Problem%20Set%205%20Solutions%20-%20Cost%20Minimization

Problem%20Set%205%20Solutions%20-%20Cost%20Minimization -...

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Unformatted text preview: Economics 3010 Fall 2009 Professor Daniel Benjamin Cornell University Problem Set 5 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 $80,000 Costs of raw materials 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). Economic profit is the difference between total revenue and the sum of its explicit and implicit costs. Hence, in addition to the accounting costs listed above, we have to consider the implicit costs, i.e. opportunity costs of the resources, to compute the economic costs as below: Interest foregone 6,000 Rent foregone 4,000 Wage foregone 40,000 Total Opportunity Costs Hence, $50,000 Economic profit = Total revenue (300,000) – Total explicit costs (290,000) – Total implicit costs (50,000) = ‐ 40,000 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. MPL = ∂Y α 1‐α = (1 − α )1 24 From Y = AK L , (a) = Y/L. Thus, MPL = (1‐α)Y/L AK α L−α 43 ∂L (a ) MPK = ∂Y = α 14243 From Y = AKαL1‐α, (b) = Y/K. Thus, MPK = αY/K AK α −1L1−α ∂K (b) Note that Y/L and Y/K are the average product of labor and capital. Hence, this confirms that 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? π = total revenue – total costs Total revenue = P∙Y = P F(K, L) Total costs = labor cost + capital cost = wL + RK Hence, π = P F(K, L) – wL – RK Fixed costs are costs that are independent of the level of output, and, in particular, they must be paid whether or not the firm produces output. In economics, the distinction between the long run and the short run is that in the long run, all the factors of production can be varied while in the short run at least one factor of the production is fixed. Hence, by definition, there is no fixed cost in the long run. (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. The profit maximization problem is: Max π = P F(K, L) – wL – RK The optimal solution of L and K satisfies: ∂π ∂F ∂F =P − w = 0 Since = MPL , P × MPL − w = 0 , or MPL × P = w ∂L ∂L ∂L ∂π ∂F ∂F =P − R = 0 Since = MPK , P × MPK − R = 0 , or MPK × P = R ∂K ∂K ∂K (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. From (c), MPL × P = w and MPK × P = R. Recall (a), MPL = (1‐α)Y/L and MPK = αY/K. Plug this result into (c), MPL × P = (1 − α ) MPK × P = α Y × P = w, By rearranging we get wL = (1‐α)PY (1) L Y × P = R, By rearranging we get RK = αPY (2) K This result shows that the total payment to workers (wL) and the total payment to the owners of capital (RK) is a fixed proportion of total value of output (PY). Combining (1) and (2), we get, wL + RK = (1‐α) PY + α PY = ((1‐α)+ α )PY = PY, and so π = PY – wL – RK = 0, i.e. Total profit = 0 (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. From the general from of C‐D production function, Y = F(K, L) = AKαL1‐α, F(φ1K, φ1L) + F(φ2K, φ2L) = A(φ1K)α (φ1L)1‐α + A(φ2K)α (φ2L)1‐α α = Aφ1α +1−α K α L1−α + Aφ2 +1−α K α L1−α = Aφ1 K α L1−α + Aφ2 K α L1−α = A(φ1 + φ2 ) K α L1−α = A(φ1 + φ2 )α + (1−α ) K α L1−α = A[(φ1 + φ2 )K ]α [(φ1 + φ2 )L]1−α = F((φ1 + φ2)K, (φ1 + φ2)L) The equation obtained tells use that the combined production of the 2 firms (the left hand side) can be represented by another production (the right hand side) which is also of the Cobb‐ Douglas form, but with different coefficients of capital and labor (namely (φ1 + φ2)). (f) Suppose the entire production side of the economy is populated by firms that have a Cobb‐ 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? In part (e), we’ve shown that if the two firms have Cobb‐Douglas production functions, then the combined production of the two firms is also Cobb‐Douglas. This logic can be extended to N firms (i.e. the economy as a whole), which gives us F(φ1K, φ1L) + F(φ2K, φ2L) + … + F(φNK, φNL) = F((φ1 + φ2+ … + φN)K, (φ1 + φ2… + φN)L) Note that when all the firms in the economy are included, we’ll have φ1 + φ2+ … + φN = 1, since φi stands for the fraction of the economy’s capital and labor that firm i employs. Therefore, the production function of the economy simplifies to F((φ1 + φ2+ … + φN)K, (φ1 + φ2… + φN)L) = F(K, L) = AKα L1‐α, a Cobb‐Douglas production function where the inputs are the total amounts of capital (K) and labor (L) in the economy as a whole. As we’ve seen in the chart, the labor share of the total income is constant around 0.7, which implies that the capital share of total income is around 0.3. Hence, a reasonable estimate for α is 0.3. 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. A production function exhibits diminishing marginal product if its partial derivative with respect to capital (or labor) is a decreasing function. Assuming diminishing technical rate of transformation, we can represent the situation graphically by convex isoquants (recall that isoquants represent the level sets of the production function). At the cost minimizing level of output, the isocost curve will be tangent to the isoquant. This can be shown graphically. The firm’s cost minimization problem resembles an individual’s utility maximization problem, but is in a sense reversed. Here, what is being optimized is the cost function (the analog of the budget constraint in consumer theory) and the constraint is that a certain output level be produced, i.e. that we be on a certain isoquant (the equivalent of an indifference curve in consumer theory). That is, we set out to find the inputs that would minimize costs while still achieving a given level of output. Finally, note that we have a minimization problem, rather than a maximization problem. Mathematically, minimizing f(x) is equivalent to maximizing –f(x), so this poses no real difficulties. Algebraically, the problem is min wL + rK subject to Y=F(L,K) i.e. max ‐ w L ‐ rK subject to Y=F(L,K), where the choice variables are L and K. Ignoring any non‐negativity constraints, the Lagrangian becomes: L = ‐ wL ‐ rK + λ( F(K,L) ‐ Y), where λ is the multiplier on the production constraint. So the first order conditions are: (1) w = λMPL (2) R = λMPK (3) Y = F(K,L) Combining conditions (1) and (2) we find that 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? In part (a) we found that the ratio of marginal products is equal to the wage ratio at an interior optimum. When the ratio of marginal products is low, the wage ratio will be low, and vice versa. Since construction equipment is traded in world markets, its price is pretty much the same everywhere, so R is essentially the same in both Nepal and the US. However, wages are much lower in Nepal than in the US. Hence w/R will be lower in Nepal. Hence a gravel‐producer in Nepal will optimally set MPL/MPK much lower than a gravel‐producer in the US will. Doing so means that the profit‐maximizing gravel‐producer in Nepal will produce the gravel using more labor input and less capital input than profit‐maximizing gravel‐producer in the US. Diminishing marginal product of labor‐‐‐the fact that MPL gets smaller when L is larger‐‐‐is the reason why choosing more labor input causes MPL/MPK to fall. (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.) Let’s consider a production function with only two inputs, skilled labor and unskilled labor. Let MP represent the marginal product of skilled labor and MP be the marginal product of unskilled labor. Let’s also assume that all union members are skilled and non‐union members are unskilled, and that both labor segments experience diminishing marginal product. At a firm’s interior optimum, the ratio of these marginal products will equal the wage ratio. Mathematically, MPS/MPU = ws/wu, where ws is the skilled wage rate and wu is the unskilled wage rate. Minimum wage laws tend to push wu up, which makes the wage ratio ws/wu go down. Assume that skilled and unskilled labor are gross technical substitutes. Now that the relative price of skilled labor has gone down, firms will have an incentive to substitute away from using unskilled labor and towards using more skilled labor. This process, which will make MPS go down and MPU go up, will continue until the TRS = wage ratio equality is restored. S U ...
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