Question

The Longheel Press produces memo pads in its local shop. The company can rent its equipment and hire workers at competitive rates. Equipment needed for this operation can be rented at $52 per hour, and labor can be hired at $12 per worker hour. The company has allocated $150,000 for the initial run of memo pads. The production function using available technology can be expressed as:

Q = 0.25K^0.25L^0.75,

where Q represents memo pads (boxes per hour), K denotes capital input (units per hour), and L denotes labor input (units of worker time per hour).

a. Construct the isocost equation and the marginal products of labor and of capital.

b. Determine the appropriate input mix to get the greatest output for an outlay of $150,000 for a production run of memo pads. Also, compute the level of output.

c. Explain what would happen in the short run (keeping capital fixed) to the appropriate input mix if production were increased. How would the input combination change from the short run to the long run? And compare the short run cost and the long run cost.

Answer 1

Q = 0.25K^0.25L^0.75

(a)

(i) Isocost line: C ($) = wL + rK

150,000 = 12L + 52K

37,500 = 3L + 13K [Dividing by 4]

(ii)

MPL = Q/L = 0.25 x 0.75 x (K/L)^0.25 = 0.1875 x (K/L)^0.25

MPK = Q/K = 0.25 x 0.25 x (L/K)^0.75 = 0.0625 x (L/K)^0.75

(b)

Input mix is optimal when MPL/MPK = w/r = 12/52 = 3/13

MPL/MPK = 3 x (K/L) = 3/13

3K/L = 3/13

L = 13K

Substituting in isocost function,

37,500 = 3L + 3L = 6L

L = 6,250

K = L/13 = 6,250/13 = 480.77

Q = 0.25 x 480.77^0.256250^0.75 = 0.25 x 4.68 x 702.93 = 822.42

(c)

In short run, keeping K fixed at 480.77, as we keep increasing output, we need to use more and more amount of labor, which will eventually start lowering the rate of increase in output from additional labor, assuming law of diminishing returns holds true. This is because capital being fixed, it cannot be changed in short run.

In the long run, both inputs are variable, so firm can increase production by varying the amount of both capital and labor. Therefore, amount of capital used will be higher than that in short run.

Because of input flexibility, long run cost will be lower than short run cost for producing a given amount of output.