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Managerial Economics & Business Strategy Chapter 5 The Production Process and Costs McGraw-Hill/Irwin Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved. Overview I. Production Analysis – – – – Total Product, Marginal Product, Average Product. Isoquants. Isocosts. Cost Minimization II. Cost Analysis – Total Cost, Variable Cost, Fixed Costs. – Cubic Cost Function. – Cost Relations. III. Multi-Product Cost Functions 5-2 Production Theory Production theory serves as the basis for other business disciplines such as human resources, operations management, managerial accounting, and strategic management. Successful businesses use an optimal amount and mix of production factors as factor and output markets change. 5-3 Production Analysis Technology – engineering know-how Production Function – Q = F(K,L) • Q is quantity of output produced. • K is capital input. • L is labor input. • F is a functional form relating the inputs to output. – The maximum amount of output that can be produced with K units of capital and L units of labor. – Manager’s responsibility – use the production function efficiently – how much of each input to use, 5-4 SR and LR Short-Run vs. Long-Run Decisions – in the SR some factors are fixed limiting input choice decisions. The mix of variable factors can be changed in the SR. In the LR is the production horizon in which all factors are variable. In the simple two factor case if K is fixed in the SR then the only use decisions apply to L. Q = f(L) = F(K*,L) 5-5 Production Function Algebraic Forms Linear production function: inputs are perfect substitutes. Q F K , L aK bL Where a and b are constants. If capital is 5 times more productive than labor then the function becomes: Q = 5K + L If F(8,3) then 5(8) + 1(3), then Q = 43 Linear production functions do not obey the law of DMR 5-6 Linear Isoquants Capital and labor are perfect substitutes – Q = aK + bL – MRTSKL = b/a – Linear isoquants imply that inputs are substituted at a constant rate, independent of the input levels employed. K Increasing Output Q1 Q2 Q3 L 5-7 Production Function Algebraic Forms Leontief production function: inputs are used in fixed proportions – a fixed proportion of K to L for output to increase. Q F K , L minbK, cL 5-8 Leontief Isoquants Capital and labor are K perfect complements. Capital and labor are used in fixed-proportions. Q = min {bK, cL} Since capital and labor are consumed in fixed proportions there is no input substitution along isoquants (hence, no MRTSKL). Q3 Q2 Q1 Increasing Output L 5-9 Production Function Algebraic Forms Cobb-Douglas production function: inputs have a degree of substitutability. Q F K , L K a Lb Here the relationship between output and inputs is not linear. 5-10 Cobb-Douglas Isoquants Inputs are not perfectly substitutable. Diminishing marginal rate of technical substitution. K Q3 Q2 Q1 Increasing Output – As less of one input is used in the production process, increasingly more of the other input must be employed to produce the same output level. Q = KaLb MRTSKL = MPL/MPK L 5-11 Productivity Measures: Total Product 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. – 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 5-12 Productivity Measures: Average Product of an Input Average Product of an Input: measure of output produced per unit of input. – 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. 5-13 Productivity Measures: Average Product of an Input – 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. 5-14 Productivity Measures: Marginal Product of an Input Marginal Product of 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 last worker. • Slope of the short-run production function (with respect to labor). 5-15 Productivity Measures: Marginal Product of an Input – 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). 5-16 Guiding the Production Process Producing on the production function Most difficult to achieve – Aligning incentives to induce maximum worker effort. Employing the right level of inputs – When labor or capital vary in the short run, to maximize profit a manager will hire: • labor until the value of marginal product of labor equals the wage: VMPL = w, where VMPL = P x MPL. • capital until the value of marginal product of capital equals the rental rate: VMPK = r, where VMPK = P x MPK . 5-17 Marginal Product of an Input in a CobbDouglas Production Function Q = F(K,L) = KaLb then MPL = bKaLb-1 and MPK = aKa-1Lb See page 167 for demonstration problem. 5-18 Increasing, Diminishing and Negative Marginal Returns Q Increasing Marginal Returns Diminishing Marginal Returns Negative Marginal Returns Q=F(K,L) MP AP L 5-19 Isoquants Assume management flexibility in the production process. Illustrates the long-run combinations of inputs (K, L) that yield the producer the same level of output. The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output. 5-20 Cobb-Douglas Isoquants Inputs are not perfectly substitutable. Diminishing marginal rate of technical substitution. K Q3 Q2 Q1 Increasing Output – As less of one input is used in the production process, increasingly more of the other input must be employed to produce the same output level. Q = KaLb MRTSKL = MPL/MPK L 5-21 Marginal Rate of Technical Substitution (MRTS) The rate at which two inputs are substituted while maintaining the same output level. The manager must determine the efficient combination of resource use to max profits. MRTSKL MPL MPK 5-22 Isocost The combinations of inputs K that produce a given level of C1/r output at the same cost: C0/r wL + rK = C Rearranging, K= (1/r)C - (w/r)L For given input prices, K isocosts farther from the C/r origin are associated with higher costs. Changes in input prices change the slope of the isocost line. New Isocost Line associated with higher costs (C0 < C1). C0 C0/w C1 C1/w L New Isocost Line for a decrease in the wage (price of labor: w0 > w1). C/w0 C/w1 L 5-23 Cost Minimization Marginal product per dollar spent should be equal for all inputs (Equi-marginal rule): MPL MPK MPL w w r MPK r But, this is just MRTS KL w r So equilibrium for the producer is the point where the slope of the isoquant = slope of the isocost. 5-24 Equi-Marginal Rule Example Suppose Ford uses 10 workers and 10 robots to produce one automobile. Suppose also the MPL = 10 cars/day and the MPK = 10 cars/day. If the wage = $40/hr and the cost of the robots is $100/hr is Ford using these two inputs in a cost-minimizing manner? C = 10($40) + 10($100) = $1400 MPL/PL = MPK/PK 10/40 is not equal to 10/100 .25 > .10 Increase the number of workers to 12 so that the MPL decreases to 8 (ratio = .20), decrease the number of robots to 8 to increase the MPK to 20 (ratio = .20) 8/40 = 20/100 C = 12($40) + 8($100) = $1280 5-25 Cost Minimization K Slope of Isocost = Slope of Isoquant Point of Cost Minimization Q L 5-26 Optimal Input Substitution A firm initially produces Q0 by employing the combination of inputs represented by point A at a cost of C0. Suppose w0 falls to w1. – The isocost curve rotates counterclockwise; which represents the same cost level prior to the wage change. – To produce the same level of output, Q0, the firm will produce on a lower isocost line (C1) at a point B. – The slope of the new isocost line represents the lower wage relative to the rental rate of capital. K A K0 B K1 Q0 0 L0 L1 C0/w0 C1/w1 C0/w1 L 5-27 Cost Analysis Types of Costs – Short-Run • Fixed costs (FC) • Sunk costs • Short-run variable costs (VC) • Short-run total costs (TC) – Long-Run • All costs are variable • No fixed costs 5-28 Cost Function The cost function provides the manager essential information needed to determine the profit maximizing level of output. Summarizes the information about the production process. 5-29 Total and Variable Costs C(Q): Minimum total cost of $ producing alternative levels of output: C(Q) = VC + FC C(Q) = VC(Q) + FC VC(Q) VC(Q): Costs that vary with output. FC: Costs that do not vary with output. FC 0 Q 5-30 Fixed and Sunk Costs FC: Costs that do not change as output changes. Sunk Cost: A cost that is forever lost after it has been paid. The amount of fixed costs that cannot be recouped. Decision makers should ignore sunk costs to maximize profit or minimize losses $ C(Q) = VC + FC VC(Q) FC Q 5-31 Some Definitions 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 Marginal Cost MC = DC/DQ MR AFC Q 5-32 Fixed Cost Q0(ATC-AVC) $ = Q0 AFC = Q0(FC/ Q0) MC ATC AVC = FC ATC AFC Fixed Cost AVC Q0 Q 5-33 Variable Cost $ Q0AVC MC ATC = Q0[VC(Q0)/ Q0] AVC = VC(Q0) AVC Variable Cost Minimum of AVC Q0 Q 5-34 Total Cost Q0ATC $ = Q0[C(Q0)/ Q0] = C(Q0) MC ATC AVC ATC Minimum of ATC Total Cost Q0 Q 5-35 Cubic Cost Function C(Q) = f + a Q + b Q2 + cQ3 Marginal Cost? – Memorize: MC(Q) = a + 2bQ + 3cQ2 – Calculus: dC/dQ = a + 2bQ + 3cQ2 5-36 An Example – 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: FC = 10 – Marginal cost function: MC(Q) = 1 + 2Q – Marginal cost of producing 2 units: MC(2) = 1 + 2(2) = 5 5-37 Another Example C(Q) = 45 + 6Q + 12Q2 Suppose Q = 20 MC = 6 + 24Q AFC = 45/20 = 2.25 AVC = 6Q + 12Q2 /Q = 6 + 12Q = 246 ATC = 45 + 6Q + 12Q2/Q = 45/Q + 6 +12Q= 248.25 5-38 LRAC Function The LRAC function is the envelope of all possible SRATC functions. As output expands the firm must adjust its scale to be sure it is operating at least cost. 5-39 Long-Run Average Costs $ LRAC Economies of Scale Diseconomies of Scale Q* Q 5-40 Multi-Product Cost Function – Economies of Scope and Cost Complementarities C(Q1, Q2): Cost of jointly producing two outputs. General functional form: CQ1, Q2 f aQ1Q2 bQ12 cQ22 MC1 = aQ2 + 2bQ1 MC2 = aQ1 + 2cQ2 If a > 0 then there are no cost complementarities in production. If a < 0 then there are cost complementarities. 5-41 Economies of Scope C(Q1, 0) + C(0, Q2) > C(Q1, Q2). – It is cheaper to produce the two outputs jointly instead of separately. Example: – It is cheaper for Time-Warner to produce Internet connections and Instant Messaging services jointly than separately. – Dunkin Donuts 5-42 Cost Complementarity The marginal cost of producing good 1 declines as more of good two is produced: DMC1Q1,Q2) /DQ2 < 0. Example: – Cow hides and steaks. 5-43 Quadratic Multi-Product Cost Function C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2 MC1(Q1, Q2) = aQ2 + 2Q1 MC2(Q1, Q2) = aQ1 + 2Q2 Cost complementarity: a<0 Economies of scope: f > aQ1Q2 C(Q1 ,0) + C(0, Q2 ) = f + (Q1 )2 + f + (Q2)2 C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2 f > aQ1Q2: Joint production is cheaper 5-44 A Numerical Example: C(Q1, Q2) = 90 - 2Q1Q2 + (Q1 )2 + (Q2 )2 Cost Complementarity? Yes, since a = -2 < 0 MC1(Q1, Q2) = -2Q2 + 2Q1 Economies of Scope? Yes, since 90 > -2Q1Q2 Example pg 189. 5-45 Conclusion To maximize profits (minimize costs) managers must use inputs such that the value of marginal product of each input reflects price the firm must pay to employ the input. The optimal mix of inputs is achieved when the MRTSKL = (w/r). Cost functions are the foundation for helping to determine profit-maximizing behavior in future chapters. 5-46