Managerial Economics & Business Strategy Chapter 5 The Production Process and Costs Production Analysis • Production Function Q = F(K,L) The maximum amount of output that can be produced with K units of capital and L units of labor. The role of the manager is to make sure the firm operates on the production function • Short-Run (SR) v. Long-Run (LR) Decisions In SR some variables fixed. The LR is a period long enough such that all inputs are variable Total Product • Linear Production Function Perfect substitution among inputs (meaning, constant) • Example: Q = F(K,L) = aK + bL Suppose: a = 10, b = 5, and K is fixed at 16 units in SR. Short run production function: Q = 10(16)+5 L = 160 + 5L Production when 100 units of labor are used? Q = 160+500 = 660 units Total Product • Cobb-Douglas Production Function Imperfect substitution (K and L are substitutable but not at a constant rate). • Example: Q = F(K,L) = Ka Lb K is fixed at 16 units, let a = b = 0.5 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 Total Product • Leontief Production Function Substitution impossible. K and L are complements. • Example: Q = F(K,L) = min(bK, cL) Let b=c=1. Let K=16 and L=100. Production when 100 units of labor are used? 16 units Marginal Product of Labor • Measures the output produced by the last worker holding K constant. • Discrete: MPL = DQ/DL (discrete) • Continuous: Slope (derivative) of the production function (continuous) Q MPL L • At times may see “increasing marginal returns” due to specialization, and “diminishing marginal returns.” Using Calculus to Find MPL Linear Production Function Q aK bL Q MPL b L Cobb-Douglas Production Function QK L a b a b 1 MPL bK L Average Product of Labor • APL = Q/L • Measures the output of an “average” worker. • Geometrically, APL is measured as the slope of a ray from the origin to a point on the TP curve. Stages of Production Q Increasing Marginal Returns Diminishing Marginal Returns Negative Marginal Returns Slope=0, MP=0 Q=F(K,L) Slope=.75, AP=.75 MP AP L Productivity in SR • Go to SR and LR production handout What’s more important? AP or MP • Suppose in SR and K is fixed. How does a firm determine the optimal level of L to hire? How many workers are needed to maximize profits? If manager knows how much an additional worker contributes to revenues and how much an additional worker costs, then manager can decide if hiring the worker will increase the firm’s profits. • Therefore, MP more important. Value Marginal Product of Labor • Represent the additional revenue generated by employing an additional unit of L. VMPL = MR MPL = dR/dQ MPL. • Note: In a perfectly competitive market, the price of the good is the marginal revenue. • VMPL=P MPL • Similarly, for the value of the MPK, VMPK=P MPK Find Profit-Maximizing Level of Labor • Q=2(K)1/2(L)1/2. • The company has already spent $10,000 on 4 units of K, and due to current economic conditions, the company doesn’t have the flexibility to buy more equipment. • W=$100 per worker. P=$200/unit. • How much L? How much Q is produced? What are profits? The Answer • L=16, Q=16, = 8400. Because $8400 is less than FC of $10,000, the firm is covering a portion of its FC ($1600 of FC). Cost Analysis • Types of Costs Fixed costs (FC) Variable costs (VC) Marginal costs (MC) Total costs (TC) Sunk costs • Need to understand costs to find profitmaximizing output Total and Variable Costs (SR) C(Q): Minimum total cost $ of producing alternative levels of output: C(Q) = VC + FC VC(Q) C(Q) = VC + FC VC(Q): Costs that vary with output FC: Costs that do not vary with output FC Q 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 Fixed and Sunk Costs •You lease a building for $5000 a month. •This is a FC. •After you pay, it is a sunk cost, unless some of it is refundable, then only the nonrefundable portion is sunk. Sunk costs are irrelevant to marginal decision-making that takes place during the month. •If at the end of the month, the firm is losing money (negative economic profit), then paying the lease becomes a marginal decision, and shouldn’t renew Fixed and Sunk Costs •Problem from the book: “A local restaurateur who had been running a profitable business for many years recently purchased a 3-way liquor license. This license gives the owner the legal right to sell…spirits in her restaurant. The cost of obtaining the 3-way license was about $75,000, since only 300 such licenses are issued by the state. While the license is transferable, only $65,000 is refundable if the owner chooses not to use the license. After selling alcoholic beverages for about 1 year, the restaurateur came to the realization that she was losing dinner customers and that her profitable restaurant was turning into a noisy, unprofitable bar. Subsequently, she spent about $6000 placing advertisements in various newspapers and restaurant magazines across the state offering to sell the license for $70,000. After a long wait, she finally received one offer to purchase the license for $66,000. What is your opinion of the restaurateur’s decisions? Would you recommend that she accept the $66,000 offer? Fixed and Sunk Costs •Assuming no re-sale value, how much of the liquor license is sunk? •Now assuming a tradable license, should she have placed $6000 ad? •Should she accept the $66,000 offer? More Cost Definitions (SR) Average Total Cost ATC = AVC + AFC ATC = C(Q)/Q Average Variable Cost AVC = VC(Q)/Q $ MC ATC AVC Average Fixed Cost AFC = FC/Q Marginal Cost dC MC = DC/DQ = dQ AFC Q Costs in SR • Go to SR cost handout Fixed Cost Q0(ATC-AVC) $ = Q0 AFC = Q0(FC/ Q0) MC ATC AVC = FC ATC AFC Fixed Cost AVC Q0 Q Variable Cost $ Q0AVC = Q0[VC(Q0)/ Q0] = VC(Q0) MC ATC AVC AVC Variable Cost Q0 Q Total Cost Q0ATC $ = Q0[C(Q0)/ Q0] = C(Q0) MC ATC AVC ATC Total Cost Q0 Q Cubic Cost Function • C(Q) = f + a Q + b Q2 + cQ3 • Marginal Cost? 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 Another Example • • • • • • • C(Q) = 5000 + 20Q2 + 10Q Find AFC, AVC, ATC and MC when Q=10. C(10) = $7100. AFC = $500 per unit AVC=$210 per unit ATC = $710 per unit MC(Q) = 40Q + 10, thus MC(10)=$410 Inputs in the LR • Goal: Minimize the costs of producing a given level of output. Finding the optimal level of K and L in LR. • An isoquant shows 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. • Go to SR and LR production handout L Linear Isoquants • Capital and labor are perfect substitutes Constant slope (e.g., -3/4 always) • Marginal Rate of Technical Substitution (MRTS): the rate of substitution between 2 inputs while maintaining the same level of output • MRTS = MPL / MPK The negative of the slope of the isoquant. K Increasing Output Q1 Q2 Q3 L Leontief Isoquants • Capital and labor are perfect complements • Capital and labor are used in fixed-proportions Q3 K Q2 Q1 Increasing Output Cobb-Douglas Isoquants • Inputs are not perfectly substitutable • Diminishing MRTS K Q3 Q2 Q1 Increasing Output Nonlinear, nonconstant slope • Most production processes have isoquants of this shape L Isocost • The combinations of inputs that yield the same total cost to the producer C wL rK C w K L r r • The above equation is just in y=mx + b form. The slope is the ratio of input prices (w/r). The ratio of input prices is the market rate of substitution. Isocost • The combinations of inputs that cost the producer the same amount of money • 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 C0 C1 L K New Isocost Line for a decrease in the wage (price of labor). L Cost Minimization • The additional output (i.e., marginal product) per dollar spent should be equal for all inputs: MPL MPK w r • Expressed differently w MRTS KL r Cost Minimization K Slope of Isocost = Slope of Isoquant Point of Cost Minimization That is, MRTS = w/r Q=100 CLower CHigher L LR versus SR K C1 > C0, where C0 represents the costs of producing Q0 in LR. C1 C0 C1: Cost of producing Q0 in SR KLR KSR* Q0 L LLR LSR The Long-run and costminimization • The important thing to remember is that the LR offers flexibility—the ability to substitute away from more expensive inputs to lower priced inputs (e.g., see the chapter Headline, “Winning the Battle versus Winning the War”). Cost-minimization problem • A firm’s production process follows a Cobb-Douglas production process. At current production levels, an engineer has determined that the MPL is 60 units per hour and the MPK is 120 units per hour. The wage rate is $8 per hour and the opportunity cost of capital is $12 per hour. Is the firm producing its output at the lowest cost? If not, what changes should they make in the LR? Relationship between LR & SR Costs $ ATC ATC ATC ATC LRAC LRAC is “envelope” of SR ATC curves 10 30 42 Output (millions) LR Costs and Economies of Scale $ LRAC Economies of Scale Diseconomies of Scale Output Economies of Scale • When AC decline as Q increases Spreading overhead or fixed costs over a larger output A reason to explain mergers • Merger between HP and Compaq Merger or Selling off Subsidiary • Merger makes sense if: Economies of scope exist (different product type) Economies of scale exist (same product type) • Selling off an unprofitable subsidiary If economies of scope exist, then may not see a large reduction in total costs. Will see an increase in average costs.