Answer to Question #158710 in Macroeconomics for Iqra

Question

You have been hired by Kia as manager for its Pakistan operations. Assume following is the short￾run production function at their assembly plant outside Karachi:

Q = 10L2 – 0.5 L3

 where L is variable input labor, Q is output of Cars assembled

a. Find the ranges of the three stages of production.

c. At the end of the year it is expected that output will double with purchase ofnew 

equipment and machinery. The production function is estimated to be

Q = 60L.30K.70 

 where L is labor and K is capital.

Suppose initial L1 = 1 and K1 = 1. When inputs are in increased to L2 = 2 and K2 = 2, 

do you observe increasing, decreasing or constant returns to scale?

Answer 1

Part A

$Q$ $=$ $10L$$^2$ $-0.5L^3$

Average Product, $AP=$ $Q/L$

$10L^2-0.5L^3/L$

$10L-0.5L^2$

At maximization we equate to zero & differentiate with respect to L.

$d(AP)/dL=0$

$10-1.5*2L=0$

$L=10$

Stage 1 is as a result of average product increase. The range is from L=0 to L=10

Marginal product,

$MP=dQ/dL$

$MP=10*2L-0.5*3L^2$

$MP=20L-1.5L^2$

At Stage 2 margin product decreases but still remains positive.

At Stage 3 marginal product is equal to zero

$MP=0$

$20L-1.5L^2=0$

$20=1.5L$

$L=20/1.5$

$L=13.33$

Part C

$60L$^0.3 $K$^0.7

Plugging $K=L=1$

$Q=60*1*1=60$

Plugging $K=L=2$

$Q=60*2$^0.3$*2$^0.7$=60*2=120$

The firm is having a constant returns to scale because the output is doubled when the inputs are doubled.