© 2009 Pearson Education Canada 6/1 Chapter 6 Production and Cost: One Variable Input.

© 2009 Pearson Education Canada6/1

Chapter 6Chapter 6

Production and Cost: One Production and Cost: One Variable InputVariable Input


Production FunctionProduction Function

The The production functionproduction function identifies identifies the maximum quantity of good y that the maximum quantity of good y that can be produced from any input can be produced from any input bundle (bundle (zz11, z, z22).).

A production function is stated as: A production function is stated as: yy=F(=F(zz11, , zz22).).


Production FunctionsProduction Functions

In a In a fixed proportions production fixed proportions production functionfunction, the ratio in which the inputs , the ratio in which the inputs are used never varies.are used never varies.

In a In a variable proportion production variable proportion production functionfunction, the ratio of inputs can vary., the ratio of inputs can vary.


Figure 6.1 Finding a production functionFigure 6.1 Finding a production function


From Figure 6.1From Figure 6.1 The production function is:The production function is:

FF((zz11zz22)=(1200)=(1200zz11zz22))1/21/2

This is a This is a Cobb-Douglas production Cobb-Douglas production function.function. The general form is given The general form is given below where A, u and v are positive below where A, u and v are positive constants.constants.

vu zAzy 21


CostsCosts Opportunity costOpportunity cost is the value of the is the value of the

highest forsaken alternative.highest forsaken alternative. Sunk costsSunk costs are costs that, once are costs that, once

incurred, cannot be recovered. incurred, cannot be recovered. Avoidable costsAvoidable costs are costs that need are costs that need

not be incurred (can be avoided).not be incurred (can be avoided). Fixed costsFixed costs do not vary with output. do not vary with output. Variable costsVariable costs change with output. change with output.


Long-Run Cost MinimizationLong-Run Cost Minimization

The goal is to choose quantities of The goal is to choose quantities of inputs inputs zz11 and and zz22 that minimize total that minimize total costs, subject to being able to costs, subject to being able to produce y units of output.produce y units of output.

That is:That is:

1.1. Minimize Minimize ww11zz11+w+w22zz22 ((ww11,w,w22 are input are input prices).prices).

2.2. Choosing zChoosing z1 1 and and zz22 subject to the subject to the constraint constraint y=F(zy=F(z11, z, z22).).


Production: One Variable InputProduction: One Variable Input

Total production functionTotal production function TP ( TP (zz11) ) ((zz22 fixed at 105) fixed at 105) defined as:defined as:

TP (TP (zz11)=F()=F(zz11, 105), 105)

Marginal productMarginal product MP( MP(zz11) is the rate ) is the rate of output change when the variable of output change when the variable input changes (given fixed amounts input changes (given fixed amounts of all other inputs).of all other inputs).

MP (MP (zz11)=slope of TP ()=slope of TP (zz11) )


Figure 6.2 A total product function Figure 6.2 A total product function


Figure 6.3 From total product to marginal productFigure 6.3 From total product to marginal product


The Free-disposal AssumptionThe Free-disposal Assumption

Because a production function gives the Because a production function gives the maximum output from any input maximum output from any input combination, we assume that increased combination, we assume that increased amounts of inputs will not be used if they amounts of inputs will not be used if they negatively impact output.negatively impact output.

This is sometimes called the This is sometimes called the free-free-disposal assumptiondisposal assumption and combined with and combined with our definition of the production function, our definition of the production function, implies that marginal product cannot be implies that marginal product cannot be negative.negative.


The Free-disposal AssumptionThe Free-disposal Assumption

Given the free-disposal assumption, Given the free-disposal assumption, the marginal product of any input is the marginal product of any input is always greater than or equal to zero.always greater than or equal to zero.

Furthermore, for any input bundle, Furthermore, for any input bundle, the marginal product of at least one the marginal product of at least one input is positive.input is positive.


Diminishing Marginal ProductivityDiminishing Marginal Productivity

Reflects the assumption that at some point the Reflects the assumption that at some point the rate of increase in total output (rate of increase in total output (marginal marginal productproduct) will begin to decline.) will begin to decline.

Suppose the quantities of all inputs except one Suppose the quantities of all inputs except one - say, input1- are fixed. There is a quantity of - say, input1- are fixed. There is a quantity of input 1- say, input 1- say, z”z”1 1 - such that whenever z- such that whenever z1 1

exceeds z”exceeds z”11, the marginal product of input 1 , the marginal product of input 1

decreases as zdecreases as z1 1 increases.increases.


Figure 6.4 From total product to Figure 6.4 From total product to

marginal product: Another illustrationmarginal product: Another illustration


Average ProductAverage Product

Average productAverage product (AP) of the variable (AP) of the variable input equals total output divided by the input equals total output divided by the quantity of the variable input.quantity of the variable input.

APAP(z(z11)=TP(z)=TP(z11)/z)/z11


Figure 6.5 From total product toFigure 6.5 From total product toaverage productaverage product


Figure 6.6 Comparing the average and Figure 6.6 Comparing the average and marginal product functionsmarginal product functions


Marginal and Average ProductMarginal and Average Product

1.1. When MP exceeds AP, AP is increasing.When MP exceeds AP, AP is increasing.

2.2. When MP is less than AP, AP declines.When MP is less than AP, AP declines.

3.3. When MP=AP, AP is constant.When MP=AP, AP is constant.


Costs of Production: One Variable InputCosts of Production: One Variable Input

The cost-minimization problem is:The cost-minimization problem is:

Minimize Minimize ww11zz11 by choice of by choice of zz1.1.

Subject to constraint Subject to constraint yy=TP(z=TP(z11).). The variable cost function, VC(The variable cost function, VC(yy) is:) is:

VC(VC(yy)=the minimum variable cost of )=the minimum variable cost of producing producing yy units of output. units of output.


Figure 6.7 Deriving the variable cost functionFigure 6.7 Deriving the variable cost function


More CostsMore Costs

Average variable costAverage variable cost is variable is variable cost per unit of output. AV(cost per unit of output. AV(yy)=VC()=VC(yy)/)/yy

Short-run marginal costShort-run marginal cost is the rate is the rate at which costs increase in the short-at which costs increase in the short-run. SMC(run. SMC(yy)=slope of VC()=slope of VC(yy))


Figure 6.8 Deriving average variable Figure 6.8 Deriving average variable cost and short-run marginal costcost and short-run marginal cost


Short-run Marginal Costs and Short-run Marginal Costs and Average Variable CostsAverage Variable Costs

1.1. When SMC is below AVC, AVC When SMC is below AVC, AVC decreases as y increases.decreases as y increases.

2.2. When SMC is equal to AVC, AVC is When SMC is equal to AVC, AVC is constant (its slope is zero).constant (its slope is zero).

3.3. When SMC is above AVC, AVC When SMC is above AVC, AVC increases as increases as yy increases. increases.


Average Product and Average CostAverage Product and Average Cost

AVC (AVC (yy’)=’)=ww11/AP(/AP(zz11’)’)

The The average variable cost functionaverage variable cost function is the inverted image of the is the inverted image of the average average product functionproduct function. .


Marginal ProductMarginal Product and and Marginal CostMarginal Cost

SMC (SMC (yy’)=(’)=(ww11ΔΔzz11)/(MP()/(MP(zz’))’))

The The short-run marginal cost short-run marginal cost functionfunction is the inverted image of the is the inverted image of the marginal product function.marginal product function.


Figure 6.9 Comparing cost and product functionsFigure 6.9 Comparing cost and product functions


Figure 6.10 Seven cost functionsFigure 6.10 Seven cost functions


Figure 6.11 The costs of commutingFigure 6.11 The costs of commuting


Figure 6.12 Total commuting costsFigure 6.12 Total commuting costs


Figure 6.13 The allocation of commuters to routesFigure 6.13 The allocation of commuters to routes


Application: The Allocation of Application: The Allocation of Output Among Different PlantsOutput Among Different Plants

Let us reinterpret the lessons from traffic Let us reinterpret the lessons from traffic congestion to solve a problem related to congestion to solve a problem related to multi-plant firms. multi-plant firms.

The firm’s problem is to allocate output The firm’s problem is to allocate output to its two plants so as to minimize its to its two plants so as to minimize its total variable cost.total variable cost.



Think of Figure 6.13 as the variable and Think of Figure 6.13 as the variable and marginal cost functions associated with marginal cost functions associated with two plants (TCCtwo plants (TCC11, TCC, TCC2 2 and MCand MC11, MC, MC22) ) respectively.respectively.

The curve TCC in Figure 6.13a tells us The curve TCC in Figure 6.13a tells us that the firm can minimize the variable that the firm can minimize the variable costs of producing 5000 units by costs of producing 5000 units by allocating 2000 units to the first plant allocating 2000 units to the first plant and 3000 units to the second plant. and 3000 units to the second plant.



Using the logic from Figure 6.13b: Using the logic from Figure 6.13b: – Total costs can be reduced by allocating Total costs can be reduced by allocating

output from the high-marginal cost plant to output from the high-marginal cost plant to the low-cost marginal plant.the low-cost marginal plant.

– To minimize the total variable cost of To minimize the total variable cost of producing a given output in two or more producing a given output in two or more plants, a firm allocates output to the plants plants, a firm allocates output to the plants so that short-run marginal cost is the same so that short-run marginal cost is the same in all plants. in all plants.

© 2009 Pearson Education Canada 6/1 Chapter 6 Production and Cost: One Variable Input.

Documents

Transcript of © 2009 Pearson Education Canada 6/1 Chapter 6 Production and Cost: One Variable Input.