Question

The production function of the Auto parts firm is given by Q-5L-L, where Q is the units of output and L is the number of labor hours. Each output sells for 100 dollars per unit. The human resources manager estimates that the marginal cost of hiring an extra worker is 50 dollars. How many labor hours should this firm hire? Hint: MPL=5-2 L 1) 2) A frim's production function is given by Q(L)-6L, where Q measures output and L is the number of labor hours. Output sells for 4 dollars per unit and cost of labor is 30 dollars per labor hour How many labor hours should this firm optimally hire? Hint: MP 6 3) A frim's production function is given by Q(L,K)-40L a) How much labor and capital should this firm optimally hire if P-10, PK 80, and the total budget is 800,000 dollars? b) Does this production function satisfy deceasing returns to scale? Hint: MPL=40K and MPK-40 L 4) A frim's production function is given by Q(LK)-5L+10K. How much labor and capital should this firm optimally hire if P P10, while the total budget is 200,000 dollars? Does this production function satisfy constant returns to scale? Hint: MPL 5 and MP10

Answer 1

Solution:

For first two questions:

Optimum level of labor hours that a firm should hire is where Marginal revenue product of labor (MRPL) equals the wage rate, w. Here, MRPL = MPL*P, where MPL is marginal product of labor and P is the price of output. Also, wage rate is the price paid for 1 unit of labor. So, with a fixed marginal cost, that is a fixed amount to hire an extra unit of labor, marginal cost, MC = wage rate, w. Then, we can answer the following:

1) Q = 5L - L², so, MPL = $\partial Q/\partial L$ = 5 - 2L (as already given in the hint)

Further we are given, P = $100 per unit of output, and w = MC = $50 per hour of labor.

Then, MRPL = MPL*P = (5 - 2L)*100 = 500 - 200L

Using, optimality condition of MRPL = w, we get

500 - 200L = 50

L = 450/200 = 2.25. Thus, this firm should hire 2.25 labor hours.

2) Q = 6L , so, MPL = $\partial Q/\partial L$ = 6 (as already given in the hint)

Further we are given, P = $4 per unit of output, and w = MC = $30 per hour of labor.

Then, MRPL = MPL*P = 6*4 = $24

Since, for any number of labor hours, additional revenue generated is always lower than the additional cost incurred by hiring an extra labor ($24 < $30), thus, in such a case, optimum decision would be to hire 0 hours of labor.

For remaining two questions:

In the long run, when output function also includes the capital, K (capital is no more fixed as in the short run) along with labor, L, optimality condition (or condition determining optimal levels inputs: capital and labor) becomes: MPL/MPK = w/r, where MPL is marginal product of labor, MPK is marginal product of capital, w is the wage rate (here given as price of labor, PL) and r is the rental rate of capital (here given as the price of capital, PK).

The total cost is given as: TC = w*L + r*K. Our aim would be to maximize profit, with a given level of budget (=TC). Further, MPL = $\partial Q/\partial L$ and MPK = $\partial Q/\partial K$ , then we have the following:

3) Q = 40LK

a) So, MPL = 40K, MPK = 40L

MPL/MPK = (40K)/(40L) = K/L

w/r = PL/PK = 10/80= 1/8

So, at optimality: K/L = 1/8 so, L = 8K

With budget of $800,000, using the total cost function and the optimality condition we have

TC = PL*L + PK*K

800000 = 10*8K + 80*K

800000 = 160*K

K = 800000/160 = 5,000 units

So, L = 8*5000 = 40,000 units

So, this firms should optimally hire 5,000 capital hours and 40,000 labor hours.

b) Returns to scale: Q(tL, tK) = tⁿ*Q(L, K), if n>1, increasing returns to scale, if n =1 then constant returns to scale, and if 0<n < 1 then decreasing returns to scale. Economically this means that when all factor inputs (here, K and L) are increased by a same factor (here, t), then if total output(here, Q) increases by a greater factor, then production function exhibits increasing returns to scale, if increases by the same factor, then exhibits constant returns to scale, and if increases by a lesser factor, then decreasing returns to scale.

With Q(L, K) = 40LK

Increasing L and K by factor t gives: Q(tL, tK) = 40(tL)(tK) = t²(40LK) = t²Q(L, K), so here n = 2 which is greater than 1. This production function exhibits increasing returns to scale (and thus, not decreasing returns to scale).

4) Q(L, K) = 5L + 10K (case of perfect substitutes, thus extreme solutions possible)

a) MPL = 5, MPK = 10

So, MPL/MPK = 5/10 = 0.5

Input price ratio, PL/PK = 10/10 = 1

Here, for any levels of L and K, the optimality condition always has MPL/MPK < PL/PK (0.5 < 1). This means that for every hour of labor hired, the relative marginal product received from labor is always costlier (that is the relative price of hiring labor unit is always greater than the relative marginal return that labor offers). Correspondingly, hiring capital is cheaper (going with this relativity). So, in this case optimally, firm should hire 0 hours of labor and spend entirely on capital hiring.

With budget or total costing of $200000, cost function gives: 200000 = 10*0 + 10*K

K = 200000/10 = 20,000 hours.

So, this firms should optimally hire 20,000 capital hours and 0 labor hours.

b) Returns to scale:

With Q(L, K) = 5L + 10K

Increasing L and K by factor t gives: Q(tL, tK) = 5(tL) + 10(tK)

Q(tL, tK) = t(5L + 10K) = t*Q(L, K), so here n = 1. So yes, this production function exhibits increasing constant to scale.