Solving the Model

Because markets are competitive, externalities are absent, and individuals are infinitely-lived, the models equilibrium must correspond to the Pareto optimum. Because of this, we can find the equilibrium either by ignoring markets and finding the social optimum directly, or by solving for the competitive equilibrium. We will take the second approach, on the grounds that it is easier to apply to variations of the model where Pareto optimality fails. Finding the social optimum is sometimes easier, however; as a result, many real-business-cycle models are solved that way.

The solution to the model focuses on two variables, labor supply per person, £, and the fraction of output that is saved, s. The basic strategy is to rewrite the equations of the model tn log-linear form, substituting (I - s)Y for whenever it appears. We will then determine how £ and s must depend on the current technology and on the capital stock inherited from the previous period to satisfy the equilibrium conditions. We will focus on the two conditions for household optimization, (4.23) and (4.26); the remaining equations follow mechanically from accounting and from competition.

We will find that s is independent of technology and the capital stock. Intuitively, the combination of logarithmic utility, Cobb-Douglas production, and 100% depreciation causes movements in both technology and capital to have offsetting income and substitution effects on saving. It is the fact that s is constant that allows the model to be solved analytically.

"With these changes, the model corresponds to a one-sector version of Long and Plossers (1983) real-business-cycle model. McCallum (1989) investigates this model. In addition, except for the assumption of 6 = 1, the model corresponds to the basic case considered by Prescott (1986). It is straightforward to assume that a constant fraction of output is purchased by the government instead of eliminating government altogether.

See Problem 4.11 for the solution using the social-optimum approach.

Specifically, we make two changes to the model: we eliminate government and we assume 100% depreciation each period. Thus equations (4.10) and (4.11) are dropped from the model, and equations (4.2) and (4.4) become

Kti = Ft - Cf, (4.27)

l + rr = a(y~\ (4.28)

V Af /

The elimination of government can be justified on the grounds that doing so allows us to isolate the effects of technology shocks. The grounds for the assumption of complete depreciation, on the other hand, are only that it allows the model to be solved.

Consider (4.23) first; this condition is I/q = e PEtld + ff+O/Cf+i]. Since Cf = (1 - St)Yt/Nt, rewriting (4.23) along the hnes just suggested gives us

(1 - St)

T( NfJ

= -p + InEf

1 + t+i

(1 - St+l)

Tf+i

(4.29)

Since the production function is Cobb-Douglas and depreciation is 100%, 1 + rt+i = aVt+i/Kt+i. In addition, 100% depreciation imphes that Kt+i = Ff - Cf = Sf Ff. Substituting these facts into (4.29) yields

ln(l -Sf)-lnFf + lnNf

= ~p -b Inif

= -p + InEf

aFfi

Xf+i(l - Sf.i)

Nf+i J

(4.30)

Lsfd -Sf+i)Ff J = -p + Ina + InAff + n - InSf - In Ff -I- InEf

L 1 - Sf+] J

where the final line uses the facts that a, Nt+\, St, and Ff are known at date t and that N is growing at rate n. Equation (4.30) simplifies to

ln5f - ln(l - St) = -p + n + Ina + InEf

Ll -s,+iJ

(4.31)

A and X-technology and capital-do not enter (4.31). Thus there is a constant value of s that satisfies this condition. To see this, note that if s is constant at some value s, Sf+i is not uncertain, and so Ef[l/(1 - Sfi)] is simply 1/(1 - s). Thus (4.31) becomes

In s = In a -b - p,

s = ae"-P.

(4.32)

(4.33)

Thus the saving rate is constant.

Now consider (4.26), which states Cf/(1 ~ £t) = / . Since Cf = Q/Nt = (1 - s)Yt/Nt, we can rewrite this condition as

ln[(l - s)Ff/Nf] - ln(l - £t) = InWf - Info.

(4.34)

Since the production function is Cobb-Douglas, w, = (1 - a)Yt/(£tNt). Substituting this fact into (4.34) yields

The discussion that follows is based on McCallum (1989).

ln(l - s) + In - InNf - ln(l - it)

(4.35)

= ln(l a) + In y, - In e, - InNf - Ink Canceling terms and rearranging gives us

Inf - Ind - £t) = Itid - a) - Ind - s) - In b. (4.36)

Finally, straightforward algebra yields

Thus labor supply is also constant. The reason this occurs despite households willingness to substitute their labor supply intertemporally is that movements in either technology or capital have offsetting impacts on the relative-wage and interest-rate effects on labor supply. An improvement in technology, for example, raises current wages relative to expected future wages, and thus acts to raise labor supply. But, by raising the amount saved, it also lowers the expected interest rate, which acts to reduce labor supply. In the specific case we are considering, these two effects exactly balance.

The remaining equations of the model do not involve optimization; they follow from technology, accounting, and competition. Thus we have found a solution to the model with s and £ constant.

As described above, any competitive equilibrium of this model is also a solution to the problem of maximizing the expected utility of the representative household. Standard results about optimization imply that this problem has a unique solution (see Stokey, Lucas, and Prescott, 1989, for example). Thus the equilibrium we have found must be the only one.

Discussion

This model provides an example of an economy where real shocks drive output movements. Because there are no market failures, the movements are the optimal responses to the shocks. Thus, contrary to the conventional \\1sdom about macroeconomic fluctuations, here fluctuations do not reflect any market failures, and government interventions to mitigate them can only reduce welfare. In short, the implication of real-business-cycle models, in their strongest form, is that observed aggregate output movements simply represent the time-varying Pareto optimum.

The specific form of the output fluctuations implied by the model is determined by the dynamics of technology and the behavior of the capital stock. In particular, the production function, Yt = KiAtLt)"", implies