ECON 412 Labor Economics and Labor Markets Homework 2
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ECON 412
Labor Economics and Labor Markets
Homework 2
Fall 2022
Problem 1
Consider a firm which produces according to the following production function: q = A(^E +^K)
where q denotes output, E denotes the number of workers hired by the firm, K denotes capital, and A > 0 is a technology parameter. The marginal product of labor is therefore given by MPE = 2A^E . In the short run, the firm rents 100 units of capital, at a rental cost of r = 10 dollars per unit. The firm sells its output at a market price of 3 dollars, and has to pay each employee a wage of 15 dollars. The current factory setup implies that the technology parameter A = 20.
1.1. In the short run, how many employees should the firm hire? How many units of output does the firm produce, and how much profit does it make? Show your work and make sure to verify all conditions for optimality.
1.2. Now suppose the firm can install a new technology which would streamline the production facility, and increase the technology parameter to A = 30, such that the new production function would become
q = 30(^E +^K)
If the firm would install this new technology, how would the optimal labor demand, output and profits change? Assume all the other parameters (w, p, r, K0 ) remain unchanged.
1.3. Suppose that the new technology described in Problem 1.2 costs 300 dollars to implement and operate. Should the firm invest in the new technology or not? Explain.
1.4. What is the short-run elasticity of labor demand (E) with respect to the wage rate (w)? Is the firm’s labor demand elastic or inelastic?
Problem 2
Suppose you obtain data on a sample of working adult individuals in the U.S. (indexed by i = 1, ..., N), who self-reported the following variables:
. labinci = annual income from labor (after taxes)
● nonlabinci = annual non-labor income (after taxes)
● hoursi = annual hours worked
● educi = total years of completed schooling
● agei = age in years
● malei = a dummy variable equal to 1 for males, 0 for females
2.1 Using these data, how would you estimate the wage elasticity of labor supply? Write down
a specific regression, and feel free to define any new variables using the ones you already have, if you think they would be relevant. Make sure to mention what your coefficient of interest is.
2.2
How would you test each of the following three hypotheses:
A) the substitution effect dominates the income effect.
B) leisure is a normal good.
C) all else equal, men work more hours than women.
Write down specific null (H0 ) and alternative (H1 ) hypotheses using parameters from the regression equation you wrote down in 2.1.
2.3 Suppose you estimate a wage elasticity of -0.2. How would you interpret this?
2.4 Suppose you are particularly worried about people misreporting their annual work hours. Do you think the true labor supply elasticity would be larger or smaller than your estimate of -0.2? Explain.
2.5 Besides measurement error, what other potential concerns do you have about this regres- sion approach (i.e. with respect to the key OLS assumption of conditional independence being potentially violated)? Which slope coefficients do you think may be biased? Explain.
Problem 3
Suppose a small competitive firm hires two types of labor: blue-collar workers (E1 ) who have to be paid a market wage w1 = 10, and white-collar workers (E2 ) who have to be paid a market wage w2 = 20. There is no capital. The firm’s production technology is
q = f (E1 , E2 ) = ln(E1 ) + 2 ln(E2 )
Assume that the firm takes the output price (p = 100) as given.
3.1 Write down the firm’s maximization problem, and characterize the optimal labor demands (E1(*) and E2(*)) as functions of the parameters (p, w1 and w2 ). How many workers of each type does the firm hire? For simplicity, just focus on solving the first order condition(s) and ignore the other optimality conditions.
3.2 Now suppose that the minimum wage in this economy increases from 7.5 dollars to 12.5 dollars. Assuming that this does not affect the wage of white collar workers nor the output price, how would the labor demand for each type of worker change?
3.3 How would your answer to the previous subquestion change if you took into account the fact that the minimum wage increase affects all firms in the industry, and not just the one small (competitive) firm we are studying? Explain your reasoning in detail. (For example, you could assume that there are millions of more or less identical, small, competitive firms out there making the same goods and hiring the same two types of labor).
3.4 (Bonus question) Try to show that for this particular type of production function, labor demand for either type of worker is neither elastic nor inelastic.
2022-11-02