The Method of Undetermined Coefficients

.As described in Section 4.6, the idea of the method of imdetermined coefficients is to guess the general fvmctional form of the solution and then to use the model to determine the precise coefficients. In the model we are considering, in period f two variables are given: the money stock, ,, and the prices set the previous period, Xf i. In addition, the model is linear. It is therefore reasonable to guess that x, is a linear function of Xf-i and

X, = fl + \Xt ] -b vrut. (6.63)

Our goal is to determine whether there are values of , A, and v that yield a solution of the model.

Although we could now proceed to find , A, and v, it simplifies the algebra if we first use knowledge of the model to restrict (6.63). The fact that we have normalized the constant in the expression for individuals desired prices to zero, so that p*, -pt = , implies that the equilibrivmr with flexible prices is for to equal zero and for each price to equal m. In light of this, consider a situation where Xf i and are equal. If period-f price-setters also set their prices to nit, the economy is at its flexible-price equilibrium. In addition, since m follows a random walk, the period-t price-setters have no reason to expect nif+i to be on average either more or less than m,, and hence no reason to expect Xf+i to depart on average from m,. Thus in this situation p,* and frPi+i are both equal to m,, and so price-setters will choose Xt = m,. In sum, it is reasonable to guess that if X( i = fn,, then X[ = irit. In terms of (6.63), this condition is

p + Kmt + vm, = m, (6.64)

for all m,.

Two conditions are needed for (6.64) to hold. The first is A - i = 1; otherwise (6.64) cannot be satisfied for all values of m,. Second, when we impose A- = 1, (6.64) implies p = 0. Substituting these conditions into (6.63) yields

Xf = AXf-i -b (1 - \) ,. (6.65)

Our goal is now to find a value of A that solves the model.

Since (6.65) holds each period, it implies Xf+i = AXf - (1 - X)m,+i. Thus the expectation as of period t of ,1 is Axt + (1 - A)£t"if+i, which equals AXf -b (1 - \) ,. Using (6.65) to substitute for Xf then gives us

EtXt+i = A[AXf i -I- (1 - \)mt] -b (1 - \)mt

(6.66)

= Axt-i -b (1 - ) ,.

(6.67)

Thus, if price-setters believe that Xf is a linear fimction of Xfi and mt of the form assumed in (6.65), then, acting to maximize their profits, they will indeed set their prices as a hnear function of these variables. If we have found a solution of the model, these two linear equations must be the same. Comparison of (6.65) and (6.67) shows that this requires

A + aa = a (6.68)

A(l - a2) -b (1 - 2A) = 1 - a. (6.69)

Consider (6.68). This is a quadratic equation in a. The solution is

1 -I- /1 - 4 2 a = . (6.70)

One can show that these two values of a also satisfy (6.69). Using the definition of A in equation (6.62), one can show that the two values of a are

Of the two values of a, only a = ai gives reasonable results. When a = ab a < 1, and so the economy is stable. When a = a2, in contrast, a > 1, and thus the economy is unstable: the slightest disturbance sends output off toward plus or minus infinity. As a result, the assumptions underlying the model-for example, that sellers do not ration buyers-break down. For that reason, we focus on a = ai.

Thus equation (6.65) with a = ai solves the model: if price-setters 11 e that others are using that rule to set their prices, they find it in their own interests to use that same rule.

We can now describe the behavior of output, yt equals mt - Pt, which in turn equals frif - (Xf i + Xf)/2. With the behavior of x given by (6.65), this imphes

yt = mt- {faxf2 + (1 - X)mt-i] + [XXti + (1 - X)mt]}

(6-73)

= frif - (a(Xf2 + Xf i) -b (1 - a)i(mf ] + mt)].

Substituting this expression into (6.62) yields

Xf = afxfi + Xxt -1 + (1 - \)mt] + (1 - 2A)mf = (A + AA2)Xf-i + [A(l - a) + (1 - 2A)]mf.

Implications

Equation (6.74) is the key result of the model. As long as Ai is positive (which is true if < 1), (6.74) implies that shocks to aggregate demand have long-lasting effects on output-effects that persist even after all price-setters have changed their prices. Suppose the economy is initially at the equilibrium with flexible prices (so is steady at zero), and consider the effects of a positive shock of size u* in some period. In the period of the shock, not all price-setters adjust their prices, and so not surprisingly, rises; from (6.74), = [(1 -b A)/2]u. In the following period, even though the remaining price-setters are able to adjust their prices, does not return to normal even in the absence of a further shock: from (6.74), is A[(l + A)/2]u°. Thereafter output returns slowly to normal, with yf = Ayf i each period.

The response of the price level to the shock is the flip side of the response of output. The price level rises by [1 - (1 + A)/2]u" in the initial period, and then fraction 1 - A of the remaining distance from u° in each subsequent period. Thus the economy exhibits price-level inertia.

The source of the long-lasting real effects of monetary shocks is again price-setters reluctance to allow variations in their relative prices. Recall that p,* = - (1 - ) ,, and that Ai > 0 only if < 1. Thus there is gradual adjustment only if desired prices are an increasing function of the price level. Suppose each price-setter adjusted fuU to the shock at the first opportunity. In this case, the price-setters who adjusted their prices in the period of the shock would adjust by the full amoimt of the shock, and the remainder would do the same in the next period. Thus would rise by «"/2 in the initial period and return to normal in the next.

To see why this rapid adjustment cannot be the equilibrium if is less than 1, consider the individuals who adjust their prices immediately. By assumption, all prices have been adjusted by the second period, and so in that period everyone is charging his or her optimal price. But since < 1, the optimal price is lower when the price level is lower, and so the price that is optimal in the period of the shock, when not all prices have been adjusted, is less than the optimal price in the next period. Thus these individuals

Using the facts that mt = nif-i + Uf and(A, i+x,- 2)/2 = pt-1, we can simplify this to ...

yt = m, 1 + Uf - [Apf i + (1 - A)mf-i + (1 - A)uf]

= A(mf i-pf i) + Uf (6.74)

1 + A = Ayf-i + - .