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
/
/,
Marginal Revenue and Elasticity
• 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
Marginal Revenue and Elasticity
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
Marginal Revenue 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
Marginal Revenue Curve
• 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
Short-Run Supply by a Price-Taking Firm
output
price SMC
SATC
SAVC
P* = MR
q*
Since P > SATC,profit > 0
Short-Run Supply by a Price-Taking Firm
output
price SMC
SATC
SAVC
P* = MR
q*
If the price risesto P**, the firmwill produce q**and > 0
q**
P**
Short-Run Supply by a Price-Taking Firm
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
Short-Run Supply by a Price-Taking Firm
• 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
Short-Run Supply by a Price-Taking Firm
• 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
Short-Run Supply by a Price-Taking Firm
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)
Profit Maximization and Input Demand
• 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
Profit Maximization and Input Demand
• These first-order conditions for profit maximization also imply cost minimization– they imply that RTS = w/v
Profit Maximization and Input Demand
• 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
Profit Maximization and Input Demand
• 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
Producer Surplus in the Short Run
• 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
Producer Surplus in the Short Run
output
price SMC
P*
q*
If the market priceis P*, the firm will produce q*
Producer Surplus in the Short Run
• In mathematical terms, producer surplus is given by
* *)*()](*[ surplus producer
q qq
qTCqPdqqMCP
0 0
)](*[*)(** surplus producer 00 TCPqTCqP
SFC * surplus producer
Producer Surplus in the Short Run
• 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
Revenue Maximization
• 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*
Revenue Maximization
output
price
d
MR
P*
q*
If the firm wishes to maximize revenues, it will produce q*
Revenue Maximization
output
price
P*
q*
d
MR
SMCIf the firm wishes to maximize profits, it will produce q**
q**
Revenue Maximization
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
Revenue Maximization
• 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
Revenue Maximization
• 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
Constrained Revenue Maximization
• 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
Constrained Revenue Maximization
• 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
The Principal-Agent Problem
• 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
The Principal-Agent Problem
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*
The Principal-Agent Problem
Benefits
Profits
Owner’s constraint
U1
B*
*
The owner-manager maximizes profit because any other owner-manager will also want B* inbenefits
B* represents a true costof doing business
The Principal-Agent Problem
• 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 Principal-Agent Problem
• 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
The Principal-Agent Problem
Benefits
Profits
Owner’s constraint
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 Principal-Agent Problem
• 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 Principal-Agent Problem
• 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
The Principal-Agent Problem
• 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