Question

1. Consider a firm with the following production function:

Q= 60 L^2 K^1/2 -4L^3

The firm is operating in the short run with a fixed capital stock of K=4

Use this to answer the following questions:

2. Suppose the firm in the previous question is able to increase its capital to K ¯ = 9.  

After what quantity of labor does diminishing marginal returns now occur?

What is the maximum output the firm can now produce?

After what number of workers does diminishing marginal returns begin to occur?

After what number of workers does negative marginal returns begin to occur?

What quantity of workers will maximize the output per worker (average product of labor).

What is the maximum output (total product) the firm is able to produce with the current level of capital?

Note: only enter the number itself in your answers. Any added text will give a wrong answer message.  

Answer 1

2) k=9,

Q=60*L^2*(9)^1/2-4L^2=180L^2- 4L^3

MP(L)=360L-12L^2

Derivative of Marginal product of labour,

∆MP(L)/∆L=360-24L=0

L=360/24=15

So at=15, Diminishing marginal return occurs.

The maximum output will be there ,where marginal product of labour will be zero.

Mp=360L-12L^2

0=360-12L^2

L^2=30

L=√30

Maximum output=180*(√30)^2-4*(√30)^3=180*30-4*30*√30=4742.7

Negative marginal return start when output reaches maximum ,so negative marginal return start at L=30.

Note;The last two questions are same as first two