Question

DW Company produces brass fittings. DW's engineers estimate the production function represented below as relevant for their long-run capital-labor decisions.

Q = 600L^0.4K^0.6,
where Q = annual output measured in pounds,
L = labor measured in person hours,
K = capital measured in machine hours.
DW's employees are relatively highly skilled and earn £20 per hour. The firm estimates a rental charge of £60 per hour on capital. DW forecasts annual costs of £500,000 per year, measured in real Pounds.

(a) Do the production function exhibits increasing, constant, or decreasing returns to scale? Explain your answer (mathematical demonstration).

(b) What are the marginal products of labor (?? ) and capital (?? )? Determine the firm's !"

optimal capital-labor ratio, given the information above.

(c) Construct the isocost equation. How much capital and labour should the firm employ, given

the £ 500,000 budget?

(d) Calculate the firm’s output at the optimal production situation.

Answer 1

Solution:

We are given the production function as: Q = 600*L0.4*K0.6, where L is the quantity of labor input used and K is the quantity of capital input used.

Further, the wage rate is given as: w = 20 pounds per hour, and capital rental rate: r = 60 pounds per hour

Annual cost or budget is of 500,000 pounds per year.

a) Finding the returns to scale exhibited by given production function:

Q(K, L) = 600*L0.4*K0.6

Increasing all inputs by a factor t: Q(tK, tL) = 600*(tL)0.4*(tK)0.6

Q(tK, tL) = 600*t0.4*L0.4*t0.6*K0.6

Q(tK, tL) = 600*t0.4+0.6*L0.4*K0.6 = t1*600*L0.4*K0.6

So, Q(tK, tL) = t*Q(K, L)

As increasing all inputs by factor t increases the final output by same factor t, this production function exhibits constant retuurns to scale.

b) Marginal product of labor, MPL = dQ/dL

MPL = 0.4*600*L0.4-1*K0.6

MPL = 240*L-0.6*K0.6 = 240*(K/L)0.6

Marginal product of capital, MPK = dQ/dK

MPK = 0.6*600*L0.4*K0.6-1

MPK = 360*L0.4*K-0.4 = 360*(L/K)0.4

Findng the optimal capital-labor ratio: Marginal rate of technical substitution, MRTS = MPL/MPK

MRTS = [240*(K/L)0.6]/[360*(L/K)0.4] = (2/3)*(K/L)

Optimal input bundle is the one where the input price line is tangent to isoquant curve, or in other words, where the slope of budget line (w/r) equals the slope of isoquant curve (MRTS):

w/r = 20/60 = 1/3

So, optimal condition gives: 1/3 = (2/3)*(K/L)

1 = 2*(K/L)

L = 2*K

Or optimal capital-labor ratio is: K/L = 1/2 or 1:2

c) Total cost/budget = r*K + w*L

With budget of 500,000 pounds, this becomes: 500000 = 60*K + 20*L

So, this is the required isocost line, which will give different combinations for (L, K) such that total cost remains 500000.

d) At optimal production situation:

With given budget: 500000 = 60*K + 20*(2*K)

500000 = 60K + 40K

So, K = 500000/100 = 5000

And then, L = 2*5000 = 10000

Then, firm's output at this optimal production situation becomes:

Q = 600*100000.4*50000.6

Q = 600*39.811*165.723 = 3958551.87 or 3,958,552 units approximately.