Problem%20Set%205%20-%20Cost%20Minimization%20-%20Profit%20Maximization

Problem%20Set%205%20-%20Cost%20Minimization%20-%20Profit%20Maximization

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Unformatted text preview: Economics 3010 Fall 2009 Professor Daniel Benjamin Cornell University Problem Set 5 This problem set is due in section on Friday, October 30. Please collect your answers into a stapled packet, and write your own name and your TA’s name on the front of each packet, as well as your netID and the time of the section you usually attend. 0. (Aplia) Please do the assigned problems on the Aplia website. These problems are due at 11:45pm on Thursday, October 29. 1. (Profit Maximization: Economic profit and opportunity cost) Suppose you own a firm that produces T‐shirts, and you want to calculate your profit over the year. Your bookkeeper provides you with the following information: Total Revenue from Selling T‐shirts $300,000 Costs of raw materials $80,000 Wages and salaries $150,000 Electricity and phone $20,000 Advertising cost $40,000 Total Explicit Cost $290,000 Accounting Profit $10,000 Suppose that in order to start your business you invested $100,000 of your own money – money that could have been earning $6,000 every year if invested elsewhere, perhaps in someone else’s business, or just put in the bank to earn interest. Also, you are using two extra rooms in your own house as a factory – rooms that could have been rented out for $4,000 per year. Finally, you are managing the business full time, without receiving a separate salary, and you could instead be working at a job earning $40,000 per year. Explain why your economic profit is actually negative $40,000 (per year). 2. (Profit Maximization: The Cobb‐Douglas production function) In 1927, economist (later Senator) Paul Douglas noticed that the division of national income between capital and income had been roughly constant over a long period. Even though the U.S. economy had been growing over time, workers (taken as a whole) and owners of capital (taken as a whole) shared in the increased output. The following figure shows that the same pattern has persisted into more recent times: Douglas asked mathematician Charles Cobb to figure out what production function would explain this observation. And so…the Cobb‐Douglas production function was born (and later incorporated into theory of the consumer as the Cobb‐Douglas utility function). In this problem, you will verify that the Cobb‐Douglas production function generates “constant factor shares” (shares of output for the factors of production). (a) Consider a firm that produces “widgets” (a classic economics term referring to a hypothetical “any‐ product”). Suppose that the number of widgets produced (Y) depends on the amount of capital (K), the amount of labor (L), and the level of technology (A) according to the following production function (F): Y = F(K, L) = AKαL1‐α, where K > 0, L > 0, and 0 < α < 1. Show that the marginal products can be written as: MPL = (1‐α)AKαL‐α = (1‐α)Y/L MPK = αAKα‐1L1‐α = αY/K. Explain why we can interpret these equations as showing that with this production function, the marginal products of the factors are always proportional to their average products. (b) Explain why the firm’s profit is π = P F(K, L) – wL – RK, where P is the price of widgets, w is the market wage, and R is the rental rate for capital. If this is a model of long‐run production, why does it make sense that we do not subtract fixed costs from profit? (c) Suppose the firm takes P, w, and R as given (because the firm is “small” relative to both the market for the output good and the market for both input goods). Show that at the profit‐maximizing levels of L and K, factors of production are paid their “marginal revenue products”: MPL × P = w MPK × P = R. (d) Show that at the firm’s profit‐maximizing choice of L and K, wL = (1‐α)PY RK = αPY. Explain why we can interpret the first equation as showing that the total payment to workers is a constant proportion of the value of output. Explain why we can interpret the second equation as showing that the total payment to owners of capital is also a constant proportion of the value of output. Explain how these two equations together imply that the firm earns zero (economic) profit. (e) Now suppose there are two firms that produce widgets, Firm 1 and Firm 2. Firm 1 employs fraction φ1 of the economy’s capital and labor, and Firm 2 employs fraction φ2 of the economy’s capital and labor. Prove that F(φ1K, φ1L) + F(φ2K, φ2L) = F((φ1 + φ2)K, (φ1 + φ2)L), where 0 < φ1, φ2 < 1. Interpret this equation as showing that if the two firms have Cobb‐Douglas production functions, then the combined production of the two firms is also Cobb‐Douglas. (f) Suppose the entire production side of the economy is populated by firms that have a Douglas production function. Using what you figured out in part (e), explain why it makes sense to describe the economy’s production process for the economy as a whole as being generated by a Cobb‐Douglas production function, where the inputs are the total amounts of capital and labor in the economy as a whole. Judging from the data shown above, what is a reasonable estimate of α? 3. (Cost Minimization: Optimal production techniques) Suppose that the production function for gravel (Y) is given by Y = F(K, L), and gravel production has diminishing marginal product for both capital (K) and labor (L). (a) Draw an isoquant/isocost curve diagram with K and L on the axes , showing that the isocost curve is tangent to the isoquant at an (interior) cost‐minimizing choice of inputs. Write down the firm’s cost‐ minimization problem, where w is the market wage and R is the rental rate of (gravel‐producing) capital equipment. Show that an interior solution to the firm’s cost‐minimization problem satisfies MPL / MPK = w / R. (b) Construction equipment is traded in world markets and, aside from shipping costs, its price does not differ much from one country to another. Wages in Nepal are among the lowest in the world (e.g., a haircut costs about 10 cents), while wages in the U.S. are among the highest in the world. Why is gravel made by hand in Nepal but by machine in the U.S.? What is the role of diminishing marginal product of labor in your explanation? (c) Now let’s apply the same logic to understand a different puzzle. American labor unions have historically been among the most outspoken proponents of a higher minimum wage. Yet almost all members of the major unions – the Teamsters, the AFL‐CIO, and the United Auto Workers – already earn substantially more than the minimum wage and so are not directly affected by minimum wage legislation. Explain why it may be in the self‐interest of the unions to devote money and effort to lobby in favor of a higher minimum wage. (Hint: Consider a production function where the two inputs are skilled labor and unskilled labor.) ...
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This note was uploaded on 12/28/2011 for the course ECON 3010 at Cornell.

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