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Section III. Business Cycles B. Rational Expectations Inflation expectations form a key part of the dynamics of the Neo-classical synthesis model that was described in the previous chapter. However, there is a view by some that the model of expectations formation is too ad hoc. In this section, we consider a more theoretically well-founded model of expectation formation in the context of a very simple version of the business cycle model. We write down a linear function of the IS expenditure curve qt q t d rt r Where qt ln(Yt ) , q ln(Y P ) and αt is a demand shock affected by changes in consumer or business confidence or fiscal policy. Consider if there is a long-run interest rate, r which we might thing of as an interest rate along the balanced growth path. Assume in the long-run, that spending cannot go above potential output. Write monetary policy in the form rt r t b tFED tgt Where t is a monetary policy shock and tFED is the central bank’s measure of the inflation rate. The parameter b is the inflation sensitivity of monetary policy. We can combine these two equations to a version of the aggregate demand curve qt q t d t b tFED tgt q t d t d b tFED tgt The central bank has many statisticians measuring the economy and then makes decisions about monetary policy on an intermittent basis. This means that when monetary policy is being made, the central bank is using information from time t-1. We will then write the Feds measure of inflation as a forecast made with information available at time t-1. tFED tE t Infot 1 must make its monetary p A final equation will represent the supply curve of the economy. 1 qt q t gtW Where is the logarithm of potential output. This equation says that firms will produce potential output if inflation matches wage growth which would keep the real wage rate constant. If inflation races ahead of nominal wage growth, real wage growth will decline, firms will hire more workers and produce more than potential. If inflation is slower than nominal wage growth, real wage growth increases and employment and production will decline. Wage growth is decided by contracts that are signed at time t-1. Therefore we can say that wage growth is equal to expected inflation plus a cost push shock, νt. gtW tE t Therefore we can write the aggregate supply curve as 1 qt q t tE t t tE qt q t We can solve for the endogenous variables {qt, πt} as a function of the exogenous variables {αt, νt, μt} and tE . The question is how to write a model of tE . In the previous model we assumed that inflation expectations were just lagged inflation tE t 1 . If this is the case, then monetary policy makers can choose the parameters of monetary policy to stabilize the economy. However, an alternative is to examine rational expectations. Rational expectations is just another way of saying model consistent expectations. We think of the exogenous variables as being shifts in the economy that are being driven by noneconomic events. These might be thought of as random shocks to the system. We know from probability and statistics class that we can think of random variables as having a distribution of possible outcomes and some probabilities associated with those outcomes. Those random variables will then have a mean/expected value which is a weighted average of the possible outcomes using the probabilities as weights. They will also have a variance which is a weighted average of squared deviations of possible outcomes from mean with the probabilities again used as the weights. Random variables also have higher moments like skew or kurtosis etc. If we were rational, the best unbiased predictor for the random variable will be the expected value. If xt is a random variable, we write the expected value as E[xt]. We can write the variance of E[ (xt -E[xt])2]. When a variable follows a time series then we can decompose it into two parts: the predictable component (which is know at time t-1) and the innovation which cannot be predicted using information available at time t-1. xt predictablet Infot 1 innovationt The expected value of the innovation term is E[innovationt]=0. This means that if we have all information available to us at time t-1 the best forecast of xt is E xt Infot 1 predictablet Infot 1 E innovationt predictablet For example, if the random variable followed an AR(1) process, xt xt 1 t where t is a white noise term then predictablet 1 xt 1 innovationt t And we could write E xt Infot 1 xt 1 . We could generalize this to the AR(p) process xt 1 xt 1 2 xt 2 ... p xt p t . Then xt Infot 1 1 xt 1 2 xt 2 ... p xt p Now consider the three exogenous variables that are impacting the economic system we are modeling. {αt, νt, μt}. We can decompose each of them into their predictable component and their unpredictable components t at Infot 1 t E t 0 E t Infot 1 at t mt Infot 1 t E t 0 E t Infot 1 mt t nt Infot 1 t E t 0 E t Infot 1 nt Think of the economic model is a numerical algorithm that accepts the exogenous variables and the expectations of endogenous variables into outcomes for the endogeous variables. Expectations of Endogenous Variables Model Endogenous Variables Exogenous Variables Then the endogenous variable, given expectations, is a function of the exogenous variables. If the exogenous variables are a random variable, the endogenous variables, as functions of random variables, are themselves random variables. Thus, endogenous variables have their own distribution, expected value and standard deviation. If we were going to create the best forecast of the endogeonous variables we would use their expected value. This expected value would be a function of the expectations that economic agents had of them E ( Endogenous Variablest ) f ( Expectations of Endogenous Variables) The theory of rational expectations suggest that if we as outside observers were to use the expected value of the endogenous as the forecast, then rational economic agents such as workers and firms should do so as well. A model consistent set of expectations should solve the equation E( Endogenous Variablest ) f (E Endogenous Variablest ) Given the dynamics of the exogenous variables, we apply the above equation to forecasts using the information at time t+1 E Endogenous Variablest Infot 1 f (E Endogenous Variablest Infot 1 ) Lets apply the theory to examine the model when This model writes the production decision of the firm as equal to tFED tE E t Infot 1 So our model is AD : qt q t d t d b E t Infot 1 tgt AS : t E t Infot 1 qt q t First examine the model under the assumption of t 0 so we could focus on demand shocks. Using the AS supply curve, if expectations are model consistent, then the expected value of the left-hand side of the equation were equal to the expected value of the right hand side of the equation AS : E t Infot 1 E E t Infot 1 qt q Infot 1 Expectations and forecasts are linear functions. By this we mean if the random variable yt were a linear function of another random variable, yt = a∙xt + b, then E[yt] = a∙E[xt]+ b. Therefore we can write the right hand side as AS : E t Infot 1 E E t Infot 1 Infot 1 E qt q Infot 1 To solve this problem, we need to apply the Law of Iterated Expectations. The Law states that the expectation is its own expected value. By this we mean that a forecast is made with the information available at time t-1. If we have that same information, we should be able to know what that forecast is. Therefore we could write L.I .E.: E E t Infot 1 Infot 1 E t Infot 1 Combine this with the AS curve to get AS : E t Infot 1 E t Infot 1 E qt q Infot 1 E qt q Infot 1 0 E qt Infot 1 q Since output is different from potential output only if inflation is different than workers expectations, then workers should expect that output is equal to potential output. This might be why we call model consistent expectations, rational expectations. It would not be rational to forecast a forecast error; if you did, why not simply improve your forecast. Since the output gap is dependent on the worker’s making forecast errors, it is not rational to forecast an output gap. We can use the AD curve to forecast inflation. The forecast of output should be equal to the forecast of the right hand side of the equation. AD : E qt q Infot 1 E t d t d b E t Infot 1 Infot 1 d b tgt 0 E t Infot 1 d E t Infot 1 d b E E t Infot 1 Infot 1 d b tgt 0 at d mt d b E t Infot 1 d b tgt a d mt E t Infot 1 tgt t d b Now that we have solved for model consistent expectations, we can now solve for the outcome of the model. The aggregate demand curve becomes a d mt AD : qt q t d t d b t d b t at d t mt tA d t Then we can plug this into the aggregate supply curve to get the level of inflation a d mt A t tgt t t d t d b We find a number of results. 1. Output is white noise. All the fluctuations are driven by the innovations in demand shocks rather than by the systematic component. This is a clear implication of the rational expectations model. If people use all relevant predictable information to form their expectations, the only reason that inflation can differ from expected inflation is unpredictable innovations. When firm’s produce output different from potential output, only when inflation is different from the inflation that people expected when labor contracts are signed, then output can only differ from potential due to unpredictable shocks. 2. Systematic monetary policy cannot impact the stability of output. Assuming no covariance between money and demand shocks, we can write the variance of the output gap as 2 d 2 2 . Note that regardless of the degree of inflation sensitivity of monetary policy, b. Since systematic policy cannot take the innovations into account when real interest rates are set, it cannot moderate the impact of these innovations. Moreover, since workers and firms can predict the systematic component of monetary policy, it will automatically be incorporated into inflation expectations and not impact output. 3. Monetary policy shocks can destabilize output. The only avenue along which monetary policy can work to reduce output volatility is to eliminate monetary policy shocks 2 0 . 4. Systematic monetary policy can impact inflation. The degree to which the expected component of demand, at d mt , pushes up prices depends on the counter-veiling monetary policy response. Examine a couple of variants of the model. First, lets assume that the central bank can set the real interest rate in light of the current inflation rate (i.e. tFED t ). Then our model becomes AD : qt q t d t d b t tgt AS : t E t Infot 1 qt q First solve for model consistent expectations. From the aggregate supply equation, we say that the expected value of the left hand side of the aggregate supply equation is equal to the expected value of the right hand side. AS : E t Infot 1 E t Infot 1 E qt q Infot 1 E qt q Infot 1 0 E qt Infot 1 q Similarly for the demand equation AD : E qt q Infot 1 E t d t d b t Infot 1 a d mt at d mt d b E t Infot 1 E t Infot 1 t d b Notice that this is the same as the previous model. The central bank can respond to inflation rather than expected inflation. Therefore, this difference may result in a difference in the actual actions of the central bank However, when we try to predict the central bank’s behavior we can only base our prediction on our own expectations of inflation, therefore there will be no predictable difference in the behavior of the central bank. Insert the model consistent expectations to solve for actual output. First insert expected inflation into the supply equation. a d mt AS : t t qt q d b Now insert the aggregate supply equation into the aggregate demand equation a d mt AD : qt q t d t d b t qt q d b qt q t at d t mt d b qt q qt q ta d t d b qt q 1 d b qt q ta d t qt q ta d t 1 d b We can also solve for inflation AS : t tgt at d mt ta d t d b 1 d b We can make two points. 1. Output is white noise. It is an integral point of the rational expectations model that if output differs from potential due to expectations errors and expectations errors can only be caused by new unpredictable information, then output must be unpredictable. 2. The central bank’s ability to stabilize output can only come from an information advantage over private agents. We see that here the volatility of the output gap is 2 d 2 2 2 given by qt q which is a negative function of the inflation 2 1 d b sensitivity of the monetary policy rule, b. When there is a demand shock, that will translate into inflation, the central bank will raise interest rates to counter-act it. Now lets examine the model with supply shocks. Continue to use the assumption that the central bank can respond to current inflation. Simplify by assuming that there are no demand shocks. AD : qt q d b t tgt AS : t E t Infot 1 qt q t Again, solve for model consistent expectations, using the aggregate supply curve so that the expected value of the right hand side is equivalent to the expected value of the left hand side. AS : E t Infot 1 E E t Infot 1 qt q t Infot 1 E E t Infot 1 Infot 1 E qt q Infot 1 E t Infot 1 E t Infot 1 E qt q Infot 1 nt n E qt Infot 1 q t If we expect a rise in the cost push variable this will raise real wages and induce firms to cut back on production. Therefore, the expected level of the cost push shock at time t, nt, has a negative effect on expected supply next period. We can also then calculate expected inflation using the aggregate demand curve. E qt q Infot 1 d b E t Infot 1 E qt q Infot 1 nt E t Infot 1 tgt tgt d b d b The cost push shock of higher wages will push through into higher inflation. Now that we have solved for model consistent inflation we can now solve the model. qt q d b t tgt d b E t tgt Infot 1 qt q t n d b t qt q t d b n qt q t d b qt q d b t n n t d b t 1 d b t d b t d b n qt q t 1 d b t 1 d b qt q nt Plug the results into the AD curve qt q d b t tgt t tgt nt nt t d b 1 d b d b 1 d b t We can say a few things about this result. 1. Cost push shocks directly impact that supply curve and therefore impact the level of production regardless of whether they are expected or not. We see that in the equilibrium both output and inflation are a function of nt in a staglationary way. That is an increase in labor market inefficiency increases inflation and reduces the level of output. 2. Because expected cost push shocks directly impact output, monetary policy has 1 no effect on the impact of cost push shocks on output. The coefficient on nt, , is not a function of the monetary policy parameter, b. We see that the only impact of monetary policy on output come through the ability of the central bank to change inflation in a way unexpected by workers when wage contracts are signed. Any way that monetary policy respond to information that the workers also have, will not affect t E t Infot 1 and will not impact output. 3. The central bank can affect the impact of the cost push shock t but must make a trade-off. An inflation sensitive monetary policy rule will lead to a bigger impact on output and a smaller impact on inflation following a cost-push shock. There will be, by definition, no difference between the predict of inflation and the prediction