Question

Write down your answers to the following question and upload a scanned copy of your answer on blackboard (e.g. using the Adobe Scan App on your phone). Only PDF documents will be accepted.

Suppose the long-run production function of Curry’s brewing company is given by: Q = 3K^0.25L^0.75.

• K is the total operating hours of all brewing machines and L is the total working hours of all workers.
• Q is the number of beers produced per day.
• The current rental rate (r) is $25/hour, wage rate (w) is $15/hour.

Part A: Production in the long run.

Find out the company’s optimal combination of labor & capital input (K^*, L^*) at Q = 300 in the long run. Keep two decimal places for your answers.

e). What is the firm’s total cost at the optimal input combination in the long run? (4 pts)

Part B: Production in the short run.

Suppose the long-run production function of Curry’s brewing company is still given by: Q = 3K^0.25L^0.75. Please answer the following questions and keep two decimal places for your answers.

f). In the short run, the firm’s number of brewing machine (K) is fixed at 16 units ( = 16). Write an equation for the short-run production function for the firm showing output as a function of labor (L). (4 pts)

g). Given short-run production function for the firm you derived in g), how many units of labor (L) will the firm need to use to produce 300 units of output (i.e., Q = 300)? (4pts)

h). Suppose the current rental rate is still $25/hour and the wage rate is still $15/hour in the short run. What will be the total cost of producing 300 units of output in the short run? Compare your answer in e), how much money will the firm save by using this optimal input combination in the long run? (6 pts)

Answer 1

Conclusions:

Part A: Long run production

The company’s optimal combination of labor & capital input at Q = 300 units in the long run is:

K* = 29.90

L* = 149.53

e.) Total cost at the optimal combination in the long-run is: $2990.45

Part B: Short run production

f.) Short run production function showing output as a function of labor is:

Q = 6L^0.75

g.) The company required labor units = 184.20 in the short run to produce 300 units of output.

h.) Total cost in the short-run = $3163

e.) Money saved by using optimal combination in the long run = $172.55