Question

A firm produces gizmos according to the production function Q =10KL , where Q is the quantity of gismos produced, K is the quantity of capital rented and L is the quantity of labour hired. The manager has been given a production target: Produce 9,000 gizmos per day. He is informed that the daily rental price of capital is $400 per unit and the wage rate is $200 per day. a) Currently, the firm has 10 units of capital. How many workers should the manager hire to meet the production target? What is the firm’s daily total cost? b) In the long run, how much K and L should the manager choose to minimize the cost of producing 9,000 gizmos per day? What is the long-run daily total cost? c) Illustrate your answers in a) and b) on the iso-quant and iso-cost diagram

Answer 1

a) Q = 10KL = 9,000
At K = 10,
10(10)L = 9000
So, 100L = 9,000
So, L = 9000/100
Thus, L = 90

Total cost = (wage*L) + (price of capital*K) = (200*90) + (400*10) = 18,000 + 4,000 = $22,000

b) Combination of L and K which minimize the cost is determined where MRTS = wage/price of capital = 200/400 = 1/2
Now, MRTS = MPL/MPK =

So, K/L = 1/2
So, L = 2K

Now, 10KL = 10K(2K) = 9,000
So, 20K2 = 9,000
So, K2 = 9000/20 = 450
So, K = 4501/2
So, K = 21.21

L = 2K = 2(21.21)
So, L = 42.42

Total cost, C = (wage*L) + (price of capital*K) = (200*42.42) + (400*21.21) = 8484 + 8484 = 16,968

(c)