Max, Inc. assembles gadgets. It can make each gadget either by hand or with a sp
ID: 1115513 • Letter: M
Question
Max, Inc. assembles gadgets. It can make each gadget either by hand or with a special gadget-making machine. Each gadget can be assembled in 15 minutes by a worker or in 5 minutes by the machine. The firm can also assemble some of the gadgets by hand and some with machines. Both types of work are perfect substitutes, and they are the only inputs necessary to produce the gadgets.
a) (12) It costs the firm $25 per hour to use the machine and $15 per hour to hire a worker. The firm wants to produce 240 gadgets. What are the cost-minimizing input quantities? Illustrate your answer with a clearly labeled graph (isoquant and isocost curves).
b) (8) Derive the expression for the long-run total cost in part (a) that the firm incurs, as a function of Q
Explanation / Answer
Max, Inc. assembles gadgets. It can make each gadget either by hand or with a special gadget-making machine. Each gadget can be assembled in 15 minutes by a worker or in 5 minutes by the machine. Both types of work are perfect substitutes which implies isoquants are straight line downward sloping.
a) (12) It costs the firm $25 per hour to use the machine and $15 per hour to hire a worker. The firm wants to produce 240 gadgets. Now that we need to find MPWorker /Wage rate and MPmachine/rental price. Here MPWorker /Wage rate = (60/15)/15 = 4/15 and MPmachine/rental price = (60/5)/25 = 12/25
Since marginal productivity per dollar is not same, we must have a corner solution. Hence we find if only worker or if only machines are needed
Only worker = 4 gadgets oer hour so we need 240/4 = 60 labor hours and cost is 60 x 15 = $900
Onlye machines = 12 gadgets and so we need 240/12 = 20 machines and cost is 20 x 25 = $500
Since cost using machines is low we hire only machines and there is a corner solution.
b) (8) Derive the expression for the long-run total cost in part (a) that the firm incurs, as a function of Q
We have Q = 4X + 12Y where Y is number of machines and X is labor hours. Each hour produces 4 units from labor and 12 units from machines. We know that MPX = 4 and MPY = 12. Px = 15 and Py = 25 so MPX/Px = 4/15 and MPY/Py = 12/25. Since MPY/Py > MPX/Px, firm substitutes workers and hire machines in the long run so lon run production function is Q = 12Y and Y = Q/12
Cost function C = 15X + 25Y
C = 25Y
C = 25Q/12
This gives the long run cost function C = (25/12)Q