Empirical Estimates of Labor Demand
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 constantelasticityofsubstitution (CES) form:
Y = ( a_{1} L^{s} +
a_{2} K^{s}
)^{1/s}
Then, MP_{L} = a_{1}
(Y/L)^{1s} and MP_{K} =
a_{2}
(Y/K)^{1s}. Thus the firstorder condition
satisfies:
( a_{1}/a_{2} )
( K/L )^{1s} = w/r
which implies
log (K/L) =  log(a_{1}/a_{2})
+ ( 1/(1s) ) 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/(1s) ).
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 crosssubstitution 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 wageemployment 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 wageemployment relationship should be positive. In any
case, while the assumption of wagetaking 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 crosssection
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 wageswhich 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.

Lecture Notes

Labor Demand
Minimum wage