제 5 장 생산공정과 비용 The Production Process and Costs

개요 Overview I. Production Analysis n Total Product, Marginal Product, Average Product n Isoquants n Isocosts n Cost Minimization II. Cost Analysis n Total Cost, Variable Cost, Fixed Costs n Cubic Cost Function n Cost Relations III. Multi-Product Cost Functions

생산분석 Production Analysis Production Function n Q = F(K,L) n 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

총생산 Total Product Cobb-Douglas Production Function Example: Q = F(K,L) = K.5 L.5 n K is fixed at 16 units. n Short run production function: Q = (16).5 L.5 = 4 L.5 n Production when 100 units of labor are used? Q = 4 (100).5 = 4(10) = 40 units

한계생산성의 측정 Marginal Productivity Measures Marginal Product of Labor: MP L =  Q/  L n Measures the output produced by the last worker. n Slope of the short-run production function (with respect to labor). Marginal Product of Capital: MP K =  Q/  K n Measures the output produced by the last unit of capital. n When capital is allowed to vary in the short run, MP K is the slope of the production function (with respect to capital).

평군생산성의 측정 Average Productivity Measures Average Product of Labor n AP L = Q/L. n Measures the output of an “average” worker. n 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 n AP K = Q/K. n Measures the output of an “average” unit of capital. n 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.

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

생산공정에의 적용 Guiding the Production Process Producing on the production function n Aligning incentives to induce maximum worker effort. Employing the right level of inputs n 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.

等量線 Isoquant Curve 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.

한계기술대체율 Marginal Rate of Technical Substitution (MRTS) The rate at which two inputs are substituted while maintaining the same output level.

Linear Isoquants Capital and labor are perfect substitutes n Q = aK + bL n MRTS KL = b/a n Linear isoquants imply that inputs are substituted at a constant rate, independent of the input levels employed. Q3Q3 Q2Q2 Q1Q1 Increasing Output L K

Leontief Isoquants Capital and labor are 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 MRTS KL ). Q3Q3 Q2Q2 Q1Q1 K Increasing Output L

Cobb-Douglas Isoquants Inputs are not perfectly substitutable. Diminishing marginal rate of technical substitution. n 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 Q1Q1 Q2Q2 Q3Q3 K L Increasing Output

등비용선 Isocost Curve 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 L C1C1 L K New Isocost Line for a decrease in the wage (price of labor: w 0 > w 1 ). C1/rC1/r C1/wC1/w C0C0 C0/wC0/w C0/rC0/r C/w0C/w0 C/w1C/w1 C/rC/r New Isocost Line associated with higher costs (C 0 < C 1 ).

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

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

최적요소대체 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. n The isocost curve rotates counterclockwise; which represents the same cost level prior to the wage change. n To produce the same level of output, Q 0, the firm will produce on a lower isocost line (C 1 ) at a point B. n The slope of the new isocost line represents the lower wage relative to the rental rate of capital. Q0Q0 0 A L K C 0 /w 1 C 0 /w 0 C 1 /w 1 L0L0 L1L1 K0K0 K1K1 B

비용분석 Cost Analysis Types of Costs n Fixed costs (FC) n Variable costs (VC) n Total costs (TC) n Sunk costs

총비용과 가변비용 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 0

고정비용 및 매몰비용 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. $ Q FC C(Q) = VC + FC VC(Q)

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 =  C/  Q $ Q ATC AVC AFC MC MR

Fixed Cost $ Q ATC AVC MC ATC AVC Q0Q0 AFC Fixed Cost Q 0  (ATC-AVC) = Q 0  AFC = Q 0  (FC/ Q 0 ) = FC

Variable Cost $ Q ATC AVC MC AVC Variable Cost Q0Q0 Q 0  AVC = Q 0  [VC(Q 0 )/ Q 0 ] = VC(Q 0 )

$ Q ATC AVC MC ATC Total Cost Q0Q0 Q 0  ATC = Q 0  [C(Q 0 )/ Q 0 ] = C(Q 0 ) Total Cost

Cubic Cost Function C(Q) = f + a Q + b Q 2 + cQ 3 Marginal Cost? n Memorize: MC(Q) = a + 2bQ + 3cQ 2 n Calculus: dC/dQ = a + 2bQ + 3cQ 2

An Example n Total Cost: C(Q) = 10 + Q + Q 2 n Variable cost function: VC(Q) = Q + Q 2 n Variable cost of producing 2 units: VC(2) = 2 + (2) 2 = 6 n Fixed costs: FC = 10 n Marginal cost function: MC(Q) = 1 + 2Q n Marginal cost of producing 2 units: MC(2) = 1 + 2(2) = 5

규모의 경제 Economies of Scale LRAC $ Q Economies of Scale Diseconomies of Scale

다품종 생산 비용함수 Multi-Product Cost Function C(Q 1, Q 2 ): Cost of jointly producing two outputs. General function form:

범위의 경제 Economies of Scope C(Q 1, 0) + C(0, Q 2 ) > C(Q 1, Q 2 ). n It is cheaper to produce the two outputs jointly instead of separately. Example: n It is cheaper for Time-Warner to produce Internet connections and Instant Messaging services jointly than separately.

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: n Cow hides and steaks.

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

A Numerical Example: C(Q 1, Q 2 ) = Q 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 2

결론 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.