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Readings Readings Baye 6th edition or 7th edition, Chapter 5 BA 445 Lesson A.7 Choosing Inputs 1 Overview Overview BA 445 Lesson A.7 Choosing Inputs 2 Overview Profit Maximization through choosing each input into production balances the value of the marginal product of each input with the marginal cost of each input. Cost Minimization balances the marginal product per dollar invested into an input across all inputs. Cost minimization makes the most of the resources devoted to charity. 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 A.7 Choosing Inputs 3 Profit Maximization Profit Maximization BA 445 Lesson A.7 Choosing Inputs 4 Profit Maximization Overview Profit Maximization through choosing each input into production balances the value of the marginal product of each input with the marginal cost of each input. BA 445 Lesson A.7 Choosing Inputs 5 Profit Maximization Production Functions allow detailed analysis • Q = F(K,L) covers the simplest case of a single output and two inputs. Q is quantity of output produced. F is a function relating two inputs to a single output. L is labor input. • Labor is rented and not consumed (except for gladiators). K is capital input. • Capital is everything that is not labor. • Capital can be disaggregated to, say, machines and land, making the function Q = F(K1,K2,L). • “Capital” is often rented, like machines or land. • F(K,L) is the maximum amount of output that can be produced with K units of capital and L units of labor. Maximum output implies a type of efficient production. BA 445 Lesson A.7 Choosing Inputs 6 Profit Maximization Short-Run vs. Long-Run Decisions • Some capital or labor is fixed in the short-run. Office space (capital) may not be completely adjustable in the short-run. Labor with contracts is not completely adjustable in the short-run. • Unless stated otherwise in a homework or exam problem, for the rest of the course just consider an extreme division of the short-run from the long-run: labor is variable in both the short-run and long-run. capital is fixed in the short-run but variable in the long-run. BA 445 Lesson A.7 Choosing Inputs 7 Profit Maximization Total Product: measures maximum output produced. • 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 A.7 Choosing Inputs 8 Profit Maximization Average Product of an Input: measures output produced per unit of input. (It is a common productivity measure, but it is less useful than marginal product.) • Average Product of Labor: APL = Q/L. Measures the output of an “average” worker. .5 .5 Example: Q = F(K,L) = K L • 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. .5 .5 Example: Q = F(K,L) = K L • 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 A.7 Choosing Inputs 9 Profit Maximization Marginal Product on an Input: measures the change in total output caused by the last unit of an input. • Marginal Product of Labor: MPL = DQ/DL Measures output produced by the last worker. .5 .5 Example: Q = F(K,L) = K L • At inputs K = 16 and L = 16, the marginal product of labor is MPL = .5(16) -0.5(16)0.5 = .5 • Marginal Product of Capital: MPK = DQ/DK Measures output produced by the last unit of capital. • Marginal Products are the essential measures of input productivity for profit maximization and cost minimization. BA 445 Lesson A.7 Choosing Inputs 10 Profit Maximization Negative Marginal Productivity should obviously be avoided. Optimal labor use is also short of that which maximizes output since labor has positive cost (even if it is volunteer labor). Q Increasing Marginal Productivity Diminishing Marginal Productivity Negative Marginal Productivity Q=F(K,L) MP BA 445 Lesson A.7 Choosing Inputs AP L 11 Profit Maximization Guiding the Production Process 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 A.7 Choosing Inputs 12 Profit Maximization Employing the right level of inputs is simplest when output sells at a competitive market price p and the inputs are chosen to maximize profit. Then, a manager will vary inputs and • hire more or less labor (which is variable in both the short run and long run) until the value of the marginal product of labor VMPL = P x MPL equals the wage, VMPL = w. – For example, if output sells for $3 per unit and employing an extra hour of labor produces 4 units of output, extra labor earns $12. So, if the wage is less than $12, you should hire extra labor. – Profits are maximized when VMPL = w. • hire more or less capital (in the long run) until the value of the marginal product of capital VMPK = P x MPK equals the rental rate, VMPK = r. BA 445 Lesson A.7 Choosing Inputs 13 Cost Minimization Cost Minimization BA 445 Lesson A.7 Choosing Inputs 14 Cost Minimization Overview Cost Minimization balances the marginal product per dollar invested into an input across all inputs. Cost minimization makes the most of the resources devoted to charity. BA 445 Lesson A.7 Choosing Inputs 15 Cost Minimization Everyone Should Minimize Cost • • • 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 focused on helping others (not on their own feelings of satisfaction) 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 A.7 Choosing Inputs 16 Cost Minimization Employing the right level of inputs is complex when either firms are in non-competitive markets or charities are not choosing inputs to maximize profit. In those cases, guiding the production process requires 1) choosing the right level of output. (For this lesson, we leave aside that decision.) 2) choosing the right level of inputs that minimize the cost of producing the chosen output. BA 445 Lesson A.7 Choosing Inputs 17 Cost Minimization Cost Minimization To minimize the cost of producing output, the marginal product per dollar spent should equal for all inputs: MPL/w = MPK/r • Example: Suppose MPL = 12 (an extra unit of labor yields 12 more units of output) Suppose w = $4 (an extra dollar to labor can hire 1/w = 1/4 more hours of labor) Suppose MPK = 15 (an extra unit of labor yields 12 more units of output) Suppose r = $3 (an extra dollar to capital can hire 1/r = 1/3 more hours of capital) Then, MPL/w = 3 (an extra dollar to labor yields 3 more units of output) Then, MPK/r = 5 (an extra dollar to capital yields 5 more units of output) Therefore, you can produce the same output at lower cost by hiring less labor and more capital. That cost savings continues until you restore the costminimization equality MPL/w = MPK/r BA 445 Lesson A.7 Choosing Inputs 18 Cost Minimization The Input Substitution Effect measures changes in the quantity demanded of inputs caused by changes in input rental rates. The effect holds output constant and so is not a complete analysis. Consider the input substitution effect of an increase in the rental rate of input X: There is a decrease in the quantity demanded of Input X There is an increase in the demand for Input Y if the two inputs are substitutes (like 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. There is a decrease for Input Y if the two are complements (like Manuel Labor and Shovels). BA 445 Lesson A.7 Choosing Inputs 19 Cost Minimization Example 1: Compute the effects of a U.S. crackdown on illegal Mexican labor Step 1: There are various ways to crack down on illegal labor: • decrease the supply of illegal labor able to cross the border. • decrease the supply of illegal labor wanting to cross the border by increasing penalties if they are caught. • decrease the demand for illegal labor by increasing penalties if they are caught. Any one of those crackdowns increases the rental rate of illegal labor. Step 2: The input substitution effect increases the demand for input substitutes (like unskilled U.S. labor) and decreases the demand for input complements (like shovels and interpreters). Since the input substitution effect holds output constant, this is not a complete analysis. BA 445 Lesson A.7 Choosing Inputs 20 Cost Minimization Example 2: Compute the effects of creating jobs for handicapped workers. Is it good to create such jobs? • In general, the input substitution effect change in the rental rate of an input X is a change in the quantity of input X demanded in the opposite direction. • In particular, one way to create jobs for handicapped workers is to decrease the cost of hiring handicapped workers. That decreased cost increases the quantity demanded for handicapped workers. But is that a good idea? • Consider a specific example: BA 445 Lesson A.7 Choosing Inputs 21 Cost Minimization Suppose handicapped Mr. H values his time at $8 per hour. Suppose In-N-Out and all other fast-food suppliers value Mr. H’s time at $4 per hour for up to 40 hours per week. (The value of Mr. H’s marginal product of labor is $4.) Will Mr. H have a job with In-N-Out? Now suppose the government subsidizes Mr. H’s employment $6 per hour. Will Mr. H now have a job with In-N-Out? How much does that $6-per-hour subsidy cost of government each week? Can you reallocate that subsidy money to make Mr. H happier? Was the subsidy a good idea? BA 445 Lesson A.7 Choosing Inputs 22 Cost Measures Cost Measures BA 445 Lesson A.7 Choosing Inputs 23 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 A.7 Choosing Inputs 24 Cost Measures Preview • Cost functions are an alternative to production functions in formulating production technology. They have some good and bad features. • 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 A.7 Choosing Inputs 25 Cost Measures Short-Run vs. Long-Run Decisions • For the rest of this lesson, just consider the extreme division of the short-run and the long-run: labor is variable in both the short-run and long-run. capital is fixed in the short-run but variable in the long-run. BA 445 Lesson A.7 Choosing Inputs 26 Cost Measures Types of Costs • Short-Run Fixed costs (FC) is a fixed amount owed for any positive output. • Since capital is fixed in the short-run, FC equals the cost of capital. Sunk costs is the part of fixed cost that is owed even if output is zero. • For example, suppose you lease a railroad car for $10,000 for a month, but can recoup $6,000 if your output is zero. • Then, FC = $10,000, and sunk cost = $4,000. Short-run total costs (C or TC) • Short-run variable costs (VC), defined by VC = C – FC. • Long-Run Define FC, Sunk Cost, TC, and VC as in the short run. Since all capital is variable, Sunk Cost = 0. BA 445 Lesson A.7 Choosing Inputs 27 Cost Measures Graphing Short Run Costs C(Q): Minimum total cost of producing alternative levels of output: C(Q) = VC(Q) + FC $ C(Q) = VC + FC VC(Q) VC(Q): Costs that vary with output. FC: Costs that do not vary with output, as long as output is positive. Sunk Cost: Cost if output is zero. FC Sunk Cost 0 BA 445 Lesson A.7 Choosing Inputs Q 28 Cost Measures Simplifying Assumption In principle, the entire cost function C(Q) could be different between the short run and the long run. But to simplify the rest of the course, assume the only difference between the short run (SR) and long run (LR) is C(0), the cost of 0. And assume SR sunk cost = fixed cost • LR C(0) = 0 • SR C(0) = Fixed cost. $ Short run C(Q) = VC + FC VC(Q) FC 0 BA 445 Lesson A.7 Choosing Inputs Q 29 Cost Measures Marginal Cost of Production: measures the change in total cost caused by the last unit of output. • Marginal Cost of output: MC = DC/DQ • Marginal Cost will be the essential measure of production cost to determine output that maximizes profit. BA 445 Lesson A.7 Choosing Inputs 30 Cost Measures A Graph of Marginal Cost • 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 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 crowded. • Unless a homework or exam problem explicitly states otherwise, draw the marginal cost curves U-shaped, as in the graph on the right. MC Decreasing productivity of inputs because of crowding Specializatio n of Labor BA 445 Lesson A.7 Choosing Inputs Q 31 Cost Measures Average Total Cost of Production: measures the total cost per unit of output. • Average Total Cost: ATC = C/Q • Average Total Cost will help compute profit. BA 445 Lesson A.7 Choosing Inputs 32 Cost Measures A Graph of Average Total Cost When on the same graph, marginal cost (MC) and average total cost (ATC) relate to each other just as the grade points you earn on your next class relate to your grade point average. • MC < ATC implies ATC decreases (if your next grade is lower than your average, then your average decreases). • MC > ATC implies ATC is increases • Thus, ATC decreases until it intersects MC. $ BA 445 Lesson A.7 Choosing Inputs MC ATC Q 33 Cost Measures Average Variable Cost of Production: measures the variable cost per unit of output. • Average Variable Cost: AVC = VC/Q • Average Variable Cost will help compute profit. BA 445 Lesson A.7 Choosing Inputs 34 Cost Measures A Graph of Average Variable Cost When on the same graph, marginal cost (MC) and average variable cost (AVC) relate to each other like MC and ATC • MC < ATC implies ATC decreases (if your next grade is lower than your average, then your average decreases). • MC > ATC implies ATC is increases • Thus, ATC decreases until it intersects MC. $ MC ATC AVC Q BA 445 Lesson A.7 Choosing Inputs 35 Cost Measures Summary Average Total Cost ATC = AVC + AFC ATC = C(Q)/Q $ MC ATC AVC Average Variable Cost AVC = VC(Q)/Q Marginal Cost MC = DC/DQ Q BA 445 Lesson A.7 Choosing Inputs 36 Cost Measures Recovering Fixed Cost Q0(ATC-AVC) $ = Q0 AFC = Q0(FC/ Q0) MC ATC AVC = FC ATC AFC Fixed Cost AVC Q0 BA 445 Lesson A.7 Choosing Inputs Q 37 Cost Measures Recovering Variable Cost $ Q0AVC MC ATC = Q0[VC(Q0)/ Q0] AVC = VC(Q0) AVC Variable Cost Minimum of AVC Q0 BA 445 Lesson A.7 Choosing Inputs Q 38 Cost Measures Recovering Total Cost Q0ATC $ MC = Q0[C(Q0)/ Q0] ATC AVC = C(Q0) ATC Minimum of ATC Total Cost Q0 BA 445 Lesson A.7 Choosing Inputs Q 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 A.7 Choosing Inputs 40 Summary Summary BA 445 Lesson A.7 Choosing Inputs 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 A.7 Choosing Inputs 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 A.7 Choosing Inputs 43 BA 445 Managerial Economics End of Lesson A.7 BA 445 Lesson A.7 Choosing Inputs 44