chapter 6 production and cost: one variable input

© 2005 Pearson Education Canada Inc.6.1

Chapter 6Chapter 6

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


Production FunctionProduction Function

The production function identifies the The production function identifies the maximum quantitymaximum quantity of good y that of good y that can be produced from any input can be produced from any input bundle (zbundle (z11, z, z22).).

Production function is stated as: Production function is stated as: y=F(zy=F(z11, z, z22).).


Production FunctionsProduction Functions

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

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

F(zF(z11zz22)=(1200z)=(1200z11zz22))1/21/2

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

vu zAzy21


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

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

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

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


Long-Run Cost Minimization Long-Run Cost Minimization

The goal is to choose quantities of The goal is to choose quantities of inputs zinputs z11 and z and z22 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 wMinimize w11zz11+w+w22zz2 2 (w(w11,w,w2 2 are input are input prices).prices).

2.2. Choosing zChoosing z1 1 and zand z2 2 subject to the subject to the constraint y=F(zconstraint y=F(z11, z, z22).).


Production: One Variable InputProduction: One Variable Input

Total production function TP (zTotal production function TP (z11) (Z) (Z22 fixed at 105)fixed at 105) defined as:defined as:

TP (zTP (z11)=F(z)=F(z11, 105), 105)

Marginal product MP(zMarginal product MP(z11)the rate of )the rate of output change when the variable output change when the variable input changes (given fixed amounts input changes (given fixed amounts of all other inputs).of all other inputs).

MP (zMP (z11)=slope of TP (z)=slope of TP (z11) )


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


Diminishing Marginal ProductivityDiminishing Marginal Productivity

As the quantity of the variable input As the quantity of the variable input is increased (all other input is increased (all other input quantities being fixed), at some point quantities being fixed), at some point the rate of increase in total output the rate of increase in total output will begin to decline.will begin to decline.


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

AP(ZAP(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 WMinimize W11ZZ11 by choice of Z by choice of Z1.1.

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

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


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


More CostsMore Costs

Average variable cost is variable cost Average variable cost is variable cost per unit of output. AV(y)=VC(y)/yper unit of output. AV(y)=VC(y)/y

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


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 y increases. increases as y increases.


Average Product and Average CostAverage Product and Average Cost

AVC (y’)=wAVC (y’)=w11/AP(z/AP(z11’)’)

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


Marginal Product and Marginal CostMarginal Product and Marginal Cost

SMC (y’)=(wSMC (y’)=(w11ΔΔzz11)/(MP(z’)))/(MP(z’))

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

chapter 6 production and cost: one variable input

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