consumer surplus · gross consumer surplus consumer buys units of good 1. consumer has different...
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Consumer Surplus
Demand Function and Demand Curve
Demand function: Demand Curve:
( )mppxx ,, 2111 =
1p
1x
Inverse Demand Function
Consider a demand function
The inverse demand function is
Cobb-Douglas example:
( )mppxx ,, 2111 =
( )111 xpp =
11
pmcx =
11
xmcp =
Inverse Demand Curve
Inverse Demand Curve
1p
1x
Optimal choice:
Suppose:(composite good)Rearrange:
12 =p
MRSpp −=
2
1
MRSp −=10 *
1x
*1p
Gross Consumer Surplus
Consumer buys units of good 1.Consumer has different willingness to pay for each extra unit.GCS: Area under demand curve. GCS tells us how much money consumer willing to pay for
1p
1x0 *1x
1*x
1*x
Consumer Surplus
Consumer buys units of good 1.
Consumer pays for each unit.
1p
1x0 *1x
1*x
*1p
*1p
*11
* xpGCSCS ×−=
The Welfare Effect of Changes in Prices
Goal: provide a monetary measure of the effects of price changes on the utility of the consumer.3 ways of doing it:
1. Compute changes in consumer’s surplus;2. Compensating variation;3. Equivalent variation.
Change in Consumer’s Surplus
Suppose a tax increases price of good 1 from
to .
Decrease in CS:
*1p **
1p*
1p
**1p
**1x *
1x 1x
1p
R T
TR +
Change in Consumer’s Surplus
In practice, to compute the change in CS we need to have an estimate of the consumer’s demand function. This can be done using statistical methods.How is change in CS related to change in utility? The two coincide when utility is quasi-linear:
( ) ( ) 2121, xxvxxu +=
Compensating Variation
CV=how much money we need to give the consumer after the price change to make him just as well off as he was before the price change.Budget line:
1x
2x
XY
Z
112 xpmx −=
CV
Equivalent Variation
EV=how much money we need to take away from the consumer before the price change to make him just as well off as he was after the price change.Budget line:
1x
2x
XYZ
112 xpmx −=
EV
Compensating and Equivalent Variations
To compute CV and EV we need to know utility function of the consumer.This can be estimated from the data by observing consumer’s demand behavior.E.g. observe consumer’s choices at different prices and income levels. Observe that expenditures shares are relatively constant:Cobb-Douglas preferences.
An Example: Increase in Oil Prices
Often, OPEC manages to restrict production and significantly increase oil prices.
What’s the effect of this increase on consumers’ welfare?
Model
Consumers’ utility function over gasoline and composite goods, :
Moreover:
( ) ( ) 221
121 10, xxxxu +=
1x2x
.2$;1$200$
***11 ==
=pp
m
Find Consumer Demand’s Before Price Increase
Consumer solves:
Optimality condition:
2*
21*
1
221
1,
200..
10max21
xpxpts
xxxx
+=
+
*
*
*
21 1
2
1
1
5 ppp
x−=−=−
Find Consumer Demand’s Before Price Increase
Since:
Demand for gasoline is:
Demand for composite good:
( ) 25125 21
1*
* ==p
x
1$*1 =p
.17525200*2 =−=x
Find Consumer Demand’s After Price Increase
Since:
Demand for gasoline goes down:
Demand for composite good:( ) 4
25125 21
1**
** ==p
x
2$**1 =p
5.1874252200**
2 =−=x
Compute Change in Consumer’s Surplus
1x
Inverse demand function:
Consumer surplus:
1p
( )21
1
15
xp =
25425
2
1.2/25
25**
*
==
CSCS
.2/25*** =−CSCS
Compute Compensating Variation
Government pays amount CV such that:
Plug in numbers:
( )
+= CVuu 5.187,
425175,25
( ) CV++
=+ 5.187
425101752510
21
21
Compute Compensating Variation
Plug in numbers:
Get:
( ) EV++
=+ 5.187
425101752510
21
21
225=EV
Compute Equivalent Variation
Government pays amount EV such that:
Plug in numbers:
( )EVuu −=
175,255.187,
425
( ) EV−+=+
17525105.187
42510 2
121
Compute Equivalent Variation
Plug in numbers:
Get:
( ) EV−+=+
17525105.187
42510 2
121
225=EV
Conclusion
In this case: change in consumer’s surplus equals compensating variation which equals equivalent variation.
In general these three measures differ.
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