**Ajax, Inc. Gadget Production Problem****Context:**Ajax, Inc. assembles gadgets. Each gadget can be made either by hand or using a gadget-making machine. A worker can assemble one gadget in 15 minutes, while the machine can do so in 5 minutes. Both methods are perfect substitutes and are the only inputs needed for production.**Tasks:****a) Production Function Equation:**- Define variables: Let $ G $ represent the number of gadgets assembled, $ M $ the number of machine hours used, and $ L $ the number of labor hours.- Formulate the equation for the firm's production function.**b) Cost-Minimizing Input Quantities:**- Costs: $30 per hour for the machine, $10 per hour for labor.- Objective: To produce 120 gadgets.- Determine the input quantities that minimize costs.- Provide a clearly labeled graph illustrating the solution.**d) Long-Run Total Cost Expression:**- Derive the expression for the firm's long-run total cost as a function of $ q $ (the quantity produced).- Represent this relationship in a diagram.

Total cost (TC) is the total of all expenses that a business or an individual must pay in order to…

3) 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. a) Write down the equation of the firm's production function for the firm. Let G be the number of gadgets assembled, M the number of hours the machines are used, and L the number of hours of labor. b) 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 is the cost-minimizing input quantities? Illustrate your answer with a clearly labeled graph.

3) 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. a) Write down the equation of the firm's production function for the firm. Let G be the number of gadgets assembled, M the number of hours the machines are used, and L the number of hours of labor. b) 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 is the cost-minimizing input quantities? Illustrate your answer with a clearly labeled graph.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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please only solve part d, e, f
3)) 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. a) Write down the equation of the firm's production function for the firm. Let G be the number of gadgets assembled, M the number of hours the machines are used, and L the number of hours of labor. b) 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 is the cost-minimizing input quantities? Illustrate your answer with a clearly labeled graph. d) Derive the expression for the long-run total cost that the firm incurs, as a function of q. Represent it in a diagram.
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can you please provide explanations and working