제 5 장
생산공정과 비용
The Production Process and Costs
개요 Overview
I. Production Analysis
nTotal Product, Marginal Product, Average Product
nIsoquants
nIsocosts
nCost Minimization
II. Cost Analysis
nTotal Cost, Variable Cost, Fixed Costs
nCubic Cost Function
nCost Relations
III. Multi-Product Cost Functions
생산분석Production Analysis
•Production Function
nQ = F(K,L)
nThe maximum amount of output that can beproduced with K units of capital and L units oflabor.
•Short-Run vs. Long-Run Decisions
•Fixed vs. Variable Inputs
총생산Total Product
•Cobb-Douglas Production Function
•Example: Q = F(K,L) = K.5 L.5
nK is fixed at 16 units.
nShort run production function:
Q = (16).5 L.5 = 4 L.5
nProduction when 100 units of labor are used?
Q = 4 (100).5 = 4(10) = 40 units
한계생산성의 측정Marginal Productivity Measures
•Marginal Product of Labor: MPL = Q/L
nMeasures the output produced by the last worker.
nSlope of the short-run production function (with respectto labor).
•Marginal Product of Capital: MPK = Q/K
nMeasures the output produced by the last unit ofcapital.
nWhen capital is allowed to vary in the short run, MPK isthe slope of the production function (with respect tocapital).
평군생산성의 측정Average Productivity Measures
•Average Product of Labor
nAPL = Q/L.
nMeasures the output of an “average” worker.
nExample: Q = F(K,L) = K.5 L.5
•If the inputs are K = 16 and L = 16, then the average product oflabor is APL = [(16) 0.5(16)0.5]/16 = 1.
•Average Product of Capital
nAPK = Q/K.
nMeasures the output of an “average” unit of capital.
nExample: Q = F(K,L) = K.5 L.5
•If the inputs are K = 16 and L = 16, then the average product oflabor is APL = [(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 NegativeMarginal Returns
생산공정에의 적용Guiding the Production Process
•Producing on the production function
nAligning incentives to induce maximum worker effort.
•Employing the right level of inputs
nWhen labor or capital vary in the short run, tomaximize profit a manager will hire
•labor until the value of marginal product of labor equalsthe wage: VMPL = w, where VMPL = P x MPL.
•capital until the value of marginal product of capitalequals the rental rate: VMPK = r, where VMPK = P xMPK .
等量線Isoquant Curve
•The combinations of inputs (K, L) thatyield the producer the same level of output.
•The shape of an isoquant reflects the easewith which a producer can substituteamong inputs while maintaining the samelevel of output.
한계기술대체율Marginal Rate of Technical Substitution (MRTS)
•The rate at which two inputs are substitutedwhile maintaining the same output level.
Linear Isoquants
•Capital and labor areperfect substitutes
nQ = aK + bL
nMRTSKL = b/a
nLinear isoquants imply thatinputs are substituted at aconstant rate, independentof the input levelsemployed.
Q3
Q2
Q1
IncreasingOutput
L
K
Leontief Isoquants
•Capital and labor are perfectcomplements.
•Capital and labor are used infixed-proportions.
•Q = min {bK, cL}
•Since capital and labor areconsumed in fixedproportions there is no inputsubstitution along isoquants(hence, no MRTSKL).
Q3
Q2
Q1
K
IncreasingOutput
L
Cobb-Douglas Isoquants
•Inputs are not perfectlysubstitutable.
•Diminishing marginal rateof technical substitution.
nAs less of one input is used inthe production process,increasingly more of the otherinput must be employed toproduce the same output level.
•Q = KaLb
•MRTSKL = MPL/MPK
Q1
Q2
Q3
K
L
IncreasingOutput
등비용선Isocost Curve
•The combinations of inputs thatproduce a given level of outputat the same cost:
wL + rK = C
•Rearranging,
K= (1/r)C - (w/r)L
•For given input prices, isocostsfarther from the origin areassociated with higher costs.
•Changes in input prices changethe slope of the isocost line.
K
L
C1
L
K
New Isocost Line fora decrease in thewage (price of labor:w0 > w1).
C1/r
C1/w
C0
C0/w
C0/r
C/w0
C/w1
C/r
New Isocost Lineassociated with highercosts (C0 < C1).
비용최소화Cost Minimization
•Marginal product per dollar spent should beequal for all inputs:
•But, this is just
Cost Minimization
Q
L
K
Point of CostMinimization
Slope of Isocost
=
Slope of Isoquant
최적요소대체Optimal Input Substitution
•A firm initially produces Q0by employing thecombination of inputsrepresented by point A at acost of C0.
•Suppose w0 falls to w1.
nThe isocost curve rotatescounterclockwise; whichrepresents the same cost levelprior to the wage change.
nTo produce the same level ofoutput, Q0, the firm willproduce on a lower isocost line(C1) at a point B.
nThe slope of the new isocostline represents the lower wagerelative to the rental rate ofcapital.
Q0
0
A
L
K
C0/w1
C0/w0
C1/w1
L0
L1
K0
K1
B
비용분석Cost Analysis
•Types of Costs
nFixed costs (FC)
nVariable costs (VC)
nTotal costs (TC)
nSunk costs
총비용과 가변비용
Total and Variable Costs
C(Q): Minimum total costof producing alternativelevels of output:
C(Q) = VC(Q) + FC
VC(Q): Costs that varywith output.
FC: Costs that do not varywith output.
$
Q
C(Q) = VC + FC
VC(Q)
FC
0
고정비용 및 매몰비용
Fixed and Sunk Costs
FC: Costs that do notchange as outputchanges.
Sunk Cost: A cost that isforever lost after it hasbeen 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
Q0
AFC
Fixed Cost
Q0(ATC-AVC)
= Q0 AFC
= Q0(FC/ Q0)
= FC
Variable Cost
$
Q
ATC
AVC
MC
AVC
Variable Cost
Q0
Q0AVC
= Q0[VC(Q0)/ Q0]
= VC(Q0)
$
Q
ATC
AVC
MC
ATC
Total Cost
Q0
Q0ATC
= Q0[C(Q0)/ Q0]
= C(Q0)
Total Cost
Cubic Cost Function
•C(Q) = f + a Q + b Q2 + cQ3
•Marginal Cost?
nMemorize:
MC(Q) = a + 2bQ + 3cQ2
nCalculus:
dC/dQ = a + 2bQ + 3cQ2
An Example
nTotal Cost: C(Q) = 10 + Q + Q2
nVariable cost function:
VC(Q) = Q + Q2
nVariable cost of producing 2 units:
VC(2) = 2 + (2)2 = 6
nFixed costs:
FC = 10
nMarginal cost function:
MC(Q) = 1 + 2Q
nMarginal 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 비용함수Multi-Product Cost Function
•C(Q1, Q2): Cost of jointly producing twooutputs.
•General function form:
범위의 경제Economies of Scope
•C(Q1, 0) + C(0, Q2) > C(Q1, Q2).
nIt is cheaper to produce the two outputs jointly insteadof separately.
•Example:
nIt is cheaper for Time-Warner to produce Internetconnections and Instant Messaging services jointly thanseparately.
Cost Complementarity
•The marginal cost of producing good 1declines as more of good two is produced:
MC1Q1,Q2) /Q2 < 0.
•Example:
nCow hides and steaks.
Quadratic Multi-Product CostFunction
•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
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
결론Conclusion
•To maximize profits (minimize costs) managersmust use inputs such that the value of marginal ofeach input reflects price the firm must pay toemploy the input.
•The optimal mix of inputs is achieved when theMRTSKL = (w/r).
•Cost functions are the foundation for helping todetermine profit-maximizing behavior in futurechapters.