Question

1. (30) Consider that you run a panini shop here on campus. You hire labor (L) at a daily wage rate w = 10, rent panini presses (K) at a daily rental rate r = 40, and produce panini with the following (daily) technology:

q = f(L, K) = 2L^0.5 K^0.5

(a) (5) Solve for the tangency condition, expressing L as a function of K.

(b) (5) For q = 100, what is the cost-minimizing, long-run choice of L and K? Label your answers L ∗ and K∗ and draw a box around them.

(c) (5) Tomorrow, you will face a once-off increase in demand and your total output for the day will be 200. Capital is fixed in the short run. What is your short-run L and K for q = 200? Label your answers L and K and draw a box around them.

(d) (5) Following the scenario in (c), how much does each sandwich cost to produce in the short run? How does this compare to the price per sandwich in the long run? Explain in two sentences or less.

(e) (10) Show your answers to (b) and (c) graphically. Put L on the horizontal axis and K on the vertical one. Include the relevant isocosts, isoquants, and input bundles. Include the production expansion path.

Answer 1

q = 2L^0.5K^0.5

(a)

Cost is minimized when MPL/MPK = w/r = 10/40 = 1/4

MPL = q/L = 2 x 0.5 x (K/L)^0.5

MPK = q/K = 2 x 0.5 x (L/K)^0.5

MPL/MPK = K/L = 1/4

L = 4K

(b)

When q = 100, substituting L = 4K in production function,

2 x (4K)^0.5K^0.5 = 100

(4)^0.5K^0.5K^0.5 = 50

2 x K = 50

K = 25

L = 4 x 25 = 100

(c)

In short run, K is fixed at 25. When q = 200,

2 x (25)^0.5L^0.5 = 200

2 x 5 x L^0.5 = 200

L^0.5 = 200/10 = 20

Squaring,

L = 400

(d)

Short run total cost = wL + rK = 10 x 400 + 40 x 25 = 4000 + 1000 = 5000

In long run, both L and K are variable. Therefore the firm can substitute cheaper labor for costlier capital in long run. As a result, long run total cost will be lower than short run total cost.

NOTE: As per Answering Policy, 1st 4 parts are answered.