Transcript Chapter 13 PROFIT MAXIMIZATION AND SUPPLY MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON Copyright ©2002 by South-Western, a division of Thomson Learning.

Chapter 13
PROFIT MAXIMIZATION
AND SUPPLY
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.
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
d
dTR dTC
 ' (q ) 

0
dq
dq
dq
dTR dTC

dq
dq
Output Choice
• To maximize economic profits, the firm
should choose the output for which
marginal revenue is equal to marginal
cost
dTR dTC
MR 

 MC
dq
dq
Second-Order Conditions
• MR = MC is only a necessary condition
for profit maximization
• For sufficiency, it is also required that
d 2
d' (q )

0
2
dq q q *
dq q q *
• “marginal” profit must be decreasing at
the optimal level of q
Profit Maximization
revenues & costs
Profits are maximized when the slope of
the revenue function is equal to the slope
of the cost function
TC
TR
But the second-order
condition prevents us
from mistaking q0 as
a maximum
q0
q*
output
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
dTR d [P (q )  q ]
dP
marginal revenue  MR(q ) 

P q
dq
dq
dq
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 is
q = 100 – 10P
• Solving for price, we get
P = -q/10 + 10
• This means that total revenue is
TR = Pq = -q2/10 + 10q
• Marginal revenue will be given by
MR = 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
eq,P
dq / q dq P



dP / P dP q
Marginal Revenue and
Elasticity
• This means that

 q dP 
q  dP
1

  P 1 
MR  P 
 P 1  
 e
dq
 P dq 
q ,P





– 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
price
As output increases from 0 to q1, total
revenue increases so MR > 0
As output increases beyond q1, total
revenue decreases so MR < 0
P1
D (average revenue)
output
q1
MR
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
price
SMC
SATC
P* = MR
SAVC
Maximum profit
occurs where
P = SMC
q*
output
Short-Run Supply by a
Price-Taking Firm
price
SMC
SATC
P* = MR
SAVC
Since P > SATC,
profit > 0
q*
output
Short-Run Supply by a
Price-Taking Firm
price
SMC
P**
SATC
P* = MR
SAVC
If the price rises
to P**, the firm
will produce q**
and  > 0
q*
q**
output
Short-Run Supply by a
Price-Taking Firm
SMC
price
If the price falls to
P***, the firm will
produce q***
SATC
P* = MR
SAVC
profit maximization
requires that P =
SMC and that SMC
is upward-sloping
P***
q***
q*
output
<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 profitmaximizing decision is to shut down and
produce no output
Short-Run Supply by a
Price-Taking Firm
price
SMC
SATC
SAVC
The firm’s short-run supply
curve is that portion of the
SMC curve that is above
minimum SAVC
output
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 is
STC/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 by
SATC = STC/q = 16/q + q/100
• SATC is minimized when
SATC/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, secondorder 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 profitmaximizing 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 in the
Short Run
SMC
price
If the market price
is P*, the firm will
produce q*
P*
Producer surplus is the
shaded area below P*
and above SMC
q*
output
Producer Surplus in the
Short Run
• In mathematical terms, producer surplus
is given by
q*
q q *
producer surplus   [P * MC(q )]dq  (P * q  TC ) q 0
0
producer surplus  P * q * TC(q *)  [P * 0  TC(0)]
producer surplus   *  SFC
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
If the firm wishes to
maximize revenues, it
will produce q*
price
P*
d
output
q*
MR
Revenue Maximization
SMC
price
If the firm wishes to
maximize profits, it will
produce q**
P*
d
q**
output
q*
MR
Revenue Maximization
SMC
price
Increasing output
beyond q** increases
revenue but lowers
economic profit
P*
d
q**
output
q*
MR
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
If the manager is also the owner
of the firm, he will maximize his
utility at profits of * and benefits
of B*
Profits
*
U1
Owner’s constraint
B*
Benefits
The Principal-Agent Problem
The owner-manager maximizes
profit because any other ownermanager will also want B* in
benefits
Profits
B* represents a true cost
of doing business
*
U1
Owner’s constraint
B*
Benefits
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
Given the manager’s budget
constraint, he will maximize utility
at benefits of B**
Profits
Agent’s constraint
*
**
U2
Profits for the firm
will be ***
U1
***
Owner’s constraint
B*
B**
Benefits
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

1

MR  P 1 
 e
q ,P





Important Points to Note:
• The supply curve for a price-taking, profitmaximizing 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