A Constant Gain Learning Explanation of U.S. Post War Inflation and Unemployment

Abstract

An adaptive learning model, similar to Primiceri (Q J Econ 121(3):867–901, 2007) is built, where policymakers use constant gain learning algorithm to update their knowledge of the model every time period. This framework is used to study post war US inflation and unemployment. The model accurately explains the Great Inflation and gives us interesting results—while the rational expectations equilibrium is characterized by low inflation, learning leads to disequilibrium dynamics when initial knowledge of the model is incorrect. Specifically, policymakers in 1960s under estimated the natural rate, persistence of inflation and slope of Phillips Curve. Hence, policy was more expansionary than optimal, resulting in inflation. The convergence of beliefs to rational expectations equilibrium explains the subsequent disinflation in 1980s. This study differs from Primiceri (2007) in the following ways: (i) The results establish the presence of time inconsistency in policy between 1963–1979. Time inconsistency and the resulting inflation bias is explained as the rational, endogenous outcome of evolving beliefs. Hence, the possibility of a repetition of 70s cannot be eliminated in the future, for similar outcomes can be expected under similar beliefs. (ii) We find evidence against the narrative of ‘Volcker disinflation’, which credits the disinflation of 80s to the appointment of Volcker, an ‘inflation hawk’, as the chairman of Federal Reserve. (iii) Further, as an extension of Primiceri (2007), who provides point estimates, time varying estimates of persistence of inflation and slope of Phillips Curve are estimated to provide quantitative precision to the narrative proposed. This extension essentially arises from the Lucas critique. (iv) Time varying sacrifice ratio is calculated for the entire sample period, which, to our knowledge, is the first attempt for the US economy.

Appendix

In this section, simpler version of the model considered in this study is solved to theoretically establish the role of ‘k’ in determining the optimal policy every period and hence future path of inflation and unemployment. Assume that policymakers choose the optimal value of the policy variable ($V_t$) every time period, by minimizing the following loss function:

$$\begin{aligned} min L_t(V_t,k)= {E}_t[(\pi _{t+2}-\pi ^*)^2+\lambda (u_{t+1}-ku^N_{t+1})^2, \end{aligned}$$

(8)

subject to the following constraints:

$$\begin{aligned} \pi _t&= \alpha \pi _{t-1}-\theta (u_{t-1}-u^N_{t-1}) + \epsilon _t,\end{aligned}$$

(9a)

$$\begin{aligned} (u_{t}-{u}^N_{t})&=\rho (u_{t-1}-{u}^N_{t-1})+ V_{t-1}+ \eta _s. \end{aligned}$$

(9b)

where the variables are defined as in the rest of the paper. In Eq. (8), loss this period depends on the expected deviation of unemployment from the natural rate one period ahead while deviation of inflation from its target two periods ahead affects loss this period. The loss function has been defined this way to throw light on the relative impact of inflation and unemployment on the optimal policy reaction, since policy at time ‘t’ affects unemployment at time ‘$t+1$’ and inflation at time ‘$t+2$’ in the New Keynesian model described in “The learning model”. Further, this simplified loss function does not contain discount rate and a policy smoothing term since the results of the analysis explored here is indifferent to these additional complexities. Proceeding to solve the model to obtain the optimal policy reaction, Eq. (9b) becomes:

$$\begin{aligned} u_{t}&= u^N_{t}+\rho (u_{t-1}-{u}^N_{t-1})+ V_{t-1}+ \eta _t \end{aligned}$$

(10a)

$$\begin{aligned} \Rightarrow (u_{t+1}-ku^N_{t+1})&= (1-k)u^N_{t+1} + \rho (u_{t}-{u}^N_{t})+ V_{t}+ \eta _{t+1} \end{aligned}$$

(10b)

Equation (9a) becomes:

$$\begin{aligned} \pi _{t+2} - \pi ^*&= \alpha \pi _{t+1}-\theta (u_{t+1}-u^N_{t+1})-\pi ^* + \epsilon _{t+2}\end{aligned}$$

(11a)

$$\begin{aligned} \Rightarrow \pi _{t+2} - \pi ^*&= \alpha \pi _{t+1}-\theta [\rho (u_t-u^N_t)+ V_t]-\pi ^* + \epsilon _{t+2} \end{aligned}$$

(11b)

Substituting Eqs. (10b) and (11b) in (8)²⁰,

$$\begin{aligned} L_t= E_t[(\alpha \pi _{t+1}-\theta \rho (u_t-u^N_t)-\theta V_t-\pi ^*)^2+\lambda [(1-k)u^N_{t+1} + \rho (u_{t}-{u}^N_{t})+ V_{t}]^2], \end{aligned}$$

(12)

Minimizing (12) wrt $V_t$,

$$\begin{aligned} 0&= \frac{dL_t}{dV_t}= E_t[ 2\lambda [(1-k)u^N_{t+1} + \rho (u_{t}-{u}^N_{t})+ V_{t}]-2\theta (\alpha \pi _{t+1}-\theta \rho (u_t-u^N_t)-\theta V_t-\pi ^*)]\end{aligned}$$

(13a)

$$\begin{aligned} 0&= E_t[ u^N_{t+1}[2\lambda (1-k)] + (u_t-u^N_t)[2\lambda \rho +2\theta ^2\rho ]+V_t(2\lambda + 2\theta ^2) - 2\theta \alpha \pi _{t+1}]+ 2\theta \pi ^* \end{aligned}$$

(13b)

Rearranging the terms,

$$\begin{aligned} \boxed {V_t = \frac{\theta }{\lambda + \theta ^2}(\alpha E_t \pi _{t+1}-\pi ^*) - \rho (u_t-u^N_t)- \frac{\lambda (1-k)}{\lambda + \theta ^2} E_t u^N_{t+1}} \end{aligned}$$

(14)

Equation (14) is the best response function of the policymaker. The last term in RHS tends to zero as $k \rightarrow 1$. Intuitively, as ‘k’ tends towards zero, the target rate of unemployment ($ku^N_t$) also tends to zero. For a given unemployment rate and natural rate, a fall in the target rate of unemployment($ku^N_t$) generates expansionary policy and inflation subsequently. From Eq. (14),

$$\begin{aligned} \frac{\partial V_t}{\partial k}= \left(\frac{\lambda }{\lambda + \theta ^2}\right){u^N_{t+1}} \end{aligned}$$

(15)

Note that k lies between 0 and 1. As k falls, policy becomes more expansionary ($V_t$ falls) and accomodative of inflation. Hence, the sign of the partial effect of ‘k’ on ‘$V_t$’ is positive in Eq. (15). Policy every period follows the rule specified in Eq. (14). When $k=1$, the last term on RHS disappears which implies that the average level of policy is lower whenever $0 \le k < 1$ in comparison to when $k=1$. To see this, let the economy be in equilibrium at time t. In equilibrium, current and expected future values of unemployment is equal to the natural rate. The equilibrium conditions translate to:

$$\begin{aligned} u_t= u^N_t, \end{aligned}$$

$$\begin{aligned} E_t u_{t+1}= E_t u^N_{t+1} \end{aligned}$$

Equilibrium in real economy implies that $E_t\pi _{t+2}= \alpha E_t \pi _{t+1}$. Then, the best response of policymaker in equilibrium becomes:

$$\begin{aligned} \boxed {V^e_t= \frac{\theta }{\lambda + \theta ^2}(\alpha E_t \pi _{t+2}-\pi ^*)- \frac{\lambda (1-k)}{\lambda + \theta ^2} E_t u^N_{t+1}}, \end{aligned}$$

(16)

where ‘$V^e_t$’ is the policy reaction in equilibrium. Note that, in equilibrium, if ‘$k=1$’, optimal policy response is higher, implying tighter monetary policy. If ‘$k<1$’, optimal policy in equilibrium gets relatively more accomodative of inflation as ‘$k \rightarrow 0$’. This increases the mean inflation—optimal policy response in equilibrium has an inflation bias. In the long run, inflation equals expectations and hence unemployment equals its natural rate. However, the long run inflation is higher as ‘k’ tends to zero.²¹