BUS 525: Managerial Economics The Production Process and Costs Lecture 5 The Production Process and Costs Going behind the supply curve

Overview I. Production Analysis II. Cost 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

Production Analysis Production Function A function that defines the maximum amount output that can be produced with a given set of inputs Q = F(K,L) 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 Factors of Production Managers are required to optimally choose the quantity and type of inputs to use in production process. E.g. employee mix, optimally substituting between labor and other inputs, as wages and other inputs prices change Short run is defined as a time frame in which there are fixed factors of production Long run is defined as the horizon over which the manager can adjust all factors of production Fixed factors are the inputs the managers cannot adjust in the short run. Variable factors are the inputs a manager can adjust to alter production. Short run Q = F(L) = F(K*, L), K is the fixed factor of production

1-4 Refer to Table 5.1 Page 158 Up to 5th unit of labor MPL is increasing 76-316 units Remains constant with 6th unit 7-10th unit declines from 316-76 units From 11th unit TP falls and MPL becomes negative Why?

Total Product The maximum level of output that can be produced with a given amount of input Cobb-Douglas Production Function Example: Q = F(K,L) = K.5 L.5 K is fixed at 16 units. Short run production function: Q = (16).5 L.5 = 4 L.5 Production when 100 units of labor are used? Q = 4 (100).5 = 4(10) = 40 units Linear production function Q= aK+ bL, e.g Q = 4k + L Leontief production function Q = min {bK, cL}, fixed proportion production functions, e.g. Word processing, Five typists Five computers: five papers; Five typists one computer: one paper

Average Productivity Measures Average Product of Labor: A measure of the output produced per unit of input APL = Q/L. Measures the output of an “average” worker. 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 labor is APL = [(16) 0.5(16)0.5]/16 = 1. Example: Q = F(K,L) = K.5 L.5If the inputs are K = 16 and L = 16, then the average product of labor is APL = [(16)0.5(16)0.5]/16 = 1.

Marginal Productivity Measures Marginal product : The 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). 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 long run, MPK is the slope of the production function (with respect to capital). Marginal product : The change in total output attributable to the last unit of an input Marginal product of a linear production function: Q= ak + bL MPk = a; MPl = b Marginal product of a Cobb-Douglas production function: Q= kaLb MPk = aKa-1Lb ; MPl = bKaLb-1 MPL of the second unit of labor is 172 units

Class Exercise Given the Cobb-Douglas Production Function Q = F(K,L) = K.5 L.5, K = 16, L=64 Find out the (i) Average Product of Labor (APl), (ii) Average Product of Capital (APk), (iii) Marginal Product of Capital (MPl) (iv) Marginal Product of Capital (MPk)

Increasing, Diminishing and Negative Marginal Returns Q Q=F(K,L) Increasing marginal returns to labor: Range of input usage over which marginal product increases. Decreasing/diminishing marginal returns to labor: Range of input usage over which marginal product declines. Negative marginal returns: Range of input usage over which marginal product is negative. AP L MP

Guiding the Production Process Producing on the production function Aligning incentives to induce maximum worker effort, tips, profit sharing. 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 . Why tips? Profit sharing

The profit maximizing usage of inputs ∏ = R- C, Q = F (K,L), C = wL + rK ∏ = P. F(K,L) - wL - rK

Refer to Table 5.2 Page 163 Up to 9th unit VMPL is greater than wage. Manager should hire upto ninth unit of labor Why some firms have 5000 and others have 2 workers? Notice that MPL is diminishing in the range

Isoquant The 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.

Slope of Isoquant

Marginal Rate of Technical Substitution (MRTS) The rate at which two inputs are substituted while maintaining the same output level. The production function satisfies the law of diminishing marginal rate of technical substitution: As a producer uses less of an input, increasingly more of the other input must be employed to produce the same level of output Add MRTS graph

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. Q3 Q2 Q1 Increasing Output L K Inputs are perfectly substitutable for each other.

Leontief Isoquants Capital and labor are perfect complements. K Increasing Output Capital and labor are perfect complements. Capital and labor are used in fixed-proportions. Q = min {bK, cL} What is the MRTSKL? Since capital and labor are consumed in fixed proportions there is no input substitution along isoquants (hence, no MRTSKL). L

Cobb-Douglas Isoquants Inputs are not perfectly substitutable. Diminishing marginal rate of technical substitution. 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 K Q3 Increasing Output Q2 Q1 L

Isocost The combinations of inputs that produce a given level of output at the same cost: wL + rK = C Rearranging, 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. K New Isocost Line associated with higher costs (C0 < C1). C1/r C0 C0/w C0/r C1 L C1/w K New Isocost Line for a decrease in the wage (price of labor: w0 > w1). C/r C/w1 L C/w0

Cost Minimization K Q L Point of Cost Minimization Slope of Isocost = Slope of Isoquant A C Q L

Cost minimization

Cost Minimization Marginal product per dollar spent should be equal for all inputs: But, this is just

Optimal Input Substitution K 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. C0/w1 A K0 C1/w1 B K1 To minimize the cost of producing a given level of output, the firm should use less of an input and more of other inputs when that input price rises. L1 Q0 L0 C0/w0 L

Cost Analysis Types of Costs Fixed costs (FC) Variable costs (VC) Total costs (TC) Sunk costs A cost that is forever lost after it has been incurred. Once paid they are irrelevant to decision making Fixed costs: Costs that do not change with changes in output: include the cost of fixed inputs used in production Variable costs: Costs that change with changes in output: include the cost of inputs that vary with output. Example of sunk costs: Feasibility studies for projects

Refer to Table 5.3 Page 179

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. $ Q C(Q) = VC + FC VC(Q) FC

Fixed and Sunk Costs FC: Costs that do not change as output changes. $ FC: Costs that do not change as output changes. Sunk Cost: A cost that is forever lost after it has been paid. C(Q) = VC + FC VC(Q) A decision maker/manager should ignore sunk costs to maximize profits or minimize costs Buy a textbook for Tk. 500 that you do not need. The book seller does not allow refund of exchange Someone wants to buy it for Tk. 200, what should you do? FC Q

Refer to Table 5.4 Page 181

1-30 Refer to Table 5.5 Page 182

Some Definitions Average Total Cost ATC = AVC + AFC ATC = C(Q)/Q $ Average Variable Cost AVC = VC(Q)/Q Average Fixed Cost AFC = FC/Q Marginal Cost MC = DC/DQ $ Q MC ATC AVC MR Average fixed cost: Fixed costs divided by the number of units of output Average variable cost: Variable costs divided by the number of units of output Marginal costs: The cost of producing an additional unit of output. Note that : MC intersects ATC and the AVC at their lowest points. This implies that when MC is below an average cost curve AC is declining and When MC is above AC is rising Grade in the course: CGPA MG AGPA B A G↑ B C G↓ Note further that ATC and AVC curves get closer as output increases. Why? AFC declines as output increases. AFC

Fixed Cost MC $ ATC AVC ATC Fixed Cost AFC AVC Q Q0 Q0(ATC-AVC) = Q0 AFC = Q0(FC/ Q0) = FC MC $ Q ATC AVC ATC Fixed Cost AFC Relationship between productivity and cost curves: MC and AVC are mirror images of the MP and AP curves respectively. When MP is maximum MC is minimum when MP is diminishing MC goes up. AVC Q0

Variable Cost MC $ ATC AVC AVC Variable Cost Q Q0 Q0AVC = Q0[VC(Q0)/ Q0] = VC(Q0) $ Q ATC AVC AVC Variable Cost Q0

Total Cost MC $ ATC AVC ATC Total Cost Q Q0 Q0ATC = Q0[C(Q0)/ Q0]

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

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

Economies of Scale $ LRAC Economies Diseconomies of Scale of Scale Q Long-run costs: All cost are variable Long-run average cost curve: A curve that defines the minimum average cost of producing alternative levels of output, allowing for optimal selection of both fixed and variable factors of production. Economies of scale: Exists when long run average costs decline as output is increased. Diseconomies of scale: Exists when long run average costs rise as output increased. Constant returns to scale: Exists when long run average costs remain constant as output increased. Economic versus accounting cost Cost of production include not only the accounting costs but also the opportunities forgone by producing a given product. Economies of Scale Diseconomies of Scale Q

Multi-Product Cost Function A function that defines the cost of producing given levels of two or more types of outputs assuming all inputs are used efficiently C(Q1, Q2): Cost of jointly producing two outputs. General multiproduct cost function

Economies of Scope C(Q1, 0) + C(0, Q2) > C(Q1, Q2). Example: It is cheaper to produce the two outputs jointly instead of separately. Example: It is cheaper for Citycell to produce Internet connections and Instant Messaging services jointly than separately.

Cost Complementarity DMC1(Q1,Q2) /DQ2 < 0. The marginal cost of producing good 1 declines as more of good 2 is produced: DMC1(Q1,Q2) /DQ2 < 0. Example: Bread and biscuits

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 f > aQ1Q2: Joint production is cheaper When a is negative an increase in Q2 will lower MC1 C(Q1 ,0) + C(0, Q2 ) > C(Q1, Q2) = f + (Q1 )2 + f + (Q2)2 - f - aQ1Q2 - (Q1 )2 - (Q2 )2>0 f - aQ1Q2 >0

A Numerical Example: C(Q1, Q2) = 100 – 1/2Q1Q2 + (Q1 )2 + (Q2 )2 Cost Complementarity? Yes, since a = -2 < 0 MC1(Q1, Q2) = -2Q2 + 2Q1 Economies of Scope? Yes, since 90 > -2Q1Q2 The firm wishes to produce 5 units of good 1 and 4 units of good 2. a=1/2<0, cost complementarities exist f-aQ1Q2>0 100- 1/2x5x4 = 90 >0, Economies of scope exists C(5,4) = 100-10+25+16 = 131 C(5,0) = 100+25 = 125 C(0,4) = 100+16 = 116 Benefit of joint production = 241-131= $110

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 MRTSKL = (w/r). Cost functions are the foundation for helping to determine profit-maximizing behavior in future chapters.

Mid1 Answers (a) p5; (b) p21; © p56; (d) p13; (e) p90; (f) p79. (a)p46; (b) p53(price floor, surplus). 3. (a)p92; (b) -0.793; © -0.91(inelastic demand); 0.156(substitute, inelastic); 0.32(normal good, inelastic);0.13(positively related, inelastic). (a) Px = 115 – ¼ Qxd; (b) 12,800; © 16,200; (d) Consumer surplus decreases. (a) p=300; (b) p=200; Q=1000; © p=225; Q=1250. (a) No, demand is inelastic, revenue will fall; (b) Income must rise by 22.22%; © Sale of shoes will increase by 2.25%.