Question

A firm discovers that when it uses K units of capital and L units of labor, it is able to produce X = L^1/4*K^3/4 units of output.

a. Draw the graph of isoquants in labor-capital plane.

b. Suppose that the firm produces 24 units of output using 16 units of capital and 81 units of labor. Compute MRTS subscript LK. Compute the MPL. Compute the MPK.

c. On the basis of your answer to part (b), is the equation MRTS sbs LK=MPL/MPK approximately true? (It would become closer to being true if we measured inputs in smaller units.)

d. Suppose that capital and labor can each be hired at $1 per unit and that the firm uses 16 units of capital in the short run. What is the short-run total cost to produce 10 units of output?

e. Continue to assume that capital and labor can each be hired at $1 per unit. Show that in the long run, if the firm produces 24 units of output, it will employ 16 units of capital and 81 units of labor. What is the long-run total cost to produce 12 units of output?

f. Does this production function exhibit constant, increasing, or decreasing return to scale?

Answer 1

Solution:

Given production function: X = L^1/4*K^3/4

b) X = 24 units, L = 81 units, and K = 16 units

MRTS_LK = MPL/MPK

MPL = $\partial X/\partial L$ = (1/4)*(K/L)^3/4

With K = 16 and L = 81, MPL = (1/4)*(16/81)^3/4 = (1/4)*(8/27) = 0.074

MPK = $\partial X/\partial L$ = (3/4)*(L/K)^1/4

With K = 16 and L = 81, MPK = (3/4)*(81/16)^1/4 = (3/4)*(3/2) = 1.125

MRTS_LK = MPL/MPK = [(1/4)*(K/L)^3/4]/[(3/4)*(L/K)^1/4] = (1/3)*(K/L)

With K = 16 and L = 81, MRTS_LK = (1/3)*(16/81) = 0.06584

c) From our answers in part (b), we know that MRTS_LK = 0.06584

And, MPL/MPK = 0.074/1.125 = 0.06578

So, yes, we can conclude that the equation MRTS = MPL/MPK is approximately true. Also, this value = 0.0658 (approx).

d) Rental rate of capital, r = $1, wage rate of labor, w = $1

With K = 16 units in short run,

Our short run production function becomes: X = L^1/4*(16)^3/4 = 8*L^1/4

In order to produce 10 units of output, that is, for X = 10, we find L required.

10 = 8*L^1/4

L^1/4 = (10/8)

On powering by 4 on bot sides, we get, L = (10/8)⁴ = 2.44 (approx)

So, the total cost = w*L + r*K

Total cost = 1*2.44 + 1*16 = $18.44. So, short run total cost to produce 10 units of output is $18.44.

e) The firms always aims at maximizing it's profits. Thus, in long run it is expected that firm is choosing the optimal labor-capital mix bundle. Furthermore, profit maximizing (or cost minimizing) condition for a firm is where MRTS_LK = w/r

Now, with w= $1 and r = $1, w/r = 1

MRTS_LK = (1/3)*(K/L)

With profit maximizing condition, we then get (1/3)*(K/L) = 1

K = 3L

So, production function becomes: X = L^1/4*(3L)^3/4 = 27^1/4L

For production of 24 units of output, L required = (24)/(27)^1/4 = 24/2.28 = 10.53 (approx)

So, units of capital required, K = 3*L = 3*10.53 = 31.59

And total cost = 1*10.53 + 1*31.59 = $42.12

(Verify the question once, as optimal choice is not coming out to be what is given: 16 units of capital and 81 units of labor; logically this input mix requires total cost of 16 + 81= $97 which is a lot higher than the cost generated by optimal bundle I found ($97 > $42.12), so I think the question is wrong here).

If X = 12 units, from above, L = (12)/(27)^1/4 = 12/2.28 = 5.26 (approx)

So, units of capital required, K = 3*L = 3*5.26 = 15.79

And total cost = 1*5.26 + 1*15.79 = $21.05

f) Returns to scale exhibited by the production function:

If both inputs are increased by a factor, say t, then by how much does the units of output increase?

X = L^1/4*K^3/4

Let the new production level be X', so X' = (tL)^1/4*(tK)^3/4

X' = t^1/4+3/4L^1/4K^3/4 = t*L^1/4*K^3/4

X' = tX

This means that increasing both the inputs by factor of t increases the output by the same factor, that is factor t. This implies that given production function exhibits constant returns to scale. (If the increase is by a factor greater than t, function exhibits increasing returns to scale; if the increase is by a factor less than t, function exhibits decreasing returns to scale)