This appendix describes the procedure used to derive a linear-quadratic approximation to the social planning problem described in the paper, solve the approximating model, and derive the implied linear systems representation for the observables in the model. The solution technique described in this section draws on the work of Hansen and Sargent (1988).
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This paper is drawn from my dissertation at the University of Chicago. I would like to thank Robert Townsend, Michael Woodford and, especially, Robert Lucas, for helpful discussions and comments. I am also grateful to an anonymous referee for very detailed comments. The views expressed in this paper do not necessarily reflect those of the IMF. This paper is forthcoming in the Canadian Journal of Economics.
In the remainder of the paper, the term aggregation bias is to be interpreted broadly to include selectivity bias and other sources of bias in aggregate relative to disaggregate data.
The average length of the workweek is not very sensitive to skill level. In the NLS data (see Keane and Prasad (1993) for a description of this micro panel dataset), the reported “usual weekly hours” for most workers tend to cluster around the figure of 40 hours a week. About 60 percent of the observations for weekly hours lie between 38 and 42 hours.
This specification provides a straightforward interpretation of output as a function of capital and the effective units of labor. For instance, if a new worker must be trained, then it is natural to assume that, of the hours supplied by that worker to the firm, some fraction is used to acquire training and only the remainder is available for the production process. Hansen and Sargent (1988) use a similar adjustment cost specification to examine the relatively higher cyclical variability of overtime relative to straight-time employment.
An alternative justification for the adjustment cost on skilled labor could be in terms of firm-specific capital. For instance, the firm-specific capital embodied in skilled workers would be lost each time a separation occurred between a firm and a skilled worker, implying a real resource cost. However, since the model does not keep track of the identities of individual workers, the justification for these adjustment costs in terms of firm-specific capital would be tenuous. Also see Oi (1962) for one of the earliest models where skilled labor is postulated to be a quasi-fixed factor.
Since all households in this economy are identical, no distinction is made between household and aggregate capital stocks in order to simplify the notation.
The hourly wage rates are given by h-1w1t and h-1w2t. Since the shift length h is fixed and identical for skilled and unskilled workers, the distinction between the shift wage and the hourly wage is not important here.
This suggests that profits may not necessarily be zero in every period. While the discounted expected profit stream of the firm will indeed equal zero, profits may be nonzero away from the steady state. This is not inconsistent with a competitive equilibrium since the adjustment cost in the model arises at the level of the firm. Profits, if any, are transferred in a lumpsum fashion to households (which, as stated earlier, own equal shares in the representative firm).
It should be noted that this could potentially overstate the ratio of average hours worked for skilled relative to unskilled workers. Workers are considered unemployed for the whole year if they report a zero wage for the survey period. This could cause more unskilled workers to be assigned zero hours in any year since they have, on average, lower employment probabilities than skilled workers.
Note that the standard deviation of output in the heterogeneous-agent economy is below that in the homogeneous-agent economy, largely because of the lower cyclical variation in one of the factor inputs—labor. To match the empirical standard deviation of output, the standard deviation of the productivity shock would have to be increased to about 0.0080.
The percentage standard deviation of skilled (unskilled) employment is 0.35 (1.16) and its correlation with output is 0.57 (0.92).
This elasticity is given by cov(ht, yt)/var(yt) or, alternatively, may be computed as corr(ht, yt)·σh / σy, where σ1 represents the standard deviation of series i. Earlier discussion in the literature has focused on the labor elasticity of output, which is simply the reciprocal of the measure discussed above. I work with the former measure since it is the measure that Kydland and Prescott (1988) report in their paper.