Question

The Pontiac Assembly is a factory which uses labour and capital to build trucks according to the technology f(L,K) = 4LK: During a war, the government demands that the Pontiac Assembly produces 2 trucks for the military this year. The factory is currently renting 1 units of capital. Due to various limitations, the amount of capital the factory uses cannot be changed. The wage paid for each unit of labour is 6 and rent for each unit of capital is 9. Assume trucks, labour and capital are continuous variables. Represent labour in the horizontal axis and capital in the vertical axis. Answer the following: If rounding is needed, round your answer to 3 decimal places. Each part is worth 0.5 point. a) Let Lo be the smallest amount of labour the factory must use to meet its production target. What is Lo? b) Suppose the factory employs Lo units of labour. What is the Marginal Product of Capital at this point? C) Suppose the factory employs Lo units of labour. What is the Technical Rate of Substitution MP) at this point? MPX d) After the war, the Pontiac Assembly sells trucks in a competitive market. The current market price for a truck is 70. The factory still cannot change the amount of capital it employs, but it controls the amount of labour it employs. How many trucks will the Pontiac Assembly produce to maximise profit? Note that the only decision variable this firm controls is the amount of labour it hires. Hint: express the profit function using the production function.

Answer 1

A)Q=4*L^0.5*K^0.5

L=(Q^2)/(16K)

Given K=1,

L=(Q^2)/16

So minimum L required to produce target is,

Target=2

L=(2^2)/16=4/16=0.25

B) Marginal product of capital=∆Q/∆K=(2*L^0.5)/(K^0.5)

MPk=[2*(0.25^0.5)]/(1^0.5)=[(2*0.5)]/(1)=1

C)MPL=∆Q/∆L=

(2*K^0.5)/(L^0.5)

MPk=[2*(1^0.5)]/(0.25^0.5)=[(2*1)]/(0.5)=4

MRTS=MPk/MPL=1/4=0.25

D)

Q=4*L^0.5*K^0.5

L=(Q^2)/(16K)

Given K=1,

L=(Q^2)/16

TC=Lw+Kr=(Q^2)/16]*6 +1*9=3(Q^2)/8 +9

MC=∆TC/∆Q=6Q/8=3Q/4

Profit Maximizing quantity is where, MC= price=70

70=3Q/4

Q=70*4/3=280/3=93.33( Profit Maximizing quantity of trucks)