Homework #6

Econ 370 – Hendrix

Fall 2002

  1. Find the marginal product with respect to both inputs, as well as the technical rate of substitution for each of the following production functions:
  1. Do the following production functions exhibit constant, increasing, or decreasing returns to scale?
  1. Suppose there is a firm that uses only a single input to produce their product. The production function is expressed , where x is the number of units of input. The price of the output is $100 and the price of the input is $50.
  1. Write a function that states the firm’s profit as a function of the amount of input.
  2. What is the profit-maximizing amount of input? What is the profit-maximizing amount of output? What is the level of profit when profits are maximized?
  3. Suppose the firm is taxed $20 per unit of its output and the price of its input is subsidized by $10. What is the new input level? What is the new output level? How much profit does the firm make?
  4. Suppose that instead of taxes and subsidies, the firm is taxed at 50% of its profits. Write down its after-tax profits as a function of the amount of input. What is the profit-maximizing amount of output? How much profit does it make after taxes?
  1. Suppose a firm has two variable factors and a production function . The price of its output is 4, the price of factor 1 is , and the price of factor 2 is .
  1. Write the two equations that say that the value of the marginal product of each factor is equal to its wage. If the factor prices are equal, what is ?
  2. For this production function, is it possible to solve the two marginal productivity equations uniquely for and ?
  1. A firm uses labor and machines to produce output according to the production function , where L is the number of units of labor used and M is number of machines. The cost of labor is $40 per unit and the cost of using a machine in $10.
  1. Draw an isocost line for thisfirm, showing combinations of machines and labor that cost $400 and another isocost line showing combinations that cost $200. What is the slope of these isocost lines?
  2. Suppose that the firm wants to produce its output in the cheapest possible way. Find the number of machines it would use per worker.
  3. On the same graph as the isocost, sketch the production isquant corresponding to an output of 40. Calculate the amount of labor and the number of machines that are used to produce 40 units of output in the cheapest way possible, give the factor prices. Calculate the cost of producing 40 units at these factor prices.
  4. How many units of labor and how many machines would the firm use to produce y units in the cheapest possible way? How much would this cost?
  1. Joe sells lemonade in a competitive market. His production function is where output is measured in gallons, factor 1 is the number of pounds of lemons he uses, and factor 2 is the number of labor hours spend squeezing them.
  1. What type of returns to scale does Joe’s production function exhibit?
  2. Where is the cost of a pound of lemons and is the wage rate for lemon-squeezers, what is the cheapest way for Joe to produce lemonade? (In other words, how many hours per pound of lemons should he use?)
  3. If he is going to produce y units in the cheapest possible way, how many pounds of lemons will he use? How many hours of labor will he use?
  4. What is Joe’s cost of producing y units?