Economics 813a
R. H. Rasche 1998
Taylor on Aggregate Dynamics and Staggered Contracts
The question that Taylor seeks to address is how can models with rational expectations and with informational advantages introduced through overlapping staggered contracts, account for serial correlation in unemployment or real output (i.e. business cycles)?
Taylor’s model includes: a) staggered multiperiod nominal wage contracts and b) current wage contracts that are based upon the wage rates at other firms (under other contracts) which will be in effect during the life of the current (new) wage contract. Taylor calls this the relative contract model.
This assumption requires workers and firms to look both backward and forward in time to see what other workers will be paid during the life of the contract that is currently being negotiated.
Features
1) all wage contracts run for N periods.
2) a constant fraction (1/N) of all firms determine their wage contracts at each point in time.
3) the wage contract specifies the nominal wage rate that will prevail for the life of the contract. Let xt = the log of the nominal wage rate set by the contract that is negotiated at time = t (and that remains in effect through t+N-1).
4) the contracted nominal wage is fixed for the entire length of the contracts.
The nominal wage contract model is:
where:
=
= expected excess demand in the labor market at t+s. et+s = 0 is “full employment
t = an independently distributed random shock (this is a critical assumption)
h > 0.
Assume that other nominal wage rates are weighted proportionally to the number of periods that they overlap the current contract. , so the weights decline linearly into the past and future and sum to 1.0.
Price Determination
where pt is the log of the current price level. Thus the average real wage rate is constant (and equal to 1.0)
Aggregate Demand
yt + pt = mt + vt; (also a critical assumption)
where yt is the (log of real output - the log of full employment output)
mt is the (log of the nominal money stock - the log of full employment money stock).
Hence this is a random velocity model analogous to that of Fischer (in particular note that there is no interest elasticity of the demand for real balances, so he doesn’t have to specify an IS curve to complete the model.
Aggregate Production Function
Monetary Policy Rule
. is a fixed money stock; allows monetary policy to react to current changes in the price level. Note, unlike the Fisher model, the monetary authorities are allowed to react to current events.
The model can be reduced to a single equation by substitution:
From the contract equation:
.
From the Aggregate Demand equation:
where and .
Apply the conditional expectations operator and certainty equivalence to the real output equation:
.
Substitute this in the expression derived from the contract equation to get:
and the model has been reduced to a single equation in xt, expectations of xt and the exogenous shock t.
With some algebra the double summation can be shown to be equal to:
.
Substitute this expression for the double summation into the equation for xt
Take conditional expectations and apply certainty equivalence to this equation:
Divide through this equation by to get:
, where .
Note the similarities to the Muth inventory model. In this model there are both leads and lags in the expectations variable and this equation must hold at all horizons t+j, j 0. Define
B(L) =, where , and b-s.= bs. Then the conditional expectation of xt is determined by the homogeneous difference equation: . Note that B(L) is symmetric in negative and positive powers of L. This facilitates the solution of the model, since under these conditions B(L) can be factored as B(L) = A(L)A(L-1), where and 0.
Decomposition of Polynomials that are Symmetric in Leads and Lags
Let B(L) =
Assume that is a root of B(L) so that .
so if is a root of B(L), then is also a root of B(L). Let , i = 1, ..., (N-1) be the largest (N-1) roots of B(L) that are outside of the unit circle. Define Q(L) = . Write
since the roots of B(L) are the roots of Q(L). Then:
Therefore
and
where A(L) is invertible.
Digression: The Muth Inventory Investment Model Once Again
The homogeneous difference equation that determines the expected price level in the Muth model with inventory investment is B(L) = , where are the reciprocals of the roots of B(L). Note that this equation is symmetric in powers of L as in the Taylor model.
Let 1 < 1 and 2 = > 1. Multiply B(L) by to get:
Thus in this case A(L) = (1- 1L) and A(L-1) = (1-1L-1). Clearly A(L-1) is invertible, so
.
Since B(L)Et-1pt = 0, , or (1-1L)
We have , so .
But, so which is the result obtained from our previous solution methods.
Rational Expectations Solution to the Taylor Model
Since A(L) is an invertible polynomial
and the problem is reduced to one that involves only and lagged x’s, since = xt-s, s = 1, ..., N-1. But xt = Et-1xt + t, so A(L)xt - t = 0 or A(L)xt = t.
The remaining question is what are the coefficients of A(L)? It is not hard to show that they are totally determined by N and , since these parameters determine the values of which in turn uniquely determine the , the coefficients of A(L).
The coefficients of A(L) are Taylor’s reduced form coefficients. Since these depend on , if the parameter of the policy rule () is changed so as to change , then the reduced form relationship between xt and its history will change.
Let . Then the price equation in operator notation is pt = D(L)xt.
Then A(L)pt = A(L)D(L)xt = D(L)A(L)xt = D(L)t, since A(L)xt = t. Hence pt = A-1(L)D(L)t which is an ARMA(N-1,N-1) process. A-1(L)D(L) is an infinite order polynomial.
yt = -pt + vt from the aggregate demand equation, so yt = -A-1(L)D(L)t + vt so yt (and unemployment) follow an ARMA process. Serial correlations of processes like A-1(L)D(L) are known, based upon the s coefficients. If is close to 1.0, substantial serial correlation persists to a very high order. Recall that the coefficients of A(L) depend upon the policy rule parameter, , so changes in the policy rule will change the time series process that generates yt.
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