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=9000

If K*=10

9000=10*10*L

L=9000/100

L*=90

Total cost= 10*400+90*200=4000+18000=$22000

b. Optimal K and L in the long run according to cost minimization method is MPL/MPK=PL/PK

MPL/MPK= 10K/10L= K/L= 200/400=1/2

2K=L

Q=900=10KL

9000= 10K*2K= 20K^2

9000/20=K^2

K= √450

K=21.2

L= 42.4

Total cost= 400*21.2+200*42.4= 16960

c.

Part a.

joqe7 50 40 30 20 10 \lsoquant (90, 10) Isocost -20 0 20 40 60 80 Iso 100 120 Capital

Part b.

Labor -70 60 50 40- 30 20 (42.4, 21.2) Isoquant 70 socost 10 20 30 40 50 60 -10 0 7-10 70 80 90 Capital