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Chapter 11
Profit Maximization
Chapter Overview
1. The Profit Maximization Condition
2. Marginal Revenue and Elasticity
3. Short Run Equilibrium
• Short Run Supply Curve for a Price Taking Firm
• Profit Function
• Producer Surplus
5. Profit Maximization and Input Demand
• Marginal Revenue product
• Substitution effect
• Output effect
2
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
• If firms are strictly profit maximizers, they will
make decisions in a “marginal” way
– examine the marginal profit obtainable from
producing one more unit of hiring one
additional laborer
Output Choice
• Total revenue for a firm is given by
R(q) = p(q)q
• In the production of q, certain economic costs
are incurred [C(q)]
• Economic profits () are the difference between
total revenue and total costs
(q) = R(q) – C(q) = p(q)q –C(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
dR dC
 ' (q ) 

0
dq
dq dq
MR 
dR dC

 MC
dq dq
• To maximize economic profits, the firm
should choose the output for which
marginal revenue is equal to marginal cost
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
Marginal Revenue, Marginal Cost,
and Profit Maximization
A firm chooses output q*,
so that profit, the
difference AB between
revenue R and cost C, is
maximized.
At that output, marginal
revenue (the slope of the
revenue curve) is equal
to marginal cost (the
slope of the cost curve).
The second-order condition
prevents us from mistaking
q which is less than q0 as a
maximum
Δπ/Δq = ΔR/Δq − ΔC/Δq = 0
MR(q) = MC(q)
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
dR d [ p(q )  q ]
dp
marginal revenue  MR(q ) 

 pq
dq
dq
dq
• 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 from linear
demand function
• Suppose that the demand curve is
q = 100 – 10p
• Solving for price, we get
p = -q/10 + 10
• This means that total revenue is
R = pq = -q2/10 + 10q
• Marginal revenue will be given by
MR = dR/dq = -q/5 + 10
Profit Maximization
• To determine the profit-maximizing output, we
must know the firm’s costs
• If output 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 the demand
curve 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
The Inverse Elasticity Rule
• Because MR = MC when the firm maximizes
profit, we can see that


1

MC  p1 
 e 
q ,p 

e q ,p
p

MC 1 e q ,p
• The “markup” will fall as the demand curve
facing the firm becomes more elastic
• If eq,p > -1, MC < 0
• This means that firms will choose to operate
only at points on the demand curve where
demand is elastic
Markup Pricing
•
•
•
•
•
What does it mean?
Where is it used? Why?
Markup on price or cost?
Should it be a constant?
How does markup relate to profit maximization?
Markup Pricing
• What does it mean?
– It is a pricing technique whereby a certain percentage
of cost of goods sold or of price is added to the cost of
goods sold in order to obtain the market price.
• Markup on price or cost?
– Suppose an item costs $50 to produce and is sold for
$100.
– Whether it is 50 percent markup or a markup of 100
percent.
– In the 1st instance the margin is a markup on price (the
proportion of the selling price that represents an
amount added to the cost of the goods sold)
– In 2nd case the markup is the markup on cost (the
proportion of the cost of goods sold that is added on
to that figure to arrive at the selling price)
Profit Maximization & Markup
• Profit is maximum when:
P1  1  
EP 

MR  MC
 EP 
 1 
  MC  MC 

P  MC 
 EP  1 
 EP  1 
Profit maximizing markup is inversely related to price
elasticity as follows:
1
markup 
EP  1
Profit Maximization & Markup
• Note: As long as Ep (which
is nearly always negative) Value of 100 x -1 / (Ep + 1) =
Elasticity
Optimal percent
has an absolute value
Coefficient markup on Cost
larger than 1, m will be
-2
100
positive.
-3
50
• Profit maximizing markup
is lower the higher the
-4
33.3
absolute value of the own
-5
25
price elasticity of demand
-9
12.5
for the good or the
service.
The formula relating to markup on price to
elasticity : Markup = -1/Ep
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
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)
Quantity per period
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
Market
price
SMC
SAC
SAVC
P* = MR
Maximum profit occurs
where P = SMC
q*
Quantity per period
Short-Run Supply by a Price-Taking Firm
Market
price
SMC
SAC
SAVC
P* = MR
Since P > SAC,
profit > 0
q*
Quantity per period
Short-Run Supply by a Price-Taking Firm
Market
price
SMC
P**
SAC
SAVC
P* = MR
If the price rises to
P**, the firm will
produce q** and  > 0
q*
q** Quantity
per period
Short-Run Supply by a Price-Taking Firm
Market
price
SMC
If the price falls to
P***, the firm will
produce q***
SAC
P* = MR
SAVC
Profit maximization
requires that P =
SMC and that SMC
is upward-sloping
P***
q***
q*
Quantity per period
<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
Market
price
SMC
SAC
SAVC
The firm’s short-run
supply curve is the SMC
curve that is above SAVC
Quantity per period
Short-Run Supply
• Suppose that the firm’s short-run total cost
curve is
SC(v,w,q,k) = vk1 + wq1/k1-/
where k1 is the level of capital held constant in
the short run
• Short-run marginal cost is
SMC(v ,w , q, k1 ) 
SC w (1 ) /   / 
 q
k1
q

Short-Run Supply
• The price-taking firm will maximize profit where
w ( 1 ) /   / 
p = SMC
SMC  q
k
P
1

• Therefore, quantity supplied will be
w
q  




  /( 1 )
  /( 1 )
k1
P
 /( 1 )
• To find the firm’s shut - down price, we need to solve
for SAVC
SVC = wq1/k1-/
SAVC = SVC/q = wq(1-)/k1-/
Profit Functions
• A firm’s economic profit can be expressed as a
function of inputs
 = Pq - C(q) = Pf(k,l) - vk - wl
• Only the variables k and l are under the firm’s
control
– the firm chooses levels of these inputs in order
to maximize profits
• treats P, v, and w as fixed parameters in its
decisions
Profit Functions
• A firm’s profit function shows its maximal profits
as a function of the prices that the firm faces
 ( P, v, w)  Max[ Pf (k , l )  vk  wl ]
k ,l
Envelope Results
• We can apply the envelope theorem to see how
profits respond to changes in output and input
prices
( P ,v ,w )
( P ,v ,w )
 k( P ,v ,w )
 q( P ,v ,w )
v
P
( P ,v ,w )
 (l P ,v ,w )
w
Producer Surplus in the Short Run
• The profit function is nondecreasing in output
prices, we know that if P2 > P1
(P2,…)  (P1,…)
• The welfare gain to the firm of this price
increase can be measured by
welfare gain = (P2,…) – (P1,…)
Mathematically, we can use the envelope
theorem results
P2
p2
P1
p1
welfare gain   q( P )dP   (  / P )dP
 ( P2 ,...)  ( P1,...)
Producer Surplus in the Short Run
Market
price
SMC
P2
P1
If the market price
is P1, the firm will
produce q1
If the market price
rises to P2, the firm
will produce q2
q1
q2
Quantity per period
Producer Surplus in the Short Run
Market
price
SMC
The firm’s profits rise
by the shaded area
P2
P1
q1
q2
Quantity per period
Producer Surplus in the Short Run
Market
price
SMC
Suppose that the
firm’s shutdown
price is P0
P1
P0
q1
Quantity per period
Producer Surplus in the Short Run
Market
price
SMC
Producer surplus at
a market price of P1
is the shaded area
P1
P0
q1
Quantity per period
Producer Surplus in the Short Run
• The extra profits available from facing a price
of P1 are defined to be producer surplus
P1
producer surplus  ( P1,...)  ( P0 ,...)   q( P )dP
P0
• Producer surplus is the extra return that producers
make by making transactions at the market price
over and above what they would earn if nothing
were produced
– the area below the market price and above the
supply curve
Producer Surplus in the Short Run
• Because the firm produces no output at the
shutdown price, (P0,…) = -vk1
– profits at the shutdown price are equal to fixed costs
• This implies that
producer surplus = (P1,…) – (P0,…)
= (P1,…) – (-vk1) = (P1,…) + vk1
producer surplus is equal to current profits plus shortrun fixed costs
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 –C(q) = Pf(k,l) – (vk + wl)
• The FOCs for a maximum are
/k = P[f/k] – v = 0
/l = P[f/l] – w = 0
Profit Maximization and Input Demand
• 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.
• MRPL = PL and MRPK = PK
• These FOCs for profit maximization also imply
cost minimization
– they imply that MRTS = w/v
Profit Maximization and Input Demand
• To ensure a true maximum, SOCs require that
kk = fkk < 0
ll = fll < 0
kk ll - kl2 = fkkfll – fkl2 > 0
– capital and labor must exhibit sufficiently
diminishing marginal productivities so that
marginal costs rise as output expands
Input Demand Functions
• In principle, the first-order conditions can
be solved to yield input demand functions
Capital Demand = k(P,v,w)
Labor Demand = l(P,v,w)
• These demand functions are
unconditional
–they implicitly allow the firm to adjust
its output to changing prices
Single-Input Case
• We expect l/w  0
– diminishing marginal productivity of labor
• The first order condition for profit maximization
was
/l = P[f/l] – w = 0
Two-Input Case
• For the case of two (or more inputs), the story is
more complex
– if there is a decrease in w, there will not only
be a change in l but also a change in k as a
new cost-minimizing combination of inputs is
chosen
• when k changes, the entire fl function
changes
• But, even in this case, l/w  0
Two-Input Case
• When w falls, two effects occur
– substitution effect
• if output is held constant, there will be a
tendency for the firm to want to substitute l
for k in the production process
– output effect
• a change in w will shift the firm’s expansion
path
• the firm’s cost curves will shift and a
different output level will be chosen
Substitution Effect
k per period
If output is held constant at q0 and w
falls, the firm will substitute L for K in the
production process
Because of diminishing MRTS
along an isoquant, the
substitution effect will
always be negative
q0
l per period
Output Effect
A decline in w will lower the firm’s MC
Price
MC
MC’
Consequently, the firm
will choose a new level
of output that is higher
P
q0 q1
Quantity per period
Output Effect
Output will rise to q1
k per period
Thus, the output effect also
implies a negative
relationship between l and
w
q0
q1
l per period
Input Demand
• For a fixed quantity,
as price of labor
decreases from $2 to
$1, firm moves along
its labor demand
curve from B to A.
Increase in output
shifts the demand
curve.
Cross-Price Effects
• No definite statement can be made about
how capital usage responds to a wage
change
– a fall in the wage will lead the firm to
substitute away from capital
– the output effect will cause more capital to be
demanded as the firm expands production
Substitution and Output Effects
• We have two concepts of demand for any
input
– the conditional demand for labor, lc(v,w,q)
– the unconditional demand for labor, l(P,v,w)
• At the profit-maximizing level of output
lc(v,w,q) = l(P,v,w)
Substitution and Output Effects
• Differentiation with respect to w yields
c
c
(l P ,v ,w ) l ( v ,w ,q ) l ( v ,w ,q ) q



w
w
q
w
substitution
effect
output effect
total effect