Minimum Wage Labor Subsidy: Suppose you run your business by using a homothetic, decreasing returns to scale production process that requires minimum wage labor ℓ and capital k where the minimum wage is w and the rental rate on capital is r.
A: The government, concerned over the lack of minimum wage jobs, agrees to subsidize your employment of minimum wage workers — effectively reducing the wage you have to pay to (1 − s)w (where 0 < s < 1). Suppose your long run profit maximizing production plan before the subsidy was (ℓ∗,k∗, x∗).
(a) Begin with an isoquant graph that contains the isoquant corresponding to x∗ and indicate on it the cost minimizing input bundle as A. What region in the graph encompasses all possible production plans that could potentially be long run profit maximizing when the effective wage falls to (1 − s)w?
(b) On your graph, illustrate the slice of the production frontier to which you are constrained in the short run when capital is fixed. Choose a plausible point on that slice as your new short run profit maximizing production plan B. What has to be true at this point?
(c) Can you conclude anything about how the marginal product of capital changes as you switch to its new short run profit maximizing production plan?
(d) Will you hire more workers in the long run than in the short run?
(e) Will you hire more capital in the long run than in the short run?
(f) Once you have located B in part (b), can you now use this to narrow down the region (that you initially indicated in part (a)) where the long run profit maximizing production plan must lie?
(a) Suppose that w = 10 = r and p = 5. What is your profit maximizing production plan before the labor subsidy?
(b) What is the short run profit maximizing plan after a subsidy of s = 0.5 is implemented.
(c) What is the new long run profit maximizing plan once capital can be adjusted?
(d) For any Cobb-Douglas function f (ℓ,k) = Aℓβαkβ(1−α), the CES production function g (ℓ,k) = A(αℓ−ρ + (1 − α)k−ρ)−β/ρ converges to f as ρ approaches 0. What values for A, α and β will do this for the production function x = 30ℓ0.2k0.6?
(e) Using a spreadsheet to program the output supply and input demand equations for a CES production function given in equation (13.38) in a footnote in the text, verify that your long run production plans mirror those you calculated for the Cobb-Douglas function when ρ approaches 0 and α and β are set appropriately.
(f) Finally, derive the first order condition for the short run profit maximization problem that fixed capital using the CES production function. Then, using your spreadsheet, check to see whether those first order conditions hold when you plug in the short run profit maximizing quantity of labor that you calculated in (b).