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.Footnote 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.Footnote 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.Footnote 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 oldFootnote 4:

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