Introduction
We have seen how to evaluate a consumer’s wtp for a market-induced
price change: we measure it (with appropriate cautions, primarily that
we treat it as an approximation) by the change in consumer’s surplus,
the area under the demand function between the two price horizontals.
Producer’s Surplus
Philip A. Viton
We now address to the remaining question in market-induced (price)
changes: how to evaluate the bene…ts to producers or …rms.
Fortunately this is easy, and has none of the complications of
consumer’s surplus.
November 12, 2014
The obviously correct measure of bene…ts to …rms is pro…ts, and we
measure a project’s impact on a …rm by the change in its pro…ts.
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
1 / 27
Pro…ts
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
2 / 27
Total Costs: Fixed and Variable
What are pro…ts?
Consider a …rm selling x1 units of output at price p1 .
Typically a …rm will have both …xed (TFC ) and variable (TVC ) costs.
Its Total Revenues (TR) are p1 x1 .
In this case we can re-write total costs as:
Its Total Costs are TC .
TC = TFC + TVC
Then pro…ts are Total Revenues minus Total Costs or:
Π = TR
Then we have:
Π = TR
TC .
(TFC + TVC )
Note that this is an empirically observable quantity.
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
3 / 27
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
4 / 27
Change in Pro…ts (I)
Change in Pro…ts (II)
So the change in pro…ts is:
Suppose a project changes revenues from TR a to TR b and costs from
TC a to TC b .
∆Π = (TR b
∆Π = Π
Π
b
= (TR
= (TR b
= (TR b
TVC a )
PS = TR
TVC
a
(TR a TVC a )
(TFC + TVC b )) (TR a
TVC b ) (TR a TVC a )
is called the …rm’s Producer Surplus.
So the change in the …rm’s pro…ts (∆Π) is the same as the change in
producer’s surplus:
TC b )
(TFC + TVC a ))
∆Π = PS b
note that TVC by de…nition is the same in the pre- and post-project
settings.
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
5 / 27
Producer’s Surplus (I)
PS a
= ∆PS
Note that since the change in pro…ts is observable, so is producer’s
surplus: unlike consumer’s surplus we don’t need to appeal to any
notions of approximation to some theoretically correct measure.
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
6 / 27
Producer’s Surplus (II)
It is important to understand that while the change in pro…ts equals
the change in producer’s surplus, pro…ts and producer’s surplus are
not the same.
The two di¤er by the amount of …xed costs:
Π = TR
= TR
= PS
costs (which will be the value of the constant or intercept in a
cost-function regression equation) is often imprecisely estimated.
(TVC + TFC )
TVC TFC
TFC
CRP 6600 — Producer’
()
s Surplus
We now have two equivalent ways of measuring a project’s impact on
…rms: the change in pro…ts (∆Π) and the change in producer’s surplus
(∆PS). Producer’s surplus has two advantages:
1. It does not require us to understand …xed costs; and empirically, …xed
2. Total variable costs are the area under the marginal cost curve from
x1 = 0 (zero output) to x1a (observed output). See the next slide. So
if we can estimate marginal costs, we are in a position to compute
TVC.
But since for project evaluation we will always be looking at the
change in pro…ts induced by the project, this won’t matter.
Philip A. Viton
(TR a
The di¤erence between total revenue and total variable cost,
Using our decomposition of costs, we have:
b
TVC b )
November 12, 2014
7 / 27
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
8 / 27
Total Variable Costs
Project
The …gure shows a competitive
…rm’s upward-sloping marginal
cost curve.
$
MC
$
If it is a pro…t-maximizing
price-taker, then, if it faces
market price p1a its output will
be x1a .
MC
p1b
Total revenue is p1a x1a
(hatched).
p1a
0
x1
x1a
Total variable costs are the area
under MC from x1 = 0 to
x1 = x1a ( dark shade).
x1
0
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
9 / 27
Change in Producer’s Surplus
$
MC
b
1
p
a
x1
0
Philip A. Viton
x1a
The new total revenue is the
hatched area, total variable
costs are darkly shaded and
producer’s surplus is lightly
shaded.
x1b
So producer’s surplus is the area
above MC but below the price
horizontal (light shade).
Philip A. Viton
p1
Suppose our project has the
e¤ect of raising the market price
to p1b . The …rm responds by
raising its output to x1b .
x1b
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
10 / 27
Multiproduct Producer’s Surplus
Comparing the previous
pictures, we can see that the
change in producer’s surplus is
the area to the left of the
marginal cost curve and between
the two price horizontals.
When a …rm produces several products and the project changes their
prices, we naturally measure the full impact as the change in
producer’s surplus in each of the product markets. So:
Note that this is valid only for a
competitive …rm that sees its
market price change with no
shift in its cost curves.
It can be shown that ∆PS is not path-dependent: that is, for a
multiproduct …rm you will get the same answer in whatever order you
do the calculation (unlike consumer’s surplus).
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
∆PS = ∆PS1 + ∆PS2 + . . .
(just as we did for consumer’s surplus).
11 / 27
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
12 / 27
Market Impacts — Full Analysis
Example — Monopolization
We are now in a position to evaluate the full impact of a policy that
changes a market price from its pre-project level p a to its post project level
pb
For the impact on consumers we use the change in consumer’s
surplus:
∆CS = CS (p b ) CS (p a )
As we know from general microeconomics, monopolization has two
impacts: it reduces the quantity produced and increases the price
consumers pay for the product.
For the impact on …rms we use the change in producer’s surplus
(aggregating over …rms, if necessary):
∆PS = PS (p b )
Intuitively, this is a bad thing ; but not so fast. We would naturally
assume that the price increase leads to an increase in pro…ts: and
might not this gain to producers outweigh the loss to consumers?
PS (p a )
By using our new tools, we will now see that this is not so.
Monopolization is an overall net “bad” to the economy.
And then the full impact is:
∆W = ∆CS + ∆PS
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
13 / 27
Monopolization — Setup
$
Σ MC=S
pb
pa
MR
xb xa
Philip A. Viton
D
x
As an example of how all this …ts together, let’s consider a policy that
allows an industry to become monopolized. So our pre-project state (a) is
competition, and our post-project state (b) is monopolization. Note that
we are assuming that this is the only change in the economy.
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
14 / 27
Monopolization – Impact on Consumers
The …gure shows the basic
position: under competition,
demand = supply, where supply
(S) is the sum of the …rms’
marginal costs ; the result is
price p a and output x a .
$
Σ MC=S
pb
pa
Under monopolization,
pro…t-maximizing output is
governed by the condition MC
= MR, leading to output x b .
The monopolist prices to sell all
output, resulting in a price p b .
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
As usual, the impact on
consumers is ∆CS = signed
area under the demand function
between the two price
horizontals.
a
b
MR
This is the shaded area, and it is
important to note that it is a
negative quantity.
D
x
xb xa
15 / 27
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
16 / 27
Producer’s Surplus under Competition
Producer’s Surplus under Monopoly
$
$
Σ MC=S
Σ MC=S
Pre-project producer’s surplus
PS a is TR ( = p a x a )
TVC
(= area under MC from x = 0
to x = x a ).
pb
pa
Similarly, relative to the
post-project monopoly
equilibrium (x b , p b ), producer’s
surplus PS b is the shaded area.
pb
pa
In the …gure, PS a is shaded.
MR
b
x
D
MR
x
a
x
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
17 / 27
Change in Producer’s Surplus
Philip A. Viton
$
c (+)
d(-)
MR
Philip A. Viton
D
x
November 12, 2014
18 / 27
$
Note that this has two parts: a
rectangle (c ), representing
where PS b is more than PS a
and hence has a positive sign ;
and a triangle (d ) representing
part of PS a not included in
PS b , so this has a negative sign.
Σ MC=S
CRP 6600 — Producer’
()
s Surplus
Net Impact (I)
Comparing the two previous
…gures, ∆PS is the shaded area
xb xa
x
xb xa
Philip A. Viton
pb
pa
D
Σ MC=S
pb
pa
Obviously the total e¤ect is
positive : as we’d expect,
monopoly increases …rm pro…ts.
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
For the overall net impact, we
compare ∆PS to ∆CS.
First, area a in the consumer’s
surplus picture exactly matches
area c in the producer’s surplus
picture. The signs are opposite
(a is negative, c is positive) so
they cancel.
b
d
MR
D
x
xb xa
19 / 27
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
20 / 27
Net Impact (II)
Net Impact (III)
Σ MC=S
pb
pa
b
d
MR
b
x
Philip A. Viton
$
Second, (negative) area b from
the consumer’s surplus picture is
not included in the producer’s
surplus picture. So we need to
include it here.
$
D
x
The result is that the overall
change in welfare ∆W is the
shaded area.
Σ MC=S
pb
pa
Third, (negative) area d from
the producer’s surplus picture is
not included in the consumer’s
surplus picture. So we need to
include that, too
Note that both portions of this
are negative quantities.
b
d
MR
So our result is that ∆W < 0 :
monopolization de…nitely
decreases social welfare.
D
x
xb xa
a
x
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
21 / 27
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
22 / 27
November 12, 2014
24 / 27
Conclusion
Computational Appendix
We conclude that going from a situation of competition to one of
monopoly has an unambiguously negative overall impact.
In the literature, this is referred to as the Deadweight Loss of
monopolization.
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
23 / 27
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
Computational Considerations (I)
Computational Considerations (II)
For example, suppose in the linear case you have:
In the context of our monopolization example, suppose you need to
calculate the pre-project equilibrium (p a , x a ). Conceptually, you want to
equate supply (= MC) and demand.
Note that the marginal cost function is MC (x ) : that is, it takes a
quantity and tells us its marginal cost, in dollars.
But the demand function is x (p ) : that is, it takes a price and tells
us the demand (quantity) at that price.
So you can’t just equate x (p ) and MC (x ) : that would be
meaningless.
Marginal cost: MC (x ) = α0 + α1 x
( α1 > 0)
Demand: x (p ) = β0 + β1 p
( β1 < 0)
The in order to …nd the competitive equilibrium you can invert the
demand function, giving p = (x β0 )/β1 and then equate demand
and supply:
(x β0 )/β1 = α0 + α1 x
which is an equation in quantity only.
Alternatively, because under competition p = MC , you can feed in
the marginal cost equation into the demand function:
Instead, you must …rst invert (solve) either the demand function or
the marginal cost function, so they both tell us the same kind of thing
(output or money). Then you can equate the two.
x
= β0 + β1 p
= β0 + β1 ( α0 + α1 x )
which is also an equation in x only; and solve.
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
25 / 27
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
26 / 27
Philip A. Viton
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
27 / 27
Computational Considerations (III)
The solution turns out to be (assuming α1 β1 6= 1) :
x=
β0 + α0 β1
1 α1 β1
And the competitive price (= marginal cost of producing this output)
is:
p = α0 + α1 x
= α0 + α1
Philip A. Viton
β0 + α0 β1
1 α1 β1
CRP 6600 — Producer’
()
s Surplus
November 12, 2014
27 / 27