Exercises on Production Function and Cost Minimization

Transcription

1 Exercises on Production Function and Cost Minimization 1) Suppose you manage a plant that produces exercise equipment for Crossfit enthusiasts. In your plant you use workers (L) and assembly line machines (K) to produce the exercise equipment. Your firm s technology is summarized by the production function: Q =100K 1/3 L 2/3 The isoquant curves for your plant are given in the graph below Q=400 Q=500 Q= Capital Labor a) What type of returns to scale is your production function exhibiting? b) Suppose you want to produce 400 pieces of exercise equipment a month and assume both inputs are variable. Price of labor is $3,000 per worker and the price of capital is $12,000 per capital. i.e. PL=$3,000 and PK=$12,000. What is the minimum cost of producing Q=400 in the long run?

2 c) Suppose your plant has a fixed installation of 4 assembly machines ( K = 4 ) as part of its design. However you can hire as many workers as you wish. You want to produce 400 units. Assume price of labor is $3,000 per worker and the price of capital is $12,000 per capital. i.e. PL=$3,000 and PK=$12,000. What is the total cost of producing Q=400 units in the short run? What is the variable cost of producing Q=400 units in the short run? What is the fixed cost of producing Q=400 units in the short run? d) Suppose again that your plant has a fixed installation of 4 assembly machines ( K = 4 ) as part of its design. However you can hire as many workers as you wish. Assume price of labor is $3,000 per worker and the price of capital is $12,000 per capital. i.e. PL=$3,000 and PK=$12,000. Suppose after a crash of your computer s operating system, you lost the production function given by the isoquants on the first page. All you are left with is your production function in the formula form. Your plant s production function was given by Q =100K 1/3 L 2/3. i) Suppose you want to produce 500 pieces of exercise equipment. Find the number of workers you need to hire to produce Q = 500? What is the total cost of producing Q=500 units in the short run? What is the variable cost of producing Q=500 units in the short run? What is the fixed cost of producing Q=500 units in the short run?

3 ii) Suppose you want to produce 600 pieces of exercise equipment. Find the number of workers you need to hire to produce Q = 600? What is the total cost of producing Q=600 units in the short run? What is the variable cost of producing Q=600 units in the short run? What is the fixed cost of producing Q=600 units in the short run? iii) Suppose you want to produce Q pieces of exercise equipment. Find the number of workers you need to hire to produce Q units? Derive the equation of the TC (total cost) as a function of Q in the short run for your plant and identify your FC (fixed cost) and VC (variable cost). e) Determine the type of returns to scale each of the following production functions given below exhibit? i) Q = K 1/3 + L 2/3 ii) Q = 2KL + L 2 iii) Q = 400K +100L

4 2) Suppose you manage production operations for UNTUCKit, a company, which makes men's shirts designed to be worn untucked. In your production you use workers (L) and assembly line machines (K) to produce the shirts. Your firm s technology is summarized by the production function: Q = KL 2 The isoquant curves for your production are given in the graph below Capital Q=100 Q=200 Q=400 Q= Labor a) What type of returns to scale is your production function exhibiting? b) Suppose you want to produce 200 shirts a month and assume both inputs are variable. Price of labor is $20,000 per worker and the price of capital is $50,000 per capital. i.e. PL=$20,000 and PK=$50,000. What is the minimum cost of producing Q=200 shirts in the long run?

5 c) Suppose your factory has a fixed installation of 8 assembly lines ( K = 8 ) as part of its design. However you can hire as many workers as you wish. You want to produce 200 shirts. Assume price of labor is $20,000 per worker and the price of capital is $50,000 per capital. i.e. PL=$20,000 and PK=$50,000. What is the total cost of producing Q=200 shirts in the short run? What is the variable cost of producing Q=200 shirts in the short run? What is the fixed cost of producing Q=200 units in the short run? d) Compare your total cost of producing 200 shirts in the long run (from part b) to your total cost of producing 200 shirts in in the short run (from part c) and comment. Which one is cheaper? Why? e) Assume you want to produce 100 shirts. Suppose a change in law requires you to pay $60,000 more per worker for unemployment insurance effectively raising the cost of labor to $80,000 per worker. Price of capital is still $50,000 per capital. i.e. PL=$80,000 and PK=$50,000. Find your optimal input choice for producing 100 shirts and write it below. What is the minimum cost of producing Q=100 shirts in the long run? f) Suppose again that your plant has a fixed installation of 8 assembly machines ( K = 8 ) as part of its design. However you can hire as many workers as you wish. Assume price of labor is $20,000 per worker and the price of capital is $50,000 per capital. i.e. PL=$20,000 and PK=$50,000. Suppose after a crash of your computer s operating system, you lost the production function given by the isoquants on the previous page. All you are left with is your production function in the formula form. Your plant s production function was given by Q = KL 2.

6 i) Suppose you want to produce 400 shirts. Find the number of workers you need to hire to produce Q = 400? What is the total cost of producing Q=400 shirts in the short run? What is the variable cost of producing Q=400 shirts in the short run? What is the fixed cost of producing Q=400 shirts in the short run? ii) Suppose you want to produce 600 shirts. Find the number of workers you need to hire to produce Q = 600? What is the total cost of producing Q=600 shirts in the short run? What is the variable cost of producing Q=600 shirts in the short run? What is the fixed cost of producing Q=600 shirts in the short run? iii) Suppose you want to produce Q many shirts. Find the number of workers you need to hire to produce Q many shirts? Derive the equation of the TC (total cost) as a function of Q in the short run for your plant and identify your FC (fixed cost) and VC (variable cost).