# Chapter 5 The Production Process and Costs

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1 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.

2 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

3 Production Analysis 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. Short-Run vs. Long-Run Decisions Fixed vs. Variable Inputs 5-3

4 Production Function Algebraic Forms Linear production function: inputs are perfect substitutes. ( ) bl Q = F K, L = ak + Leontief production function: inputs are used in fixed proportions. ( K, L) = min{ bk cl} Q = F, Cobb-Douglas production function: inputs have a degree of substitutability. ( ) b K L K a L Q = F, = 5-4

5 Productivity Measures: Total Product Total Product (TP): maximum output produced with given amounts of inputs. Example: Cobb-Douglas Production Function: K is fixed at 16 units. Q = F(K,L) = K.5 L.5 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 5-5

6 Productivity Measures: Average Product of an Input Average Product of an Input: measure of output produced per unit of input. Average Product of Labor: AP L = 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 AP L = [(16) 0.5 (16) 0.5 ]/16 = 1. Average Product of Capital: AP K = 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 AP K = [(16) 0.5 (16) 0.5 ]/16 =

7 Productivity Measures: Marginal Product of an Input Marginal Product on an Input: change in total output attributable to the last unit of an input. Marginal Product of Labor: MP L = Q/ L Measures the output produced by the last worker. Slope of the short-run production function (with respect to labor). Marginal Product of Capital: MP K = Q/ K Measures the output produced by the last unit of capital. When capital is allowed to vary in the short run, MP K is the slope of the production function (with respect to capital). 5-7

8 Increasing, Diminishing and Negative Marginal Returns Q Increasing Marginal Returns Diminishing Marginal Returns Negative Marginal Returns Q=F(K,L) MP L AP 5-8

9 Guiding the Production Process Producing on the production function 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: VMP L = w, where VMP L = P x MP L. capital until the value of marginal product of capital equals the rental rate: VMP K = r, where VMP K = P x MP K. 5-9

10 Isoquant 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. 5-10

11 Marginal Rate of Technical Substitution (MRTS) The rate at which two inputs are substituted while maintaining the same output level. MRTS = KL MP MP L K 5-11

12 Linear Isoquants Capital and labor are perfect substitutes Q = ak + bl MRTS KL = b/a Linear isoquants imply that inputs are substituted at a constant rate, independent of the K Increasing Output input levels employed. Q 2 Q 3 Q 1 L 5-12

13 Leontief Isoquants Capital and labor are perfect complements. Capital and labor are used in fixed-proportions. Q = min {bk, cl} K Q 1 Q 2 Q 3 Increasing Output Since capital and labor are consumed in fixed proportions there is no input substitution along isoquants (hence, no MRTS KL ). L 5-13

14 Cobb-Douglas Isoquants Inputs are not perfectly substitutable. Diminishing marginal rate of technical substitution. K Q 1 Q 2 Q 3 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 = K a L b MRTS KL = MP L /MP K L 5-14

15 Isocost The combinations of inputs that produce a given level of output at the same cost: Rearranging, wl + rk = C K= (1/r)C - (w/r)l For given input prices, isocosts farther from the origin are associated with higher costs. Changes in input prices change the slope of the isocost line. C 1 /r C 0 /r K C/r K C 0 C 0 /w New Isocost Line associated with higher costs (C 0 < C 1 ). C 1 C 1 /w L New Isocost Line for a decrease in the wage (price of labor: w 0 > w 1 ). C/w 0 C/w 1 L 5-15

16 Cost Minimization Marginal product per dollar spent should be equal for all inputs: MP w But, this is just MP r MP MP L K L = = MRTS KL = K w r w r 5-16

17 Cost Minimization K Slope of Isocost = Slope of Isoquant Point of Cost Minimization Q L 5-17

18 Optimal Input Substitution A firm initially produces Q 0 by employing the combination of inputs represented by point A at a cost of C 0. Suppose w 0 falls to w 1. The isocost curve rotates counterclockwise; which represents the same cost level prior to the wage change. To produce the same level of output, Q 0, the firm will produce on a lower isocost line (C 1 ) at a point B. The slope of the new isocost line represents the lower wage relative to the rental rate of capital. K K 0 K 1 0 A L 0 L 1 B C 0 /w 0 C 1 /w 1 Q 0 C 0 /w 1 L 5-18

19 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 5-19

20 Total and Variable Costs C(Q): Minimum total cost of producing alternative levels of output: C(Q) = VC(Q) + FC VC(Q): Costs that vary with output. FC: Costs that do not vary with output. \$ C(Q) = VC + FC VC(Q) FC 0 Q 5-20

21 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. Decision makers should ignore sunk costs to maximize profit or minimize losses \$ C(Q) = VC + FC VC(Q) FC Q 5-21

22 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 MR Marginal Cost MC = DC/DQ AFC Q 5-22

23 Fixed Cost \$ Q 0 (ATC-AVC) = Q 0 AFC MC ATC = Q 0 (FC/ Q 0 ) AVC = FC AFC ATC AVC Fixed Cost Q 0 Q 5-23

24 Variable Cost \$ Q 0 AVC = Q 0 [VC(Q 0 )/ Q 0 ] = VC(Q 0 ) MC ATC AVC AVC Variable Cost Minimum of AVC Q 0 Q 5-24

25 Total Cost \$ Q 0 ATC = Q 0 [C(Q 0 )/ Q 0 ] MC ATC = C(Q 0 ) AVC ATC Total Cost Minimum of ATC Q 0 Q 5-25

26 Cubic Cost Function C(Q) = f + a Q + b Q 2 + cq 3 Marginal Cost? Memorize: Calculus: MC(Q) = a + 2bQ + 3cQ 2 dc/dq = a + 2bQ + 3cQ

27 An Example Total Cost: C(Q) = 10 + Q + Q 2 Variable cost function: VC(Q) = Q + Q 2 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) =

28 Long-Run Average Costs \$ LRAC Economies of Scale Q* Diseconomies of Scale Q 5-28

29 Multi-Product Cost Function C(Q 1, Q 2 ): Cost of jointly producing two outputs. General function form: ( Q Q ) = f + aq Q + bq 2 2 C +, cq

30 Economies of Scope C(Q 1, 0) + C(0, Q 2 ) > C(Q 1, Q 2 ). 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. 5-30

31 Cost Complementarity The marginal cost of producing good 1 declines as more of good two is produced: MC 1 (Q 1,Q 2 ) / Q 2 < 0. Example: Cow hides and steaks. 5-31

32 Quadratic Multi-Product Cost Function C(Q 1, Q 2 ) = f + aq 1 Q 2 + (Q 1 ) 2 + (Q 2 ) 2 MC 1 (Q 1, Q 2 ) = aq 2 + 2Q 1 MC 2 (Q 1, Q 2 ) = aq 1 + 2Q 2 Cost complementarity: a < 0 Economies of scope: f > aq 1 Q 2 C(Q 1,0) + C(0, Q 2 ) = f + (Q 1 ) 2 + f + (Q 2 ) 2 C(Q 1, Q 2 ) = f + aq 1 Q 2 + (Q 1 ) 2 + (Q 2 ) 2 f > aq 1 Q 2 : Joint production is cheaper 5-32

33 A Numerical Example: C(Q 1, Q 2 ) = 90-2Q 1 Q 2 + (Q 1 ) 2 + (Q 2 ) 2 Cost Complementarity? Yes, since a = -2 < 0 MC 1 (Q 1, Q 2 ) = -2Q 2 + 2Q 1 Economies of Scope? Yes, since 90 > -2Q 1 Q

34 Conclusion To maximize profits (minimize costs) managers must use inputs such that the value of marginal of each input reflects price the firm must pay to employ the input. The optimal mix of inputs is achieved when the MRTS KL = (w/r). Cost functions are the foundation for helping to determine profit-maximizing behavior in future chapters. 5-34

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