Question

Provide a convincing argument that a firm using with g(K,L) =min{K/4, L/5} would select K*= 128 in the Long Run given P=$96, w=$16 and r=$4??

Answer 1

The production function is given as , which is a Leontiff production function having L-shaped kinked isoquant. The cost of production is given as
c=WL + TK
or .

The optimal combination of inputs would be where the corner of the isoquant lies, which is where or
K= 0.81
. Putting this in the production function as or or or
L = 50
, and since
K= 0.81
, we have or or . These are the required conditional input demand.

In the long run, the firm must have zero economic profit. For K*=128, since we have or . Also, since
L = 50
, we have or . For the given price, the total revenue at this quantity would be dollars. The total cost would be as dollars. The profit here would be $\pi = TR - TC$ or $\pi = 3072 - 3072 = 0$ . Hence, the firm would employ K*=128 in the long run for the given price and input prices.

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OPTIONAL : Note however that the question asks for a convincing arguement which is provided above, but this would be true for any Q>0. The reason being that the cost function is 96Q, and marginal cost of $96 is equal to price of $96 for all Q>0. For example, take Q=10, we would have K*=40, L*=50, and hence $\pi = TR - TC = PQ - (wL^* + rK^*) = 960 - (16*50 + 4*40) = 960 - 960 = 0$ .