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Readings Readings Baye 6th edition or 7th edition, Chapter 5 BA 445 Lesson I.7 Minimizing Cost 1 Overview Overview BA 445 Lesson I.7 Minimizing Cost 2 Overview Productivity Measures including total product and marginal product help firms and charities find the right allocation of inputs to produce output cheaply. They determine the value of buying additional inputs. Cost Minimization by firms and charities predicts effects of a crackdown on immigration, of outsourcing jobs, of minimum wage legislation, and of encouraging handicapped employment. Cost Measures including total cost and marginal cost are alternatives to productivity measures to help firms chose output to maximize profit. They also determine when it is best to shut down production. BA 445 Lesson I.7 Minimizing Cost 3 Productivity Measures Productivity Measures BA 445 Lesson I.7 Minimizing Cost 4 Productivity Measures Overview Productivity Measures including total product and marginal product help firms and charities find the right allocation of inputs to produce output cheaply. They determine the value of buying additional inputs. BA 445 Lesson I.7 Minimizing Cost 5 Productivity Measures Everyone Should Minimize Cost • All firms in all market environments Monopoly (1 firm with restricted entry) Duopoly (2 firms with restricted entry) Oligopoly (a few firms with restricted entry) Monopolistic competition (a few firms but with free entry) Perfect Competition (many firms with free entry) • Charities Toys for Tots should minimize the cost of getting toys to tots • They should often solicit cash over donated toys or labor. (Instead of having 1000 people each buying 1 toy for $20, people could each donate $20 and Toys for Tots could buy more than 1000 toys for $20,000 by getting a quantity discount.) • Donors Perfectly altruistic people helping kids (not feeling warm and fuzzy) should donate money rather than toys or labor, and so minimize their own cost of giving. (see http://faculty.pepperdine.edu/jburke2/giving.pdf) BA 445 Lesson I.7 Minimizing Cost 6 Productivity Measures • Production Function (allows detailed analysis) Q = F(K,L) • Q is quantity of output produced. • K is capital input. – Capital is everything that is not labor. – Capital can be disaggregated to, say, machines and land. – “Capital” is often rented (used for a time) but not consumed in production. • L is labor input. – Labor is always rented and not consumed (except for Cannibalism, which is frowned upon in polite society). • F is a function relating inputs to a single output. F(K,L) is the maximum amount of output that can be produced with K units of capital and L units of labor. • There is thus an assumption of a type of efficiency implicit in the definition of the production function. BA 445 Lesson I.7 Minimizing Cost 7 Productivity Measures Short-Run vs. Long-Run Decisions • Some capital or labor is fixed in the short-run. Office space may not be adjustable in the short-run. Labor with contracts is not completely adjustable in the short-run. • It takes time to terminate labor. • Unless stated otherwise in a homework or exam problem, just consider the case where labor is variable in both the short-run and long-run, and capital is fixed in the short-run but flexible in the long-run. BA 445 Lesson I.7 Minimizing Cost 8 Productivity Measures Production Function Algebraic Forms • Linear production function: inputs are perfect substitutes. Machines and unskilled labor as alternatives on an assembly line. Other examples? Q F K , L aK bL • Leontief production function: inputs are used in fixed proportions. Example: K = an artificial heart, and L = surgeon’s labor. Q F K , L min bK , cL • Cobb-Douglas production function: inputs have a degree of substitutability. Example: Producing houses with K = wood and L = labor in U.S. vs. India. --- Other examples where labor can substitute for materials? Q F K , L K L a b BA 445 Lesson I.7 Minimizing Cost 9 Productivity Measures • Total Product (TP): maximum output produced with given amounts of inputs. • Example: Cobb-Douglas Production Function: Q = F(K,L) = K.5 L.5 K is fixed at 16 units in the short-run. Short-run Cobb-Douglass production function: Q = (16).5 L.5 = 4 L.5 Total Product when 100 units of labor are used? Q = 4 (100).5 = 4(10) = 40 units BA 445 Lesson I.7 Minimizing Cost 10 Productivity Measures Average Product of an Input: measures output produced per unit of input. (It is a common measure of productivity, but it is less useful than marginal product, explained later.) Average Product of Labor: APL = Q/L. • Measures the output of an “average” worker. • Example: Q = F(K,L) = K.5 L.5 – If the inputs are K = 16 and L = 16, then the average product of labor is APL = [(16) 0.5(16)0.5]/16 = 1. Average Product of Capital: APK = Q/K. • Measures the output of an “average” unit of capital. • Example: Q = F(K,L) = K.5 L.5 – If the inputs are K = 16 and L = 16, then the average product of capital is APK = [(16)0.5(16)0.5]/16 = 1. BA 445 Lesson I.7 Minimizing Cost 11 Productivity Measures Marginal Product on an Input: change in total output attributable to the last unit of an input. Marginal Product of Labor: MPL = DQ/DL • Measures the output produced by the marginal (the last) worker. • Slope of the short-run production function (with respect to labor). • Example: Q = F(K,L) = K.5 L.5 – If the inputs are K = 16 and L = 16, then the marginal product of labor is MPL = .5(16) -0.5(16)0.5 = .5. Marginal Product of Capital: MPK = DQ/DK • Measures the output produced by the last unit of capital. • When capital is allowed to vary in the short run, MPK is the slope of the production function (with respect to capital). • As we will see, MP is the essential measure of input productivity for profit maximization and cost minimization. BA 445 Lesson I.7 Minimizing Cost 12 Productivity Measures Increasing, Diminishing and Negative Marginal Productivity Increasing Diminishing Negative Q Marginal Marginal Marginal Productivity Productivity Productivity Q=F(K,L) AP L MP BA 445 Lesson I.7 Minimizing Cost 13 Productivity Measures Guiding the Production Process In terms of the production function and marginal product, profit maximization requires two areas of management: • Use the best technology so that, for any input levels (K,L) of capital and labor, the firm produces maximal output, F(K,L). • Employ the right level of inputs In the case where output sells at a competitive price p, to maximize profit a manager will vary inputs (if possible) and should • hire labor until the value of the marginal product of labor VMPL = P x MPL equals the wage, VMPL = w. – For example, if employing an extra hour of labor earns $12 more revenue and the wage is $8, you should hire that hour. – Equilibrium (no more hires, and no less) requires VMPL = w. • hire capital until the value of marginal product of capital VMPK = P x MPK equals the rental rate, VMPK = r. BA 445 Lesson I.7 Minimizing Cost 14 Cost Minimization Cost Minimization BA 445 Lesson I.7 Minimizing Cost 15 Cost Minimization Overview Cost Minimization by firms and charities predicts effects of a crackdown on immigration, of outsourcing jobs, of minimum wage legislation, and of encouraging handicapped employment. BA 445 Lesson I.7 Minimizing Cost 16 Cost Minimization Guiding the Production Process In terms of the production function and marginal product, profit maximization requires two areas of management: • Use the best technology so that, for any input levels (K,L) of capital and labor, the firm produces maximal output, F(K,L). • Employ the right level of inputs. BA 445 Lesson I.7 Minimizing Cost 17 Cost Minimization Employing the right level of inputs is simplest in the case where output sells at a competitive market price p and the inputs are chosen to maximize profit a manager will vary inputs (if possible) and should • hire labor until the value of the marginal product of labor VMPL = P x MPL equals the wage, VMPL = w. – For example, if employing an extra hour of labor earns $12 more revenue and the wage is $8, you should hire that hour. – Equilibrium (no more hires, and no less) requires VMPL = w. • hire capital until the value of marginal product of capital VMPK = P x MPK equals the rental rate, VMPK = r. BA 445 Lesson I.7 Minimizing Cost 18 Cost Minimization Employing the right level of inputs is harder in the case where either firms are in non-competitive markets or charities are not choosing inputs to maximize profit. In those cases, the manager or charity director has two choices to make. 1) Choose the right level of output. We leave aside that decision for now. 2) Choose the right level of inputs that minimize the cost of producing the chosen output. BA 445 Lesson I.7 Minimizing Cost 19 Cost Minimization Cost Minimization To minimize cost, the marginal product per dollar spent should be equal for all inputs: MPL/w = MPK/r • For example, if using an extra dollar to employ 1/w more hours of labor yields 12 more units of output (MPL/w = 12) and if using an extra dollar to employ 1/r more hours of labor yields 10 more units of output (MPK/r = 10) , you should hire more labor and less capital until you restore the cost-minimization equality MPL/w = MPK/r BA 445 Lesson I.7 Minimizing Cost 20 Cost Minimization Optimal Input Substitution The general case of substitution (holding output constant) when there may be more than two production inputs is like the substitution effect for consumption goods (holding purchasing power constant). Decomposing the Effects of a Rental Rate Decrease in Input X In the input substitution effect (holding output constant), the rental rate decrease in Input X makes the quantity demanded for Input X increase, and makes demand decrease for Input Y if the two inputs are substitutes (Wood and Labor in building houses). If there are only two inputs (that is, if the production process uses only two inputs, like Capital and Labor), then they must be substitutes. increase for Input Y if the two are complements (Manuel Labor and Shovels). BA 445 Lesson I.7 Minimizing Cost 21 Cost Minimization Example 1: Effects of a Crackdown on Illegal Labor Step 1: There are various ways to crack down on illegal labor: • decrease the number of illegal aliens successfully crossing the boarder • decrease the supply of illegal aliens wanting to cross the boarder by increasing penalties if they are caught, increase the penalties on U.S. producers that employ illegal labor • increase the likelihood of catching and penalizing a U.S. employer of illegal labor. Any one of those crackdowns increases the rental rate of illegal labor. Step 2: The input substitution effect increases the demand for substitutes (like unskilled domestic labor) and decreases the demand for complements (like shovels and interpreters). BA 445 Lesson I.7 Minimizing Cost 22 Cost Minimization Example 2: Effects of Outsourcing Jobs to Other Countries Step 1: Outsourcing jobs to India is caused by a decrease in the rental rate of Indian labor. (For example, the rental rate includes the cost of phone and internet service so that Indians can provide customer service in America.) Step 2: The input substitution effect of a decrease in the rental rate of Indian labor is a decreases the demand for substitutes and increases the demand for complements: • Demand decreases for domestic substitute labor • Customer service, computer programming, radiologists. • Demand increases for domestic complement labor • Managers. BA 445 Lesson I.7 Minimizing Cost 23 Cost Minimization Example 3: Minimum Wages The input substitution effect of any input cost change is always a change in the quantity of input demanded in the opposite direction. • Increasing minimum wages decreases the quantity demanded for unskilled labor, which is often replaced by machines (like in McDonalds). BA 445 Lesson I.7 Minimizing Cost 24 Cost Minimization Example 4: Jobs for Handicapped Workers The input substitution effect of any input cost change is always a change in the quantity of input demanded in the opposite direction. • Decreasing the cost of hiring handicapped workers increases the quantity demanded for handicapped workers. But is that good? • Thinking like a perfect altruist, ask: what is best for a handicapped person? Consider the following example: • Suppose handicapped Mr. H values his time at $9 per hour. • Suppose In-N-Out and all other fast-food places value Mr. H’s time at $7 per hour up to 40 hours per week. • Will Mr. H have a job? • Suppose the government subsidizes Mr. H’s employment $3 per hour. Will Mr. H now have a job? • Can you reallocate the subsidy money to make Mr. H happier? BA 445 Lesson I.7 Minimizing Cost 25 Cost Measures Cost Measures BA 445 Lesson I.7 Minimizing Cost 26 Cost Measures Overview Cost Measures including total cost and marginal cost are alternatives to productivity measures to help firms chose output to maximize profit. They also determine when it is best to shut down production. BA 445 Lesson I.7 Minimizing Cost 27 Cost Measures Preview • Cost functions are an alternative to production functions in formulating production technology. • Bad: They do not contain as much information as production functions. • Good: Cost functions contain enough information to help determine profit-maximizing output under various market conditions (monopoly, oligopoly, …) discussed later. • Good: Cost functions help accounting for cost, revenue, and profit. BA 445 Lesson I.7 Minimizing Cost 28 Cost Measures • Types of Costs Short-Run • Fixed costs (FC) is a fixed amount owed for any positive output. – If capital is fixed in the short-run, then FC equals the cost of capital. • Sunk costs is the part of fixed cost that is owed even if output is zero. – For example, you lease a railroad car for $10,000 for a month, but can recoup $6,000 if you do not use the car. – FC = $10,000, and sunk cost = $4,000. • Short-run total costs (C) • Short-run variable costs (VC). Defined by VC = C – FC. Long-Run • All costs are variable • No fixed costs BA 445 Lesson I.7 Minimizing Cost 29 Cost Measures C(Q): Minimum total cost of producing alternative levels of output: $ C(Q) = VC(Q) + FC C(Q) = VC + FC VC(Q): Costs that vary with VC(Q) output. FC: Costs that do not vary with output, as long as output is positive. FC Sunk Cost: Cost if output is zero. 0 Q BA 445 Lesson I.7 Minimizing Cost 30 Cost Measures FC: Costs that do not change as output changes. Since sunk cost is forever $ lost after it has been paid, C(Q) = VC + FC decision makers should ignore sunk costs to VC(Q) maximize profit or minimize losses. • Being rational in your personal life is a challenge. The extra $120,000 I paid for FC my house in 2007 (compared with prices 1 year later) is like a sunk cost. Q BA 445 Lesson I.7 Minimizing Cost 31 Cost Measures Marginal Cost MC = DC/DQ • Marginal cost curves often initially decrease with output Q because of specialization of labor. • Example: Workers at the Malibu subway work best when there are $ MC several customers (Q midsize) because the workers specialize. • Marginal cost curves eventually increase with output when inputs crowd. •Example: Workers at the Malibu subway work become less productive and costs increase if Q is so large that workers are Decreasing crowded. productivity • Unless a homework or exam problem of inputs explicitly states otherwise, draw the because of marginal cost curves U-shaped, as on crowding the right. Specializatio n of Labor Q BA 445 Lesson I.7 Minimizing Cost 32 Cost Measures Average Total Cost ATC = AVC + AFC ATC(Q) = C(Q)/Q $ MC ATC dATC/dQ = (dC/dQ)/Q – C/Q2 = (MC-ATC)/Q From that derivative calculation, • MC > ATC implies ATC is increasing • MC < ATC implies ATC is decreasing (There is an analogy to your GPA: If your next grade > GPA, your GPA is increasing.) Thus, ATC is decreasing until it intersects MC, then it is increasing. Q BA 445 Lesson I.7 Minimizing Cost 33 Cost Measures Average Variable Cost AVC = VC(Q)/Q dAVC/dQ = (dVC/dQ)/Q – $ MC ATC VC/Q2 = (MC-AVC)/Q AVC From that derivative calculation, • MC > AVC implies AVC is increasing • MC < AVC implies AVC is decreasing Q BA 445 Lesson I.7 Minimizing Cost 34 Cost Measures Summary Average Total Cost ATC = AVC + AFC ATC = C(Q)/Q $ MC ATC AVC Average Variable Cost AVC = VC(Q)/Q Average Fixed Cost AFC = FC/Q = ATC-AVC Marginal Cost AFC MC = DC/DQ Q BA 445 Lesson I.7 Minimizing Cost 35 Cost Measures Recovering Fixed Cost Q0(ATC-AVC) MC $ = Q0 AFC ATC = Q0(FC/ Q0) AVC = FC ATC AFC Fixed Cost AVC Q0 Q BA 445 Lesson I.7 Minimizing Cost 36 Cost Measures Recovering Variable Cost Q0AVC MC $ ATC = Q0[VC(Q0)/ Q0] AVC = VC(Q0) AVC Variable Cost Minimum of AVC Q0 Q BA 445 Lesson I.7 Minimizing Cost 37 Cost Measures Recovering Total Cost Q0ATC MC $ = Q0[C(Q0)/ Q0] ATC = C(Q0) AVC ATC Total Cost Minimum of ATC Q0 Q BA 445 Lesson I.7 Minimizing Cost 38 Cost Measures Cubic Cost Function • C(Q) = f + a Q + b Q2 + cQ3 • Marginal Cost? Calculus: MC(Q) = dC/dQ = a + 2bQ + 3cQ2 BA 445 Lesson I.7 Minimizing Cost 39 Cost Measures Quadratic Cost Function Total Cost: C(Q) = 10 + Q + Q2 Variable cost function: VC(Q) = Q + Q2 Variable cost of producing 2 units: VC(2) = 2 + (2)2 = 6 Fixed costs (all of which are sunk): FC = 10 Marginal cost function: MC(Q) = 1 + 2Q Marginal cost of producing 2 units: MC(2) = 1 + 2(2) = 5 BA 445 Lesson I.7 Minimizing Cost 40 Summary Summary BA 445 Lesson I.7 Minimizing Cost 41 Summary Summary of Choosing the Right Level of Inputs 1) For managers in perfectly-competitive, for-profit industries, to maximize profits (including minimize costs), must use inputs such that the value of marginal of each input equals the price the firm must pay to employ the input. • For example, w = VMPL = P x MPL 2) For any industry and for charities, to minimize cost of producing any level of output, the marginal product per dollar spent should equal for all inputs: • For example, MPL/w = MPK/r Note: Equation 2) does not determine the level of output, and Equation 1) does determine output but only in the special case of perfectly-competitive, for-profit industries. BA 445 Lesson I.7 Minimizing Cost 42 Review Questions Review Questions You should try to answer some of the review questions (see the online syllabus) before the next class. You will not turn in your answers, but students may request to discuss their answers to begin the next class. Your upcoming Exam 1 and cumulative Final Exam will contain some similar questions, so you should eventually consider every review question before taking your exams. BA 445 Lesson I.7 Minimizing Cost 43 BA 445 Managerial Economics End of Lesson I.7 BA 445 Lesson I.7 Minimizing Cost 44

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