Eaton Micro 6e Ch07

7/27/2019 Eaton Micro 6e Ch07

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2005 Pearson Education Canada Inc.7.1

Chapter 7

Production and Cost: Many

Variable Inputs


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Figure 7.1 Isoquants for courier services


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Marginal Rate of Technical

Substitution (MRTS)

The MRTS measures the rate atwhich one input can be substitutedfor the other, with output remaining

constant.

The MRTS is the absolute value ofthe slope of the isoquant.


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Perfect Substitutes and Perfect

Compliments

Inputs are perfect substitutes whenone output can always be substitutedfor the other on fixed terms and the

MRTS is constant.

With perfect compliments,substitution is impossible and the

MRTS cannot be defined for thebundle at the kink in the isoquant.


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Figure 7.2 Some illustrative isoquants


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Diminishing Rate of Technical

Substitution

Most cases fall between perfect substitutesand perfect compliments. In these cases,one input can be substituted for the other

but the MRTS is not constant. In such cases, it becomes increasingly

difficult to substitute one input for theother.

This means the MRTS diminishes movingfro left to right along the isoquant.


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Figure 7.3 The marginal rate of

technical substitution, MRTS


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MRTS as a Ratio of Marginal Products

When the quantity of input 1 isdecreased by Z1, the change in y is(approx) the marginal product of the

input times the change in thequantity of input 1.

Therefore: y =MP1 yz1Similarly: y =MP2 yz2


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MRTS as a Ratio of Marginal Products

When Z1 is very small, MRTS canapproximated by z2/z1

Solving for Z1 & Z2 and substituting from

above yields MRTS = (y/MP2)(y/MP1)Reducing gives MRTS = MP1/MP2Therefore MRTS is equal to the marginal

product of input 1 divided by themarginal product of input 2.


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Returns to Scale

Increasing returns to scale occurswhen increasing all inputs by X%increases output by more than X%.

Constant returns to scale occurswhen an increase in all inputs of X%increases output by X%.

Decreasing returns to scale occurswhen an increasing all inputs by X%increases output by less than X%.


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Figure 7.4 Constant returns to scale


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The Cost Minimization Problem:

A Perspective

The cost function shows the minimumcost of producing any level of output inthe long-run.

The long-run cost minimizing problem is:minimize w1z1+w2+z2choosing z1 and z2

subject to constraint y=F(z1, z2)


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Conditional Input Demand Functions

The solution to the cost minimizationproblem gives the values of theendogenous variables (z1

* & z2*) as a

function of the exogenous variables(y, w1 and w2).

Since z1* & z2

* are dependent uponthe level of y

chosen, the input

demand functions are described asconditional demand functions.


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The Long-run Cost Function

Once we know the input demandfunctions, the long-run cost functionis the sum of the input quantities and

their respective prices.

TC(y,w1,w2) = w1z1* +w2z2

*


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Solving Cost Minimization Problems

The isocostline shows all bundles ofinputs that cost the same. It can beexpressed as: c=w1z1

+w2z2.

The absolute value of the slope of theisocost line is w1/w2.

This slope says that w1/w2 of input 2 mustbe given up to get an additional unit of

input 1. The slope is the opportunity cost of input

1 in terms of input 2.


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Figure 7.5 The cost-minimizing bundle


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Principles of Cost Minimization

1. The cost minimizing input bundle is onthe isoquant: y F(z1

* +z2*).

2. The MRTS is equal to w1/w2 at the cost

minimizing bundle:MRTS(z1

*z2*) w1/w2

The second principle can be generalized

by stating the marginal product per dollar

must be identical for all inputs.


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Comparative Statics for Input Prices

If all input prices change by thesame factor of proportionality (a):

1. The cost of minimizing the inputbundle for y units of output doesnot change.

2. The minimum cost pf producing yunits of output changes by (a).


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Figure 7.7 Costs and input prices


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From Figure 7.7

If the cost-minimizing quantity of bothinputs (i and j) is positive and there isdiminishing MRTS, if pi increases and pj

does not, the cost minimizing quantity of iincreases and j decreases.

If the price of an input increases and thequantity demanded of that input is positive,

the minimum cost of producing any level ofoutput rises.


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Comparative Statics: Level of Output

The expansion path connects the costminimizing bundles that are generatedas output increases.

A normal inputis one where the quantitydemanded increases when output rises.

An inferior input is one where thequantity demanded decreases whenoutput rises.


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Figure 7.8 The output expansion path


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Homothetic Production Functions

A homothetic production function is atype of function where the expansionpath is a ray through the origin.

For these types of functions theMRTS is constant along any ray fromthe origin.


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Long-run Costs and Output

Long-run average costs (LAC) isequal to the total cost of output (TC)divided by the quantity of output (y):

LAC(y)=TC(y)/y

As output rises, LAC is constant,

decreasing, or increasing as thereare constant, increasing, ordecreasing returns to scale.


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Figure 7.9 Costs and returns to scale


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Figure 7.10 More on costs and returns to scale


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Long-run Marginal Cost

Long-run marginal cost (LMC) is therate at which costs increase asoutput increases (the slope of TC).

When LMC lies below LAC, LAC isdecreasing, when LMC exceeds LAC,LAC is rising, LMC intersects LAC at

the LAC minimum.


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Figure 7.11 Deriving LAC and LMC from TC


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Figure 7.12 Comparing TC and STC


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Figure 7.13 Relationships between

long-run and short-run cost functions


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Figure 7.14 A cost-based theory of market structure


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2005 Pearson Education Canada Inc7 33

From Figure 7.14

U-shaped cost curves reflect initialincreasing and subsequent decreasingreturns to scale.

If LAC attains its minimum at a relativelylarge level of output, we expect to see amonopoly or oligopoly.

If LAC attains its minimum at a relativelysmall level of output, we expect to see acompetitive market.

Eaton Micro 6e Ch07

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