Fairfax County Public Schools



Cost-Minimizing Input CombinationProblem SetA firm produces its output using only capital and labor. Labor costs $100 per worker per day and capital costs $200 per unit per day. If the marginal product of the last worker employed is 500 and the marginal product of the last unit of capital employed is 1,000 is the firm employing the cost-minimizing combination of inputs? Explain.Yes, the firm is employing the cost-minimizing combination of inputs because the marginal product per dollar is equal for capital and labor: 500/$100 = 1,000/$200 = 5 units of output per dollar.Bernie can spend his income on two different goods: Taylor Swift CDs and notebooks for his class notes. The accompanying table shows Bernie’s utilities from notebooks and CDs. The price of a notebook is $5, the price of a CD is $10, and Bernie has $55 of income to spend.Calculate the marginal utility of each notebook and the marginal utility of each DC. Then calculate the marginal utility per dollar spent on notebooks and the marginal utility per dollar spent on DCs.Quantity of NotebooksUtility from NotebooksMarginal UtilityMarginal Utility Per $Quantity of CDsUtility from CDsMarginal UtilityMarginal Utility Per $00--------00--------135357180808265306215070739025532106064110204426050551351535300404What is the utility maximizing combination of notebooks and CDs for Bernie? The optimal consumption rule states that the optimal bundle, from all those on a consumer’s budget line, is the one at which the marginal utility per dollar spent on each good is equal. When Bernie consumes 3 notebooks and 4 CDs, the marginal utility per dollar spent on notebooks is the same as the marginal utility per dollar spent on CDs, so this is the optimal consumption bundle.Answer the following questions under the assumption that firms use only two inputs and seek to maximize profit.Would it be wise for a firm that does not have the cost-minimizing combination of inputs to hire more of the input with the highest marginal product and less of the input with the lowest marginal product? Explain. No. The input with the highest marginal product might be much more expensive than the input with the lowest marginal product, making the marginal product per dollar higher for the input with the lowest marginal product. When that is the case, costs would be lower if the firm hired more of the input with the lowest marginal product (but the highest marginal product per dollar) and less of the input with the highest marginal product (but the lowest marginal product per dollar.)What is the cost-minimization rule? The cost-minimization rule says that firms should adjust their hiring of inputs to equalize the marginal product per dollar spent on each input.When a firm hires more labor and less capital, what happens to the marginal product of labor per dollar and the marginal product of capital per dollar? Explain. The marginal product of labor per dollar decreases and the marginal product of capital per dollar increases. Each factor has diminishing marginal returns. So when more labor is hired, the marginal product of labor (and thus the marginal product of labor per dollar) decreases. Likewise, when less capital is hired, the marginal product of capital (and thus the marginal product of capital per dollar) increases because the units of capital that are given up had a lower marginal product than those that remain.Quantity of Labor (workers)Quantity of Pencils Produced0014029031204140515061607166Assume that the wage is $10 per day and the price of pencils is $1.What is the MPL of the 4th worker? 20What is the MP per dollar of the 5th worker? 10/$10 = 1 pencil per dollarHow many workers would the firm hire if it hired every worker for whom the marginal product per dollar is greater than or equal to 1 pencil per dollar? The firm would hire 6 workersIf the marginal product per dollar spent on labor is 1 pencil per dollar, the marginal product of the last unit of capital hired is 100 pencils per dollar, and the rental rate is $50 per day, is the firm minimizing its cost? Explain. No. The marginal product per dollar spent on capital is 100/$50 = 2 pencils per dollar. Thus, the firm is not following the cost-minimization rule because the marginal product per dollar spent on labor (1) is less than the marginal product per dollar spent on capital (2).Pride Textiles produces and sells towels in a perfectly competitive market. Pride Textiles hires its workers in a perfectly competitive labor market. Assume that the market wage rate for workers is $80 per day.State the conditions necessary for hiring the profit-maximizing amount of labor. (1 point)One point is earned for stating that MRP = MFCAt the profit-maximizing level of output, suppose that the marginal product of the last worker hired is 20 towels per day. Calculate the price of a towel. (1 point)One point is earned for calculating the price: $80/20 = $4Draw a correctly labeled graph of the labor supply and demand curves for Pride Textiles, and show the equilibrium amount of labor hired. (3 points)One point is earned for a correctly labeled graph with downward-sloping demand curve.One point is earned for drawing a horizontal supply curve.One point is earned for showing equilibrium amount of labor.Wage rate Per daySLMRP$800Amount of laborL*Given your answer to part (b), if the price of a towel increases, explain how Pride’s profit-maximizing quantity of labor will be affected (2 points)One point is earned for stating that the amount of labor will increase.One point is earned for explaining that MRP > W.The John Lamb Company, a profit-maximizing firm producing widgets, is in a perfectly competitive widget market. Assume John Lamb employs a fixed number of employees and rents a machine for a variable number of hours from a perfectly competitive market.Using correctly labeled side-by-side graphs of the factor market for machines and the John Lamb Company, show each of the following. (2 points)The equilibrium rental price of machines in the factor market, labeled as PR.John Lamb’s equilibrium rental quantity of machines, labeled as QK.QQUANTITY OF MACHINESRENTAL PRICESDMRPM = DMFC = SMQKQUANTITY OF MACHINESOne point is earned for the correct side-by-side graphs with a horizontal machine supply curve for John Lamb (S, D, PR, SM).One point is earned for showing the equilibrium rental quantity of mahcines, QK, at the intersection of MRP and the horizontal supply curve.Assume that the popularity of widgets declines, decreasing the demand for widgets. What will happen to each of the following?Marginal product curve for machine-hours (1 point)Marginal revenue product curve for machine-hours. Explain. (1 point)One point is earned for stating that there will be no change to the marginal product curve for machine-hours.One point is earned for explaining that the MRP curve for machine-hours will decrease (shift to the left) because the decrease in demand decreases the price of widgets.John Lamb is employing the cost-minimizing combination of inputs. The marginal product of labor is 28 widgets per worker hour and the wage rate is $14 per hour. The marginal product of the machine is 60 widgets per machine-hour. What is the hourly rental price of a machine? (1 point)One point is earned for correctly calculating the rental price of a machine:MPL/w = MPK/r 28/14 = 60/r Therefore, r=$30. ................
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