production and cost

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

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

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

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

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.

Q

L

Q=F(K,L)

Increasing

Marginal

Returns

DiminishingMarginalReturns

NegativeMarginalReturns

MP

AP

Stages of Production

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

Linear IsoquantsLinear Isoquants

Capital and Capital and labor are labor are perfect perfect substitutessubstitutes

Q3Q2Q

1

Increasing Output

L

K

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

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

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).

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

Cost MinimizationCost Minimization

Q

L

K

Point of Cost Minimization

Slope of Isocost

= Slope of Isoquant

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

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

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 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

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

Economies of Scale

LRAC

$

Output

Economiesof Scale

Diseconomiesof Scale

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

Multi-Product Cost Multi-Product Cost FunctionFunction

C(QC(Q11, Q, Q22): Cost of producing ): Cost of producing

two outputs jointlytwo outputs jointly

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.

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.

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

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?