Pro…t Maximization
Philip A. Viton
May 8, 2012
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Introduction
So far we have studied the problem of cost minimization: the problem
of choosing inputs to produce a given (target) output at least cost.
But we have not addressed the problem of where that target output
comes from.
For private-sector producers (…rms), clearly the appropriate behavioral
assumption is that it selects output to maximize pro…ts.
We now study the decisions of these private-sector producers, in a
non-spatial setting.
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Pro…ts
We have:
Pro…t = Total Revenue
Total Cost
So maximizing pro…ts involves (1) making total revenues as large as
possible; and (2) making total costs as small as possible.
Therefore a pro…t-maximizer will also engage in cost minimization.
So our previous results on cost functions apply to a pro…t-maximizing
…rm. We have:
Pro…t = Total Revenue
C (q; r )
where C (q; r ) is the minimum Total Cost function.
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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The Competitive Firm
Behavioral motivation: pro…t maximization.
The …rm is a price-taker in all input markets.
The competitive …rm is a price-taker in the output market as well: it
acts as though, no matter how much output it produces, the market
price p will not change.
Usual justi…cation: the …rm is a small participant in a large industry.
Clearly this is not always applicable: you need to decide in any speci…c
case if the model of the competitive …rm is appropriate or not.
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Pro…t Maximization by the Competitive Firm
Suppose the …rm is producing a small quantity of output, say q1 units.
Suppose it increases output by 1 unit.
Total revenues increase by p (because it is a price-taker in the output
market and all output is sold at price p).
Total costs increase by MC (q1 ).
So increasing output by 1 unit will increase pro…ts if p > MC (q1 ).
So the …rm should go on increasing output as long as p > MC .
It should stop increasing output when p = MC .
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
5/1
Pro…t Maximization by the Competitive Firm
The same reasoning works in the other direction as well.
If the …rm is producing a lot of output, we ask, should it cut back by
one unit?
If it does so, revenues fall by p.
But costs fall by MC .
So pro…ts go up as long as the cost savings ( = MC ) are greater than
the revenue loss (p ).
So the …rm should cut back on output as long as MC > p.
And it should stop cutting back when MC = p.
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Pro…t Maximization by the Competitive Firm
Either way, we see that for the competitive …rm, the pro…t-maximizing
level of output is where MC = p (p is the given market price).
For a technical reason, we need to check that MC is rising at the
pro…t-maximizing output.
In the long run, the …rm will shut down (go out of business) if
p < AC .
And if the …rm is operating in the short-run, it will only stay in
business if p > AVC .
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Pro…t-Maximization (Calculus)
Price-taking behavior in the output market implies that total revenues
from output q are p q.
So pro…ts are
Π=p q
C (q; r )
First-order condition for a maximum is d Π/dq = 0 so
p dC /dq = 0 or p MC (q ) = 0 : so p = MC .
Second-order condition requires d 2 Π/dq 2 < 0 so dMC (q )/dq < 0
or dMC (q )/dq > 0 : at a pro…t maximum, marginal cost must be
increasing with output. This is the “technical reason” referred to
previously.
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Pro…t-Maximizing Choice of Output: Long Run
$
MC
AC
At output (market) price p1 the
pro…t-maximizing output is q1 .
At market price p2 it is q2 (and
not q3 , since MC is falling at
q3 ) .
p1
p2
p3
At price p3 the …rm shuts down
(p3 is below AC).
q3
Philip A. Viton
q2 q1
q
CRP 781()— Pro…ts
May 8, 2012
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Pro…t-Maximizing Choice of Output: Long Run
$
At price p1 :
MC
AC
p1
Total Cost = q1
hatched area.
AC(q1)
q1 =
AC (q1 ) =
Di¤erence is the …rm’s pro…ts at
price p1 .
q1
Philip A. Viton
Total Revenue = p1
shaded area.
q
CRP 781()— Pro…ts
May 8, 2012
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Pro…t-Maximizing Choice of Output: Short Run
At price p1, pro…t-maximizing
output is p1 .
$
SRMC
SRAC
p1
p2
p3
SRAVC
SRAFC
q2 q1
Philip A. Viton
q
At price p2 , output is p2 but the
…rm makes a loss. However, it
more than covers variable costs
and hence makes a contribution
to the unavoidable …xed costs.
Therefore it should accept the
loss and stay in business.
At price p3 , the …rm shuts
down, since it cannot even cover
its variable costs.
CRP 781()— Pro…ts
May 8, 2012
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Supply Function
The supply function of the pro…t-maximizing competitive …rm shows,
for each market price p, how much the …rm will produce.
So the supply function is given by the relevant portion of the …rm’s
marginal cost function.
If the …rm is operating in the short run, the supply function is that
portion of SRMC above SRAVC.
In the long run, it is the portion of (LR)MC above (LR)AC.
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Monopoly
The monopolist is the sole seller in a market.
Clearly it makes no sense to suppose that the monopolist is a price
taker in the output market: the monopolist is the market.
Another way to put it is that the monopolist perceives the full
(market) downward-sloping demand function.
So if the monopolist want to sell more, it must reduce the price it
charges.
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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The Monopolist’s Revenue Function
The demand function represents the monopolist’s Average Revenue
Function: it tells him what price he must charge per unit, if he wishes to
sell some quantity of output.
If the Average Revenue ( = Demand) Function is p (q ) (reading up
and across, from quantity to price) then the Total Revenue function is
R (q ) = q p (q ).
Suppose the monopolist wants to sell one more unit of output. To do
so, it must reduce the price it charges to all customers. The change
in total revenue resulting from this is the monopolist’s Marginal
Revenue (MR) function.
Formally, for small changes in output, Marginal Revenue is the slope
(derivative) of the Total Revenue function.
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Monopoly — Pro…t Maximization
Suppose the monopolist is selling some small quantity of output (and
it is pro…table). Should it increase output by one unit?
If it does so, then its revenue increases by its Marginal Revenue
And its costs increase by its Marginal Cost
So its pro…ts will increase as long as MR > MC
Clearly it should go on increasing output as long as MR > MC, ie
until MR = MC
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Monopoly — Marginal Revenue with Linear Demand
If the demand function is linear, then
to construct the marginal revenue
function:
$
Pick any price, say p1 : draw a
horizontal line to the demand
curve (= AR).
p1
Bisect that line (point A).
A
MR
AR
q
Philip A. Viton
MR connects the mid-point (A)
with the intercept on the price
axis.
CRP 781()— Pro…ts
May 8, 2012
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Proof Using Calculus
If the demand (= average revenue) function is linear, its equation is
AR = α + βq
So Total Revenue is TR = AR q = q (α + βq ) = αq + βq 2
Marginal Revenue is the derivative of Total Revenue, so
MR = α + 2βq. This is the equation of a straight line.
MR has the same intercept (α) when q = 0 and has twice the slope
(2βq vs βq) as the demand function.
This is just what our geometric construction has shown.
Philip A. Viton
CRP 781()— Pro…ts
May 8, 2012
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Monopoly — Pro…t Maximization
Pro…t-maximizing output is qM
where MC = MR
MC
pM
MR
qM
Philip A. Viton
To sell this output, the
monopolist charges price pM ,
according to the demand (=
AR) function
AR
q
CRP 781()— Pro…ts
May 8, 2012
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The Competitive Industry I
The competitive industry is
made up of many competitive
…rms.
$
S1=MC1
S1+2
S2=MC2
Industry supply is the horizontal
summation of the individual
supply functions.
p1
Firm 1 has supply (=MC)
function S1 . At price p1 is
produces output q1
q1
Philip A. Viton
q2
q1+q2
q
Firm 2 has supply function S2 :
at p1 it produces q2
CRP 781()— Pro…ts
May 8, 2012
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The Competitive Industry II
$
S1=MC1
S1+2
S2=MC2
At price p1 the total output of
these two …rms (horizontal
summation) is q1 + q2
p1
This determines the (aggregate,
industry) supply function S.
q1
Philip A. Viton
q2
q1+q2
q
CRP 781()— Pro…ts
May 8, 2012
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Equilibrium in the Competitive Industry
Industry equilibrium is
determined by equality of
aggregate demand (D) and
industry supply (S)
$
S
S2=MC2
S1=MC1
As shown, the equilibrium price
is p .
*
p
Industry equilibrium output is q
D
q1
Philip A. Viton
q2
q*
q
Each …rm takes p as given, so
we can determine the outputs of
(here) …rms 1 and 2.
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Monopoly vs Competition
Suppose we have a competitive
industry with price pc and
output qc .
$
S
pM
This leads to the monopolist
producing output qM and
charging price pM .
pc
MR
qM
Philip A. Viton
qc
Now suppose it is monopolized.
The monopolist bases its output
on the MC=MR condition.
D
q
Compared to competition,
monopoly involves less output
and higher prices.
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May 8, 2012
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