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ECON 500
ECON 500Microeconomic Theory
Producer Theory
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ECON 500
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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ECON 500
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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Part I. Production Functions
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Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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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ECON 500
Part I. Production Functions
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 I. Production Functions
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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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ECON 500
Part II. Cost Functions
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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ECON 500
Part II. Cost Functions
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Part II. Cost Functions
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ECON 500
Part II. Cost Functions
Total Cost Function
Average Cost Function
Marginal Cost Function
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ECON 500
Part II. Cost Functions
CRS Production Function and its Costs
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ECON 500
Part II. Cost Functions
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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ECON 500
Part II. Cost Functions
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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ECON 500
Part II. Cost Functions
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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ECON 500
Part II. Cost Functions
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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ECON 500
Part II. Cost Functions
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ECON 500
Part II. Cost Functions
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Part II. Cost Functions
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ECON 500
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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ECON 500
Part III. Profit Maximization
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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ECON 500
Part III. Profit Maximization
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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Part III. Profit Maximization
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Part III. Profit Maximization
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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ECON 500
Part III. Profit Maximization
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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ECON 500
Part III. Profit Maximization
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Part III. Profit Maximization
Marginal Revenue and Elasticity
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Part III. Profit Maximization
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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ECON 500
Part III. Profit Maximization
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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ECON 500
Part III. Profit Maximization
CobbDouglas Example
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ECON 500
Part III. Profit Maximization
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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ECON 500
Part III. Profit Maximization
Cobb Douglas Example
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Part III. Profit Maximization
Input Demand
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Part III. Profit Maximization
Comparative Statics for Input DemandSingle Input
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Part III. Profit Maximization
Comparative Statics for Input DemandTwo Inputs
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Part III. Profit Maximization
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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.