chapter 13 profit maximization and supply copyright ©2002 by south-western, a division of thomson...

Chapter 13

PROFIT MAXIMIZATION

AND SUPPLY

Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONS

EIGHTH EDITION

WALTER NICHOLSON

The Nature of Firms

• A firm is an association of individuals who have organized themselves for the purpose of turning inputs into outputs

• Different individuals will provide different types of inputs– the nature of the contractual relationship

between the providers of inputs to a firm may be quite complicated

Contractual Relationships

• Some contracts between providers of inputs may be explicit– may specify hours, work details, or

compensation

• Other arrangements will be more implicit in nature– decision-making authority or sharing of

tasks

Modeling Firms’ Behavior

• Most economists treat the firm as a single decision-making unit– the decisions are made by a single

dictatorial manager who rationally pursues some goal

• profit-maximization

Profit Maximization

• A profit-maximizing firm chooses both its inputs and its outputs with the sole goal of achieving maximum economic profits– seeks to maximize the difference between

total revenue and total economic costs

Output Choice

• Total revenue for a firm is given by

TR(q) = P(q)q

• In the production of q, certain economic costs are incurred [TC(q)]

• Economic profits () are the difference between total revenue and total costs

= TR(q) – TC(q) = P(q)q – TC(q)

Output Choice• The necessary condition for choosing

the level of q that maximizes profits can be found by setting the derivative of the function with respect to q equal to zero

0

dq

dTC

dq

dTRq

dq

d)('

dq

dTC

dq

dTR

Output Choice

• To maximize economic profits, the firm should choose the output for which marginal revenue is equal to marginal cost

MCdq

dTC

dq

dTRMR

Second-Order Conditions• MR = MC is only a necessary condition

for profit maximization

• For sufficiency, it is also required that

02

2

**

)('

qqqqdq

qd

dq

d

• “marginal” profit must be decreasing at the optimal level of q

Profit Maximization

output

revenues & costs

TRTC

q*

Profits are maximized when the slope ofthe revenue function is equal to the slope of the cost function

But the second-ordercondition prevents usfrom mistaking q0 asa maximum

q0

Marginal Revenue

• If a firm can sell all it wishes without having any effect on market price, marginal revenue will be equal to price

• If a firm faces a downward-sloping demand curve, more output can only be sold if the firm reduces the good’s price

dq

dPqP

dq

qqPd

dq

dTRqMR

])([)( revenue marginal

Marginal Revenue

• If a firm faces a downward-sloping demand curve, marginal revenue will be a function of output

• If price falls as a firm increases output, marginal revenue will be less than price

Marginal Revenue• Suppose that the demand curve for a sub

sandwich isq = 100 – 10P

• Solving for price, we getP = -q/10 + 10

• This means that total revenue isTR = Pq = -q2/10 + 10q

• Marginal revenue will be given byMR = dTR/dq = -q/5 + 10

Profit Maximization

• To determine the profit-maximizing output, we must know the firm’s costs

• If subs can be produced at a constant average and marginal cost of $4, then

MR = MC

-q/5 + 10 = 4

q = 30

Marginal Revenue and Elasticity

• The concept of marginal revenue is directly related to the elasticity of demand facing the firm

• The price elasticity of demand is equal to the percentage change in quantity that results from a one percent change in price

q

P

dP

dq

PdP

qdqe Pq

/

/,


• This means that

PqeP

dq

dP

P

qP

dq

dPqPMR

,

111

– if the demand curve slopes downward, eq,P < 0 and MR < P

– if the demand is elastic, eq,P < -1 and marginal revenue will be positive

• if the demand is infinitely elastic, eq,P = - and marginal revenue will equal price


eq,P < -1 MR > 0

eq,P = -1 MR = 0

eq,P > -1 MR < 0

Average Revenue Curve

• If we assume that the firm must sell all its output at one price, we can think of the demand curve facing the firm as its average revenue curve– shows the revenue per unit yielded by

alternative output choices

Marginal Revenue Curve

• The marginal revenue curve shows the extra revenue provided by the last unit sold

• In the case of a downward-sloping demand curve, the marginal revenue curve will lie below the demand curve


output

price

D (average revenue)

MR

q1

P1

As output increases from 0 to q1, totalrevenue increases so MR > 0

As output increases beyond q1, totalrevenue decreases so MR < 0


• When the demand curve shifts, its associated marginal revenue curve shifts as well– a marginal revenue curve cannot be

calculated without referring to a specific demand curve

Short-Run Supply by a Price-Taking Firm

output

price SMC

SATC

SAVC

P* = MR

q*

Maximum profitoccurs whereP = SMC


output

price SMC

SATC

SAVC

P* = MR

q*

Since P > SATC,profit > 0


output

price SMC

SATC

SAVC

P* = MR

q*

If the price risesto P**, the firmwill produce q**and > 0

q**

P**


output

price SMC

SATC

SAVC

P* = MR

q*

If the price falls to P***, the firm will produce q***

q***

P***profit maximizationrequires that P = SMC and that SMCis upward-sloping

< 0


• The positively-sloped portion of the short-run marginal cost curve is the short-run supply curve for a price-taking firm– it shows how much the firm will produce at

every possible market price– firms will only operate in the short run as

long as total revenue covers variable cost• the firm will produce no output if P < SAVC


• Thus, the price-taking firm’s short-run supply curve is the positively-sloped portion of the firm’s short-run marginal cost curve above the point of minimum average variable cost– for prices below this level, the firm’s profit-

maximizing decision is to shut down and produce no output


output

price SMC

SATC

SAVC

The firm’s short-run supply curve is that portion of the SMC curve that is above minimum SAVC

Short-Run Supply

• Suppose that the firm’s short-run total cost curve is

STC = 4v + wq2/400

• If w = v = $4, then the cost curve becomes

STC = 16 + q2/100

• This implies that short-run marginal cost isSTC/q = 2q/100 = q/50

Short-Run Supply

• Profit maximization requires that price is equal to marginal cost

P = SMC = q/50

• This means that the supply curve (with q as a function of P) is

q = 50P

Short-Run Supply

• To find the firm’s shut-down price, we need to solve for SAVC

SVC = q2/100

SAVC = SVC/q = q/100

• SAVC is minimized when q = 0– the firm will only shut down when the price

falls to 0

Short-Run Supply

• Short-run average costs are given bySATC = STC/q = 16/q + q/100

• SATC is minimized whenSATC/q = -16/q2 + 1/100 = 0

q = 40

SATC = SMC = $0.80

• For any price below $0.80, the firm will incur a loss

Profit Maximization and Input Demand

• A firm’s output is determined by the amount of inputs it chooses to employ– the relationship between inputs and

outputs is summarized by the production function

• A firm’s economic profit can also be expressed as a function of inputs

(K,L) = Pq – TC(q) = Pf(K,L) – (vK + wL)


• The first-order conditions for a maximum are

/K = P[f/K] – v = 0

/L = P[f/L] – w = 0

• A profit-maximizing firm should hire any input up to the point at which its marginal contribution to revenues is equal to the marginal cost of hiring the input


• These first-order conditions for profit maximization also imply cost minimization– they imply that RTS = w/v


• To ensure a true maximum, second-order conditions require that

KK < 0

LL < 0

KK LL - KL2 > 0

– Capital and labor must exhibit sufficiently diminishing marginal productivities so that marginal costs rise as output expands


• The first-order conditions can generally be solved for the optimal input combination

K* = K*(P,v,w)

L* = L*(P,v,w)

• These input choices can be substituted into the production function to get q*

q* = f(K,L) = f [K*(P,v,w),L*(P,v,w)] = q*(P,v,w)

Supply Function

• The supply function for a profit-maximizing firm that takes both output price (P) and input prices (v,w) as fixed is written as

quantity supplied = q*(P,v,w)– this indicates the dependence of output

choices on these prices

Supply Function

• The supply function provides a convenient reminder of two key points– the firm’s output decision is fundamentally

a decision about hiring inputs– changes in input costs will alter the hiring

of inputs and hence affect output choices as well

Producer Surplus in the Short Run

• A profit-maximizing firm that decides to produce a positive output in the short run must find that decision to be more favorable than a decision to produce nothing

• This improvement in welfare is termed (short-run) producer surplus– what the firm gains by being able to participate

in market transactions


• If the firm was prevented from making such transactions, output would be zero and profits would equal -SFC

• Production of the profit-maximizing output would yield profits of *

• The firm gains *+ SFC– this is producer surplus

Producer surplus is theshaded area below P*and above SMC


output

price SMC

P*

q*

If the market priceis P*, the firm will produce q*


• In mathematical terms, producer surplus is given by

* *)*()](*[ surplus producer

q qq

qTCqPdqqMCP

0 0

)](*[*)(** surplus producer 00 TCPqTCqP

SFC * surplus producer


• Because SFC is constant, changes in producer surplus as a result of changes in market price are reflected as changes in short-run profits– these can be measured by the changes in

the area below market price above the short-run supply curve

Producer Surplus in the Long Run

• By definition, long-run producer surplus is zero– fixed costs do not exist in the long run– equilibrium profits under perfect competition

with free entry are zero

• In long-run analysis, attention is focused on the prices of the firm’s inputs and how they relate to what they would be in the absence of market transactions

Revenue Maximization

• When firms are uncertain about the demand curve they face or when they have no reliable notion of the marginal costs of their output, the decision to maximize revenues may be a reasonable rule of thumb for ensuring their long-term survival


• A revenue-maximizing firm would choose to produce that level of output for which marginal revenue is zero

• Because we know that MR = P[1+(1/eq,P)], MR=0 implies that eq,P = -1

– demand will be unit elastic at q*


output

price

d

MR

P*

q*

If the firm wishes to maximize revenues, it will produce q*


output

price

P*

q*

d

MR

SMCIf the firm wishes to maximize profits, it will produce q**

q**


output

price

P*

q*

Increasing output beyond q** increases revenue but lowers economic profit

d

MR

SMC

q**

Constrained Revenue Maximization

• A firm that chooses to maximize revenue is paying no attention to its costs– it is possible that maximizing revenues

could result in negative profits for the firm

• It may be more realistic to assume that these firms must meet some minimal level of profitability


• Suppose that a firm faces the following demand curve

q = 100 - 10P

• Total revenues (as a function of q) is

TR = Pq = 10q - q2/10

• Marginal revenue is

MR = dTR/dq = 10 - q/5


• Total revenues are maximized when MR = 0– this means that q = 50

• If output is 50, total revenues are $250

• If we assume that AC = MC = $4, total costs are $200 and profits equal $50


• Suppose that the firm’s owners require a profit of at least $80

• Then the firm might seek to maximize revenue subject to the constraint that

= TR - TC = 10q - q2/10 - 4q = 80


• Rearranging the constraint, we get

q2 - 60q +800 = 0 or

(q - 40)(q - 20)=0

• The solution q = 40 yields higher revenues than any other output level between 20 and 40– all of these options yield at least $80 profit

The Principal-Agent Problem

• In many cases, firm managers do not actually own the firm but instead act as agents for the owners

• An agent is a person who makes economic decisions for another party


• Assume that we can show a graph of the owner’s (or manager’s) preferences in terms of profits and various benefits (such as fancy offices or use of the corporate jet)

• The owner’s budget constraint will have a slope of -1– each $1 of benefits reduces profit by $1


Benefits

Profits

Owner’s constraint

U1

B*

*

If the manager is also the ownerof the firm, he will maximize hisutility at profits of * and benefitsof B*


Benefits

Profits


U1

B*

*

The owner-manager maximizes profit because any other owner-manager will also want B* inbenefits

B* represents a true costof doing business


• Suppose that the manager is not the sole owner of the firm– suppose there are two other owners who

play no role in operating the firm

• $1 in benefits only costs the manager $0.33 in profits– the other $0.67 is effectively paid by the

other owners in terms of reduced profits


• The new budget constraint continues to include the point B*, *– the manager could still make the same

decision that a sole owner could)

• For benefits greater than B*, the slope of the budget constraint is only -1/3


Benefits

Profits


U1

B*

*

U2

Given the manager’s budgetconstraint, he will maximize utilityat benefits of B**

**

B**

Agent’s constraint

***

Profits for the firmwill be ***


• The firm’s owners are harmed by having to rely on an agency relationship with the firm’s manager

• The smaller the fraction of the firm that is owned by the manager, the greater the distortions that will be induced by this relationship


• The firm’s owners will not be happy about accepting lower profits on their investments– they may refuse to invest in the firm if they

know the manager will behave in this manner

• The manager could work out some contractual arrangement to induce the would-be owners to invest


• One possible contract would be for the manager to agree to finance all of the benefits out of his share of the profits– results in lower utility for the manager– would be difficult to enforce

• They may instead try to give managers an incentive to economize on benefits and to pursue higher profits

Important Points to Note:

• In order to maximize profits, the firm should choose to produce that output level for which the marginal revenue is equal to the marginal cost

• If a firm is a price taker, its output decisions do not affect the price of its output– marginal revenue is equal to price

Important Points to Note:• If the firm faces a downward-sloping

demand for its output, it can only sell more at a lower price– marginal revenue will be less than price and

may be negative

• Marginal revenue and the price elasticity of demand are related by the following

PqePMR

,

11

Important Points to Note:

• The supply curve for a price-taking, profit-maximizing firm is given by the positively sloped portion of its marginal cost curve above the point of minimum average variable cost– if price falls below minimum AVC, the firm’s

profit-maximizing choice is to shut down and produce nothing

Important Points to Note:• The firm’s profit-maximization problem can

also be approached as a problem in optimal input choice– this yields the same results as does an

approach based on output choices

• In the short run, firms obtain producer surplus in the form of short-run profits and coverage of fixed costs that would not be earned if the firm produced zero output

Important Points to Note:• In situations of imperfect knowledge, firms

may opt to maximize revenues– this means that the firm expands output until

marginal revenue is zero– sometimes these decisions may be

constrained by minimum profit requirements

• Because managers act as agents for a firm’s owners, they may not always make decisions that are consistent with profit maximization

chapter 13 profit maximization and supply copyright ©2002 by south-western, a division of thomson...

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