The Intratemporal First-Order Condition

To see how the method works in this case, begin by considering households first-order condition for the tradeoff between current consumption and labor supply, Ct/(1 - £,) = / (equation [4.26]). Using equation (4.3) to substitute for the wage and taking logs, we can write this condition as

InCf - ln(l - £,) = In l-r) +(1- a)InAf + alnKt - alnlf. \ b /

(4.45)

"See Problem 4.10 for the balanced growth path of the model in the absence of shocks.

Log-Linearizing the Model around the Balanced Growth Patho

In any period, the state of the economy is described by the capital stock inherited from the previous period and by the current values of technology and government purchases. The two variables that are endogenous each period are consumption and employment.

If we log-hnearize the model around the nonstochastic balanced growth path, the rules for consumption and employment must take the form

Cf = acKKt + acAAt + OccGt, (4.43)

If aKt + flLAAf + flLcGf, (4.44)

where the as will be functions of the underlying parameters of the model. As before, a tilde (~) over a variable denotes the difference between the log of that variable and the log of its balanced-growth-path value. Thus, for example, At denotes InAf -(A + gt). Equations (4.43) and (4.44) state that log consumption and log employment are linear functions of the logs of a:. A, and g, and that consumption and employment are equal to their balanced-growth-path values when K, A, and G are all equal to theirs. Since we are building a version of the model that is log-linear around the balanced growth path by construction, we know that these conditions must hold. To solve the model, we must determine the values of the as.

As with the simple version of the model, we will focus on the two conditions for household optimization, (4.23) and (4.26). For a set of as to be a solution to the model, they must imply that households are satisfying these conditions. It turns out that the restrictions that this requirement puts on the as fully determine them, and thus tell us the solution to the model.

This solution method is known as the method of undetermined coefficients. The idea is to use theory (or, in some cases, educated guesswork) to find the general functional form of the solution, and then to determine what values the coefficients m the functional form must take to satisfy the equations of the model. This method is useful in many situations.

acA +

ocg +

( e*

/ £* 1- *

+ a\aiK = a, (4.48)

Ala = 1 - a, (4.49)

Qlg = 0. (4.50)

To understand these conditions, consider first (4.50), which relates the responses of consumption and employment to movements in government purchases. Government purchases do not directly enter (4.45); that is, they do not affect the wage for a given level of labor supply. If households increase their labor supply in response to an increase in government purchases, the wage falls and the marginal disutility of working rises. Thus, they will do this only if the marginal utility of consumption is higher-that

We want to find a first-order Taylor-series approximation to this expression in the logs of the variables of the model around the balanced growth path the economy would follow if there were no shocks. Approximating the right-hand side is straightforward: the difference between the actual value of the right-hand side and its balanced-growth-path value is (1 - a)At -+- akt - alt. To approximate the left-hand side, note that since population growth is not affected by the shocks, Q = q : log total consumption differs from its balanced-growth-path value only to the extent that log consumption per worker differs from its balanced-growth-path value. Similarly, it = Lf The derivative of the left-hand side of (4.45) with respect to InCf is simply 1. The derivative with respect to In, at £t = f* is -£*), where 4* is the value of £ on the balanced growth path. Thus, log-linearizing (4.45) around the balanced growth path yields

C, + Yt = (1 - a)At + akt - clt. (4.46)

We can now use the fact that Q and U are linear functions of kt,At, and Gf Substituting (4.43) and (4.44) into (4.46) yields

( £* \

acv.Kt + , -t- accGt + ( ZTp + «1 iauJt + « -i- flLcGf)

= akt+{\~a)At. (4.47)

Equation (4.47) must hold for all values of jK;,A, and G. If it does not, then for some combinations oi k,A, and G, households can raise their utihty by changing their current consumption and labor supply. Thus the coefficients on on the two sides of (4.47) must be equal, and similarly for the coefficients on A and on G. Thus the as must satisfy:

The Intertemporal First-Order Condition

The analysis of the first-order condition relating current consumption and next periods consumption, 1/ct = e"p£f[(l + ff+il/Q+i] (equation [4.23]), is more complicated. The basic idea is the following. Begin by defining Zt+i as the difference between the log of (1 + ff+il/Cf+i and the log of its balanced-growth-path value. Now note that since (4.43) holds at each date, it implies

Cf+i flcKf+i -I- flcAAf+i + occGf+i. (4.51)

We can then use this expression for C,+i and equation (4.4) for r,+i to express t+i in terms of f+i,Af+i, and Gti- Since f+i is an endogenous \ ariable, we need to eliminate it from this expression. Specifically, we can log-linearize the equation of motion for capital, (4.2), to write +i in terms of kt, At, Gt, It, and Q; we can then use (4.43) and (4.44) to substitute for If and Cf. This yields an expression of the form

f+i - + b .At + bKcGt, (4.52)

where the bs are complicated fimctions of the parameters of the model and ofthe asP

Substituting (4.52) into the expression for Zf+i in terms of kt+i,At+i, and Gf+i then gives us an expression for Zt+i in terms of Af+i,Gf+i,Kf,Af, and Gf. The final step is to use this to find £f[Zf+i] in terms of f,Af, and Gf. Substituting this into (4.23) gives us three additional restrictions on

Equation (4.44) for L is used to substitute for L,+i in the expression for r,+i. 22See Problem 4.15.

There is one complication here. As emphasized in Section 4.4, (4.23) involves not just the expectations of next-period values, but their entire distribution. That is, what is appro-

is, if consumption is lower. Thus if labor supply and consumption respond to changes in government purchases, they must move in opposite directions. Equation (4.50) tells us not only this qualitative result, but also how the movements in labor supply and consumption must be related.

Now consider an increase in A (equation [4.49]). An improvement in technology raises the wage for a given level of labor supply. Thus if neither labor supply nor consumption respond, households can raise their utility by working more and increasing their current consumption. Thus households must increase either labor supply or consumption (or both); this is what is captured in (4.49).

Finally, the restrictions that (4.45) puts on the responses of labor supply and consumption to movements in capital are similar to the restrictions it puts on their responses to movements in technology. The only difference is that the elasticity of the wage with respect to capital, given I, is a rather than 1 ~ a. This is what is shown in (4.48).