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Methods of estimation

• The simplest method is just to postulate some simple functional form (e.g., log L = b log W + ... ) and estimate the implied elasticity of labor demand.
• Another approach is to assume some functional form for the production function and derive the implied demand elasticity. For example, suppose the production function takes the constant-elasticity-of-substitution (CES) form: Y = ( a₁ L^s + a₂ K^s )^1/s Then, MP_L = a₁ (Y/L)^1-s and MP_K = a₂ (Y/K)^1-s. Thus the first-order condition satisfies: ( a₁/a₂ ) ( K/L )^1-s = w/r which implies log (K/L) = - log(a₁/a₂) + ( 1/(1-s) ) log(w/r) The above is a simple regression. The estimated coefficient on log(w/r) is the estimate of the elasticity of substitution ( 1/(1-s) ). Once we obtain this estimate, we can derive the labor demand elasticity using the fact that labor demand elasticity is equal to - ( 1 - share of labor cost in total cost ) * elasticity of substitution
• Still another approach is to use some flexible function form for the cost function (or indirect profit function), and then derive all the factor demand equations using Shephard's lemma (or Hotelling's lemma in the case of indirect profit function). The system of factor demand equations are then estimated jointly.
  • In principle, estimates using this approach utilize all the information available in the data and are consistent with the producer theory. They should therefore produce more efficient estimates.
  • Estimating a system of equations instead of just the labor demand equation also allows us to estimate the cross-substitution across different factors.
  • In practice, because of aggregation problems, producer heterogeneity, and adjustments on the extensive margin, an approach applicable to a single firm is not always best for the entire industry. Many of the estimates based on the system of equations approach do not satisfy the restrictions implied by producer theory (e.g., homogeneity and symmetry).

Perhaps a far more troublesome aspect of empirical demand estimation is the identification problem. The estimate the labor demand response to changes in wage, the wage variations must be exogenous. When are observed wage changes exogenous?

• Suppose you look at time series data and estimate the relationship between wages and employment for the economy. Are you estimating a labor demand curve or a labor supply curve? According to Keynesian theories, aggregate demand for output determines the demand for labor. Fluctuations in aggregate demand generate fluctuations in employment levels, and wages clear the labor market. Therefore, observed wage-employment pairs are on the same labor demand curve, and the two variables should be negatively related. According to real business cycle theories, productivity shocks shifts the labor demand curve, and wage and employment levels adjusts along the same labor supply curve. So observed wage-employment relationship should be positive. In any case, while the assumption of wage-taking behavior may be appropriate for an individual firm, it is probably inappropriate for the economy. When wage changes are not exogenous (e.g., when they are driven by productivity shocks), you can't consistently estimate labor demand.
• The same problem arises if you look at time series data on wage and employment within a single firm. Wage growth may be due to productivity growth. Ignoring this fact will lead to a downward bias in the magnitude of the labor demand elasticity.
• A common approach to estimating labor demand is to use cross-section data across different firms in the same industry. To estimate labor demand using this approach, there must be wage variations across firms. In this case, we have to ask why different firms pay different wages. In a competitive model, they should not. If they do, this is probably because they are not really paying for the same kind of labor. A high wage firm is probably hiring better quality workers. Again, ignoring this fact will lead to a downward bias in the magnitude of the labor demand elasticity.

These caveats notwithstanding, here are some "stylized facts" from labor demand studies:

• the conditional labor demand elasticity is in the range from -0.15 to -0.75.
• labor and energy are substitutes
• capital and skilled labor are complements
• technical progress is complementary to skilled labor
• labor demand is less elastic for skilled labor than for unskilled labor

Other studies uses more special situations to help solve the identification problem. One instance when you will find a convincing case of exogenous wage change is when the minimum wage changes (though one might argue that legislators don't raise the minimum wage when labor demand curve is slack). Thus, studying the employment effects of minimum wage is an important part of empirical labor demand studies.

Quite often, instead of estimating a labor demand curve, it is more useful to estimate an inverse labor demand curve. That is, instead of regressing employment on wages, we may regress wages on employment. To estimate a labor demand curve, we need exogenous variations in wages--which are hard to find. To estimate an inverse labor demand curve, we need exogenous variations in employment. Exogenous variations in employment may arise because of immigration flows, so people may study labor demand by looking at the wage response to increased immigration. This approach, however, does not always give us a representative picture of the whole economy because immigrants typically are only a small fraction of the labor force.

Others have exploited the fact that population often changes for reasons other than wage changes. For example, there was a baby boom in the U.S. shortly after World War II. These baby boomers entered the labor market some twenty years later. So in the late sixties and early seventies, there is a especially large cohort of new labor market entrants. To clear the market, the wage of these baby boomers must fall relative to the wages of other cohorts. Finis Welch (1973) finds that this was indeed the case.

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Lecture Notes

Labor Demand Minimum wage