© 2005 pearson education canada inc. 7.1 chapter 7 production and cost: many variable inputs

© 2005 Pearson Education Canada Inc.7.1

Chapter 7Chapter 7

Production and Cost: Many Production and Cost: Many Variable InputsVariable Inputs


Isoquants and Input SubstitutionIsoquants and Input Substitution

An isoquant is a curve composed of An isoquant is a curve composed of all bundles that produce some fixed all bundles that produce some fixed quantity of output.quantity of output.

An example: Y=(1200ZAn example: Y=(1200Z11ZZ22))1/21/2

Setting y =120 and simplifying gives Setting y =120 and simplifying gives 12=Z12=Z11ZZ22 (see Figure 7.1). (see Figure 7.1).


Figure 7.1 Isoquants for courier servicesFigure 7.1 Isoquants for courier services


Marginal Rate of Technical Marginal Rate of Technical Substitution (MRTS)Substitution (MRTS)

The MRTS measures the rate at The MRTS measures the rate at which one input can be substituted which one input can be substituted for the other, with output remaining for the other, with output remaining constant.constant.

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


Perfect Substitutes and Perfect Perfect Substitutes and Perfect ComplimentsCompliments

Inputs are perfect substitutes when Inputs are perfect substitutes when one output can always be substituted one output can always be substituted for the other on fixed terms and the for the other on fixed terms and the MRTS is constant.MRTS is constant.

With perfect compliments, With perfect compliments, substitution is impossible and the substitution is impossible and the MRTS cannot be defined for the MRTS cannot be defined for the bundle at the kink in the isoquant. bundle at the kink in the isoquant.


Figure 7.2 Some illustrative isoquantsFigure 7.2 Some illustrative isoquants


Diminishing Rate of Technical Diminishing Rate of Technical SubstitutionSubstitution

Most cases fall between perfect Most cases fall between perfect substitutes and perfect compliments. In substitutes and perfect compliments. In these cases, one input can be substituted these cases, one input can be substituted for the other but the MRTS is not constant.for the other but the MRTS is not constant.

In such cases, it becomes increasingly In such cases, it becomes increasingly difficult to substitute one input for the difficult to substitute one input for the other.other.

This means the MRTS diminishes moving This means the MRTS diminishes moving fro left to right along the isoquant.fro left to right along the isoquant.


Figure 7.3 The marginal rate of Figure 7.3 The marginal rate of technical substitution, technical substitution, MRTSMRTS


MRTS as a Ratio of Marginal ProductsMRTS as a Ratio of Marginal Products

When the quantity of input 1 is When the quantity of input 1 is decreased by decreased by ΔΔZZ11, the change in y is , the change in y is (approx) the marginal product of the (approx) the marginal product of the input times the change in the input times the change in the quantity of input 1.quantity of input 1.

Therefore: Therefore: ΔΔy =MPy =MP1 1 ΔΔyzyz11

Similarly: Similarly: ΔΔy =MPy =MP2 2 ΔΔyzyz22


MRTS as a Ratio of Marginal ProductsMRTS as a Ratio of Marginal Products

When ZWhen Z11 is very small, MRTS can is very small, MRTS can approximated by approximated by ΔΔzz22//ΔΔzz11

Solving for ZSolving for Z11 & Z & Z22 and substituting from and substituting from above yields MRTS = (above yields MRTS = (ΔΔy/MPy/MP22)()(ΔΔy/MPy/MP11))

Reducing gives MRTS = MPReducing gives MRTS = MP11/MP/MP22

Therefore MRTS is equal to the marginal Therefore MRTS is equal to the marginal product of input 1 divided by the product of input 1 divided by the marginal product of input 2.marginal product of input 2.


Returns to ScaleReturns to Scale

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

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

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


Figure 7.4 Constant returns to scaleFigure 7.4 Constant returns to scale


The Cost Minimization Problem: The Cost Minimization Problem: A PerspectiveA Perspective

The The cost functioncost function shows the minimum shows the minimum cost of producing any level of output in cost of producing any level of output in the long-run.the long-run.

The long-run cost minimizing problem is:The long-run cost minimizing problem is:

minimize wminimize w11zz11+w+w22+z+z22

choosing zchoosing z1 1 and zand z22

subject to constraint y=F(zsubject to constraint y=F(z11, z, z22))


Conditional Input Demand FunctionsConditional Input Demand Functions

The solution to the cost minimization The solution to the cost minimization problem gives the values of the problem gives the values of the endogenous variables (zendogenous variables (z11

* * & z& z22**) as a ) as a

function of the exogenous variables function of the exogenous variables (y, w(y, w11 and w and w22).).

Since zSince z11* * & z& z22

** are dependent upon are dependent upon the level of ythe level of y chosen, the input chosen, the input demand functions are described as demand functions are described as conditional demand functions.conditional demand functions.


The Long-run Cost FunctionThe Long-run Cost Function

Once we know the input demand Once we know the input demand functions, the long-run cost function functions, the long-run cost function is the sum of the input quantities and is the sum of the input quantities and their respective prices.their respective prices.

TC(y,wTC(y,w11,w,w22) = w) = w11zz11* * +w+w22zz22

**


Solving Cost Minimization ProblemsSolving Cost Minimization Problems

The The isocostisocost line shows all bundles of inputs line shows all bundles of inputs that cost the same. It can be expressed as: that cost the same. It can be expressed as: c=wc=w11zz11

+w+w22zz22.. The absolute value of the slope of the The absolute value of the slope of the

isocost line is wisocost line is w11/w/w22.. This slope says that wThis slope says that w11/w/w2 2 of input 2 must of input 2 must

be given up to get an additional unit of be given up to get an additional unit of input 1.input 1.

The slope is the opportunity cost of input 1 The slope is the opportunity cost of input 1 in terms of input 2.in terms of input 2.


Figure 7.5 The cost-minimizing bundleFigure 7.5 The cost-minimizing bundle


Principles of Cost MinimizationPrinciples of Cost Minimization

1.1. The cost minimizing input bundle is on The cost minimizing input bundle is on the isoquant: y the isoquant: y ΞΞ F( F(zz11

* * +z+z22**).).

2.2. The MRTS is equal to wThe MRTS is equal to w11/w/w22 at the cost at the cost minimizing bundle: minimizing bundle: MRTS(zMRTS(z11

**zz22**) ) ΞΞ w w11/w/w22

The second principle can be generalizedThe second principle can be generalized

by stating the marginal product per dollar by stating the marginal product per dollar

must be identical for all inputs.must be identical for all inputs.


Comparative Statics for Input PricesComparative Statics for Input Prices

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

1.1. The cost of minimizing the input The cost of minimizing the input bundle for y units of output does bundle for y units of output does not change.not change.

2.2. The minimum cost pf producing y The minimum cost pf producing y units of output changes by (a).units of output changes by (a).


Figure 7.7 Costs and input pricesFigure 7.7 Costs and input prices


From Figure 7.7From Figure 7.7

If the cost-minimizing quantity of both If the cost-minimizing quantity of both inputs (i and j) is positive and there is inputs (i and j) is positive and there is diminishing MRTS, if pdiminishing MRTS, if pii increases and p increases and pjj does not, the cost minimizing quantity of i does not, the cost minimizing quantity of i increases and j decreases.increases and j decreases.

If the price of an input increases and the If the price of an input increases and the quantity demanded of that input is positive, quantity demanded of that input is positive, the minimum cost of producing any level of the minimum cost of producing any level of output rises.output rises.


Comparative Statics: Level of OutputComparative Statics: Level of Output

The The expansion pathexpansion path connects the cost connects the cost minimizing bundles that are generated minimizing bundles that are generated as output increases.as output increases.

A A normal inputnormal input is one where the quantity is one where the quantity demanded increases when output rises.demanded increases when output rises.

An inferior input is one where the An inferior input is one where the quantity demanded decreases when quantity demanded decreases when output rises.output rises.


Figure 7.8 The output expansion pathFigure 7.8 The output expansion path


Homothetic Production FunctionsHomothetic Production Functions

A homothetic production function is a A homothetic production function is a type of function where the expansion type of function where the expansion path is a ray through the origin.path is a ray through the origin.

For these types of functions the For these types of functions the MRTS is constant along any ray from MRTS is constant along any ray from the origin.the origin.


Long-run Costs and OutputLong-run Costs and Output

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

LAC(y)=TC(y)/yLAC(y)=TC(y)/y

As output rises, LAC is constant, As output rises, LAC is constant, decreasing, or increasing as there decreasing, or increasing as there are constant, increasing, or are constant, increasing, or decreasing returns to scale. decreasing returns to scale.


Figure 7.9 Costs and returns to scaleFigure 7.9 Costs and returns to scale


Figure 7.10 More on costs and returns to scaleFigure 7.10 More on costs and returns to scale


Long-run Marginal CostLong-run Marginal Cost

Long-run marginal cost (LMC) is the Long-run marginal cost (LMC) is the rate at which costs increase as rate at which costs increase as output increases (the slope of TC).output increases (the slope of TC).

When LMC lies below LAC, LAC is When LMC lies below LAC, LAC is decreasing, when LMC exceeds LAC, decreasing, when LMC exceeds LAC, LAC is rising, LMC intersects LAC at LAC is rising, LMC intersects LAC at the LAC minimum.the LAC minimum.


Figure 7.11 Deriving LAC and LMC from TCFigure 7.11 Deriving LAC and LMC from TC


Figure 7.12 Comparing TC and STCFigure 7.12 Comparing TC and STC


Figure 7.13 Relationships between Figure 7.13 Relationships between long-run and short-run cost functionslong-run and short-run cost functions


Figure 7.14 A cost-based theory of market structureFigure 7.14 A cost-based theory of market structure


From Figure 7.14From Figure 7.14

U-shaped cost curves reflect initial U-shaped cost curves reflect initial increasing and subsequent decreasing increasing and subsequent decreasing returns to scale.returns to scale.

If LAC attains its minimum at a relatively If LAC attains its minimum at a relatively large level of output, we expect to see a large level of output, we expect to see a monopoly or oligopoly.monopoly or oligopoly.

If LAC attains its minimum at a relatively If LAC attains its minimum at a relatively small level of output, we expect to see a small level of output, we expect to see a competitive market.competitive market.

© 2005 pearson education canada inc. 7.1 chapter 7 production and cost: many variable inputs

Documents

pearson education canada

mrts slide

scale slide

gives mrts

input substitution

marginal product of

quantity of input

input times