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64 the Great Depression were probably not due to technology shocks. Bernanke and Parkinson therefore propose to compare how the measured Solow residual moves with output in the Great Depression with how it moves with output in the postwar period. If technology shocks are a central source of fluctuations in the postwar period but not in the depression, the Solow residual and output will move together only in the postwar period. Because of a lack of reliable data on capital, Bernanke and Parkinson do not follow precisely this procedure. Instead of looking at the relation between movements m the Solow residual and in output, they look at the relation between movements m output and in labor Input. Their basic regression is Alay,t = a + bjAlnLn + £,f, (4.58) where Iny is the change in log output, In I is the change in the log of the number of personhours, and indexes industries and t indexes time. If the capital stock exhibits little shortrun variation (which is true in the postwar period), then the Solow residual is approximately equal to the percentage change in output minus the product of labors share and the percentage change in personhours (see equation [1.29]). Since the realbusinesscycle view is that output movements not arising from technology shocks do not affect the Solow residual, it therefore predicts that the estimated b, s for the depression sample will roughly equal labors share (which averages about 0.5 for the industries considered by Bernanke and Parkinson). For a period like the postwar sample, where the realbusinesscycle \lew is that the fluctuations in labor input arise largely from technology shocks, the estimated b, s should be higher. The alternative view predicts that the estimated b, s will be roughly the same in the two periods. Bernanke and Parkinson estimate (4.58) using quarterly data for each of ten industries for two sample periods, 19291939 and 19551988. Table 4.5 summarizes their results. In the depression sample, the estimated b, s exceed 1 for eight of the ten industries, with an average value of 1.07. The average for the postwar sample is 0.96. Eight of the ten fo, s are actually larger in the depression than in the postwar period. Thus it appears that supporters of realbusinesscycle theory must argue either that the depression was caused by large negative technological shocks, or that for some reason the Solow residual is a poor measure of technological change in the depression but not m other periods. 4.10 Extensions and Limitations Extensions This chapter focuses on a specific realbusinesscycle model. Research in this area, however, has considered many variations and extensions of this basic model. Here we discuss a few of the most important.
TABLE 4.5 results Bernanke and Parkinsons Estimate of b Industry  19291939  19551988  Steel  1.51  1.66   (0.17)  (0.10)  Lumber  1.07  0.86   (0.05)  (0.05)  Autos  1.21  1.05   (0.15)  (0.06)  Petroleum  0.42  0.04   (0.07)  (0.03)  Textiles  1.09  1.03   (0.17)  (0.13)  Leather  0.58  0.83   (0.08)  (0.03)  Rubber  1.21  0.98   (0.07)  (0.06)  Pulp  1.11  1.04   (0.10)  (0.38)  Stone, clay, and glass  1.11  0.94   (0.07)  (0.10)  Nonferrous metals  1.38  1.23   (0.03)  (0.07) 
Standard errors are in parentheses. SourceBernanke and Parkinson (1991). One variation of the model that has attracted considerable attention is the indivisiblelabor version. Changes in labor input come not just from continuous changes in hours, but also from movements into and out of employment. To investigate the implications of this fact, Rogerson (1988) and Hansen (1985) consider the extreme case where £ for each individual has only two possible values, zero (which corresponds to not being employed) and some positive value, (which corresponds to being employed). Rogerson and Hansen justify this assumption by arguing that there are hxed costs of working. This change in the model greatly increases the responsiveness of labor input to shocks; this in turn increases both the size of output fluctuations and the share of changes in labor input in those fluctuations. From the results of the calibration exercise described in the previous section, we know that these changes improve the ht of the model. To see why assuming allornothing employment increases fluctuations in labor input, assume that once the number of workers employed is determined, individuals are divided between employment and unemployment
"Because the instantaneous utility function, (4.7), is separable between consumption and leisure, expected utility is maximized when employed and unemployed workers have the same consumption. Thus the indivisiblelabor model implies that the unemployed are better off than the employed. See Problem 10.6 and Rogerson and Wright (1988). randomly. The mmiber of workers employed in period t, E,, must satisfy £(0 = it; thus the probability that any given individual is employed in period f is {Lt/f!Q)/Nt. Each individuals expected utility from leisure in period f is therefore .Wl4).M„l. ,4.59) Nt Nt This expression is linear in Lti individuals are not averse to employment fluctuations. In contrast, when all Individuals work the same amount, utility from leisure in period t is [1  (Lt/Nt)\. This expression has a negative second derivative with respect to If: there is Increasing marginal disutility of working. As a result, Ir varies less in response to a given amount of variation in wages in the conventional version of the model than in the mdivisiblelabor version. Hansen and Wright (1992) report that Introducing indivisible labor into the Prescott model discussed in the previous section raises the standard deviation of output from 1.30% to 1.73% (versus 1.92% in the data), and the ratio of the standard deviation of total hours to the standard deviation of output from 0.49 to 0.76 (versus 0.96 m the data). A second major extension is to include distortionary taxes (see Greenwood and Huffman, 1991; Baxter and King, 1993; Campbell, 1994; Braun, 1994; and McGrattan, 1994). A particularly appealing case is proportional output taxation, so Tf = , Yf, where Tf is the tax rate in period . Output taxation corresponds to equal tax rates on capital and labor, which is a reasonable first approximation for many countries. With output taxation, a change in 1  is, from the point of view of private agents, just like a change m technology, A °: it changes the amount of output they obtain from a given amoimt of capital and labor. Thus for a given process for 1  , aftertax output behaves just as total output does in a model without taxation in which A ° follows that same process. This makes the analysis of distortionary taxation straightforward (Campbell, 1994). Since tax revenues are used to finance government purchases, it is natural to analyze the effects of distortionary taxation and government purchases together. Doing this can change our earlier analysis of the effects of government purchases significantly. Baxter and King (1993) show, for example, that in response to a temporary increase in government purchases financed by a temporary increase in distortionary taxation, the taxinduced Incentives for intertemporal substitution typically outweigh the interestrate effects, so that aggregate output falls rather than rises.
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