Appendix I: A Production Function with Capital-Skill Complementarity
54. This appendix describes a production structure that allows for capital-skill complementarity and formalizes the notions discussed in the concluding section of the paper on the relationship between the rate of capital accumulation and measured labor productivity growth.21
55. Krusell et al. (2000) propose the following production function with four inputs—capital structures, capital equipment, skilled labor and unskilled labor.22 The production function is Cobb-Douglas over capital structures (Ks) and a constant elasticity of substitution (CES) aggregate of the three remaining inputs. The specification they find to be consistent with U.S. data is as follows:
56. The parameters a and p determine the elasticities of substitution among capital equipment (Ke), skilled labor (S) and unskilled labor (U). This production function is also flexible enough to incorporate changes in relative efficiency of different skill categories. Inputs of skilled and unskilled labor may be considered as the products of aggregate hours (h;) and an efficiency index (Ψi), where i is an index for skill type. Exogenous factor neutral productivity is represented by θt.
57. There are a couple of reasons for splitting capital into two types. First, the phenomenon of capital deepening in many industrial economies in recent years is largely attributable to equipment investment (including computers) rather than investment in structures. Second, it is not obvious that skilled and unskilled labor would have different degrees of substitutability with structures, while differences in substitutability with capital equipment are more plausible.
58. Note that this production function specification implies that the elasticity of substitution between equipment and unskilled labor is the same as that between skilled and unskilled labor. This restriction follows from the symmetry property of the CES aggregation and is consistent with empirical estimates of these elasticities. The elasticity of substitution between capital equipment and unskilled labor is l/(l-σ) and that between capital and skilled labor is 1/(1-ρ). Hence, setting σ > ρ implies capital-skill complementarity.
59. The notion of adjustment costs could be most relevant for equipment investment in the second half of the 1990s. In particular, the surge in investment in information and communication technologies (ICT) during this period has been enormous and unprecedented. Indeed, as shown in another paper in this volume23, this increase could be even larger, in real terms, at correctly measured prices. In addition to standard time-to-build considerations, the adjustment costs at such high levels of investment could be rationalized on a number of grounds. As firms change the nature of their production processes and their mix of different vintages of capital, there could be significant lags in optimizing production processes. In addition, the use of new high-tech capital could imply higher depreciation and obsolescence rates for older vintages of equipment. Further, there could be a substantial lag in training even relatively-skilled workers in the use of new capital goods.
60. In terms of modeling, these adjustment costs could easily be added to the production function as a quadratic of the change in the stock of equipment investment. This would yield a modified version of equation (Al):
61. This modification would not add any state variables to the model; hence, calibrating and simulating such a model would impose no additional computational constraints. Intuitively, it is clear that periods with high levels of equipment investment would then have lower levels of output and labor productivity, compared to a steady state with a higher level of equipment capital. In other words, the transition path to a new steady state with higher productivity (both in terms of labor productivity and TFP) might in fact be one with lower measured productivity growth.
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This chapter was prepared by Eswar Prasad (Research Department).
See Prasad (1996) for a review of the empirical literature and a discussion of the implications of this type of aggregation bias for matching moments from theoretical models to moments of actual data. Using U. K. data, Blundell, Reed and Stoker (1999) argue that composition effects arising from endogenous labor force participation decisions create an additional source of bias.
See Dickens (2000) for more details on the NES dataset and for a comparison of this dataset with Labour Force Survey data on low-wage workers.
The use of the RPI-X or the private consumption deflator made little difference to the results reported in this paper.
Note that, since the dataset used in this paper ends in 1999, this analysis may not fully capture the effects of the National Minimum Wage (which was introduced in April 1999) on wage inequality. There is some evidence that, in 2000, hourly wage rates of workers in the bottom decile of the bottom decile of the wage distribution rose significantly faster than those of workers in the top decile.
The downward level shift after 1991 in between-group inequality across occupational groups is the artifact of a change in occupational coding that occurred in that year. It has not yet been possible to consistently match the pre- and post–1991 occupational codes. This does not affect any of the results shown below that are restricted to the post–1990 period.
To abstract from year-to-year variation, the percentile differentials reported in this table are 3-year averages, centered on the years shown.
This approach was popularized by Juhn, Murphy and Pierce (1993). Using this technique, these authors show that, in the U.S., both within- and between-group inequality rose sharply among men during the 1980s.
See Prasad (2000) for more discussion of this point and for an interesting contrast provided by the German experience.
This weighting procedure is similar to that employed by Fortin and Lemieux (2000). The kernel density estimates for log hourly wages were computed using an Epanechnikov kernel with bandwidth set to 0.05. These density estimates were also computed using optimal bandwidths computed separately for each year—these bandwidths were typically in the range of 0.04–0.06. Using optimal bandwidths had little effect on the shape of the distributions. The use of a fixed bandwidth is solely to maintain consistency when comparing distributions across different years.
Fortin and Lemieux (2000) document a similar phenomenon in the U.S. They argue that this reconciles two findings. One is that male wage inequality has increased sharply in the U.S. in the 1980s and, although at a slower rate, also in the 1990s, with both within- and between-group inequality among men contributing to this increase (see, e.g., Juhn, Murphy and Pierce, 1993). The second result, documented by Lee (1999), is that the overall wage distribution in the U.S., including both men and women, was in fact quite stable in the 1980s and 1990s once the effects of the decline in the real value of the minimum wage are excluded.
Arguably, even in the case of a permanent shock, hours would be adjusted before, or in tandem with, employment. However, the discussion later in the paper notes that capital accumulation is a relatively slow process. Hence, there would be little incentive, given the gradual increase in the requirement of labor inputs in concert with slow changes in the scale of production, to adjust labor at the intensive margin.
In addition, the dispersion of annual earnings could differ from that of monthly earnings. However, the NES does not have information on annual earnings (or on the number of months of employment per year).
A variety of human capital and implicit contracting theories have been developed to explain this stylized fact. Keane and Prasad (1993) review this literature and provide some empirical evidence for the United State on the cyclical variability of employment, hours and wages for workers of different skill types.
This 4-group classification is based on 1 -digit industry codes as follows: Manufacturing (metal manufacturing; textiles, leather, clothing; other manufacturing); Construction, utilities and transportation (construction; gas, electricity and water; transport and communications); Services (retail and wholesale trade; financial and professional services; other services). Excluded from this classification are agriculture, forestry and fishing; mining and quarrying; and food, drink and tobacco. Together, these 3 industries account for only about 6 percent of total employment during the 1990s.
This 4-group classification is based on regional codes as follows: North (North East, North West, Merseyside); Midlands, Eastern (East Midlands, West Midlands, Eastern); London, South (London, South East, South West); and Wales and Scotland.
As emphasized earlier, the exercise conducted here is purely an accounting one and sets aside some general equilibrium considerations. Changes in the observed levels of employment and wages for different skill groups are the result of shifts in relative demands and supplies for different types of labor. The sort of conditional exercise conducted here ignores behavioral responses of workers and firms to these shifts.
See the paper “The ’New Economy’ in the United Kingdom” in this volume.
The mechanism discussed here requires a country-specific productivity shock in an open economy, which the United Kingdom clearly is.
Cotis and Rignols (1998) make a similar point in the context of France, although the circumstances there, and the mechanism suggested by these authors, are rather different.
On the former, see, e.g., the papers in the Barrell (1994) volume and references therein. Olivier Blanchard is one of the principal proponents of the latter argument. Blanchard and Wolfers (2000), for instance, argue for that interactions of institutions and shocks are important for explaining cross-country differences in labor market performance.
For direct evidence on capital-skill complementarity, see, e.g., Griliches (1969) and Goldin and Katz (1998). Keane and Prasad (1996) provide more references and a discussion of substitutability/complementarity in an extended production function that includes physical capital, energy, and skilled and unskilled labor.
These authors use this framework to examine the effects of relative demand shifts (for different types of skill) and capital-skill complementarity on the equilibrium skill premium. In an analysis of the West German labor market, Prasad (2000) adapts this model to study the potential employment effects of rigidities that prevent adjustment in the skill premium in response to different shocks.
See the paper “The ’New Economy’ in the United Kingdom” in this volume.