Consumer’s surplus
CV/EV
Consumer’s surplus
Intermediate Micro
Lecture 9
Chapter 14 of Varian
Examples
Consumer’s surplus
CV/EV
Welfare analysis
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Last few lectures: how does ∆pi affect demand?
Today: how does ∆pi (or other change) affect consumers’s
well-being?
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Bad news: multiple methods
Good news: Agree on good/bad
Good news: Similar magnitude
Examples
Consumer’s surplus
CV/EV
Consumer’s surplus: a micro principles review
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Start with discrete
goods model
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Reservation price: rj ,
price (per unit) at
which consumer is
indifferent between
buying j and j − 1
units of the good
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Gives $-value of
consumption of 1 unit
Examples
Consumer’s surplus
CV/EV
Consumer’s surplus: a micro principles review
Suppose price is p̄ (purple
line)
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Choice: x = 2
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Gross consumer surplus:
$-valuation of total
benefit from chosen
x.
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Area of 2 left-most
bars
r1 + r2
Examples
Consumer’s surplus
CV/EV
Consumer’s surplus: a micro principles review
Suppose price is p̄ (purple
line)
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Choice: x = 2
Consumer’s surplus:
$-valuation of benefit
from chosen x net of
foregone consumption
of other goods.
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Area above the
price line
r1 + r2 − 2p̄
Examples
Consumer’s surplus
CV/EV
Non-discrete goods
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Can find CS when x
is not discrete good
Use demand function:
xi (pi , p−i , m)
Z ∞
CS =
x(p, p−i , m)dp
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p̄
Examples
Consumer’s surplus
CV/EV
Non-discrete goods
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Or, use
inverse demand function
pi (xi , p−i m)
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Z
pi so that
xi (pi , p−i , m) = xi
x∗
p(x, p−i , m)−p̄dx
CS =
0
Examples
Consumer’s surplus
CV/EV
Welfare analysis
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CS alone not very
interesting
∆CS with policy
change is informative
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$-value (-cost) of
price change
Example: ↑ p, from p̄
to p 0
Examples
Consumer’s surplus
Multiple consumers
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Units for CS are $s
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We can sum CSs
across many
individuals
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Can also add in
producer surpluses
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Note PS is income for
households
CV/EV
Examples
Consumer’s surplus
CV/EV
Great, let’s stop here
Issues with CS
1. It requires taking integrals
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It can be done!
2. How well do ”Purchase costs” represent utility?
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Income effect
Quasilinear utility - perfectly
Other utility functions - well, maybe not
Examples
Consumer’s surplus
CV/EV
Great, let’s stop here
Issues with CS
1. It requires taking integrals
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It can be done!
2. How well do ”Purchase costs” represent utility?
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Income effect
Quasilinear utility - perfectly
Other utility functions - well, maybe not
Examples
Consumer’s surplus
CV/EV
Using the indifference curve
Other ways to measure impact of price increase
1. Compensating variation: Increase in m needed, after price
increase to restore utility to pre-change level
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Price change and ↑ m happen
How much to compensate for price change?
2. Equivalent variation: Decrease in m sufficient, before price
change, to bring utility to post-change level
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↓ m happens instead of price change
What cost is equivalent to effect of price change?
CV and EV can also be used for price decreases
Examples
Consumer’s surplus
CV/EV
Examples
Compensating variation
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p1 ↑ from p to p 0
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p2 = 1
CV is ∆m so that
budget line
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with slope −p 0
tangent to old
indifference curve
∆x2max =
∆m
p2
= ∆m
Consumer’s surplus
CV/EV
Examples
Equivalent variation
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p1 ↑ from p to p 0
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p2 = 1
EV is −∆m so that
budget line
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with slope −p
tangent to new
indifference curve
∆x2max =
∆m
p2
= ∆m
Consumer’s surplus
CV/EV
Examples
What we’re doing
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How far has the
indifference curve
moved?
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Like, in $ terms
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Use budget lines to
measure
Consumer’s surplus
CV/EV
Examples
CV vs EV
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CV 6= EV (Almost always)
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CV: in post-change $s
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EV in pre-change $s
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They don’t have same
buying power
For ↑ p
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CV ≥ EV , usually >
Equal with quasilinear
utility
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= ∆CS, too
Equal if CV, or EV,
=0
Consumer’s surplus
CV/EV
Example: Cobb-Douglas utility
Example: Cobb-Douglas utility
u(x, y ) = x 2 y
m = 300, py = 1
px changes from p = 5 to p 0 = 4
Find the CV and EV
Examples
Consumer’s surplus
CV/EV
Example: Cobb-Douglas utility
Examples
Consumer’s surplus
CV/EV
Example: Quasilinear utility
Example: Quasilinear utility
√
u(x, y ) = 4 x + y
m = 40, py = 1
px changes from p = 0.8 to p 0 = 1
Find the CV and EV
Examples
Consumer’s surplus
CV/EV
Quasilinear utility - a general result
Compensating variation
v (x ∗ 0 ) + [m + CV − p 0 x ∗ 0 ] = v (x ∗ ) + [m − px ∗ ]
CV
CV
CV
= [v (x ∗ ) − px ∗ ]
−
[v (x ∗ 0 ) − p 0 x ∗ 0 ]
0
= [v (x ∗ ) − v (x ∗ )] −
[px ∗ − p 0 x ∗ 0 ]
= [∆utility from x] − [∆expenditure on x]
Examples
Consumer’s surplus
CV/EV
Quasilinear utility - a general result
Equivalent variation
v (x ∗ 0 ) + [m − p 0 x ∗ 0 ] = v (x ∗ ) + [m − EV − px ∗ ]
EV
EV
EV
= [v (x ∗ ) − px ∗ ]
−
[v (x ∗ 0 ) − p 0 x ∗ 0 ]
0
= [v (x ∗ ) − v (x ∗ )] −
[px ∗ − p 0 x ∗ 0 ]
= [∆utility from x] − [∆expenditure on x]
So, CV = EV
Examples
Consumer’s surplus
Example: Quasilinear utility
CV/EV
Examples
Consumer’s surplus
CV/EV
Examples
Quasilinear utility and CS
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FOC for x: p = v 0 (x)
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Loss of direct utility
when x falls from x ∗
∗0
to
R x∗ 0 x ∗is 0
x
v (x)dx
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= v (x ∗ ) − v (x ∗ 0 )
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Blue area
For ↑ p
Consumer’s surplus
CV/EV
Quasilinear utility and CS
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Reduced spending
from ↓ x goes to y
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Green area
Increased spending
from ↑ p comes from
y
Red area
Total effect:
∆CS = CV = EV
Only for quasilinear
utility
−∆CS = v (x ∗ ) − v (x ∗ 0 )− p[x ∗ − x ∗ 0 ] +x ∗ 0 [p 0 − p]
blue area
−green box
+red box
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Examples