Question

A firm wishes to produce Q = 150 as cheaply as possible using only labor (L) and capital (K) with a relationship explained by the following production technology: Q = 15L + 25K. The prevailing market wage is $w/hr. You will need to work carefully to determine the wage rate but know that the rental rate of capital, r = 50.

1. Find w so that the firm can optimally employ 5 workers and 3 units of capital to produce output: Q = 150 while still minimizing costs. Show all work to report w and the total minimum cost associated with this level of production.

2. Suppose it currently costs the firm $300 to produce Q = 150 units of output optimally while using at least some capital. What’s the lower bound on how large the wage can be given these circumstances?

3. Now suppose the wage rate w falls to be strictly lower than the bound you identified in (2). Find the upper bound on the minimum cost to produce Q = 150. In other words, find the largest amount that the cost-minimizing firm would be paying to produce Q = 150 assuming it is still optimizing after the decrease in w.

Answer 1

Q = 150 Q=15L+25K 8=50 Page No : (19 k) = ( 5, 7) at Cost Minimization, MRTS = Wo MRTS = MPL = 15 = 315 on ha - MPK 25 so 3/5= w/50 W = 3x5015 - 30 Min Cost, C= 5wt38 = (5x30) +(3x 50) = 150+150 = 300. Tas sund funch is pest substitutes, so either comor soin exist, or any combination of two inputs can be used) 2.) c=300, Q=150, MRTS = 315=.6, at least some Kis used, then MPK >MPL or w. W > MRTS W> o6x50=30 Lower bound on wages = $30 8 I w<30, then MRTS > W18. oron MPL | W7 MPK18 Only Labor is used in prodo, 80 Q=150 = 15L + 0. L = 150 115 = 1o. cost = 106. - 16 80 upper bound on Cost = 10x30 = $300 ms