Download Student Number: QUIZ 4 Section D Question 1. [5 marks] Consider

Student Number:
QUIZ 4
Section D
Question 1. [5 marks] Consider the production function
Q = 10 K 2 L2
a) [3 marks] Letting w be the wage rate and r be the rental rate of capital, derive the
cost-minimizing quantities of labor and capital as a function of w , r , and Q in the long
run.
Answer:
Set up the tangency condition to determine the relationship between capital and labor.
MPL w
=
MPK r
10 K 2 L w
=
10 KL2 r
K w
=
L r
w
K= L
r
Plug this result into the production function to determine the optimal quantities of
capital and labor.
2
w 
Q = 10 L  L2
r 
 w2 4 
Q = 10
L 
r


L4 =
Qr
 
10  w 
1
2
 Q  4 r 
L=   
 10   w 
The optimal level of capital is
1
2
1
 w  Q  4  r 
K =     
 r  10   w 
1
 Q  4 w
K =   
 10   r 
1
1
2
2
b) [2 marks] Derive the long-run total cost function as a function of w , r , and Q .
Answer:
To find the long-run total cost function, plug the optimal quantities of capital and labor
from part a) into the total cost function.
TC = wL + rK
1
 Q  4 r 
TC = w   
 10   w 
1
2
1
 Q  4 w
+ r   
 10   r 
1
2
1
Q 4
TC =   rw
5
Question 2. [5 marks] In a certain market in the long-run, each firm and potential entrant
has a long-run total cost curve, TC = 8Q 3 − 4Q 2 + 20Q where Q is thousands of units
per year. Market demand is given by D( P) = 20,000 − 1000( P ) .
a) [3 marks] Find the long-run equilibrium price and quantity per firm.
Answer:
In the long-run equilibrium, each firm will produce where P = AC = MC . Thus,
24Q 2 − 8Q + 20 = 8Q 2 − 4Q + 20
⇒ Q = 0.25
Since each firm produces where P = MC , price will be
P = MC
⇒
P = 24(0.25)^ 2 − 8(0.25) + 20 = 19.5
b) [2 marks] Find the number of firms operating in the long-run in this market?
Answer:
First find the total market demand.
D ( P) = 20,000 − 1000(19.5) = 500
Since total market demand is 250 and each firm is produce 0.25 units, the total number of
firms in the market in equilibrium will be
N=
500
= 2000
.25