Chapter 21. Stabilization policy with rational expectations

ECON320Prof Mike Kennedy

The role of expectations• Activity today will depend importantly on what people expect to happen in

the future• The convention is to assume that expectations are formed by looking at the

past – for example, our assumption that πe = π-1

• Such an assumption is reasonable when things are normal, but there are lots of cases when it is not a reasonable assumption– A change in the monetary policy regime – Visible supply shocks that impinge on the structure of the economy– A change in government with a new policy agenda

• The logical limit of forward looking expectations is the Rational Expectations Hypothesis or REH– Here it is assumed that expectations are based on all the information available today– That includes information on the structure of the economy and policy changes

• The equation below is a formal way of saying that expectations for any variable X today (time t) are based on all the available information, I, at time t-1

Rational expectations and policy ineffectiveness

• Suppose that the central bank makes its interest rate decisions on expected or forecast values of inflation and output, based on information at time t-1

• The monetary policy rule is now

• Goods market equilibrium is still the same

• The SRAS curve is now

• We need to specify the stochastic properties of the exogenous variables, which are assumed to be white noise

Policy ineffectiveness con’t

• Assume the central bank cannot observe current πt and yt

– Step 1: Express the endogenous variables yt and πt as functions of the exogenous variables and expectations of y and π

Next substitute this expression into SRAS equation

– Step 2: Use the above two equations to calculate the rationally expected values of yt and πt, remembering that E[st] = E[vt] = 0, their respective means

The rationally expected values of y and π are

If we use this final condition we get the following for the forecasts of y and π:

Policy ineffectiveness con’t– Step 3: If we go back to the first two equations in Step 1, we see that

since then

– Notice that the policy parameters π, h and b do not appear in the equation for yt leading to the conclusion that systematic monetary policy stabilisation is ineffective!

– Systematic demand management cannot influence real output and employment when expectations are rational

– To see how this works, rewrite SRAS curve in terms of yt

– Since potential output and the shock are exogenous, the only way to influence output is to create surprise inflation – create errors in the forecasts of the private sector!

How robust is the Policy Ineffectiveness Proposition (PIP)?

• A problem with PIP is that it assumes that the central bank cannot act on the basis of new information as it becomes available

• While it may be true that the private sector cannot (due to fixed contracts or a desire to have infrequent price changes) the central can react and often does react to new information if it is important

• When the central bank can act after prices and wages have been set then we get the familiar monetary policy rule

• While central bank may not know actual inflation, the above is a convenient way of noting that the central bank can react to new information

• With this policy rule, we will again proceed in three steps to solve the model

Policy effectiveness under rational expectations

• Step 1: Solve the model for yt and πt in terms of exogenous variables– Inserting the new policy rule into the goods market equation

(2nd equation slide 3)

– From the aggregate supply equation (3rd equation slide 3) we have

– This can be re-inserted into the 1st equation above to get

– Next we substitute in the 1st equation on this slide into the 2nd to get

Policy effectiveness under rational expectations con’t

• Step 2: Find πet, t-1 by taking the expected value of the final equation based on

information available at t-1 and remembering that E[st] = E[vt] = 0

• Agents expect the central bank to hit its target• Step 3: Using this condition and the final two equations in Step 1 we get

• The key point here is that the parameters of the policy function now appear as determinants of yt and πt

• The reason for the effectiveness of policy is that the central bank can react to vt and st after the private sector has been locked into contracts and prices

The Lucas Critique

• An econometric macro model with “backward-looking” expectations and which was estimated under a previous policy regime cannot be used to predict economic behaviour under a new policy regime

• The past is not a good guide to the future as the parameters of the model are likely to change

• An example is a lowering of the inflation target:– If expectations are modelled as π-1 then the model will not be able to

predict what will happen if people have rational expectations

• This critique applies to structural policies as well– If the level of unemployment benefits or the degree of competition in

the economy changes then so will the natural unemployment rate

• The solution is to build macro models based on micro foundations and with rational expectations

Optimal stabilisation under rational expectations

• Policy can influence real output and unemployment even when expectations are rational as long as nominal wages and prices are fixed or slow to adjust

• The question becomes: What are the optimal values of h and b in the Taylor rule?

• We start by assuming that the bank wants to minimize the following

• The parameter κ measures the social loss of inflation relative to output• We can find the two variances from the final two equations in slide 8

Optimal stabilisation under rational expectations con’t

• From the above we see that if the economy were hit by only demand shocks then the variance of both output and inflation would be lower, the higher are both b and h

• The economy is however hit by both types of shocks which will present trade-offs due to supply shocks – they go in the opposite directions

• We can see this by taking the first order conditions of SL wrt h and b

• To lower the computational burden we will assume that b = 0; the central bank focuses on just the inflation gap which changes the SL to:

Optimal stabilisation under rational expectations con’t• Next we calculate the first order conditions to get

• The final expression says that the optimal value of h will depend importantly on the relative variances of the demand and supply shocks– When demand shocks are large, a strong interest rate response will serve to

close both the inflation and output gaps – In the opposite case, the central bank should respond only moderately to an

inflation shock– The coefficients α2 and γ as well as κ are also important

Note: The 3rd equation simplifies to the 4th because the terms in ovals cancel

Monetary policy in a liquidity trap• Can forward-looking expectations be manipulated to get out of a liquidity

trap• Noting that the demand shock (vt) represents future expected levels of

consumption and investment we can re-write the AD curve as:

• The goods market equilibrium is still given by:

• It follows from the above that the rational expectation of next period’s output gap is given by

• Substituting the first and third equation into the second we get

Monetary policy in a liquidity trap, con’t• The final equation on the previous slide may be written as

• Continuing to eliminate the output gap we get the following for the current output gap

• This expression shows that under rational expectations the current output gap depends not only on current real interest rates but also on the future path of monetary policy

• When the nominal interest rate is stuck at zero then the above becomes

• Now the central bank can influence the output gap by promising both lower interest rates but as well high inflation in the future: e.g., lower real rates

Announcement effects

• Under rational expectations, announcement effects can play an important role in affecting the economy

• An announcement of a new policy can influence the economy even before it is implemented

• To see how this works we return to our model of equity prices

• Importantly here define D as after-tax dividends and before-tax dividends as d or (where ε is a stochastic ‘white noise’ variable)

• If dividends are tax proportionally at a rate τ0, then

Announcement effects con’t• As long as the tax rate, τ0, stays constant then

• We now assume that at time t = t1 the government will lower the tax rate to τ1

• If we insert this into the first equation on the previous slide we get

• The above shows the price of equity today and how it will be affected by policy changes in the future

Announcement effects con’t

• Between the time of the announcement and the actual tax cut, the value of equity is given by the final equation on the previous slide

• Once the tax cuts occurs, we get

• Note that the market value of equity jumps immediately when the policy is announced

• Thereafter the price will rise and the speed will depend on how long it takes to implement the policy

The effect of an announced dividend tax cut