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 −=1 0 *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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