Quiz 9: Production Functions

Business

Production function shows the functional relationship between the levels of output obtained from each feasible combination of inputs. a) The production function characterized by constant img is a fixed-proportions production function. Here capital and labor must always be used in a fixed ratio. The isoquant for this production function are L-shaped. The mathematical form of the fixed-proportions production function is as follows: img The first production function is as follows: img Here, hold img and img Then the production function for img is as follows: img As here img img img img Then the total capital requirement would be img units and labor requirement would be img units. The isoquant for the production function is as follows: img Figure 1 The X axis shows the quantity of labor employed and Y axis shows the quantity of capital and the isoquant curve shows the combination of capital and labor that is used to produce 40,000 units. b. The second production function is as follows: img Here, hold img and img Then the production function for img is given below: img As here, img img img img img Then the total capital requirement would be img units and labor requirement would be img units. Represent the graph showing the isoquant as follows: img Figure 2 The X axis shows the quantity of labor employed and Y axis shows the quantity of capital and the isoquant curve shows the combination of capital and labor that is used to produce 40,000 units. c) The two production functions are as follows: img img If half of the lawn is cut by first method, then the required capital and labor would be calculated as shown below: img If half of the lawn is cut by second method, then the required capital and labor would be calculated as shown below: img img Then the total capital requirement would be img units and labor requirement would be img units. The two production functions are given below: img img If one-fourth of the lawn is cut by first method, then the required capital and labor would be calculated as shown below: img If three-fourth of the lawn is cut by second method, then the required capital and labor would be calculated as shown below: img img Then the total capital requirement would be img units and labor requirement would be img units. The fraction of K and l means the amount of capital input used per unit of labor. d) Let the fraction p of the 40,000-square-foot lawn or img -square-foot lawn is cut by first method, then the required capital and labor would be calculated as shown below: img If img of the 40,000-square-foot lawn or img -square-foot of the lawn is cut by second method, then the required capital and labor would be calculated as shown below: img img Then the total labor requirement is calculated below: img The total capital requirement is calculated below: img Therefore, the two equations derived above are given below: img Solving first equation for p and putting the value of p in second as shown below: img img This is an equation of a negatively sloping straight line. Thus, the combined isoquant will be negative sloping straight line giving a fixed output img . Represent the graph as follows: img Figure 3 The X axis shows the quantity of labor employed and Y axis shows the quantity of capital and the isoquant curve shows the combination of capital and labor that is used to produce 40,000 units by the combined production method.

a. The production function shows all the combinations of labor and capital inputs that can be used to produce the maximum amount of the good. Production function is given below: img The average product of labor is defined as the quantity of output produced by per unit of labor inputof the firm. Average product of labor can be shown as given below: img The production function for widgets is given below: img Where, img quantity of widgets produced annually. img Annual capital input img Annual labor input The production function for img is given below: img This gives the total productivity of the labor. Total value of output at different level of labor input is calculated as shown below: img The total productivity of labor curve is given below: img Figure 1 X axis shows the quantity of labor and Y axis shows the total productivity of labor. The average productivity of labor is given below: img This gives the average productivity of the labor. Average productivity of labor is calculated at different level of labor input is shown below: img The average productivity of labor curve is given below: img Figure 2 X axis shows the quantity of labor and Y axis shows the average productivity of labor. For average product to reach the maximum the partial derivative of AP with respect to l must be zero. The derivative of average productivity of labor is given below: img Evaluate it to zero to get the value of l as shown below: img img That is at labor input level img average productivity reach a maximum. At this level the widgets production would be calculated below: img 40 widgets will be produced at that point where average productivity of labor reaches a maximum. b. The marginal productivity of labor is given below: img Marginal productivity of labor is calculated at different level of labor input is shown below: img The marginal productivity of labor curve is given below: img Figure 3 X axis shows the quantity of labor and Y axis shows the marginal productivity of labor. The level at which marginal labor of productivity is zero is calculated below: img c. The production function for widgets is given below: img Where, img quantity of widgets produced annually. img Annual capital input img Annual labor input The production function for img is given below: img This gives the total productivity of the labor. Total value of output at different level of labor input is calculated as shown below: img The total productivity of labor curve is given below: img Figure 4 X axis shows the quantity of labor and Y axis shows the total productivity of labor. The average productivity of labor is given below: img This gives the average productivity of the labor. Average productivity of labor is calculated at different level of labor input is shown below: img The average productivity of labor curve is given below: img Figure 5 X axis shows the quantity of labor and Y axis shows the average productivity of labor. For average product to reach the maximum the partial derivative of AP with respect to l must be zero. The derivative of average productivity of labor is given below: img Evaluate it to zero to get the value of l as shown below: img That is at labor input level img average productivity reach a maximum. At this level the widgets production would be img 160 widgets will be produced at that point where average productivity of labor reaches a maximum. The marginal physical product (MP) of an input is the additional output that can be produced by employing one more unit of that input while holding all other inputs constant. The marginal productivity of labor is given below: img Marginal productivity of labor is calculated at different level of labor input is shown below: img The marginal productivity of labor curve is given below: img Figure 6 X axis shows the quantity of labor and Y axis shows the marginal productivity of labor. The level at which marginal labor of productivity is zero is calculated below: img d. The production function exhibits constant returns, when a proportionate increase in input leads to increase in output by the same proportion. The function exhibits diminishing returns to scale if output increases less than proportionately. However, if output increases more than proportionately, there are increasing returns to scale. If img is production function and if all inputs are multiplied by the same positive constant t (where t 1), then return to scale of the production function will be: img Constant returns to scale img Diminishing returns to scale img Increasing returns to scale img The production function for widgets is given below: img Multiply the inputs by the some positive constant t as shown below: img As img ; thus, output increases more than proportionately, the function exhibits increasing returns to scale.

Sam Malone is considering renovating the bar stools at Cheers. The production function for new bar stools is given by img Where q is the number of bar stools produced during the renovation week, k represents the number of hours of bar stool lathes used during the week, and l represents the number of worker hours employed during the period. Sam would like to provide 10 new bar stools, and he has allocated a budget of $10,000 for the project. a. Sam reasons that because bar stool lathes and skilled bar stool workers both cost the same amount ($50 per hour), he might as well hire these two inputs in equal amounts. If Sam proceeds in this way, his production function would be img Here img ; putting this on the production function we have img As he will hire two inputs in equal amounts then; img as well. Both the inputs cost the same amount ($50 per hour), the total cost is, img Therefore, he will hire 100units of both the inputs and the cost of the project will be $10,000. b. Norm asserts that Sam should choose quantities of inputs so that their marginal productivities are equal. The marginal physical product (MP) of an input is the additional output that can be produced by employing one more unit of that input while holding all other inputs constant. Mathematically, img img Here the production function is given as img Therefore; img img Sam should choose quantities of inputs so that their marginal productivities are equal. Thus at equilibrium img Putting img in the production function we get img Here img ; putting this on the production function we have img Hence img As both the input costs $50; the total cost of renovation is img If Sam opts for this plan instead, 33.003 units of capital and 132.012 units of labor will he hire and the renovation project will cost $8250.75. c. Upon hearing that Norm's plan will save money, Cliff argues that Sam should put the savings into more bar stools in order to provide seating to more of his USPS colleagues. If Sam chooses quantities of inputs so that their marginal productivities are equal, the total cost of renovation is $8250.75. Here, he will save $1749.75. In this method one tool will cost him $825.075. Then in the saved money he can make approximately 2 more tools. Therefore, 2 more bar stools can Sam get for his budget if he follows Cliff's plan. d. Carla worries that Cliff's suggestion will just mean more work for her in delivering food to bar patrons. This could decrease the marginal product of Cliff. She can prove unprofitable for the bar, and add cost to bar renovation. This could make Sam think twice about adding more stool at bar. In this way she can convince Sam to stick to his original 10-bar stool plan.