Eaton Micro 6e Ch06

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2005 Pearson Education Canada Inc.6.1

Chapter 6

Production and Cost: One

Variable Input


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Production Function

The production function identifies themaximum quantityof good y thatcan be produced from any input

bundle (z1, z2).

Production function is stated as:y=F(z1, z2).


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Production Functions

In a fixed proportions productionfunction, the ratio in which the

inputs are used never varies. In a variable proportion production

function, the ratio of inputs can vary.


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Figure 6.1 Finding a production function


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From Figure 6.1

The production function is:

F(z1z2)=(1200z1z2)1/2

This is a Cobb-Douglas production

function. The general form is givenbelow where A, u and v are positiveconstants.

vu

zAzy21


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Costs

Opportunity cost is the value of thehighest forsaken alternative.

Sunk costs are costs that, once

incurred, cannot be recovered.

Avoidable costs are costs that neednot be incurred (can be avoided).

Fixed costs do not vary with output.

Variable costs change with output.


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Long-Run Cost Minimization

The goal is to choose quantities ofinputs z1 and z2 that minimize totalcosts subject to being able to

produce y units of output. That is:

1. Minimize w1z1+w2z2 (w1,w2 are

input prices).2. Choosing z1 and z2 subject to the

constraint y=F(z1, z2).


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Production: One Variable Input

Total production function TP (z1)(Z2 fixed at 105) defined as:

TP (z1)=F(z

1, 105)

Marginal product MP(z1)the rate ofoutput change when the variableinput changes (given fixed amountsof all other inputs).

MP (z1)=slope of TP (z1)


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Figure 6.3 From total product to marginal product


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Diminishing Marginal Productivity

As the quantity of the variable inputis increased (all other inputquantities being fixed), at some point

the rate of increase in total outputwill begin to decline.


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Figure 6.4 From total product to

marginal product: another illustration


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Average Product

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

AP(Z1)=TP(Z1)/Z1


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Figure 6.5 From total product to

average product


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Figure 6.6 Comparing the average and

marginal product functions


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Marginal and Average Product

1. When MP exceeds AP, AP is increasing.

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

3. When MP=AP, AP is constant.


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Costs of Production: One Variable Input

The cost-minimization problem is:

Minimize W1Z1 by choice of Z1.

Subject to constraint y=TP(z1).The variable cost, VC(y) function is:

VC(y)=the minimum variable cost of

producing y units of output.


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Figure 6.7 Deriving the variable cost function


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More Costs

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

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


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Figure 6.8 Deriving average variable

cost and short-run marginal cost


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Short-run Marginal Costs and

Average Variable Costs

1. When SMC is below AVC, AVCdecreases as y increases.

2. When SMC is equal to AVC, AVC isconstant (its slope is zero).

3. When SMC is above AVC, AVC

increases as y increases.


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Average Product and Average Cost

AVC (y)=w1/AP(z1)

The average variable cost function isthe inverted image of the averageproduct function.


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Marginal Product and Marginal Cost

SMC (y)=(w1z1)/(MP(z))

The short-run marginal cost functionis the inverted image of the marginalproduct function.


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Figure 6.9 Comparing cost and product functions


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Figure 6.10 Seven cost functions


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Figure 6.11 The costs of commuting


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Figure 6.12 Total commuting costs


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Figure 6.13 The allocation of commuters to routes

Eaton Micro 6e Ch06

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