Question

Ajax, 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.

1. It costs the firm $30 per hour to use the machine and $10 per hour to hire a worker. The firm wants to produce 120 gadgets. What are the cost-minimizing input quantities? Illustrate your answer with a clearly labeled graph.

2. What are the cost-minimizing input quantities if it costs the firm $20 per hour to use the machine, and $10 per hour to hire a worker? Illustrate your answer with a graph.

3. Write down the equation of the firm’s production function for the firm. Let  be the number of gadgets assembled,  the number of hours the machines are used, and  the number of hours of labor.

Answer 1

Ans.a)So the Isoquants for the production function are straight lines. At the given input prices slope of
an isoquant is equal to the ratio of the input prices. Hence, all positive input quantities (measured
in work hours) such that 4L + 12M = 120 are cost-minimizing.

Ans.B) So now When one hour of the machine’s work costs $20 cost-minimizing firm does not use manual work at all. The cost-minimizing quantity of the machine’s work necessary to produce So the 120 widgets is equal to M = 120/12 = 10 hours. The firm spends $200. (Note that if the firm were to use only manual labor, the cost would be $300 = 30 hours x $10 per hour).

Ans.C) let x is no. Of gadgets assembled so the equation will be

X= 4L + 12M.