Monetary Policy, Incomplete Information, and the Zero Lower Bound

Abstract

In the context of a stylized New Keynesian model, we explore the interaction between imperfect knowledge about the state of the economy and the zero lower bound. We show that optimal policy under discretion near the zero lower bound responds to signals about an increase in the equilibrium real interest rate by less than it would when far from the zero lower bound. In addition, we show that Taylor-type rules that either include a time-varying intercept that moves with perceived changes in the equilibrium real rate or respond aggressively to deviations of inflation and output from their target levels perform similarly to optimal discretionary policy. Our analysis of first-difference rules highlights that rules with interest rate smoothing terms carry forward current and past misperceptions about the state of the economy and can lead to suboptimal performance.

Appendix

1.1 A Detailed Description of the Model

The economy consists of a representative household, a continuum of monopolistically competitive firms producing differentiated intermediate goods, perfectly competitive final goods firms, and a central bank that is responsible for monetary policy.

1.2 Households

There is a representative household that values streams of consumption $C_{t}$ and hours worked $H_{t}$ according to preferences given by the utility function

$$E_{t}\sum _{\ell =0}^{\infty }\delta _{t+\ell -1}\left( \log \left[ C_{t+\ell }\right] -H_{t+\ell }\right)$$

(12)

where $E_{t}$ represents the mathematical expectation conditional on all exogenous shocks and endogenous prices and quantities up to time t. The household maximizes expected utility flows subject to a sequence of budget constraints,

$$P_{t}C_{t}+R_{t}^{-1}B_{t}=W_{t}H_{t}+B_{t-1}+T_{t}.$$

(13)

where household nominal expenditures on consumption at date t are given by $P_{t}C_{t}$, where $P_{t}$ denotes the aggregate price level for consumption goods, $B_{t}$ are risk-less nominal bonds sold at the price $R_{t}^{-1}$ in period $t-1$, and $R_{t}=(1+i_{t})$ denotes the gross nominal return on these bonds. A household receives income from any bonds carried over from last period in addition to its labor income, $W_{t}H_{t}$. A household also pays lump-sum taxes and receives dividends from intermediate goods firms, the sum of which is denoted by $T_{t}$.

To capture exogenous changes in the household’s desire to save, we allow for stochastic time discounting. The rate of time discounting at time t is given by $\delta _{t}=\beta ^{t}\left( \prod \nolimits _{s=0}^{t}1+\eta _{t+s}\right) ^{-1},$ where $0<\beta <1$. Hence, ${\frac{\delta _{t+1}}{\delta _{t}}}={\frac{\beta }{1+\eta _{t}}}$, and $\eta _{t}$ is the shock to the rate of time discounting and follows an AR(1) process in logs, such that

$$\log (1+\eta _{t})=\rho _{\eta }\log (1+\eta _{t-1})+\varepsilon _{\eta, t}$$

(14)

where $\rho _{\eta }$ denotes the persistence of the shock and $\sigma _{\eta }$ denotes the standard deviation of the innovations, $\varepsilon _{\eta, t}$. A decrease in $\eta _{t}$ represents an exogenous factor that induces a temporary rise in a household’s propensity to save, and reduces current household demand for goods.

The first-order necessary conditions for optimality from the household’s problem can be written as follows

$$1={\frac{\beta }{1+\eta _{t}}}R_{t}E_{t}\left\{ {\frac{C_{t}}{C_{t+1}}}\Pi _{t+1}^{-1}\right\},$$

(15)

$$w_{t}=C_{t}$$

(16)

where $w_{t}\equiv {\frac{W_{t}}{P_{t}}}$ denotes the real wage, and $\Pi _{t+1}\equiv {\frac{P_{t+1}}{P_{t}}}$ is the inflation rate between t and $t+1$. The linear specification for labor services in Eq. (12) along with a competitive labor market implies that there is a perfectly elastic supply of labor available to the economy’s firms.

1.3 Firms

There is a continuum of monopolistically competitive firms producing differentiated intermediate goods. The latter are used as inputs by perfectly competitive firms that produce a single final good, $Y_{t}$, using a constant returns to scale production technology, $Y_{t}=\left( \int _{0}^{1}Y_{t}(j)^{\frac{\epsilon -1}{\epsilon }}\ dj\right) ^{\frac{\epsilon }{\epsilon -1}}$, where $Y_{t}(j)$ is the quantity of intermediate good j used as an input, and $\epsilon >1$ is the elasticity of substitution. Profit maximization and perfect competition yield the set of demand schedules, $Y_{t}(j)=\left( {\frac{P_{t}(j)}{P_{t}}}\right) ^{-\epsilon }Y_{t}$ and aggregate price index, $P_{t}=\left( \int _{0}^{1}P_{t}(j)^{1-\epsilon }\ dj\right) ^{\frac{1}{1-\epsilon }}$.

The production function for intermediate good j is given by

$$Y_{t}(j)=H_{t}(j).$$

(17)

Since intermediate goods are imperfect substitutes, the intermediate goods-producing firms sell their output in a monopolistically competitive market. During period t, the firm sets its nominal price $P_{t}(j)$, subject to the requirement that it satisfies the demand of the representative final goods producer at that price.

A fraction $1-\theta$ of firms are able to change their price optimally in any given period. The remaining firms change their price at a rate consistent with the target level of inflation, which is determined exogenously. Firms that change their price solve the following problem

$$\max _{\tilde{P}_{t}}E_{t}\sum _{\ell =0}^{\infty }\theta ^{\ell }{\frac{\delta _{t-1+\ell }}{\delta _{t-1}C_{t+\ell }}}Y_{t+\ell }\left\{ \left( {\frac{\tilde{P}_{t}{\bar{\Pi }}^{\ell }}{P_{t+\ell }}}\right) ^{1-\epsilon }-\left( {\frac{\tilde{P}_{t}{\bar{\Pi }}^{\ell }}{P_{t+\ell }}}\right) ^{-\epsilon }(1-\tau _{t})w_{t+\ell }\right\},$$

where ${\frac{\delta _{t-1+\ell }}{\delta _{t-1}C_{t+\ell }}}$ measures the marginal value—in time t utility terms—to the representative household of an additional unit of real profits received in the form of dividends during period $t+\ell$, $w_{t+\ell }$ is the firm’s real marginal cost at time $t+\ell$, and $\tau _{t}$ is an employment subsidy that is financed out of lump-sum taxes paid by the household. We assume that

$$1-\tau _{t}={\frac{\epsilon -1}{\epsilon }}(1+\mu _{t})$$

(18)

and

$$\log (1+\mu _{t})=\rho _{\mu }\log (1+\mu _{t-1})+\epsilon _{\mu, t}$$

(19)

where $\epsilon _{\mu, t}\sim N(0,\sigma _{\mu }^{2})$ and can be interpreted as an innovation in the price-over-wage markup. The first-order condition for this problem is

$$E_{t}\sum _{\ell =0}^{\infty }\theta ^{\ell }{\frac{\delta _{t-1+\ell }}{\delta _{t-1}C_{t+\ell }}}Y_{t+\ell }\left( {\frac{{\bar{\Pi }}^{\ell }}{P_{t+\ell }}}\right) ^{-\epsilon }\left\{ {\frac{\tilde{P}_{t}{\bar{\Pi }}^{\ell }}{P_{t+\ell }}}-{\frac{\epsilon }{\epsilon -1}}(1-\tau _{t})w_{t+\ell }\right\} =0,$$

(20)

Define $\tilde{p}_{t}\equiv \tilde{P}_{t}/P_{t}$ and $\Pi _{t,t+\ell }\equiv {\frac{P_{t+\ell }}{P_{t}}}$. Then the first-order condition becomes

$$\tilde{p}_{t}= {\frac{E_{t}\sum _{\ell =0}^{\infty }\theta ^{\ell }{\frac{\delta _{t-1+\ell }}{\delta _{t-1}C_{t+\ell }}}Y_{t+\ell }\left( {\frac{\Pi _{t,t+\ell }}{{\bar{\Pi }}^{\ell }}}\right) ^{\epsilon }{\frac{\epsilon }{\epsilon -1}}(1-\tau _{t})w_{t+\ell }}{E_{t}\sum _{\ell =0}^{\infty }\theta ^{\ell }{\frac{\delta _{t-1+\ell }}{\delta _{t-1}C_{t+\ell }}}Y_{t+\ell }\left( {\frac{\Pi _{t,t+\ell }}{{\bar{\Pi }}^{\ell }}}\right) ^{\epsilon -1}}}\equiv {\frac{K_{t}}{F_{t}}}$$

(21)

$$K_{t}= {\frac{\epsilon }{\epsilon -1}}(1-\tau _{t})w_{t}+{\frac{\beta }{1+\eta _{t}}}\theta E_{t}\left( {\frac{\Pi _{t+1}}{{\bar{\Pi }}}}\right) ^{\epsilon }K_{t+1}$$

(22)

$$F_{t}= {\frac{Y_{t}}{C_{t}}}+{\frac{\beta }{1+\eta _{t}}}\theta E_{t}\left( {\frac{\Pi _{t+1}}{{\bar{\Pi }}}}\right) ^{\epsilon -1}F_{t+1}$$

(23)

1.4 Price Dispersion and Market Clearing

Prices evolve so that

$$P_{t}^{1-\epsilon }=\theta {\bar{\Pi }}^{1-\epsilon }P_{t-1}^{1-\epsilon }+(1-\theta )\tilde{P}_{t}^{1-\epsilon }$$

(24)

meaning that

$$\tilde{p}_{t}=\left[ {\frac{1-\theta {\bar{\Pi }}^{1-\epsilon }\Pi _{t}^{\epsilon -1}}{1-\theta }}\right] ^{\frac{1}{1-\epsilon }}$$

(25)

From demand curves,

$$Y_{t}=p_{t}^{*}H_{t} \quad {\text{where}}\quad p_{t}^{*}\equiv \left( {\frac{P_{t}^{*}}{P_{t}}}\right) ^{\epsilon }\quad {\text{and}}\quad P_{t}^{*}\equiv \left[ \int _{0}^{1}P_{t}(j)^{-\epsilon }dj\right] ^{-\frac{1}{\epsilon }}$$

(26)

which means that

$$\left(p_{t}^{*}\right)^{-1}=(1-\theta )\tilde{p}_{t}^{-\epsilon }+\theta {\bar{\Pi }}^{-\epsilon }\Pi _{t}^{\epsilon }\left(p_{t-1}^{*}\right)^{-1}$$

(27)

The aggregate resource constraint can be written as

$$p_{t}^{*}H_{t}=Y_{t}=C_{t}.$$

(28)

Finally, bonds are in zero net supply so bond market clearing implies $B_{t}=0$.

1.5 Linearization

The log-linearization of Eqs. (25) and (27) implies that, to first order, $p_{t}^{*}\approx 1$. Equations (21), (22), (23), and (25) become

$$\pi _{t}={\frac{(1-\theta )(1-\beta \theta )}{\theta }}\left[ y_{t}+\mu _{t}\right] +\beta E_{t}\left\{ \pi _{t+1}\right\}$$

(29)

where $x_{t}\equiv \log (X_{t}/X)$ where X denotes the steady-state value of $X_{t}$. The intertemporal Euler equation is

$$y_{t}=E_{t}\left\{ y_{t+1}\right\} -E_{t}\left\{ i_{t}-\pi _{t+1}-\eta _{t}\right\}$$

(30)

where, in a slight abuse of notation, we have used $i_{t}\equiv \log (R_{t}/R)$.

1.6 Natural and Efficient Levels

The natural rate of output and the real interest rate are defined as the levels in an associated economy without price rigidities. The flexible price equilibrium is characterized by the following two equations

$$\begin{aligned}&&1={\frac{\beta R_{t}^{n}}{\eta _{t}}}E_{t}\left\{ {\frac{Y_{t}^{n}}{Y_{t+1}^{n}}}\right\} \\ &&(1-\tau _{t})Y_{t}^{n}={\frac{\epsilon -1}{\epsilon }} \end{aligned}$$

where $R_{t}^{n}$ is the real natural rate of interest.

We assume that, in the nonstochastic steady state, the employment subsidy is set such that it offsets the monopoly distortion. A log-linearized approximation around the natural equilibrium steady state yields

$$\begin{aligned} r_{t}^{n}&=\eta _{t}-E_{t}\{\mu _{t+1}-\mu _{t}\}\\ y_{t}^{n}&=-\mu _{t} \end{aligned}$$

A decrease in the desired to save (an increase in $\eta _{t}$) and expected decrease in marginal costs cause the natural rate of interest to rise. The efficient equilibrium corresponds to that of flexible prices and no exogenous variation in marginal costs. The efficient allocations can be described as follows

$$\begin{aligned} 1&= {\frac{\beta R_{t}^{e}}{\eta _{t}}}E_{t}\left\{ {\frac{Y_{t}^{e}}{Y_{t+1}^{e}}}\right\} \\ Y_{t}^{e}&= 1 \end{aligned}$$

where $R_{t}^{e}$ is the efficient real interest rate. The log-linear approximation around the efficient nonstochastic steady state implies that

$$r_{t}^{e}=\eta _{t}$$

and the efficient level of output is constant.

1.7 Solution Method

To solve the nonlinear model, under optimal discretionary policy, we use a projection method similar to Gust, López-Salido, and Smith (2012) and Lawrence and Fisher (2000). The solution algorithm involves parameterizing the unknown functions $E_{t}x_{t+1}=f^{x}\left( \xi _{t},s_{t},\xi _{t-1}\right)$, $E_{t}\pi _{t+1}=f^{\pi }\left( \xi _{t},s_{t},\xi _{t-1}\right),$ where $E_{t}$ denotes rational expectations based on private-sector information. Then we can write the central bank’s first-order condition, using (1), (2), and (10), as

$$i_{t}-{\frac{\phi _{t}}{\lambda +\kappa ^{2}}}=E\left[ \left( E_{t}x_{t+1}+E_{t}\pi _{t+1}+r_{t}^{e}\right) +\left( \beta E_{t}\pi _{t+1}+\kappa \mu _{t}\right) {\frac{\kappa }{\lambda +\kappa ^{2}}}\Big |\left\{ s_{t},\xi _{t-1},\xi _{t-2}, \ldots \right\} \right]$$

We parameterize a function $f^{r}\left( s_{t},\xi _{t-1}\right)$, defined so that

$$i_{t}-{\frac{\phi _{t}}{\lambda +\kappa ^{2}}}=f^{r}\left( s_{t},\xi _{t-1}\right) +E\left\{r_{t}^{e}+\left( {\frac{\kappa ^{2}}{\lambda +\kappa ^{2}}}\right) \mu _{t}|\left\{ s_{t},\xi _{t-1}\right\} \right\}$$

To solve for the optimal policy we use the fact that $\phi _{t}=0$ when $i_{t}$ is greater than or equal to the lower bound constraint. Notice that this is essentially an application of the approach used in Lawrence and Fisher (2000); however, in our model we also include unknown time t expectations of private agents in the parameterized expectations of the central bank.

To be consistent with the information structure, the function $f^{r}$ does not take $\xi _{t}$ as an argument. This ensures that the solution imposes that the monetary authority is unable to use information encoded in the private-sector decisions in the current period. To solve for the unknown parameters of these functions, we conjecture a guess and iterate until the parameters satisfy the private-sector equilibrium conditions and the central bank’s first-order condition at a finite number of points. Note that the above expression can be used to determine the short-term interest rate and the Lagrange multiplier $\phi _{t}$ as follows. If the right-hand side is positive ($f^{r}\left( s_{t},\xi _{t-1}\right) +E\left\{r_{t}^{e}+\left( {\frac{\kappa ^{2}}{\lambda +\kappa ^{2}}}\right) \mu _{t}|\left\{ s_{t},\xi _{t-1}\right\} \right\}>0$) then $i_{t}=f^{r}\left( s_{t},\xi _{t-1}\right) +E\left\{r_{t}^{e}+\left( {\frac{\kappa ^{2}}{\lambda +\kappa ^{2}}}\right) \mu _{t}|\left\{ s_{t},\xi _{t-1}\right\} \right\}$; otherwise $i_{t}=0$ and $\phi _{t}=-\left[f^{r}\left( s_{t},\xi _{t-1}\right) +E\left\{r_{t}^{e}+\left( {\frac{\kappa ^{2}}{\lambda +\kappa ^{2}}}\right) \mu _{t}|\left\{ s_{t},\xi _{t-1}\right\} \right\}\right]$.