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An in-depth analysis of consumer surplus, explaining the demand function, demand curve, inverse demand function, and the calculation of consumer surplus, compensating variation, and equivalent variation. It also includes an example of how to compute these measures for a consumer facing a price increase.
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Consumer Surplus
Demand Function and DemandCurve Demand function:
Demand Curve:
,
,
2
1
1
1
=
1
p
1 x
Inverse Demand Curve
Inverse Demand Curve
1
p
1 x
Optimal choice: Suppose:(composite good) Rearrange:
1
2
=
p
MRS
p p
−
=
1 2
MRS
p
−
=
1
0
Gross Consumer Surplus Consumer buysunits of good 1. Consumer has differentwillingness to pay for eachextra unit. GCS: Area under demandcurve. GCS tells us how muchmoney consumer willingto pay for
1
p
1
x
0
The Welfare Effect of Changes inPrices
Goal: provide a monetary measure of theeffects of price changes on the utility ofthe consumer. 3 ways of doing it:
Compute changes in consumer’s surplus;
Compensating variation;
Equivalent variation.
Change in Consumer’s Surplus Suppose a taxincreases price ofgood 1 from
to
.
Decrease in CS:
1 p
1 p
1 p
1 p
1 x
1 x
1 x
1
p
R
T
Compensating Variation CV=how much moneywe need to give theconsumer
after
the
price change to makehim just as well off ashe was
before
the
price change. Budget line:
1 x
2 x
X
Y
Z
1
1
2
x
p
m
x
−
=
CV
Equivalent Variation EV=how much moneywe need to take awayfrom the consumer before
the price
change to make himjust as well off as hewas
after
the price
change. Budget line:
1 x
2 x
X
Y
Z
1
1
2
x
p
m
x
−
=
EV
An Example: Increase in OilPrices Often, OPEC managesto restrict productionand significantlyincrease oil prices. What’s the effect ofthis increase onconsumers’ welfare?
Model Consumers’ utility function over gasolineand composite goods,
:
Moreover:
(
)
(
)
2
1 2
1
2
1
x
x
x
x
u
1
x
2
x
1
1
p
p m
Find Consumer Demand’s BeforePrice Increase Since: Demand for gasoline is: Demand for composite good:
(
)
2
1
1
p
x
1
p
2
x
Find Consumer Demand’s AfterPrice Increase Since: Demand for gasoline goes down: Demand for composite good:
(
)
2
1
1
p
x
1
p
2
x
Compute CompensatingVariation Government pays amount CV such that: Plug in numbers:
=
CV
u
u
5 .
187 ,
4 25
175 ,
25
(
)
1 2
1 2
Compute CompensatingVariation Plug in numbers: Get:
(
)
1 2
1 2