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57 Afi = (yflayf2). 1  a (4.41) Recall that (4.9) states that Af = pAAfi + eA,t Substituting this fact and (4.41) into (4.40), we obtain Yt = aYti + (1  a){pAAti + £A.t) = ,1 + pAiYti  aYt2) + (1  a)fA,t (4.42) = (" + PA)Ytl  %2 + (1  a)fA,f Thus, departures of log output from its normal path follow a secondorder autoregressive processthat is, can be written as a linear combination of its two previous values plus a whitenoise disturbance. Readers who are familiar with the use of lag operators can derive (4.42) using that approach. In lagoperator notation, y, i is LY,, where L maps variables to their previous periods value. Thus (4.40) can be written as Y, = alY, +(1  a)A,, or (1  aL)Y, = (1  a)A,. Similarly, we can rewrite (4.9) as (1  paL)A, = ,,, or A, = (1  pI) ,. Thus we have (t  aL)Y, = (1  a)(l  paL) Sa,,. "Multiplying" through by 1  paL yields (1  aL)(i  paL)Y, = (1  )£ (, or [I  (a + pa)L + apAL]Y, = (1  a)ea,, This is equivalent to Y, = (a + pa)LY, apLY, +{1 a)ea,,, which corresponds to (4.42). (See Section 6.8 for a discussion of lag operators and of the legitimacy of manipulating them in these kinds of ways.) In Ff = a \ , +a a)(ln Af + Inlf). (4.38) We know that Kt = s Yti and If = £ (\ thus InTf =alnsbalnyfi +(la)(lnAf+lnllnNf) " = alnsbalnTfibd  a)(A + 0f) (4.39) + {1 a)At +(1 a)(lnf + N + nt), where the last line uses the facts that In Af = A + gt + At and InTVf = N + nt (see [4.6] and [4.8]). The two components of the righthand side of (4.39) that do not follow deterministic paths are a In Yfi and (1  a)At. It must therefore be possible to rewrite (4.39) in the form Yt = aYti+aa)At, (4.40) where Yt is the difference between In Tf and the value it would take if In Af equaled A + gt each period (see Problem 4.14 for details). To see what (4.40) implies concerning the dynamics of output, note that since it holds each period, it implies Tfi = a %2 +(1 ~ a)Afi, or
**This result is sensitive to the detrending, however. Tfie combination of a positive coefficient on the first lag of Yf and a negative coefficient on the second lag can cause output to have a "humpshaped" response to disturbances. Suppose, for example, that a = 5 and pa = 0.9. Consider a onetime shock of 1/(1  a) to Ea Using (4.42) iteratively shows that the shock raises log output relative to the path it would have otherwise followed by 1 in the period of the shock (1  a times the shock), 1.23 in the next period (a pa times 1), 1.22 in the following period {a + pA times 1.23, minus a times times 1), then 1.14,1.03, 0.94, 0.84, 0.76, 0.68, ... in subsequent periods. Because a is not large, the dynamics of output are determined largely by the persisfence of the technology shocks, . If = 0, for example, (4.42) simplifies to Yf = a Yti t (1  a)5A,t. If « = . this implies that almost ninetenths of the initial effect of a shock disappears after only two periods. Even if PA = I, twothirds of the initial effect is gone after three periods. Thus the model does not have any mechanism that translates transitory technology disturbances into significant longlasting output movements. We will see that the same is true of the more general version of the model. Nonetheless, these results show that this model yields interesting output dynamics. Indeed, if actual U.S. log output is detrended linearly, it follows a process similar to the humpshaped one described above (Blanchard, 1981).! In other ways, however, this special case of the model does not do a good job of matching major features of fluctuations. Most obviously, the saving rate is constantso that consumption and investment are equally volatileand labor input does not vary. In practice, as we saw in Section 4.1, investment varies much more than consumption, and employment and hours are strongly procyclicalthat is, they move in the same direction as aggregate output. In addition, the model predicts that the real wage is highly procyclical. Because of the CobbDouglas production function, the real wage is {la)Y/L; since I does not respond to technology shocks, this means that the real wage rises oneforone with Y. In actual fluctuations, in contrast, the real wage appears to be at most only moderately procychcal. Thus the model must be modified if it is to capture many of the major features of observed output movements. The next section shows that introducing depreciation of less than 100% and shocks to government purchases improves the models predictions concerning movements in employment, saving, and the real wage. To see intuitively how lower depreciation improves the fit of the model, consider the extreme case of no depreciation and no growth, so that investment is zero in the absence of shocks. In this situation, a positive technology shock, by raising the marginal product of capital in the next period, makes it optimal for households to undertake some investment. Thus the saving
Kimball (1991) employs a similar approach. rate rises. The fact that saving is temporarily high means that expected consumption growth is higher than it would be with a constant saving rate; from consumers intertemporal optimization condition, (4.23), this requires the interest rate to be higher. But we know that a higher interest rate increases current labor supply. Thus introducing incomplete depreciation causes investment and employment to respond more to shocks. The reason that introducing shocks to government purchases improves the fit of the model is straightforward: it breaks the tight link between output and the real wage. Since an increase in government purchases increases households lifetime tax liability, it reduces their lifetime wealth. This causes them to consume less leisurethat is, to work more. When labor supply rises without any change in technology, the real wage falls; thus output and the real wage move in opposite directions. With output fluctuations coming from changes in L instead of changes in A, real wages move in the opposite direction from output. It follows that with shocks to both government purchases and technology, the model can generate an overall pattern of real wage movements that is not strongly procyclical or countercyclical. 4.6 Solving the Model in the General Case Overview As discussed above, the full model of Section 4.3 cannot be solved analytically. This is true of almost all realbusinesscycle models. Papers in this area generally address this difficulty by solving the models numerically. That is, once a model is presented, parameter values are chosen, and the models quantitative implications for the variances and correlations of various macroeconomic variables are discussed. As Campbell (1994) emphasizes, this procedure provides little guidance concerning the sources of the models implications. He argues that one should instead take firstorder Taylor approximations of the equations of the models in the logs of the relevant variables aroimd the models balanced growth paths in the absence of shocks, and then investigate the properties of these approximate models. He also argues that one should focus on how the variables of a model respond to shocks instead of merely describing the models implications for variances and correlations. This section apphes Campbells method to the model of Section 4.3. Unfortunately, even though taking a loglinear approximation to the model allows it to be solved analytically, the analysis remains cumbersome. For that reason, we will only describe the broad features of the derivation and results without going through the specifics in detail.
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