ECON 500 Producer Theory Final

8/13/2019 ECON 500 Producer Theory Final

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ECON 500Microeconomic Theory

Producer Theory


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Producer Theory

A theory of how firms coordinate the transformation of inputs into outputs

with a goal of maximizing profits in the meantime.

Part I Production Functions

Part II Cost FunctionsPart III Profit Maximization


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Part I. Production Functions

The principal activity of any firm is to turn inputs into outputs.In the theory of producer behavior the relationship between inputs and

outputs is formalized by a production function of the form

where qrepresents the firms output of a particular good during a period, k

represents the machine (that is, capital) usage during the period, l

represents hours of labor input, mrepresents raw materials used, and represents the possibility of other variables affecting the production

process.


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Most of our analysis will involve two inputs kand l, capital and laborrespectively

Where qis the maximum amount of a good that can be produced by using

alternative combinations of kand l.

Since it is possible to produce the same amount of a good using differentcombinations of capital and labor, it is important to understand their

respective contributions to the production process at various levels of

utilization.


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Diminishing Marginal Productivity:

We assume that the marginal physical product of an input depends on how

much of that input is used. We also postulate that inputs cannot be added

indefinitely to a production process without eventually exhibiting some

deterioration in its productivity. Mathematically:


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Diminishing Marginal Productivity

Changes in the marginal productivity of labor depend not only on how

labor input is growing, but also on changes in capital.

Therefore, we must also be concerned with MPl / kIn most cases, this cross partial derivative is positive indicating that the

marginal productivity of an input increasing in the other input.

Diminishing marginal productivity of a single input can be offset by the

increases in other inputs. The gloomy prediction of Malthus that ledeconomics to be called a dismal science is misplaced.


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Average Physical Productivity

APlis easily measured and is of much empirical significance. The

relationship between average and marginal physical productivity is also of

significance.

WhenAPlis at its maximum, it equalsMPl


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Isoquants

An isoquant shows those combinations of kand lthat can produce a given

level of output. Mathematically, an isoquant records the set of kand lthat

satisfies:


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Marginal Rate of Technical Substitution

The marginal rate of technical substitution (RTS) shows the rate at which

labor can be substituted for capital while holding output constant along an

isoquant.


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RTS and Marginal Productivities

The total differential of the production function is

Along and isoquant, we know that dq=0, therefore

Hence


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Diminishing RTS

Isoquants naturally have a negative slope but they are also as convex

curves. This means along any one of the isoquants, the RTS is diminishing

However, it is notpossible to derive a diminishing RTS from theassumption of diminishing marginal productivity alone since Marginal

Physical Product depends on the level of both inputs and cross

productivity effects are present.

To show that isoquants are convex, we would like to show thatdRTS/dl< 0. Since RTS = fl/fk, we have


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Diminishing RTS

Because fland fkare functions of both k and l, we must be careful in

taking the derivative of this expression

Using the fact that dk/dl = - fl/fkalong an isoquant and

Youngs theorem (fkl

= flk

), we have


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Diminishing RTS

Because we have assumed fk> 0, the denominator of this function is

positive. Hence the whole fraction will be negative if the numerator is

negative. Because flland fkkare both assumed to be negative, the

numerator definitely will be negative if fklis positive.

When we assume a diminishing RTS we are assuming that marginal

productivities diminish rapidly enough to compensate for anypossible

negative cross-productivity effects.


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Returns to Scale

If the production function is given by q = f (k,l)and if all inputs are

multiplied by the same positive constant t (where t > 1), then we classify

the returns to scale of the production function by


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Constant Returns to Scale

CRS production functions are homogenous of degree 1:

Derivatives of a CRS function are homogenous of degree 0:

And CRS functions are homothetic


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Elasticity of Substitution

If the RTS does not change for changes in k/l, we say that substitution is

easy because the ratio of the marginal productivities of the two inputs does

not change as the input mix changes.

Alternatively, if the RTS changes rapidly for small changes in k/l, we

would say that substitution is difficult because minor variations in the

input mix will have a substantial effect on the inputs relative

productivities.

A scale-free measure of this responsiveness is


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Part II. Cost Functions

Accounting Costs vs. Economics Costs

Accountants emphasize out-of-pocket expenses, historical costs,

depreciation, and other bookkeeping entries.

Economists on the other hand define cost by the size of the payment

necessary to keep the resource in its present employment, or alternatively,

by the remuneration that input would earn in its next best use.

Labor costs = hourly wage = wCapital costs = rental rate of capital = v

Cost of entrepreneurial services = forgone earnings


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In order to minimize costs, the firm should choose inputs such that the rateat which k can be traded for l in production to the rate at which they can

be traded in the marketplace.

Alternatively, the firm should chose inputs such that the marginal

productivity per dollar spent should be the same for all inputs.


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Total Cost Function

Average Cost Function

Marginal Cost Function


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CRS Production Function and its Costs


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Properties of Cost Functions

Non-decreasing in v, w, and q

Homogenous of degree 1 in input prices

Concave in input prices


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Contingent Demand for Inputs and Shephards Lemma

Shephards lemma claimsthat C/w =l

Suppose that the price of labor (w) were to increase slightly. Costs would

rise by approximately the amount of labor lthat the firm was currentlyhiring.

Along the pseudo cost function all inputs are heldconstant, so an

increase in the wage increases costs in direct proportion to the amount of

labor used.

Because the true cost function is tangent to the pseudo-function at the

current wage, its slope (that is, its partial derivative) also will show the

current amount of labor input demanded.


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Short run vs. Long run

In the short run, the firm is able to alter only its labor input while capital

input is fixed at some level k1.

With this formulation, payments to capital attain a fixed cost nature in the

short run and do not vary with the amount produced. Short run cost

function becomes:


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Short run vs. Long run

In the short run, to change output, firms are forced to use input

combinations that are non-optimal.

Unavailability of input substitution prevents the firm from finding theinput mix where RTS equals the ratio of input prices.

Hence, in the short run, costs are not necessarily minimized.


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Part III. Profit Maximization

A profit-maximizing firm chooses both its inputs and its outputs with thesole goal of achieving maximum economic profits. That is, the firm seeks

to make the difference between its total revenues and its total economic

costs as large as possible.

This holistic approach that treats the firm as a single decision-makingunit and sweeps away all the complicated behavioral issues about the

relationships (contractual or implicit) among input providers.

The profit maximizing assumption has a long history in economic

literature. It is plausible because firm owners may indeed seek to maketheir asset as valuable as possible and because competitive markets may

punish firms that do not maximize profits. The assumption also yields

interesting theoretical results that can explain actual firms decisions.


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Marginalism and Profit Maximization

Firms perform the conceptual experiment of adjusting those variables that

can be controlled until it is impossible to increase profits further. This

involves, say, looking at the incremental, or marginal, profit obtainable

from producing one more unit of output. As long as this incremental profitis positive, the extra output will be produced. When the incremental profit

of an activity becomes zero, the firm has pushed output far enough, and it

would not be profitable to go further.


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Firms choose the level of output that maximizes

The first order condition for a maximum is

In order to maximize economic profits, the firm should choose that output

for which marginal revenue is equal to marginal cost


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Marginal Revenue

If the firm can sell all it wishes without having any effect on market price,

the market price will indeed be the extra revenue obtained from selling

one more unit. Total revenue will be linear in output, marginal and

average revenue will be equal to price.

However, a firm may not always be able to sell all it wants at the

prevailing market price. If it faces a downward-sloping demand curve for

its product, the revenue obtained from selling one more unit will be less

than the price of that unit because, in order to get consumers to take theextra unit, the price of all other units must be lowered. Marginal revenue

be below price for any quantity q>1.


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Marginal Revenue

If price does not change as quantity increases dp/dq = 0, marginal revenue

will be equal to price. In this case we say that the firm is a price taker

because its output decisions do not influence the price it receives.

On the other hand, if price falls as quantity increases dp/dq < 0, marginal

revenue will be less than price.


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Marginal Revenue and Elasticity


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Price Marginal Cost Markup

Using the MR elasticity relationship and equating MR to MC

The more elastic demand becomes, the lower the markup over MC


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Supply Decision of a Price Taking Firm

If a firm is sufficiently small such that its output choice has no market on

the market price, the marginal revenue becomes equal to the market price.

The profit maximizing level of output q*can be found by

P =MR = MC

As long as at this level of output average variable cost is below the market

price, the firm would continue to operate in the short run.

The short run supply curve for a price taking firm is the positively sloped

segment of the firms short-run marginal cost above the point of minimum

average variable cost.


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CobbDouglas Example


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I - The supply curve is positively sloped - increases in P cause the firm to

produce more because it is willing to incur a higher marginal cost

II - The supply curve is shifted to the left by increases in the wage rate,

III - The supply curve is shifted outward by increases in capital input

IV - The rental rate of capital, v, is irrelevant to short-run supply decisions


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Cobb Douglas Example

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Input Demand

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Comparative Statics for Input DemandSingle Input

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Comparative Statics for Input DemandTwo Inputs

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Demand and Marginal Revenue for a Competitive Firm


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Demand and Marginal Revenue for a Competitive Firm

A competitive firm supplies only a small portion of the total output of all the firms in anindustry. Therefore, the firm takes the market price of the product as given, choosingits output on the assumption that the price will be unaffected by the output choice.In (a) the demand curve facing the firm is perfectly elastic,even though the market demand curve in (b) is downward sloping.

DEMAND CURVE FACED BY A COMPETITIVE FIRM


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The demand curve d facing an individual firm in a competitive market is

both its average revenue curve and its marginal revenue curve. Along thisdemand curve, marginal revenue, average revenue, and price are all equal.

Profit Maximization by a Competitive Firm

MC(q) = MR= P

Because each firm in a competitive industry sells only a small fractionof the entire industry output, how much output the firm decides to sellwill have no effect on the market price of the product.

Because it is a price taker, the demand curve d facing an individual competitivefirm is given by a horizontal line.

A perfectly competitive firm should choose its output so that marginal costequals price:

Choosing Output in the Short Run


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Short-Run Profit Maximization by a Competitive Firm

A COMPETITIVE FIRM

MAKING A POSITIVE

PROFIT

In the short run, thecompetitive firm maximizesits profit by choosing anoutput q* at which itsmarginal cost MC is equalto the price P(or marginalrevenue MR) of its product.

The profit of the firm ismeasured by the rectangle

ABCD.Any change in output,whether lower at q1 orhigher at q2, will lead tolower profit.

Output Rule: If a firm is producing anyoutput, it should produce at the level at whichmarginal revenue equals marginal cost.

Choosing Output in the Short Run


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The Competitive Firms Short-run


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The Competitive Firm s Short run

Supply Curve

THE SHORT-RUN SUPPLY

CURVE FOR A COMPETITIVE

FIRM

The firms supply curve is the portion of the marginal cost curve for which

marginal cost is greater than average variable cost.

In the short run, the firmchooses its output so thatmarginal cost MC is equal toprice as long as the firmcovers its average variablecost.

The short-run supply curve isgiven by the crosshatchedportion of the marginal costcurve.

The Firms Response to an Input Price Change


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THE RESPONSE OF A FIRM

TO A CHANGE IN INPUT

PRICE

When the marginal cost ofproduction for a firmincreases (from MC1 to MC2),

the level of output thatmaximizes profit falls (from q1to q2).

The Firm s Response to an Input Price Change

The Short-Run Market Supply Curve


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The Short Run Market Supply Curve

INDUSTRY SUPPLY IN THE

SHORT RUN

The short-run industrysupply curve is thesummation of the supplycurves of the individualfirms.

Because the third firm has

a lower average variablecost curve than the first twofirms, the market supplycurve S begins at price P1and follows the marginalcost curve of the third firmMC3 until price equals P2,

when there is a kink.For P2 and all prices aboveit, the industry quantitysupplied is the sum of thequantities supplied by eachof the three firms.

Elasticity of Market Supply

Es = (Q/Q)/(P/P)

Producer Surplus in the Short Run


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Producer Surplus in the Short Run

producer surplus Sum over all units produced by a firm ofdifferences between the market price of a good and the marginal cost of

production.

PRODUCER SURPLUS FOR

A FIRM

The producer surplus for a

firm is measured by theyellow area below the marketprice and above the marginalcost curve, between outputs 0and q*, the profit-maximizingoutput.

Alternatively, it is equal torectangleABCD because thesum of all marginal costs upto q* is equal to the variablecosts of producing q*.


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PRODUCER SURPLUS VERSUS PROFIT

PRODUCER SURPLUS FOR

A MARKET

Producer surplus = PS =R VC

Profit = =R VC FC

The producer surplus for amarket is the area below themarket price and above themarket supply curve, between0 and output Q*.

Choosing Output in the Long Run


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Long-Run Profit Maximization

OUTPUT CHOICE IN THE

LONG RUN

The firm maximizes its profitby choosing the output atwhich price equals long-run

marginal cost LMC.In the diagram, the firmincreases its profit from

ABCD to EFGD byincreasing its output in thelong run.

The long-run output of a profit-maximizing competitive firm is the point at which

long-run marginal cost equals the price.

Choosing Output in the Long Run

Long-Run Competitive Equilibrium


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o g u Co pet t e qu b u

ACCOUNTING PROFIT AND ECONOMIC PROFIT

=R wL rK

ZERO ECONOMIC PROFIT

zero economic profit A firm is earning a normal return on its investmenti.e., it is doing as well as it could by investing its money elsewhere.

ENTRY AND EXIT

In a market with entry and exit, a firm enters when it can earn a positive long-run profit and exits when it faces the prospect of a long-run loss.

Economic profit takes into account opportunity costs. One such opportunity costis the return to the firms owners if their capital were used elsewhere. Accountingprofit equals revenues R minus labor cost wL, which is positive. Economic profit, however, equals revenues R minus labor cost wL minus the capital cost, Rk.


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long-run competitive equilibrium All firms in an industry aremaximizing profit, no firm has an incentive to enter or exit, and price issuch that quantity supplied equals quantity demanded.

When a firm earns zero economic profit, it has no incentive to exit the industry.Likewise, other firms have no special incentive to enter.

A long-run competitive equilibrium occurs when three conditions hold:

1. All firms in the industry are maximizing profit.

2. No firm has an incentive either to enter or exit the industry because allfirms are earning zero economic profit.

3. The price of the product is such that the quantity supplied by theindustry is equal to the quantity demanded by consumers.


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LONG-RUN COMPETITIVE

EQUILIBRIUM

Initially the long-run equilibriumprice of a product is $40 per unit,shown in (b) as the intersection ofdemand curve D and supply curveS1.

In (a) we see that firms earn

positive profits because long-runaverage cost reaches a minimumof $30 (at q2).

Positive profit encourages entry ofnew firms and causes a shift tothe right in the supply curve to S2,as shown in (b).

The long-run equilibrium occurs ata price of $30, as shown in (a),where each firm earns zero profitand there is no incentive to enteror exit the industry.


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FIRMS HAVING IDENTICAL COSTS

To see why all the conditions for long-run equilibrium must hold,assume that all firms have identical costs.

Now consider what happens if too many firms enter the industry inresponse to an opportunity for profit. The industry supply curve will shiftfurther to the right, and price will fall.

Only when there is no incentive to exit or enter can a market be in long-run equilibrium.

FIRMS HAVING DIFFERENT COSTS

Now suppose that all firms in the industry do not have identical cost curves.Perhaps one firm has a patent that lets it produce at a lower average cost thanall the others. In that case, it is consistent with long-run equilibrium for that firm

to earn a greater accountingprofit and to enjoy a higher producer surplus thanother firms.

If the patent is profitable, other firms in the industry will pay to use it. Theincreased value of the patent thus represents an opportunity cost to the firmthat holds it. It could sell the rights to the patent rather than use it. If all firmsare equally efficient otherwise, the economicprofit of the firm falls to zero.


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THE OPPORTUNITY COST OF LAND

There are other instances in which firms earning positive accounting profit maybe earning zero economic profit.

Suppose, for example, that a clothing store happens to be located near a largeshopping center. The additional flow of customers can substantially increasethe stores accounting profit because the cost of the land is based on its

historical cost. When the opportunity cost of land is included, the profitability ofthe clothing store is no higher than that of its competitors.

Economic Rent

In competitive markets, in both the short and the long run, economic rent is

often positive even though profit is zero.

economic rent Amount that firms are willing to pay for an input less theminimum amount necessary to obtain it. A payment for the services of aneconomic resource which is not necessary as an incentive for its production

In the long run, in a competitive market, the producer surplus that a firm earnson the output that it sells consists of the economic rent that it enjoys from all its

scarce inputs.

Producer Surplus in the Long Run


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FIRMS EARN ZERO PROFIT IN LONG-RUN EQUILIBRIUM

In long-run equilibrium, all firms earn zero economic profit.

In (a), a baseball team in a moderate-sized city sells enough tickets so that price ($7) isequal to marginal and average cost.

In (b), the demand is greater, so a $10 price can be charged. The team increases salesto the point at which the average cost of production plus the average economic rent isequal to the ticket price.

When the opportunity cost associated with owning the franchise is taken into account,the team earns zero economic profit.

The Industrys Long-Run Supply Curve


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y g pp y

Constant-Cost Industry

constant-cost industry Industry whose long-run supply curve is horizontal.

In (b), the long-run supplycurve in a constant-costindustry is a horizontal line SL.

When demand increases,initially causing a price rise,

the firm initially increases itsoutput from q1 to q2, as shownin (a).

But the entry of new firmscauses a shift to the right in

industry supply.

Because input prices areunaffected by the increasedoutput of the industry, entryoccurs until the original price isobtained (at point B in (b)).

The long-run supply curve for a constant-cost industry is,

therefore, a horizontal line at a price that is equal to the

long-run minimum average cost of production.

LONG-RUN SUPPLY IN A

CONSTANT COST INDUSTRY

The Industrys Long-Run Supply Curve


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Increasing-Cost Industry

increasing-cost industry Industry whose long-run supply curve is upward sloping.

LONG-RUN SUPPLY IN AN

INCREASING COST INDUSTRY

In (b), the long-run supplycurve in an increasing-costindustry is an upward-sloping

curve SL.When demand increases,initially causing a price rise,

the firms increase their outputfrom q1 to q2 in (a).

In that case, the entry of new

firms causes a shift to the rightin supply from S1 to S2.

Because input prices increaseas a result, the new long-runequilibrium occurs at a higherprice than the initialequilibrium.

In an increasing-cost industry, the long-run

industry supply curve is upward sloping.

The Effects of a Tax


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An output tax raises the firms

marginal cost curve by the

amount of the tax.The firm will reduce its outputto the point at which themarginal cost plus the tax isequal to the price of theproduct.

EFFECT OF AN OUTPUT TAX

ON A COMPETITIVE FIRMS

OUTPUT


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EFFECT OF AN OUTPUT TAX

ON INDUSTRY OUTPUT

An output tax placed on allfirms in a competitive marketshifts the supply curve for the

industry upward by theamount of the tax.

This shift raises the marketprice of the product andlowers the total output of theindustry.

ECON 500 Producer Theory Final

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