $\hat{R}_{t}^{*}$ is given by:

$$\begin{aligned} \hat{R}_{t}^{*} = \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *} + \hat{R}_{t}^{g*}, \end{aligned}$$

(A64)

where $\hat{R}_{t}^{z*}$, $\hat{R}_{t}^{\xi *}$, and $\hat{R}_{t}^{g*}$ are given by:

$$\begin{aligned} \hat{R}_{t}^{z*}= & {} -\sigma \left( \bar{y}/\bar{c}\right) \left( \frac{1+\varphi }{\sigma \left( \bar{y}/\bar{c}\right) + \varphi }\right) \left( 1-\rho _{z}\right) \hat{z}_{t}, \end{aligned}$$

(A65)

$$\begin{aligned} \hat{R}_{t}^{\xi *}= & {} \left( 1-\rho _{\xi }\right) \hat{\xi }_{t}, \end{aligned}$$

(A66)

$$\begin{aligned} \hat{R}_{t}^{g*}= & {} \sigma \left( \bar{y}/\bar{c}\right) \left( \frac{\left( \bar{g}/\bar{y}\right) \varphi }{\sigma \left( \bar{y}/\bar{c}\right) + \varphi }\right) \left( \hat{g}_{t} - E_{t}\left[ \hat{g}_{t+1}\right] \right) , \end{aligned}$$

(A67)

1.9 Appendix B.9. The Fiscal Regime

For the RANK-model, I will only investigate the stability conditions for the fiscal regime. To do so, I substitute the government spending rule from the main text (12) into the aggregate demand equation, which together with the New Keynesian Phillips curve gives the following system of two-by-two equations (while keeping $\hat{R}_{t}^{n}=0$):

$$\begin{aligned} \hat{\pi }_{t}= & {} \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t},\\ -Bg_{\pi }\hat{\pi }_{t} + \left( \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y}\right) \tilde{y}_{t}= & {} \left( 1 - Bg_{\pi }\right) E_{t}\left[ \hat{\pi }_{t+1}\right] \\{} & {} + \left( \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y}\right) E_{t}\left[ \tilde{y}_{t+1}\right] + \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *}, \end{aligned}$$

where B is given by:

$$\begin{aligned} B = \frac{\sigma \left( \bar{g}/\bar{c}\right) \varphi }{\sigma \left( \bar{y}/\bar{c}\right) +\varphi }, \end{aligned}$$

(A68)

Now I can write the above system of equations into the following matrix equation:

$$\begin{aligned} \begin{pmatrix} 1 &{} -\kappa \\ -Bg_{\pi } &{} \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y} \end{pmatrix} \begin{pmatrix} \hat{\pi }_{t} \\ \tilde{y}_{t} \end{pmatrix} = \begin{pmatrix} \beta &{} 0 \\ 1-Bg_{\pi } &{} \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y} \end{pmatrix} \begin{pmatrix} E_{t}\left[ \hat{\pi }_{t+1}\right] \\ E_{t}\left[ \tilde{y}_{t+1}\right] \end{pmatrix} + \begin{pmatrix} 0 \\ \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *} \end{pmatrix}, \end{aligned}$$

(A69)

I first establish the inverse of the matrix in front of current inflation and output gap:

$$\begin{aligned} \begin{pmatrix} 1 &{} -\kappa \\ -Bg_{\pi } &{} \sigma \bar{y}/\bar{c} - Bg_{y} \end{pmatrix}^{-1} = \frac{1}{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }\begin{pmatrix} -\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) &{} -\kappa \\ -Bg_{\pi } &{} -1 \end{pmatrix}. \end{aligned}$$

Now I can write the system of Eqs. (A69) in the following way:

$$\begin{aligned} \begin{pmatrix} \hat{\pi }_{t} \\ \tilde{y}_{t} \end{pmatrix} = M\begin{pmatrix} E_{t}\left[ \hat{\pi }_{t+1}\right] \\ E_{t}\left[ \tilde{y}_{t+1}\right] \end{pmatrix} + N, \end{aligned}$$

(A70)

where the matrices M and N are given by:

$$\begin{aligned} M= & {} \frac{1}{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }\begin{pmatrix} -\kappa \left( 1-Bg_{\pi }\right) - \beta \left( \sigma \bar{y}/\bar{c}-Bg_{y}\right) &{} -\kappa \left( \sigma \bar{y}/\bar{c}-Bg_{y}\right) \\ -\beta Bg_{\pi }-\left( 1-Bg_{\pi }\right) &{} -\left( \sigma \bar{y}/\bar{c}-Bg_{y}\right) \end{pmatrix}, \end{aligned}$$

(A71)

$$\begin{aligned} N= & {} \frac{1}{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }\begin{pmatrix} -\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) &{} -\kappa \\ -Bg_{\pi } &{} -1 \end{pmatrix}\begin{pmatrix} 0 \\ \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *} \end{pmatrix}, \end{aligned}$$

(A72)

Now I determine trace and determinant of M:

$$\begin{aligned} {{\,\textrm{trace}\,}}{M}= & {} \frac{-\kappa \left( 1-Bg_{\pi }\right) - \left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }, \end{aligned}$$

(A73)

$$\begin{aligned} \det {M}= & {} \frac{-\beta \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }. \end{aligned}$$

(A74)

As I have two forward-looking variables, I need two roots of the characteristic equation of matrix M that are inside the unit circle (Bullard and Mitra, 2002). First, I calculate the characteristic equation, and find that it is given by:

$$\begin{aligned} \lambda ^2 - {{\,\textrm{trace}\,}}{M} \lambda + \det {M} = 0. \end{aligned}$$

Bullard and Mitra (2002) take the following characteristic equation:

$$\begin{aligned} \lambda ^2 + a_{1}\lambda + a_{0} = 0. \end{aligned}$$

Hence in my case $a_{1}$ and $a_{0}$ are given by:

$$\begin{aligned} a_{1}= & {} - {{\,\textrm{trace}\,}}{M} = \frac{\kappa \left( 1-Bg_{\pi }\right) + \left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }, \end{aligned}$$

(A75)

$$\begin{aligned} a_{0}= & {} \det {M} = \frac{-\beta \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }. \end{aligned}$$

(A76)

1.10 Appendix B.10. Proof of Proposition 4

Proof

For $g_{y}<0$, we see that $\sigma \bar{y}/\bar{c} - Bg_{y} > 0$. With $g_{\pi }<0$, it immediately follows that the denominators of (A75) and (A76) are negative, i.e. $\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) < 0$. Next, I calculate the absolute value of $a_{0}$:

$$\begin{aligned} |a_{0}| = \frac{-\beta \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) } = \frac{\beta }{1 + \frac{ - \kappa Bg_{\pi }}{\sigma \bar{y}/\bar{c} - Bg_{y}}} < 1, \end{aligned}$$

as $\beta \le 1$ while $1 + \frac{ - \kappa Bg_{\pi }}{\sigma \bar{y}/\bar{c} - Bg_{y}}>1$. Hence the first condition of Bullard and Mitra (2002) is satisfied. Now I look at the second condition, i.e. $|a_{1}| < 1 + a_{0}$. To do that, I need to compute $|a_{1}|$. Since $g_{\pi }<0$, we immediately see that the numerator of (A75) is always positive. Hence the absolute value of $a_{1}$ is given by:

$$\begin{aligned} |a_{1}| = \frac{- \kappa \left( 1-Bg_{\pi }\right) - \left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }. \end{aligned}$$

The condition that $|a_{1}| < 1 + a_{0}$ then boils down to:

$$\begin{aligned} \frac{- \kappa \left( 1-Bg_{\pi }\right) - \left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) } < \frac{\kappa Bg_{\pi }-\left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }. \end{aligned}$$

Multiplication of both sides of the inequality with the negative denominator $\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) < 0$ transforms the inequality into the following way (where I flip the inequality sign):

$$\begin{aligned} - \kappa \left( 1-Bg_{\pi }\right) - \left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) > \kappa Bg_{\pi }-\left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) . \end{aligned}$$

After canceling equal terms on the left and right hand side of the inequality, I find the condition $-\kappa > 0$. This condition obviously does not hold, since $\kappa > 0$. Hence the second condition of Bullard and Mitra (2002), i.e. $|a_{1}| < 1 + a_{0}$, is violated. Therefore, there are not two roots inside the unit circle and hence no unique stable equilibrium exists for the case where $g_{\pi }<0$ and $g_{y}<0$. This concludes the proof.

1.11 Appendix B.11. Proof of Proposition 6

Proof

For $g_{y}>\sigma \left( \bar{y}/\bar{c}\right) /B$, we see that $\sigma \bar{y}/\bar{c} - Bg_{y} < 0$. With $g_{\pi }>1/B$, it immediately follows that the denominators of (A75) and (A76) are positive, i.e. $\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) > 0$. Next, I show that $|a_{0}|<1$ implies:

$$\begin{aligned} |a_{0}| = \frac{-\beta \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) } < 1, \end{aligned}$$

which can be rewritten as:

$$\begin{aligned} \kappa Bg_{\pi } > \left( 1 - \beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) . \end{aligned}$$

As the right hand side of the inequality is negative, the inequality automatically holds since I assume that $g_{\pi }>1/B>0$. Hence the first condition of Bullard and Mitra (2002) is satisfied. Now I look at the second condition, i.e. $|a_{1}| < 1 + a_{0}$. To do that, I need to compute $|a_{1}|$. Since $g_{y}>\sigma \left( \bar{y}/\bar{c}\right) /B$ and $g_{\pi }>1/B$, we immediately see that the numerator of (A75) is always negative. Hence the absolute value of $a_{1}$ is given by:

$$\begin{aligned} |a_{1}| = \frac{- \kappa \left( 1-Bg_{\pi }\right) - \left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }. \end{aligned}$$

The condition that $|a_{1}| < 1 + a_{0}$ then boils down to:

$$\begin{aligned} \frac{- \kappa \left( 1-Bg_{\pi }\right) - \left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) } < \frac{\kappa Bg_{\pi }-\left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }{\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) }. \end{aligned}$$

Multiplication of both sides of the inequality with the positive denominator $\kappa Bg_{\pi }-\left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) > 0$ transforms the inequality into the following way:

$$\begin{aligned} - \kappa \left( 1-Bg_{\pi }\right) - \left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) < \kappa Bg_{\pi }-\left( 1+\beta \right) \left( \sigma \bar{y}/\bar{c} - Bg_{y}\right) . \end{aligned}$$

After canceling equal terms on the left and right hand side of the inequality, I find the condition $-\kappa < 0$. This condition obviously holds, since $\kappa > 0$. Hence the second condition of Bullard and Mitra (2002), i.e. $|a_{1}| < 1 + a_{0}$, is satisfied, and a unique stable equilibrium exists for the case where $g_{\pi }>1/B$ and $g_{y}>\sigma \bar{y}/\bar{c}/B$. This concludes the proof.

1.12 Appendix B.12. Analytical Expressions for Impulse Response Functions

In this section I calculate analytical expressions for the impulse response functions to the productivity and preference shocks, and show that there exists an isomorphic mapping between the coefficients of the monetary and fiscal policy reactions such that the impulse response functions are identical.

1.12.1 Appendix B.12.1. Monetary Regime

I start by writing down the two-system equations for the monetary regime, where I replace $\hat{R}_{t}^{x*} = \mathcal {R}^{x*}\hat{x}_{t}$, where $x=\lbrace z,\xi \rbrace$.

$$\begin{aligned} \hat{\pi }_{t}= & {} \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t},\\ \sigma \left( \bar{y}/\bar{c}\right) \tilde{y}_{t}= & {} \sigma \left( \bar{y}/\bar{c}\right) E_{t}\left[ \hat{y}_{t+1}\right] - \left( \kappa _{\pi }\hat{\pi }_{t} + \kappa _{y}\tilde{y}_{t} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \mathcal {R}^{z*}\hat{z}_{t} - \mathcal {R}^{\xi *}\hat{\xi }_{t}\right) , \end{aligned}$$

Since there are no endogenous backward-looking state variables, I know that the only state variables are $\hat{z}_{t}$ and $\hat{\xi }_{t}$. Hence I can employ the method of undetermined coefficients to find the analytical solution to productivity and preference shocks. I assume that $\hat{\pi }_{t}$ and $\tilde{y}_{t}$ are given by the following solutions:

$$\begin{aligned} \hat{\pi }_{t}= & {} \alpha _{\pi ,z}\hat{z}_{t} + \alpha _{\pi ,\xi }\hat{\xi }_{t}, \end{aligned}$$

(A77)

$$\begin{aligned} \tilde{y}_{t}= & {} \alpha _{y,z}\hat{z}_{t} + \alpha _{y,\xi }\hat{\xi }_{t}. \end{aligned}$$

(A78)

Since both shocks are given by exogenous AR(1) shocks, I know that their expected value is given by:

$$\begin{aligned} E_{t}\left[ \hat{\pi }_{t+1}\right]= & {} \rho _{z}\alpha _{\pi ,z}\hat{z}_{t} + \rho _{\xi }\alpha _{\pi ,\xi }\hat{\xi }_{t}, \end{aligned}$$

(A79)

$$\begin{aligned} E_{t}\left[ \tilde{y}_{t+1}\right]= & {} \rho _{z}\alpha _{y,z}\hat{z}_{t} + \rho _{\xi }\alpha _{y,\xi }\hat{\xi }_{t}. \end{aligned}$$

(A80)

Substitution of the above expressions into the New Keynesian Phillips curve gives the following relations between the inflation and output gap coefficients:

$$\begin{aligned} \alpha _{y,z}= & {} \left( \frac{1-\beta \rho _{z}}{\kappa }\right) \alpha _{\pi ,z}, \end{aligned}$$

(A81)

$$\begin{aligned} \alpha _{y,\xi }= & {} \left( \frac{1-\beta \rho _{\xi }}{\kappa }\right) \alpha _{\pi ,\xi } \end{aligned}$$

(A82)

Substitution of the guessed solutions for the output gap and inflation, and the relation between the output gap coefficients and the inflation coefficients into the aggregate demand equation generate the following expressions for the coefficients:

$$\begin{aligned} \alpha _{\pi ,z}^{m}= & {} \frac{\kappa \mathcal {R}^{z*}}{\kappa \left( \kappa _{\pi }-\rho _{z}\right) +\left[ \kappa _{y} + \sigma \left( \bar{y}/\bar{c}\right) \left( 1 - \rho _{z}\right) \right] \left( 1-\beta \rho _{z}\right) },\ \end{aligned}$$

(A83)

$$\begin{aligned} \alpha _{\pi ,\xi }^{m}= & {} \frac{\kappa \mathcal {R}^{\xi *}}{\kappa \left( \kappa _{\pi }-\rho _{\xi }\right) +\left[ \kappa _{y} + \sigma \left( \bar{y}/\bar{c}\right) \left( 1 - \rho _{\xi }\right) \right] \left( 1-\beta \rho _{\xi }\right) }, \end{aligned}$$

(A84)

$$\begin{aligned} \alpha _{y,z}^{m}= & {} \frac{\left( 1-\beta \rho _{z}\right) \mathcal {R}^{z*}}{\kappa \left( \kappa _{\pi }-\rho _{z}\right) +\left[ \kappa _{y} + \sigma \left( \bar{y}/\bar{c}\right) \left( 1 - \rho _{z}\right) \right] \left( 1-\beta \rho _{z}\right) }, \end{aligned}$$

(A85)

$$\begin{aligned} \alpha _{y,\xi }^{m}= & {} \frac{\left( 1-\beta \rho _{\xi }\right) \mathcal {R}^{\xi *}}{\kappa \left( \kappa _{\pi }-\rho _{\xi }\right) +\left[ \kappa _{y} + \sigma \left( \bar{y}/\bar{c}\right) \left( 1 - \rho _{\xi }\right) \right] \left( 1-\beta \rho _{\xi }\right) }, \end{aligned}$$

(A86)

where $\mathcal {R}^{z*}$ and $\mathcal {R}^{\xi *}$ are given by:

$$\begin{aligned} \mathcal {R}^{z*}= & {} - \sigma \left( \bar{y}/\bar{c}\right) \left( \frac{1+\varphi }{\sigma \left( \bar{y}/\bar{c}\right) +\varphi }\right) \left( 1-\rho _{z}\right) , \end{aligned}$$

(A87)

$$\begin{aligned} \mathcal {R}^{\xi *}= & {} \left( 1-\rho _{\xi }\right) \end{aligned}$$

(A88)

1.12.2 Appendix B.12.2. Fiscal Regime

Next I solve for the impulse response fucnctions under the fiscal regime. The two equation system is again given by:

$$\begin{aligned} \hat{\pi }_{t}= & {} \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t},\\ - Bg_{\pi }\hat{\pi }_{t} + \left[ \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y}\right] \tilde{y}_{t}= & {} \left( 1 - Bg_{\pi }\right) E_{t}\left[ \hat{\pi }_{t+1}\right] + \left[ \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y}\right] E_{t}\left[ \hat{y}_{t+1}\right] + \mathcal {R}^{z*}\hat{z}_{t} + \mathcal {R}^{\xi *}\hat{\xi }_{t}, \end{aligned}$$

Again employing the method of undetermined coefficients generates the same relationship between the inflation coefficients and the output gap coefficients, and eventually results in the following expressions for the coefficients:

$$\begin{aligned} \alpha _{\pi ,z}^{f}= & {} \frac{\kappa \mathcal {R}^{z*}}{\kappa \left[ -Bg_{\pi }\left( 1 - \rho _{z}\right) -\rho _{z}\right] +\left[ \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y}\right] \left( 1 - \rho _{z}\right) \left( 1-\beta \rho _{z}\right) }, \end{aligned}$$

(A89)

$$\begin{aligned} \alpha _{\pi ,\xi }^{f}= & {} \frac{\kappa \mathcal {R}^{\xi *}}{\kappa \left[ -Bg_{\pi }\left( 1 - \rho _{\xi }\right) -\rho _{\xi }\right] +\left[ \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y}\right] \left( 1 - \rho _{\xi }\right) \left( 1-\beta \rho _{\xi }\right) }, \end{aligned}$$

(A90)

$$\begin{aligned} \alpha _{y,z}^{f}= & {} \frac{\left( 1-\beta \rho _{z}\right) \mathcal {R}^{z*}}{\kappa \left[ -Bg_{\pi }\left( 1 - \rho _{z}\right) -\rho _{z}\right] +\left[ \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y}\right] \left( 1 - \rho _{z}\right) \left( 1-\beta \rho _{z}\right) }, \end{aligned}$$

(A91)

$$\begin{aligned} \alpha _{y,\xi }^{f}= & {} \frac{\left( 1-\beta \rho _{\xi }\right) \mathcal {R}^{\xi *}}{\kappa \left[ -Bg_{\pi }\left( 1 - \rho _{\xi }\right) -\rho _{\xi }\right] +\left[ \sigma \left( \bar{y}/\bar{c}\right) - Bg_{y}\right] \left( 1 - \rho _{\xi }\right) \left( 1-\beta \rho _{\xi }\right) }, \end{aligned}$$

(A92)

Comparing the solutions (A83) - (A86) under the monetary regime with those under the fiscal regime (A89)–(A92), we see that there is an isomorphic mapping under which the equilibrium paths for inflation and the output gap are identical under both regimes. This is the case for the productivity shock when:

$$\begin{aligned} \kappa _{\pi }= & {} -B\left( 1- \rho _{z}\right) g_{\pi }\Longrightarrow g_{\pi } = -\frac{\kappa _{\pi }}{B\left( 1- \rho _{z}\right) }, \end{aligned}$$

(A93)

$$\begin{aligned} \kappa _{y}= & {} -B\left( 1 - \rho _{z}\right) g_{y}\Longrightarrow g_{y} = -\frac{\kappa _{y}}{B\left( 1- \rho _{z}\right) }, \end{aligned}$$

(A94)

while we have the following mapping for the preference shock:

$$\begin{aligned} \kappa _{\pi }= & {} -B\left( 1 - \rho _{\xi }\right) g_{\pi }\Longrightarrow g_{\pi } = -\frac{\kappa _{\pi }}{B\left( 1 - \rho _{\xi }\right) }, \end{aligned}$$

(A95)

$$\begin{aligned} \kappa _{y}= & {} -B\left( 1 - \rho _{\xi }\right) g_{y}\Longrightarrow g_{y} = -\frac{\kappa _{y}}{B\left( 1 - \rho _{\xi }\right) }, \end{aligned}$$

(A96)

Appendix C. Overlapping Generations Model

1.1 Appendix C.1. Households

A generation lives for two periods. The first period they are young, and in the second period of their existence they are old, after which each generation dies. I assume that each generation has a constant mass of 1 that does not change over time. In the first period, the young earn income $w_{t}h_{t}(i)$ from providing labor, and from ownership of the production firms $\omega _{t}(i)$ (in terms of final goods). This income is spent on consumption $c_{t}^{1}(i)$, lump sum taxes $\tau _{t}^{1}(i)$, and savings in the form of government bonds $b_{t}(i)$. Their budget constraint is then (in terms of final goods) given by:

$$\begin{aligned} c_{t}^{1}(i) + \tau _{t}^{1}(i) + b_{t}(i) = w_{t}h_{t}(i) + \omega _{t}(i), \end{aligned}$$

(A97)

When turning from young to old, the old receive income from gross repayment of the government bonds that were purchased when young as well as a pension income $s_{t}(i)$ provided by the government. This income is then used for consumption $c_{t}^{2}(i)$ and lump sum taxes $\tau _{t}^{2}(i)$. The budget constraint for the old (in terms of final goods) is then given by:

$$\begin{aligned} c_{t}^{2}(i) + \tau _{t}^{2}(i) = \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1}(i) + s_{t}(i), \end{aligned}$$

(A98)

The old’s maximization problem is given by maximizing current consumption subject to the budget constraint (A98):

$$\begin{aligned} \max _{\left\{ c_{t}^{2}(i)\right\} }\ \xi _{t}\cdot \frac{\left( c_{t}(i)^{2}\right) ^{1-\sigma }-1}{1-\sigma }, \end{aligned}$$

where $\xi _{t}$ is a preference shock. The Lagrangian for this problem is given by:

$$\begin{aligned} \mathcal {L}= & {} \xi _{t}\cdot \frac{\left( c_{t}(i)^{2}\right) ^{1-\sigma }-1}{1-\sigma } + \lambda _{t}^{2}\left[ \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1}(i) + s_{t}(i) - c_{t}^{2}(i) - \tau _{t}^{2}(i)\right] \end{aligned}$$

The first order conditions are given by:

$$\begin{aligned} c_{t}^{2}(i)&:&\lambda _{t}^{2} = \xi _{t}\left( c_{t}^{2}(i)\right) ^{-\sigma }, \end{aligned}$$

(A99)

$$\begin{aligned} \lambda _{t}^{2}&:&\left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1}(i) + s_{t}(i) - c_{t}^{2}(i) - \tau _{t}^{2}(i) = 0. \end{aligned}$$

(A100)

Now I move to the young’s optimization problem, which is given by:

$$\begin{aligned} \max _{\left\{ c_{t}^{1}(i),h_{t}(i),b_{t}(i)\right\} }\ \xi _{t} \left( \frac{\left( c_{t}^{1}(i)\right) ^{1-\sigma }-1}{1-\sigma } - \chi _{h}\frac{\left( h_{t}(i)\right) ^{1+\varphi }}{1+\varphi }\right) + E_{t}\left[ \beta \xi _{t+1}\cdot \frac{\left( c_{t+1}^{2}(i)\right) ^{1-\sigma }-1}{1-\sigma }\right] , \end{aligned}$$

subject to the budget constraints (A97) and (A98). This results in the following Lagrangian:

$$\begin{aligned} \mathcal {L}= \, & {} \xi _{t} \left( \frac{\left( c_{t}^{1}(i)\right) ^{1-\sigma }-1}{1-\sigma } - \chi _{h}\frac{\left( h_{t}(i)\right) ^{1+\varphi }}{1+\varphi }\right) + E_{t}\left[ \beta \xi _{t+1}\cdot \frac{\left( c_{t+1}^{2}(i)\right) ^{1-\sigma }-1}{1-\sigma }\right] \\+ \quad & {} \lambda _{t}^{1}\left[ w_{t}h_{t}(i) + \omega _{t}(i) - c_{t}^{1}(i) - \tau _{t}^{1}(i) - b_{t}(i)\right] \\+ \quad & {} E_{t}\left\{ \beta \lambda _{t+1}^{2}\left[ \left( \frac{R_{t}^{n}}{\pi _{t+1}}\right) b_{t}(i) + s_{t+1}(i) - c_{t+1}^{2}(i) - \tau _{t+1}^{2}(i)\right] \right\} . \end{aligned}$$

After taking the derivatives with respect to $c_{t}^{1}(i)$, $h_{t}(i)$, $b_{t}(i)$, and $\lambda _{t}^{1}$, I obtain the following first order conditions:

$$\begin{aligned} c_{t}^{1}(i)&:&\lambda _{t}^{1} = \xi _{t}\left( c_{t}^{1}(i)\right) ^{-\sigma }, \end{aligned}$$

(A101)

$$\begin{aligned} h_{t}(i)&:&\chi _{h} \left( h_{t}(i)\right) ^{\varphi } = w_{t}\left( c_{t}^{1}(i)\right) ^{-\sigma }, \end{aligned}$$

(A102)

$$\begin{aligned} b_{t}(i)&:&E_{t}\left[ \beta \Lambda _{t,t+1}^{1,2}\left( \frac{R_{t}^{n}}{\pi _{t+1}}\right) \right] = 1, \end{aligned}$$

(A103)

$$\begin{aligned} \lambda _{t}^{1}&:&w_{t}h_{t}(i) + \omega _{t}(i) - c_{t}^{1}(i) - \tau _{t}^{1}(i) - b_{t}(i) = 0, \end{aligned}$$

(A104)

where $\beta \Lambda _{t,t+1}^{1,2}\equiv \beta \lambda _{t+1}^{2}/\lambda _{t}^{1}$ denotes the young generation’s stochastic discount factor.

1.2 Appendix C.2. Production Firms

In this subsection I only discuss the changes that I make to the production sector, which turn out to be few. The structure with final goods producers, retail goods producers, and intermediate goods producers remains the same as before, as well as all assumptions regarding their production technologies and the type of markets they operate in. The only change that I have to incorporate is the fact that the old generation is now the owner of all the production firms, rather than the infinitely-lived household in the RANK model.

1.2.1 Appendix C.2.1. Final Goods Producers

As final goods producers face a static optimization problem, and do not make any profits in equilibrium as they operate in a perfectly competitive market for final goods, the optimization problem is exactly the same as in the case of a representative infinitely-lived household.

1.2.2 Appendix C.2.2. Retail Goods Producers

The production technology of retail goods producers, as well as the fact that they operate under monopolistic competition makes that their optimization problem is the same as under the representative infinitely-lived household. The only difference is that they are owned in period t by the generation that was born in period t, while they will be owned in period $t+1$ by the generation born in period $t+1$, etc. Therefore, the stochastic discount factor with which they discount future expected profits will differ. I assume that they will value a cash flow in period $t+s$ with the marginal utility $\beta \lambda _{t+s}^{y}$ of the generation that will be young in period $t+s$, where future profits are discounted with the subjective discount factor $\beta$ with which they discount next period’s utility relative to today’s utility. Therefore, the retail goods producers’ maximization problem changes into:

$$\begin{aligned} \max _{P_{t}^{*}} E_{t}\left\{ \sum _{s=0}^{\infty }\psi ^{s}\beta ^{s}\Lambda _{t,t+s}^{y}\left[ \frac{P_{t}^{*}}{P_{t+s}} - \phi _{t+s}\right] y_{t+s}^{f}\right\} , \end{aligned}$$

subject to the demand curve (A7), and where $\beta ^{s}\Lambda _{t,t+s}^{y}\equiv \beta \lambda _{t+s}^{1}/\lambda _{t}^{1}$. Note that this stochastic discount factor differs from the stochastic discount factor that the young employ to discount the future cash flow from the government bond!

Apart from the change in the discount factor, the retail goods producers’ optimization problem is identical to the optimization problem when a representative households are infinitely lived. Therefore, all the first order conditions are the same, except for the replacement of the stochastic discount factor in first order conditions (A9) and (A10), which are now given by:

$$\begin{aligned} \Xi _{1,t}= & {} \phi _{t}y_{t} + E_{t}\left[ \psi \beta \Lambda _{t,t+1}^{y}\pi _{t+1}^{\epsilon }\Xi _{1,t+1}\right] , \end{aligned}$$

(A105)

$$\begin{aligned} \Xi _{2,t}= & {} y_{t} + E_{t}\left[ \psi \beta \Lambda _{t,t+1}^{y}\pi _{t+1}^{\epsilon -1}\Xi _{2,t+1}\right] . \end{aligned}$$

(A106)

However, it will be relevant to calculate period t profits $\omega _{t}^{f}$ (in terms of the final good) for retail goods producer $f\in [0,1]$:

$$\begin{aligned} \omega _{t}^{f} = \left( \frac{P_{t}^{f}}{P_{t}}-\phi _{t}\right) y_{t}^{f} = \left( \frac{P_{t}^{f}}{P_{t}}\right) ^{1-\epsilon }y_{t} - \left( \frac{P_{t}^{f}}{P_{t}}\right) ^{-\epsilon }\phi _{t}y_{t}. \end{aligned}$$

(A107)

where I substituted the demand schedule (A7) for retail goods $f\in [0,1]$.

1.2.3 Appendix C.2.3. Intermediate Goods Producers

The optimization problem of intermediate goods producers is exactly the same as in the RANK-model.

1.3 Appendix C.3. Government

1.3.1 Appendix C.3.1. Fiscal Authority

The government budget constraint is now extended by a pension payment $s_{t}$ to the old. Otherwise the budget constraint is the same as in the RANK-model, and is therefore given by:

$$\begin{aligned} \tau _{t} + b_{t} = g_{t} + \tau ^ {w}w_{t}h_{t} + s_{t} + \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1}, \end{aligned}$$

(A108)

Lump sum taxes are now not raised on a representative household, but on both the young and old generation, i.e. $\tau _{t} = \tau _{t}^{1} + \tau _{t}^{2}$, where $\tau _{t}^{1}$ and $\tau _{t}^{2}$ denotes aggregate lump sum taxes on the young and old, respectively. In order to derive at a system with only inflation and the output gap, I need an analytical expression for the output gap, something I cannot achieve in a model which still features endogenous state variables. In order to eliminate these endgenous state variables, I assume that lump sum taxes on the old $\tau _{t}^{2}$ are exactly equal to the gross interest payments on the bonds they purchased when they were young:

$$\begin{aligned} \tau _{t}^{2} = \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1}. \end{aligned}$$

(A109)

Substitution of (A109) results in the following government budget constraint:

$$\begin{aligned} \tau _{t}^{1} + b_{t} = g_{t} + \tau ^ {w}w_{t}h_{t} + s_{t}, \end{aligned}$$

(A110)

For the young, I assume that the level of lump sum taxes $\tau _{t}^{1}$ is linear in last period’s stock of government debt $b_{t-1}$:

$$\begin{aligned} \tau _{t}^{1} = \bar{\tau }_{1} + \kappa _{b}\left( b_{t-1}-\bar{b}\right) . \end{aligned}$$

(A111)

As in the RANK-model, government purchases will be equal to its steady state value under the monetary regime, while it will endogenously respond to inflation and the output gap under the fiscal regime. Finally, I assume that the pension payment $s_{t}$ is linear in $y_{t} - \delta g_{t}$ with $0\le \delta \le 1$. The intuition behind this assumption is the following. When the economy is in a boom, pensioners get paid more than when the economy is in recession, everything else equal. However, when more government spending is employed to stabilize the business cycle, the government prevents debt from increasing too much by reducing pension payments by a factor $\delta g_{t}$:

$$\begin{aligned} s_{t} = \left( \frac{\bar{s}}{\bar{y} - \delta \bar{g}}\right) \left( y_{t} - \delta g_{t} \right) , \end{aligned}$$

(A112)

1.3.2 Appendix C.3.2. Central Bank

Monetary policy is exactly the same as in the RANK-model.

1.4 Appendix C.4. Aggregation

I assume that each member of the young is identical, and chooses the same level of consumption and labor supply in equilibrium. Integrating over all young $i\in [0,1]$ gives the following expressions for aggregate consumption and labor supply of the young.

$$\begin{aligned} c_{t}^{1}\equiv \int _{0}^{1}c_{t}^{1}(i)di = c_{t}^{1}(i)\int _{0}^{1}di = c_{t}^{1}(i). \end{aligned}$$

Similarly, I find aggregate labor supply by the young to be equal to $h_{t}^{1} = h_{t}^{1}(i)$, as well as government debt holdings $b_{t} = b_{t}(i)$, aggregate lump sum taxes $\tau _{t}^{1} = \tau _{t}^{1}(i)$, and aggregate profits from production firms $\omega _{t}^{1} = \omega _{t}^{1}(i)$. Aggregation over young member $i\in [0,1]$ budget constraint (A97) then gives the aggregate young’s budget constraint:

$$\begin{aligned} c_{t}^{1} + \tau _{t}^{1} + b_{t} = w_{t}h_{t} + \omega _{t}, \end{aligned}$$

(A113)

Substitution of the government budget constraint (A110) then reads:

$$\begin{aligned} c_{t}^{1} + g_{t} + \tau ^ {w}w_{t}h_{t} + s_{t} = w_{t}h_{t} + \omega _{t}, \end{aligned}$$

(A114)

Similarly, I can integrate over member $i\in [0,1]$ of the old generation to obtain the aggregate old budget constraint:

$$\begin{aligned} c_{t}^{2} + \tau _{t}^{2} = \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1} + s_{t}, \end{aligned}$$

(A115)

Substitution of the old’s lump sum taxes (A109) gives:

$$\begin{aligned} c_{t}^{2} = s_{t}. \end{aligned}$$

(A116)

In other words, consumpton of the old is equal to the pension payment from the government.

To find aggregate profits of the retail goods producers, I integrate the profits (A107) of retail goods producer $f\in [0,1]$

$$\begin{aligned} \omega _{t} = \int _{0}^{1}\omega _{t}^{f}df = \int _{0}^{1}\left[ \left( \frac{P_{t}^{f}}{P_{t}}\right) ^{1-\epsilon }y_{t} - \left( \frac{P_{t}^{f}}{P_{t}}\right) ^{-\epsilon }\phi _{t}y_{t}\right] df = y_{t} - \mathcal {D}_{t}\phi _{t}y_{t}, \end{aligned}$$

(A117)

where I employed equations (A8) and (A13). Now I aggregate over the left hand side of production technology (A14) of intermediate goods producer $i\in [0,1]$:

$$\begin{aligned} \int _{0}^{1}y_{t}^{i}di = \int _{0}^{1}y_{t}^{f}df = y_{t}\int _{0}^{1}\left( \frac{P_{t}^{f}}{P_{t}}\right) ^{-\epsilon }df = \mathcal {D}_{t}y_{t}. \end{aligned}$$

Integration over the right hand side of equation (A14) gives:

$$\begin{aligned} \int _{0}^{1}z_{t}h_{t}^{i}di = z_{t}\int _{0}^{1}h_{t}^{i}di = z_{t}h_{t}. \end{aligned}$$

Combining the aggregated left and right hand side gives:

$$\begin{aligned} \mathcal {D}_{t}y_{t} = z_{t}h_{t}. \end{aligned}$$

(A118)

Substitution of this relation into the expression for the profits of retail goods producers gives:

$$\begin{aligned} \omega _{t} = y_{t} - \phi _{t}z_{t}h_{t} = y_{t} - \left( 1 - \tau ^{w}\right) w_{t}h_{t}, \end{aligned}$$

(A119)

where I used first order condition (A15). Substitution of Eq. (A119) into the young’s aggregate budget constraint (A114) gives:

$$\begin{aligned} c_{t}^{1} + g_{t} + s_{t} = y_{t}, \end{aligned}$$

(A120)

Substitution of the aggregate budget constraint of the old generation (A116) gives the aggregate resource constraint of the economy:

$$\begin{aligned} c_{t}^{1} + c_{t}^{2} + g_{t} = y_{t}. \end{aligned}$$

(A121)

1.5 Appendix C.5. Overview First Order Conditions (OLG)

A compeititve equilibrium is a series of quantities $\left\{ c_{t}^{1},c_{t}^{2},h_{t},y_{t},g_{t},b_{t},\tau _{t}^{1},\tau _{t}^{2},s_{t},\omega _{t}\right\}$, (shadow) prices $\left\{ \lambda _{t}^{1},\lambda _{t}^{2},R_{t}^{n},\phi _{t},w_{t},\pi _{t},\pi _{t}^{*},\mathcal {D}_{t},\Xi _{1,t},\Xi _{2,t}\right\}$, and exogenous processes $\left\{ z_{t},\xi _{t}\right\}$ satisfying the following equations:

$$\begin{aligned} \lambda _{t}^{1}= & {} \xi _{t}\left( c_{t}^{1}\right) ^{-\sigma }, \end{aligned}$$

(A122)

$$\begin{aligned} \lambda _{t}^{2}= & {} \xi _{t}\left( c_{t}^{2}\right) ^{-\sigma }, \end{aligned}$$

(A123)

$$\begin{aligned} \chi _{h} h_{t}^{\varphi }= & {} w_{t}\left( c_{t}^{1}\right) ^{-\sigma }, \end{aligned}$$

(A124)

$$\begin{aligned} E_{t}\left[ \beta \Lambda _{t,t+1}^{1,2}\left( \frac{R_{t}^{n}}{\pi _{t+1}}\right) \right]= & {} 1, \end{aligned}$$

(A125)

$$\begin{aligned} c_{t}^{2} + \tau _{t}^{2}= & {} \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1} + s_{t}, \end{aligned}$$

(A126)

$$\begin{aligned} \omega _{t}= & {} y_{t} - \phi _{t}\mathcal {D}_{t}y_{t} = y_{t} - \left( 1 - \tau ^{w}\right) w_{t}h_{t}, \end{aligned}$$

(A127)

$$\begin{aligned} \pi _{t}^{*}= & {} \left( \frac{\epsilon }{\epsilon -1}\right) \frac{\Xi _{1,t}}{\Xi _{2,t}},\end{aligned}$$

(A128)

$$\begin{aligned} \Xi _{1,t}=\,\, & {} \phi _{t}y_{t} + E_{t}\left[ \psi \beta \Lambda _{t,t+1}^{y}\pi _{t+1}^{\epsilon }\Xi _{1,t+1}\right] ,\end{aligned}$$

(A129)

$$\begin{aligned} \Xi _{2,t}=\,\, & {} y_{t} + E_{t}\left[ \psi \beta \Lambda _{t,t+1}^{y}\pi _{t+1}^{\epsilon -1}\Xi _{2,t+1}\right] ,\end{aligned}$$

(A130)

$$\begin{aligned} 1=\,\, & {} \left( 1-\psi \right) \left( \pi _{t}^{*}\right) ^{1-\epsilon } + \psi \pi _{t}^{\epsilon -1},\end{aligned}$$

(A131)

$$\begin{aligned} \mathcal {D}_{t}= \,\,& {} \left( 1-\psi \right) \left( \pi _{t}^{*}\right) ^{-\epsilon } + \psi \pi _{t}^{\epsilon }\mathcal {D}_{t-1},\end{aligned}$$

(A132)

$$\begin{aligned} \mathcal {D}_{t}y_{t}= & {} z_{t}h_{t},\end{aligned}$$

(A133)

$$\begin{aligned} \left( 1 - \tau ^{w}\right) w_{t}= & {} \phi _{t}z_{t},\end{aligned}$$

(A134)

$$\begin{aligned} y_{t}= & {} c_{t}^{1} + c_{t}^{2} + g_{t},\end{aligned}$$

(A135)

$$\begin{aligned} \tau _{t}^{1} + \tau _{t}^{2} + b_{t}= & {} g_{t} + \tau ^ {w}w_{t}h_{t} + s_{t} + \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1},\end{aligned}$$

(A136)

$$\begin{aligned} \tau _{t}^{1}= & {} \bar{\tau }_{1} + \kappa \left( b_{t-1} - \bar{b}\right) ,\end{aligned}$$

(A137)

$$\begin{aligned} \tau _{t}^{2}= & {} \left( \frac{R_{t-1}^{n}}{\pi _{t}}\right) b_{t-1},\end{aligned}$$

(A138)

$$\begin{aligned} s_{t}= & {} \left( \frac{\bar{s}}{\bar{y} - \delta \bar{g}}\right) \left( y_{t} - \delta g_{t} \right) ,\end{aligned}$$

(A139)

$$\begin{aligned} R_{t}^{n}= & {} ....,\end{aligned}$$

(A140)

$$\begin{aligned} g_{t}= & {} ......,\end{aligned}$$

(A141)

$$\begin{aligned} \log {\left( z_{t}\right) }= & {} \rho _{z}\log {\left( z_{t-1}\right) } + \varepsilon _{z,t},\end{aligned}$$

(A142)

$$\begin{aligned} \log {\left( \xi _{t}\right) }= & {} \rho _{\xi }\log {\left( \xi _{t-1}\right) } + \varepsilon _{\xi ,t}, \end{aligned}$$

(A143)

where $\beta \Lambda _{t,t+1}^{1,2}=\beta \lambda _{t+1}^{2}/\lambda _{t}^{1}$ and $\beta \Lambda _{t,t+1}^{y}=\beta \lambda _{t+1}^{1}/\lambda _{t}^{1}$.

1.6 Appendix C.6. Linearized FOCs

$$\begin{aligned} \varphi \hat{h}_{t}= & {} -\sigma \hat{c}_{t}^{1} +\hat{w}_{t}, \end{aligned}$$

(A144)

$$\begin{aligned} \hat{\xi }_{t} - \sigma \hat{c}_{t}^{1}= & {} E_{t}\left[ \hat{\xi }_{t+1} - \sigma \hat{c}_{t+1}^{2}\right] +\hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] ,\end{aligned}$$

(A145)

$$\begin{aligned} \bar{c}_{2}\hat{c}_{t}^{2} + \bar{\tau }_{2}\hat{\tau }_{t}^{2}= & {} \left( \frac{\bar{R}_{n}}{\bar{\pi }}\right) \bar{b}\left( \hat{R}_{t-1}^{n} - \hat{\pi }_{t} + \hat{b}_{t-1}\right) + \bar{s}\hat{s}_{t}, \end{aligned}$$

(A146)

$$\begin{aligned} \bar{\omega }\hat{\omega }_{t}= & {} \bar{y}\hat{y}_{t} - \left( 1 - \tau ^{w}\right) \bar{w}\bar{h}\left( \hat{w}_{t} + \hat{h}_{t}\right) , \end{aligned}$$

(A147)

$$\begin{aligned} \hat{\pi }_{t}^{*}= & {} \hat{\Xi }_{1,t} - \hat{\Xi }_{2,t}, \end{aligned}$$

(A148)

$$\begin{aligned} \hat{\Xi }_{1,t}= & {} \left( 1-\psi \beta \bar{\pi }^{\epsilon }\right) \left( \hat{\phi }_{t}+\hat{y}_{t}\right) \nonumber \\{} & {} + \psi \beta \bar{\pi }^{\epsilon }\left( -\sigma E_{t}\left[ \hat{c}_{t+1}^{1}\right] + \sigma \hat{c}_{t}^{1} + \epsilon E_{t}\left[ \hat{\pi }_{t+1}\right] +E_{t}\left[ \hat{\Xi }_{1,t+1}\right] \right) , \end{aligned}$$

(A149)

$$\begin{aligned} \hat{\Xi }_{2,t}=\,\, & {} \left( 1-\psi \beta \bar{\pi }^{\epsilon -1}\right) \hat{y}_{t}\nonumber \\+ \quad & {} \psi \beta \bar{\pi }^{\epsilon -1}\left( -\sigma E_{t}\left[ \hat{c}_{t+1}^{1}\right] + \sigma \hat{c}_{t}^{1} + \left( \epsilon -1\right) E_{t}\left[ \hat{\pi }_{t+1}\right] +E_{t}\left[ \hat{\Xi }_{2,t+1}\right] \right) ,\end{aligned}$$

(A150)

$$\begin{aligned} \left( 1-\psi \right) \left( \bar{\pi }^{*}\right) ^{1-\epsilon }\hat{\pi }_{t}^{*}= & {} \psi \left( \bar{\pi }\right) ^{\epsilon -1}\hat{\pi }_{t},\end{aligned}$$

(A151)

$$\begin{aligned} \hat{\mathcal {D}}_{t}= & {} \left( 1-\psi \left( \bar{\pi }\right) ^{\epsilon }\right) \left( -\epsilon \hat{\pi }_{t}^{*}\right) +\psi \left( \bar{\pi }\right) ^{\epsilon }\left( \epsilon \hat{\pi }_{t}+\hat{\mathcal {D}}_{t-1}\right) , \end{aligned}$$

(A152)

$$\begin{aligned} \hat{\mathcal {D}}_{t} + \hat{y}_{t}= & {} \hat{z}_{t} + \hat{h}_{t}, \end{aligned}$$

(A153)

$$\begin{aligned} \hat{w}_{t}= & {} \hat{\phi }_{t} + \hat{z}_{t}, \end{aligned}$$

(A154)

$$\begin{aligned} \hat{y}_{t}= & {} \left( \bar{c}_{1}/\bar{y}\right) \hat{c}_{t}^{1} + \left( \bar{c}_{2}/\bar{y}\right) \hat{c}_{t}^{2}+\left( \bar{g}/\bar{y}\right) \hat{g}_{t}, \end{aligned}$$

(A155)

$$\begin{aligned} \bar{\tau }_{1}\hat{\tau }_{t}^{1} + \bar{\tau }_{2}\hat{\tau }_{t}^{2} + \bar{b}\hat{b}_{t}= & {} \bar{g}\hat{g}_{t} + \tau ^{w}\bar{w}\bar{h}\left( \hat{w}_{t} + \hat{h}_{t}\right) + \bar{s}\hat{s}_{t} + \left( \frac{\bar{R}_{n}}{\bar{\pi }}\right) \bar{b}\left( \hat{R}_{t-1}^{n} - \hat{\pi }_{t} + \hat{b}_{t-1}\right) ,\end{aligned}$$

(A156)

$$\begin{aligned} \bar{\tau }_{1}\hat{\tau }_{t}^{1}= & {} \kappa \bar{b}\hat{b}_{t-1}, \end{aligned}$$

(A157)

$$\begin{aligned} \hat{\tau }_{t}^{2}= & {} \hat{R}_{t-1}^{n} - \hat{\pi }_{t} + \hat{b}_{t-1}, \end{aligned}$$

(A158)

$$\begin{aligned} \hat{s}_{t}= & {} \frac{\bar{y}}{\bar{y} - \delta \bar{g}}\left[ \hat{y}_{t} - \delta \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] , \end{aligned}$$

(A159)

$$\begin{aligned} \hat{R}_{t}^{n}= & {} ...., \end{aligned}$$

(A160)

$$\begin{aligned} \hat{g}_{t}= & {} ...., \end{aligned}$$

(A161)

$$\begin{aligned} \hat{z}_{t}= & {} \rho _{z}\hat{z}_{t-1} + \varepsilon _{z,t}, \end{aligned}$$

(A162)

$$\begin{aligned} \hat{\xi }_{t}= & {} \rho _{\xi }\hat{\xi }_{t-1} + \varepsilon _{\xi ,t}, \end{aligned}$$

(A163)

Going forward, I assume that gross steady state inflation is equal to one: $\bar{\pi }=1$. Next, I substitute Eqs. (A149) and (A150) into (A148) to obtain:

$$\begin{aligned} \hat{\pi }_{t}^{*} = \left( 1-\psi \beta \right) \hat{\phi }_{t} + \psi \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \psi \beta E_{t}\left[ \hat{\pi }_{t+1}^{*}\right] , \end{aligned}$$

(A164)

Substitution of $\hat{\pi }_{t}^{*}=\left( \psi /\left( 1-\psi \right) \right) \hat{\pi }_{t}$ (A151) delivers the traditional New Keynesian Phillips-curve:

$$\begin{aligned} \hat{\pi }_{t} = \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \zeta \hat{\phi }_{t}, \end{aligned}$$

(A165)

where $\zeta \equiv \left( 1-\psi \beta \right) \left( 1-\psi \right) /\psi$.

1.7 Appendix C.7. Further Derivations: $\delta = 0$

Next, I substitute expression (A158) into expression (A146) and find that:

$$\begin{aligned} \hat{c}_{t}^{2} = \hat{s}_{t} = \hat{y}_{t}, \end{aligned}$$

(A166)

where I employed Eq. (A159) with $\delta =0$ and the knowledge that $\bar{c}_{2}=\bar{s}$. Substitution of the above expression into Eq. (A155) gives me the following expression for $\hat{c}_{1}$:

$$\begin{aligned} \hat{c}_{t}^{1} = \frac{\bar{y}}{\bar{c}_{1}}\left[ \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] , \end{aligned}$$

(A167)

1.7.1 Appendix C.7.1. The Flexible Prices Equilibrium

Now I aim to derive the flexible prices equilibrium. To do so, I set $\psi =0$ in Eq. (A164)

$$\begin{aligned} \hat{\pi }_{t}^{*} = \hat{\phi }_{t}, \end{aligned}$$

Since I know from Eq. (A151) that $\hat{\pi }_{t}^{*}=0$ when $\psi =0$, I find that $\hat{\phi }_{t}=0$. Next, I substitute expression (A151) into Eq. (A152), and find that $\hat{\mathcal {D}}_{t}=0$, irrespective of whether $\psi =0$ or not.

Now I consider Eq. (A144), and substitute Eq. (A154) for $\hat{w}_{t}$, expression (A153) for $\hat{h}_{t}$, and Eq. (A167) to obtain:

$$\begin{aligned} \varphi \left( \hat{y}_{t} - \hat{z}_{t}\right) = -\sigma \left( \bar{y}/\bar{c}_{1}\right) \left[ \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] +\hat{z}_{t}, \end{aligned}$$

Solving for output delivers the flexible prices level of output, or the natural level of output:

$$\begin{aligned} \hat{y}_{t}^{n} = \left( \frac{1+\varphi }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \varphi }\right) \hat{z}_{t} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( \frac{\left( \bar{g}/\bar{y}\right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \varphi }\right) \hat{g}_{t}, \end{aligned}$$

(A168)

1.7.2 Appendix C.7.2. The Sticky Prices Equilibrium

Again I consider Eq.  (A144), and substitute Eq. (A154) for $\hat{w}_{t}$, expression (A153) for $\hat{h}_{t}$, and Eq. (A167). However, the difference with respect to the flexible prices equilibrium is that $\hat{\phi }_{t}$ is no longer zero in Eq. (A154). I thus obtain:

$$\begin{aligned} \varphi \left( \hat{y}_{t} - \hat{z}_{t}\right) = -\sigma \left( \bar{y}/\bar{c}_{1}\right) \left[ \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] +\hat{z}_{t} + \hat{\phi }_{t}, \end{aligned}$$

Rearranging gives:

$$\begin{aligned} \hat{\phi }_{t}= & {} \left( \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \varphi \right) \hat{y}_{t} - \left( 1+\varphi \right) \hat{z}_{t} - \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( \bar{g}/\bar{y}\right) \hat{g}_{t}\nonumber \\= & {} \left( \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \varphi \right) \left( \hat{y}_{t} - \hat{y}_{t}^{n}\right) = \left( \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \varphi \right) \tilde{y}_{t}, \end{aligned}$$

(A169)

where $\tilde{y}_{t}\equiv \hat{y}_{t} - \hat{y}_{t}^{n}$ denotes the output gap, the difference between the level of output under the sticky prices equilibrium and the flexible prices equilibrium.

Substitution of expression (A169) into Eq. (A165) delivers the familiar New Keynesian Phillips curve in the output gap $\tilde{y}_{t}$:

$$\begin{aligned} \hat{\pi }_{t} = \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t}, \end{aligned}$$

(A170)

where $\kappa$ is given by:

$$\begin{aligned} \kappa = \left( \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \varphi \right) \zeta , \end{aligned}$$

(A171)

Finally, I substitute expressions (A166) and (A167) into the Euler Eq. (A145):

$$\begin{aligned} \hat{\xi }_{t} - \sigma \left( \bar{y}/\bar{c}_{1}\right) \left[ \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] = E_{t}\left[ \hat{\xi }_{t+1} - \sigma \hat{y}_{t+1}\right] + \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] , \end{aligned}$$

(A172)

Now I substitute $\hat{y}_{t}=\hat{y}_{t}^{n} + \tilde{y}_{t}$, and substitute expression (A168) to get the aggregate demand equation:

$$\begin{aligned} \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \tilde{y}_{t} = \sigma E_{t}\left[ \tilde{y}_{t+1}\right] - \left( \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \hat{R}_{t}^{*}\right) , \end{aligned}$$

(A173)

where $\hat{R}_{t}^{*}$ is given by:

$$\begin{aligned} \hat{R}_{t}^{*} = \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *} + \hat{R}_{t}^{g*}, \end{aligned}$$

(A174)

where $\hat{R}_{t}^{z*}$, $\hat{R}_{t}^{\xi *}$, and $\hat{R}_{t}^{g*}$ are given by:

$$\begin{aligned} \hat{R}_{t}^{z*}= & {} -\left( \frac{\sigma \left( 1+\varphi \right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \varphi }\right) \left[ \frac{\bar{y}}{\bar{c}_{1}}\left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) -\rho _{z}\right] \hat{z}_{t}, \end{aligned}$$

(A175)

$$\begin{aligned} \hat{R}_{t}^{\xi *}= & {} \left( 1-\rho _{\xi }\right) \hat{\xi }_{t}, \end{aligned}$$

(A176)

$$\begin{aligned} \hat{R}_{t}^{g*}= & {} \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( \frac{\left( \bar{g}/\bar{y}\right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \varphi }\right) \left( \varphi \hat{g}_{t} + \sigma E_{t}\left[ \hat{g}_{t+1}\right] \right) , \end{aligned}$$

(A177)

1.8 Appendix C.8. Further Derivations for $\delta = 1$: Proof of Proposition 1

Next, I substitute expression (A158) into expression (A146) and find that:

$$\begin{aligned} \hat{c}_{t}^{2} = \hat{s}_{t} = \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) \left[ \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] , \end{aligned}$$

(A178)

where I employed Eq. (A159) with $\delta =1$ and the knowledge that $\bar{c}_{2}=\bar{s}$. Substitution of the above expression into Eq. (A155) gives me the following expression for $\hat{c}_{1}$:

$$\begin{aligned} \hat{c}_{t}^{1} = \frac{\bar{y}}{\bar{y} - \bar{g}}\left[ \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] , \end{aligned}$$

(A179)

Comparing Eqs. (A178) and (A179), we immediately see that $\hat{c}_{t}^{1} = \hat{c}_{t}^{2}$.

1.8.1 Appendix C.8.1. The Flexible Prices Equilibrium

Now I aim to derive the flexible prices equilibrium. To do so, I set $\psi =0$ in Eq. (A164)

$$\begin{aligned} \hat{\pi }_{t}^{*} = \hat{\phi }_{t}, \end{aligned}$$

Since I know from Eq. (A151) that $\hat{\pi }_{t}^{*}=0$ when $\psi =0$, I find that $\hat{\phi }_{t}=0$. Next, I substitute expression (A151) into Eq. (A152), and find that $\hat{\mathcal {D}}_{t}=0$, irrespective of whether $\psi =0$ or not.

Now I consider Eq. (A144), and substitute Eq. (A154) for $\hat{w}_{t}$, expression (A153) for $\hat{h}_{t}$, and Eq. (A179) to obtain:

$$\begin{aligned} \varphi \left( \hat{y}_{t} - \hat{z}_{t}\right) = -\sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) \left[ \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] +\hat{z}_{t}, \end{aligned}$$

Solving for output delivers the flexible prices level of output, or the natural level of output:

$$\begin{aligned} \hat{y}_{t}^{n} = \left( \frac{1+\varphi }{\sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) + \varphi }\right) \hat{z}_{t} + \sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) \left( \frac{\left( \bar{g}/\bar{y}\right) }{\sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) + \varphi }\right) \hat{g}_{t}, \end{aligned}$$

(A180)

Substitution of $\bar{c} = \bar{y} - \bar{g}$ into Eq. (A58) immediately shows that the natural level of output in the RANK model is equal to the natural level of output in the OLG-NK model with $\delta =1$.

1.8.2 Appendix C.8.2. The Sticky Prices Equilibrium

Again I consider Eq. (A144), and substitute Eq. (A154) for $\hat{w}_{t}$, expression (A153) for $\hat{h}_{t}$, and Eq. (A179). However, the difference with respect to the flexible prices equilibrium is that $\hat{\phi }_{t}$ is no longer zero in Eq. (A154). I thus obtain:

$$\begin{aligned} \varphi \left( \hat{y}_{t} - \hat{z}_{t}\right) = -\sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) \left[ \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] +\hat{z}_{t} + \hat{\phi }_{t}, \end{aligned}$$

Rearranging gives:

$$\begin{aligned} \hat{\phi }_{t}= & {} \left[ \sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) + \varphi \right] \hat{y}_{t} - \left( 1+\varphi \right) \hat{z}_{t} - \sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) \left( \bar{g}/\bar{y}\right) \hat{g}_{t}\nonumber \\= & {} \left[ \sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) + \varphi \right] \left( \hat{y}_{t} - \hat{y}_{t}^{n}\right) = \left[ \sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) + \varphi \right] \tilde{y}_{t}, \end{aligned}$$

(A181)

where $\tilde{y}_{t}\equiv \hat{y}_{t} - \hat{y}_{t}^{n}$ denotes the output gap, the difference between the level of output under the sticky prices equilibrium and the flexible prices equilibrium.

Substitution of expression (A181) into Eq. (A165) delivers the familiar New Keynesian Phillips curve in the output gap $\tilde{y}_{t}$:

$$\begin{aligned} \hat{\pi }_{t} = \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t}, \end{aligned}$$

(A182)

where $\kappa$ is given by:

$$\begin{aligned} \kappa = \left[ \sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) + \varphi \right] \zeta , \end{aligned}$$

(A183)

Substitution of $\bar{c} = \bar{y} - \bar{g}$ into Eq. (A61) immediately shows that the slope of the New Keynesian Phillips curve in the RANK model is equal to the slope in the OLG-NK model with $\delta =1$. Therefore, the New Keynesian Phillips curves in the RANK model and the OLG-NK model with $\delta =1$.

Finally, I substitute expressions (A178) and (A179) into the Euler equation (A145):

$$\begin{aligned} \hat{\xi }_{t} - \sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) \left[ \hat{y}_{t} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t}\right] = E_{t}\left\{ \hat{\xi }_{t+1} - \sigma \left( \frac{\bar{y}}{\bar{y} - \bar{g}}\right) \left[ \hat{y}_{t+1} - \left( \frac{\bar{g}}{\bar{y}}\right) \hat{g}_{t+1}\right] \right\} + \hat{R}_{t}^{n} - E_{t}\left[ \hat{\pi }_{t+1}\right] , \end{aligned}$$

(A184)

Substitution of $\bar{c} = \bar{y} - \bar{g}$ into Eq. (A62) immediately shows that the aggregate demand equation in the RANK model is equal to the aggregate demand equation in the OLG-NK model with $\delta =1$. Furthermore, since the natural level of output in the RANK model is equal to the natural level of output in the OLG-NK model with $\delta =1$, substitution of $\hat{y}_{t}=\hat{y}_{t}^{n} + \tilde{y}_{t}$ with $\hat{y}_{t}^{n}$ equal to (A180) will deliver the exact same aggregate demand Eq. (A63) as in the RANK model (after substitution of $\bar{c} = \bar{y} - \bar{g}$), with the exact same expressions for the natural rate of interest components (A65)–(A67). As a result, we can conclude that the OLG-NK model with $\delta =1$ is exactly equal to the RANK model. Therefore, references to the OLG-NK model will refer to the model version with $\delta =0$ in the remainder of the appendix.

1.9 Appendix C.9. Stability Conditions Under the OLG-Model

In this subsction I will investigate the conditions under which unique stable equilibria are possible in the OLG New Keynesian model. I start by inspecting the stability conditons under the monetary regime, after which I investigate the stability conditions for the fiscal regime.

1.9.1 Appendix C.9.1. The Monetary Regime

As in the main text, I employ a standard Taylor rule (11):

$$\begin{aligned} \hat{R}_{t}^{n} = \kappa _{\pi }\hat{\pi }_{t} + \kappa _{y}\tilde{y}_{t}, \end{aligned}$$

while I set government spending equal to steady state, i.e. $\hat{g}_{t}=0$. After substitution of the Taylor rule into the aggregate demand equation, and combining this with the New Keynesian Phillips curve (A182), I get the following two by two system of equations:

$$\begin{aligned} \hat{\pi }_{t}= & {} \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t},\\ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \tilde{y}_{t}= & {} \sigma E_{t}\left[ \hat{y}_{t+1}\right] - \left( \kappa _{\pi }\hat{\pi }_{t} + \kappa _{y}\tilde{y}_{t} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \hat{R}_{t}^{z*} - \hat{R}_{t}^{\xi *}\right) , \end{aligned}$$

I write this in the following way:

$$\begin{aligned} \begin{pmatrix} 1 &{} -\kappa \\ \kappa _{\pi } &{} \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \end{pmatrix}\begin{pmatrix} \hat{\pi }_{t} \\ \tilde{y}_{t} \end{pmatrix} = \begin{pmatrix} \beta &{} 0 \\ 1 &{} \sigma \end{pmatrix}\begin{pmatrix} E_{t}\left[ \hat{\pi }_{t+1}\right] \\ E_{t}\left[ \tilde{y}_{t+1}\right] \end{pmatrix} + \begin{pmatrix} 0 \\ \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *} \end{pmatrix}, \end{aligned}$$

This can be rewritten in the following way:

$$\begin{aligned} \begin{pmatrix} \hat{\pi }_{t} \\ \tilde{y}_{t} \end{pmatrix} = M\begin{pmatrix} E_{t}\left[ \hat{\pi }_{t+1}\right] \\ E_{t}\left[ \tilde{y}_{t+1}\right] \end{pmatrix} + N, \end{aligned}$$

where M and N are given by:

$$\begin{aligned} M= & {} \frac{1}{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }}\begin{pmatrix} \beta \left( \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \right) + \kappa &{} \kappa \sigma \\ -\beta \kappa _{\pi }+1 &{} \sigma \end{pmatrix}, \end{aligned}$$

(A185)

$$\begin{aligned} N= & {} \frac{1}{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }}\begin{pmatrix} \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) &{} \kappa \\ -\kappa _{\pi } &{} 1 \end{pmatrix}\begin{pmatrix} 0 \\ \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *} \end{pmatrix}, \end{aligned}$$

(A186)

and where I note that:

$$\begin{aligned} \begin{pmatrix} 1 &{} -\kappa \\ \kappa _{\pi } &{} \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \end{pmatrix}^{-1} = \frac{1}{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }}\begin{pmatrix} \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) &{} \kappa \\ -\kappa _{\pi } &{} 1 \end{pmatrix}. \end{aligned}$$

The trace and the determinant of M are given by:

$$\begin{aligned} {{\,\textrm{trace}\,}}{M}= & {} \frac{\beta \left( \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \right) + \kappa + \sigma }{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }}, \end{aligned}$$

(A187)

$$\begin{aligned} \det {M}= & {} \frac{\beta \sigma }{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }}, \end{aligned}$$

(A188)

Now I have to inspect the sign of the two roots of the matrix M to determine under which conditions I have a unique stable equilibrium. As I have two forward-looking variables, I need two eigenvalues that are smaller in absolute value than one. I start by calculating the characteristic equation of M.

$$\begin{aligned} \lambda ^2 - {{\,\textrm{trace}\,}}{M}\lambda +\det {M} = 0. \end{aligned}$$

I now employ Bullard and Mitra (2002) to determine whether this is the case. They start from the following characteristic equation: $\lambda ^2 + a_{1}\lambda + a_{0}=0$. In this case I have:

$$\begin{aligned} a_{1}= & {} -{{\,\textrm{trace}\,}}{M} = - \frac{\beta \left( \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \right) + \kappa + \sigma }{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }}, \end{aligned}$$

(A189)

$$\begin{aligned} a_{0}= & {} \det {M} = \frac{\beta \sigma }{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }}, \end{aligned}$$

(A190)

The first condition that needs to be satisfied according to Bullard and Mitra (2002) is $|a_{0}|<1$:

$$\begin{aligned} |a_{0}| = \frac{\beta \sigma }{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }} < 1, \end{aligned}$$

where I assume in line with the literature that $\kappa _{\pi }\ge 0$ and $\kappa _{y}\ge 0$. This condition can be rewritten as:

$$\begin{aligned} \kappa \kappa _{\pi } + \kappa _{y} + \sigma \left[ \frac{\bar{y}}{\bar{c}_{1}}\left( 1-\frac{\bar{c}_{2}}{\bar{y}}\right) - \beta \right] > 0. \end{aligned}$$

The second condition is that $|a_{1}|<1+a_{0}$. Again assuming that $\kappa _{\pi }\ge 0$ and $\kappa _{y}\ge 0$, I find that:

$$\begin{aligned} |a_{1}| = \frac{\beta \left( \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \right) + \kappa + \sigma }{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }}, \end{aligned}$$

Back to the condition that $|a_{1}|<1+a_{0}$, which boils down to:

$$\begin{aligned} \frac{\beta \left( \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \right) + \kappa + \sigma }{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }} < \frac{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi } + \beta \sigma }{\kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \kappa _{\pi }} \end{aligned}$$

(A191)

Multiplying by the denominator and rearranging, I can write this condition as:

$$\begin{aligned} \kappa \left( \kappa _{\pi } - 1\right) + \left( 1-\beta \right) \left[ \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) -\sigma \right] > 0, \end{aligned}$$

(A192)

1.9.2 Appendix C.9.2. The Fiscal Regime

Next, I study stability under the fiscal regime. As in the main text, I employ the government spending rule from the main text (12):

$$\begin{aligned} \hat{g}_{t} = g_{\pi }\hat{\pi }_{t}+g_{y}\tilde{y}_{t}. \end{aligned}$$

while I set the nominal interest rate equal to steady state, i.e. $\hat{R}_{t}^{n}=0$. After substitution of the government spending rule into the aggregate demand equation, and combining this with the New Keynesian Phillips curve (A182), I get the following two by two system of equations:

$$\begin{aligned} \hat{\pi }_{t}= & {} \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t},\\ -\varphi B^{*}g_{\pi }\hat{\pi }_{t} + \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}g_{y}\right] \tilde{y}_{t}= & {} \left( 1+B^{*}\sigma g_{\pi }\right) E_{t}\left[ \hat{\pi }_{t+1}\right] + \sigma \left( 1+B^{*}g_{y}\right) E_{t}\left[ \hat{y}_{t+1}\right] \\+ & {} \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *}, \end{aligned}$$

I write this in the following way:

$$\begin{aligned} \begin{pmatrix} 1 &{} -\kappa \\ -\varphi B^{*}g_{\pi } &{} \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}g_{y} \end{pmatrix}\begin{pmatrix} \hat{\pi }_{t} \\ \tilde{y}_{t} \end{pmatrix}= & {} \begin{pmatrix} \beta &{} 0 \\ 1+B^{*}\sigma g_{\pi } &{} \sigma \left( 1+B^{*}g_{y}\right) \end{pmatrix}\begin{pmatrix} E_{t}\left[ \hat{\pi }_{t+1}\right] \\ E_{t}\left[ \tilde{y}_{t+1}\right] \end{pmatrix}\\+ & {} \begin{pmatrix} 0 \\ \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *} \end{pmatrix}, \end{aligned}$$

This can be rewritten in the following way:

$$\begin{aligned} \begin{pmatrix} \hat{\pi }_{t} \\ \tilde{y}_{t} \end{pmatrix} = M\begin{pmatrix} E_{t}\left[ \hat{\pi }_{t+1}\right] \\ E_{t}\left[ \tilde{y}_{t+1}\right] \end{pmatrix} + N, \end{aligned}$$

where M and N are given by:

$$\begin{aligned} M= & {} \frac{1}{D}\begin{pmatrix} \beta \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}g_{y}\right] + \kappa \left( 1+B^{*}\sigma g_{\pi }\right) &{} \kappa \sigma \left( 1+B^{*}g_{y}\right) \\ \beta \varphi B^{*}g_{\pi }+1 +B^{*}\sigma g_{\pi } &{} \sigma \left( 1+B^{*}g_{y}\right) \end{pmatrix}, \end{aligned}$$

(A193)

$$\begin{aligned} N= & {} \frac{1}{D}\begin{pmatrix} \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}g_{y} &{} \kappa \\ \varphi B^{*}g_{\pi } &{} 1 \end{pmatrix}\begin{pmatrix} 0 \\ \hat{R}_{t}^{z*} + \hat{R}_{t}^{\xi *} \end{pmatrix}, \end{aligned}$$

(A194)

where D is given by:

$$\begin{aligned} D = \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}\left( g_{y} + \kappa g_{\pi }\right) , \end{aligned}$$

(A195)

and where I note that:

$$\begin{aligned} \begin{pmatrix} 1 &{} -\kappa \\ -\varphi B^{*}g_{\pi } &{} \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}g_{y} \end{pmatrix}^{-1} = \frac{1}{D}\begin{pmatrix} \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}g_{y} &{} \kappa \\ \varphi B^{*}g_{\pi } &{} 1 \end{pmatrix}. \end{aligned}$$

The trace and the determinant of M are given by:

$$\begin{aligned} {{\,\textrm{trace}\,}}{M}= & {} \frac{\beta \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}g_{y}\right] + \kappa \left( 1+B^{*}\sigma g_{\pi }\right) + \sigma \left( 1+B^{*}g_{y}\right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}\left( g_{y} + \kappa g_{\pi }\right) }, \end{aligned}$$

(A196)

$$\begin{aligned} \det {M}= & {} \frac{\beta \sigma \left( 1+B^{*}g_{y}\right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}\left( g_{y} + \kappa g_{\pi }\right) }, \end{aligned}$$

(A197)

As such, I now obtain the following Bullard and Mitra (2002) coefficients:

$$\begin{aligned} a_{1}= & {} -{{\,\textrm{trace}\,}}{M} = - \frac{\beta \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}g_{y}\right] + \kappa \left( 1+B^{*}\sigma g_{\pi }\right) + \sigma \left( 1+B^{*}g_{y}\right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}\left( g_{y} + \kappa g_{\pi }\right) }, \end{aligned}$$

(A198)

$$\begin{aligned} a_{0}= & {} \det {M} = \frac{\beta \sigma \left( 1+B^{*}g_{y}\right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}\left( g_{y} + \kappa g_{\pi }\right) }, \end{aligned}$$

(A199)

1.9.3 Appendix C.9.3. Proof of Proposition 5

Proof

To prove the proposition, we need to prove that $|a_{0}|<1$ and $|a_{1}|<1+a_{0}$. Looking at expression (A199), we immediately see that the denominator is positive for $g_{\pi }<0$ and $g_{y}=0$. Therefore, I can write the condition $|a_{0}|<1$ as:

$$\begin{aligned} \beta \sigma < \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \kappa \varphi B^{*}g_{\pi }, \end{aligned}$$

which can be rewritten as:

$$\begin{aligned} \kappa \varphi B^{*}g_{\pi } < \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \beta \sigma = \sigma \left( 1 + \bar{g}/\bar{c}_{1} - \beta \right) , \end{aligned}$$

where I used that $\left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \bar{c}_{2}/\bar{y}\right) =1+\bar{g}/\bar{c}_{1}>0$ with the help of the aggregate resource constraint (A121). We immediately see that the inequality holds for any $g_{\pi }<0$ and $g_{y}=0$, as the right hand side of the inequality is larger than zero.

Looking at the second condition, $|a_{1}|<1+a_{0}$, we immediately see from expression (A198) that the denominator is positive for $g_{\pi }<0$ and $g_{y}=0$. Now let us consider the case where the numerator of the fraction is positive, which is the case when the following inequaltiy holds:

$$\begin{aligned} g_{\pi } > - \frac{\beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa + \sigma }{\kappa B^{*}\sigma }. \end{aligned}$$

(A200)

Hence, a lower bound exists which is a negative number. Next, I consider the condition $|a_{1}|<1+a_{0}$, which boils down to:

$$\begin{aligned} \beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa \left( 1+B^{*}\sigma g_{\pi }\right) + \sigma < \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \kappa B^{*}\varphi g_{\pi } + \beta \sigma , \end{aligned}$$

after multiplication of the left and right hand side with the positive denominator $\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \kappa B^{*}\varphi g_{\pi }$. The above inequality can be rewritten as:

$$\begin{aligned} g_{\pi } < \frac{\left( 1-\beta \right) \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \right] - \kappa }{\kappa B^{*}\left( \sigma +\varphi \right) }. \end{aligned}$$

(A201)

Finally, I need to check that the upper bound in (A201) is above the lower bound in (A200):

$$\begin{aligned} - \frac{\beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa + \sigma }{\kappa B^{*}\sigma } < \frac{\left( 1-\beta \right) \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \right] - \kappa }{\kappa B^{*}\left( \sigma +\varphi \right) }, \end{aligned}$$

which can be rewritten as:

$$\begin{aligned}- & {} \kappa B^{*}\left( \sigma +\varphi \right) \left[ \beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa + \sigma \right] \\< & {} \kappa B^{*}\sigma \left\{ \left( 1-\beta \right) \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \right] - \kappa \right\} . \end{aligned}$$

Working out the brackets, I can rewrite the above inequaltiy as:

$$\begin{aligned}- & {} \kappa B^{*}\sigma \left[ \beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa + \sigma \right] - \kappa B^{*}\varphi \left[ \beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa + \sigma \right] \\< & {} \kappa B^{*}\sigma \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma + \beta \sigma - \kappa \right] . \end{aligned}$$

Factoring out the negative terms on the right hand side gives:

$$\begin{aligned}- & {} \kappa B^{*}\sigma \left[ \beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa + \sigma \right] - \kappa B^{*}\varphi \left[ \beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa + \sigma \right] \\< & {} - \kappa B^{*}\sigma \left[ \beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \sigma + \kappa \right] + \kappa B^{*}\sigma \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \beta \sigma \right] . \end{aligned}$$

From the above inequality, we clearly see that the first terms on the left and right hand side cancel, as a result of which we end up with the following inequality:

$$\begin{aligned} - \kappa B^{*}\varphi \left[ \beta \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \kappa + \sigma \right] < \kappa B^{*}\sigma \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \beta \sigma \right] , \end{aligned}$$

which is true, since the left hand side is a negative number and the right hand side a positive number (where we remember that $\left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \bar{c}_{2}/\bar{y}\right) =1+\bar{g}/\bar{c}_{1}>0$). Hence the upper bound is above the lower bound, which in turn is a negative number, see (A200). Therefore, a unique stable countercyclical equilibrium exists, and part of the region for which this unique stable equilibrium exists will have $g_{\pi } < 0$. This concludes the proof.

1.10 Appendix C.10. Analytical Expressions for Impulse Response Functions

In the previous section I showed that there is an isomorphic mapping from the stability conditions for the monetary regime to the stability conditions for the fiscal regime. In this section I calculate analytical expressions for the impulse response functions to the productivity and preference shocks, and show that there also exists an isomorphic mapping between the coefficients of the monetary and fiscal policy reactions such that the impulse response functions are identical.

1.10.1 Appendix C.10.1. Monetary Regime

I start by writing down the two-system equations for the monetary regime, where I replace $\hat{R}_{t}^{x*} = \mathcal {R}^{x*}\hat{x}_{t}$, where $x=\lbrace z,\xi \rbrace$.

$$\begin{aligned} \hat{\pi }_{t}= & {} \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t},\\ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) \tilde{y}_{t}= & {} \sigma E_{t}\left[ \hat{y}_{t+1}\right] - \left( \kappa _{\pi }\hat{\pi }_{t} + \kappa _{y}\tilde{y}_{t} - E_{t}\left[ \hat{\pi }_{t+1}\right] - \mathcal {R}^{z*}\hat{z}_{t} - \mathcal {R}^{\xi *}\hat{\xi }_{t}\right) , \end{aligned}$$

Since there are no endogenous backward-looking state variables, I know that the only state variables are $\hat{z}_{t}$ and $\hat{\xi }_{t}$. Hence I can employ the method of undetermined coefficients to find the analytical solution to productivity and preference shocks. I assume that $\hat{\pi }_{t}$ and $\tilde{y}_{t}$ are given by the following solutions:

$$\begin{aligned} \hat{\pi }_{t}= & {} \alpha _{\pi ,z}\hat{z}_{t} + \alpha _{\pi ,\xi }\hat{\xi }_{t}, \end{aligned}$$

(A202)

$$\begin{aligned} \tilde{y}_{t}= & {} \alpha _{y,z}\hat{z}_{t} + \alpha _{y,\xi }\hat{\xi }_{t}. \end{aligned}$$

(A203)

Since both shocks are given by exogenous AR(1) shocks, I know that their expected value is given by:

$$\begin{aligned} E_{t}\left[ \hat{\pi }_{t+1}\right]= & {} \rho _{z}\alpha _{\pi ,z}\hat{z}_{t} + \rho _{\xi }\alpha _{\pi ,\xi }\hat{\xi }_{t}, \end{aligned}$$

(A204)

$$\begin{aligned} E_{t}\left[ \tilde{y}_{t+1}\right]= & {} \rho _{z}\alpha _{y,z}\hat{z}_{t} + \rho _{\xi }\alpha _{y,\xi }\hat{\xi }_{t}. \end{aligned}$$

(A205)

Substitution of the above expressions into the New Keynesian Phillips curve gives the following relations between the inflation and output gap coefficients:

$$\begin{aligned} \alpha _{y,z}= & {} \left( \frac{1-\beta \rho _{z}}{\kappa }\right) \alpha _{\pi ,z}, \end{aligned}$$

(A206)

$$\begin{aligned} \alpha _{y,\xi }= & {} \left( \frac{1-\beta \rho _{\xi }}{\kappa }\right) \alpha _{\pi ,\xi } \end{aligned}$$

(A207)

Substitution of the guessed solutions for the output gap and inflation, and the relation between the output gap coefficients and the inflation coefficients into the aggregate demand equation generate the following expressions for the coefficients:

$$\begin{aligned} \alpha _{\pi ,z}^{m}= & {} \frac{\kappa \mathcal {R}^{z*}}{\kappa \left( \kappa _{\pi }-\rho _{z}\right) +\left[ \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \rho _{z}\right] \left( 1-\beta \rho _{z}\right) }, \end{aligned}$$

(A208)

$$\begin{aligned} \alpha _{\pi ,\xi }^{m}= & {} \frac{\kappa \mathcal {R}^{\xi *}}{\kappa \left( \kappa _{\pi }-\rho _{\xi }\right) +\left[ \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \rho _{\xi }\right] \left( 1-\beta \rho _{\xi }\right) }, \end{aligned}$$

(A209)

$$\begin{aligned} \alpha _{y,z}^{m}= & {} \frac{\left( 1-\beta \rho _{z}\right) \mathcal {R}^{z*}}{\kappa \left( \kappa _{\pi }-\rho _{z}\right) +\left[ \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \rho _{z}\right] \left( 1-\beta \rho _{z}\right) }, \end{aligned}$$

(A210)

$$\begin{aligned} \alpha _{y,\xi }^{m}= & {} \frac{\left( 1-\beta \rho _{\xi }\right) \mathcal {R}^{\xi *}}{\kappa \left( \kappa _{\pi }-\rho _{\xi }\right) +\left[ \kappa _{y} + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \rho _{\xi }\right] \left( 1-\beta \rho _{\xi }\right) }, \end{aligned}$$

(A211)

where $\mathcal {R}^{z*}$ and $\mathcal {R}^{\xi *}$ are given by:

$$\begin{aligned} \mathcal {R}^{z*}= & {} -\left( \frac{\sigma \left( 1+\varphi \right) }{\sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) + \varphi }\right) \left( \frac{\bar{y}}{\bar{c}_{1}}\left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) -\rho _{z}\right) , \end{aligned}$$

(A212)

$$\begin{aligned} \mathcal {R}^{\xi *}= & {} \left( 1-\rho _{\xi }\right) \end{aligned}$$

(A213)

1.10.2 C.10.2. Fiscal Regime

Next I solve for the impulse response fucnctions under the fiscal regime. The two equation system is again given by:

$$\begin{aligned} \hat{\pi }_{t}= & {} \beta E_{t}\left[ \hat{\pi }_{t+1}\right] + \kappa \tilde{y}_{t},\\ -\varphi B^{*}g_{\pi }\hat{\pi }_{t} + \left[ \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \varphi B^{*}g_{y}\right] \tilde{y}_{t}= & {} \left( 1+B^{*}\sigma g_{\pi }\right) E_{t}\left[ \hat{\pi }_{t+1}\right] + \sigma \left( 1+B^{*}g_{y}\right) E_{t}\left[ \hat{y}_{t+1}\right] \\{} & {} + \mathcal {R}^{z*}\hat{z}_{t} + \mathcal {R}^{\xi *}\hat{\xi }_{t}, \end{aligned}$$

Again employing the method of undetermined coefficients generates the same relationship between the inflation coefficients and the output gap coefficients, and eventually results in the following expressions for the coefficients:

$$\begin{aligned} \alpha _{\pi ,z}^{f}= & {} \frac{\kappa \mathcal {R}^{z*}}{\kappa \left[ -B^{*}g_{\pi }\left( \varphi +\sigma \rho _{z}\right) -\rho _{z}\right] +\left[ -B^{*}g_{y}\left( \varphi +\sigma \rho _{z}\right) + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \rho _{z}\right] \left( 1-\beta \rho _{z}\right) }, \end{aligned}$$

(A214)

$$\begin{aligned} \alpha _{\pi ,\xi }^{f}= & {} \frac{\kappa \mathcal {R}^{\xi *}}{\kappa \left[ -B^{*}g_{\pi }\left( \varphi +\sigma \rho _{\xi }\right) -\rho _{\xi }\right] +\left[ -B^{*}g_{y}\left( \varphi +\sigma \rho _{\xi }\right) + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \rho _{\xi }\right] \left( 1-\beta \rho _{\xi }\right) }, \end{aligned}$$

(A215)

$$\begin{aligned} \alpha _{y,z}^{f}= & {} \frac{\left( 1-\beta \rho _{z}\right) \mathcal {R}^{z*}}{\kappa \left[ -B^{*}g_{\pi }\left( \varphi +\sigma \rho _{z}\right) -\rho _{z}\right] +\left[ -B^{*}g_{y}\left( \varphi +\sigma \rho _{z}\right) + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \rho _{z}\right] \left( 1-\beta \rho _{z}\right) }, \end{aligned}$$

(A216)

$$\begin{aligned} \alpha _{y,\xi }^{f}= & {} \frac{\left( 1-\beta \rho _{\xi }\right) \mathcal {R}^{\xi *}}{\kappa \left[ -B^{*}g_{\pi }\left( \varphi +\sigma \rho _{\xi }\right) -\rho _{\xi }\right] +\left[ -B^{*}g_{y}\left( \varphi +\sigma \rho _{\xi }\right) + \sigma \left( \bar{y}/\bar{c}_{1}\right) \left( 1 - \frac{\bar{c}_{2}}{\bar{y}}\right) - \sigma \rho _{\xi }\right] \left( 1-\beta \rho _{\xi }\right) }, \end{aligned}$$

(A217)

Comparing the solutions (A208)–(A211) under the monetary regime with those under the fiscal regime (A214)–(A217), I see that there is an isomorphic mapping under which the equilibrium paths for inflation and the output gap are identical under both regimes. This is the case for the productivity shock when:

$$\begin{aligned} \kappa _{\pi }= & {} -B^{*}\left( \sigma \rho _{z}+\varphi \right) g_{\pi }\Longrightarrow g_{\pi } = -\frac{\kappa _{\pi }}{B^{*}\left( \sigma \rho _{z}+\varphi \right) }, \end{aligned}$$

(A218)

$$\begin{aligned} \kappa _{y}= & {} -B^{*}\left( \sigma \rho _{z}+\varphi \right) g_{y}\Longrightarrow g_{y} = -\frac{\kappa _{y}}{B^{*}\left( \sigma \rho _{z}+\varphi \right) }, \end{aligned}$$

(A219)

while I have the following mapping for the preference shock:

$$\begin{aligned} \kappa _{\pi }= & {} -B^{*}\left( \sigma \rho _{\xi }+\varphi \right) g_{\pi }\Longrightarrow g_{\pi } = -\frac{\kappa _{\pi }}{B^{*}\left( \sigma \rho _{\xi }+\varphi \right) }, \end{aligned}$$

(A220)

$$\begin{aligned} \kappa _{y}= & {} -B^{*}\left( \sigma \rho _{\xi }+\varphi \right) g_{y}\Longrightarrow g_{y} = -\frac{\kappa _{y}}{B^{*}\left( \sigma \rho _{\xi }+\varphi \right) }, \end{aligned}$$

(A221)

Appendix D. Calibration

The numerical values for the relevant parameters can be found in Table 1.

Table 1 List of calibrated parameter values and source of calibration