Question

A firm produces gizmos according to the production function Q=10KL, where 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 I 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)

Given Q=10*KL

We are given Q=9000 and K=10. So,

9000=10*10*L

L=90

We know that

Wage rate=w=$200

Rent of capital=r=400

Total Cost=w*L+rK=200*90+400*10=$22000

b)

Marginal Product of labor=MPL=dQ/dL=10K

Marginal Product of capital=MPK=dQ/dK=10L

Cost minimization requires that

MPL/MPK=w/r

10K/10L=200/400

K/L=0.5

K=0.5*L

Set K=0.5*L in production function

Q=10*0.5*L*L=5L2

L=(0.2Q)0.5

K=0.5L=0.5*(0.2Q)0.5

Total Cost is given as

TC=wL+rK=200*(0.2Q)0.5+400*0.5*(0.2Q)0.5=400*(0.2Q)-0.5

TC=400*(0.2*9000)-0.5=16970.56

L=(0.2Q)0.5=(0.2*9000)0.5=42.4264

K=0.5L=0.5*42.42641=21.2132

c)

Isoquant (Q=9000) is given by following schedule

L	K=9000/(10*L)
10.00	90.00
20.00	45.00
30.00	30.00
40.00	22.50
42.43	21.21
60.00	15.00
70.00	12.86
80.00	11.25
90.00	10.00

Isocost ($22000) is given by

L	K=(22000-200*L)/400
0.00	55
20.00	45
40.00	35
60.00	25
80.00	15
90.00	10
100.00	5
110.00	0

Isocost ($16970.56) is given by

L	K=(16970.56-200*L)/400
0.00	42.43
20.00	32.43
40.00	22.43
42.43	21.21
50.00	17.43
84.85	0.00

Isoquant -TC=22000 - TC=16970.56 42.43, 2221 90.00, 10 OHHHHHHHHAA 0 10 20 30 40 50 60 70 80 90 100110120