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OverviewOverview
I. Production AnalysisI. Production AnalysisTotal Product, Marginal Total Product, Marginal
Product, Average ProductProduct, Average Product IsoquantsIsoquants IsocostsIsocostsCost MinimizationCost Minimization
II. Cost AnalysisII. Cost AnalysisTotal Cost, Variable Cost, Total Cost, Variable Cost,
Fixed CostsFixed CostsCubic Cost FunctionCubic Cost FunctionCost RelationsCost Relations
III. Multi-Product Cost III. Multi-Product Cost FunctionsFunctions
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Production AnalysisProduction Analysis
Production FunctionProduction FunctionQ = F(K,L)Q = F(K,L)The maximum amount of The maximum amount of
output that can be produced output that can be produced with K units of capital and L with K units of capital and L units of labor.units of labor.
Short-Run vs. Long-Run Short-Run vs. Long-Run DecisionsDecisions
Fixed vs. Variable InputsFixed vs. Variable Inputs
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Total ProductTotal Product
Cobb-Douglas Production Cobb-Douglas Production FunctionFunction
Example: Q = F(K,L) = KExample: Q = F(K,L) = K.5 .5 LL.5.5
K is fixed at 16 units. K is fixed at 16 units. Short run production function:Short run production function:
Q = (16)Q = (16).5 .5 LL.5 .5 = 4 L= 4 L.5.5
Production when 100 units of Production when 100 units of labor are used?labor are used?
Q = 4 (100)Q = 4 (100).5.5 = 4(10) = 40 = 4(10) = 40 unitsunits
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Marginal Product of Marginal Product of LaborLabor
MPMPLL = = Q/Q/LLMeasures the output produced Measures the output produced
by the last worker.by the last worker.Slope of the production Slope of the production
functionfunction
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Average Product of Average Product of LaborLabor
APAPLL = Q/L = Q/LThis is the accounting measure This is the accounting measure
of productivity. of productivity.
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Q
L
Q=F(K,L)
Increasing
Marginal
Returns
DiminishingMarginalReturns
NegativeMarginalReturns
MP
AP
Stages of Production
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IsoquantIsoquant
The combinations of inputs The combinations of inputs (K, L) that yield the producer (K, L) that yield the producer the same level of output.the same level of output.
The shape of an isoquant The shape of an isoquant reflects the ease with which reflects the ease with which a producer can substitute a producer can substitute among inputs while among inputs while maintaining the same level maintaining the same level of output.of output.
L
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Linear IsoquantsLinear Isoquants
Capital and Capital and labor are labor are perfect perfect substitutessubstitutes
Q3Q2Q
1
Increasing Output
L
K
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Leontief IsoquantsLeontief Isoquants
Capital and Capital and labor are perfect labor are perfect complementscomplements
Capital and Capital and labor are used labor are used in fixed-in fixed-proportionsproportions
Q
3Q
2Q
1
K
Increasing Output
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Cobb-Douglas Cobb-Douglas IsoquantsIsoquants
Inputs are not Inputs are not perfectly perfectly substitutablesubstitutable
Diminishing Diminishing marginal rate of marginal rate of technical technical substitution substitution
Most Most production production processes have processes have isoquants of isoquants of this shapethis shape
Q
1
Q
2
Q
3
K
L
Increasing Output
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IsocostIsocost The combinations The combinations
of inputs that cost of inputs that cost the producer the the producer the same amount of same amount of moneymoney
For given input For given input prices, isocosts prices, isocosts farther from the farther from the origin are origin are associated with associated with higher costs.higher costs.
Changes in input Changes in input prices change the prices change the slope of the slope of the isocost lineisocost line
K
LC1
C0
L
KNew Isocost Line for a decrease in the wage (price of labor).
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Cost MinimizationCost Minimization
Marginal product per dollar Marginal product per dollar spent should be equal for all spent should be equal for all inputs: inputs:
Expressed differentlyExpressed differentlyr
MP
w
MP KL
r
wMRTSKL
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Cost MinimizationCost Minimization
Q
L
K
Point of Cost Minimization
Slope of Isocost
= Slope of Isoquant
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Cost AnalysisCost Analysis
Types of CostsTypes of CostsFixed costs Fixed costs
(FC)(FC)Variable costs Variable costs
(VC)(VC)Total costs Total costs
(TC)(TC)Sunk costsSunk costs
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Total and Variable CostsC(Q): Minimum total cost of producing alternative levels of output:
C(Q) = VC + FC
VC(Q): Costs that vary with output
FC: Costs that do not vary with output
$
Q
C(Q) = VC + FC
VC(Q)
FC
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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)
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Some DefinitionsAverage Total Cost
ATC = AVC + AFCATC = C(Q)/Q
Average Variable Cost
AVC = VC(Q)/Q
Average Fixed Cost
AFC = FC/Q
Marginal CostMC = C/Q
$
Q
ATC
AVC
AFC
MC
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Fixed Cost
$
Q
ATC
AVC
MC
ATC
AVC
Q0
AFC Fixed Cost
Q0(ATC-AVC)
= Q0 AFC
= Q0(FC/ Q0)
= FC
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Variable Cost
$
Q
ATC
AVC
MC
AVC
Variable Cost
Q0
Q0AVC
= Q0[VC(Q0)/ Q0]
= VC(Q0)
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$
Q
ATC
AVC
MC
ATC
Total Cost
Q0
Q0ATC
= Q0[C(Q0)/ Q0]
= C(Q0)
Total Cost
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Economies of Scale
LRAC
$
Output
Economiesof Scale
Diseconomiesof Scale
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An ExampleAn Example
Total Cost: C(Q) = 10 + Q + QTotal Cost: C(Q) = 10 + Q + Q22
Variable cost function:Variable cost function:VC(Q) = Q + QVC(Q) = Q + Q22
Variable cost of producing 2 Variable cost of producing 2 units:units:
VC(2) = 2 + (2)VC(2) = 2 + (2)22 = 6 = 6
Fixed costs:Fixed costs:FC = 10FC = 10
Marginal cost function:Marginal cost function:MC(Q) = 1 + 2QMC(Q) = 1 + 2Q
Marginal cost of producing 2 Marginal cost of producing 2 units:units:
MC(2) = 1 + 2(2) = 5MC(2) = 1 + 2(2) = 5
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Multi-Product Cost Multi-Product Cost FunctionFunction
C(QC(Q11, Q, Q22): Cost of producing ): Cost of producing
two outputs jointlytwo outputs jointly
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Economies of ScopeEconomies of Scope
C(QC(Q11, Q, Q22) < C(Q) < C(Q11, 0) + C(0, Q, 0) + C(0, Q22)) It is cheaper to produce the It is cheaper to produce the
two outputs jointly instead of two outputs jointly instead of separately.separately.
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Cost ComplementarityCost Complementarity
The marginal cost of producing The marginal cost of producing good 1 declines as more of good 1 declines as more of good two is produced:good two is produced:
MCMC11//QQ22 < 0. < 0.
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Quadratic Multi-Product Quadratic Multi-Product Cost FunctionCost Function
C(QC(Q11, Q, Q22) = f + aQ) = f + aQ11QQ2 2 + (Q+ (Q1 1 ))22 + +
(Q(Q2 2 ))22
MCMC11(Q(Q11, Q, Q22) = aQ) = aQ2 2 + 2Q+ 2Q11
MCMC22(Q(Q11, Q, Q22) = aQ) = aQ1 1 + 2Q+ 2Q22
Cost complementarity: a < 0Cost complementarity: a < 0 Economies of scope: f > Economies of scope: f >
aQaQ11QQ22
C(QC(Q11 ,0) + C(0, Q ,0) + C(0, Q2 2 ) = f + (Q) = f + (Q1 1 ))22 + + ff
+ (Q+ (Q22))22
C(QC(Q11, Q, Q22) = f + ) = f + aQaQ11QQ22 + (Q+ (Q1 1 ))22 + +
(Q(Q2 2 ))22
ff > > aQaQ11QQ22: Joint production is : Joint production is
cheapercheaper
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A Numerical Example:A Numerical Example:
C(QC(Q11, Q, Q22) = 90 - 2Q) = 90 - 2Q11QQ2 2 + (Q+ (Q1 1 ))22
+ (Q+ (Q2 2 ))22 Cost Complementarity?Cost Complementarity?
Yes, since a = -2 < 0Yes, since a = -2 < 0
MCMC11(Q(Q11, Q, Q22) = -2Q) = -2Q2 2 + +
2Q2Q11
Economies of Scope?Economies of Scope?
Yes, since 90 > -Yes, since 90 > -2Q2Q11QQ22
Implications for Merger?Implications for Merger?