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

p

mcx

11

x

mcp

Inverse Demand Curve

Inverse Demand Curve

1p

1x

Optimal choice:

Suppose:

(composite good)

Rearrange:

12 p

MRSp

p

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

XY

Z

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:

22

1

121 10, xxxxu

1x2x

.2$;1$

200$***

11

pp

m

Find Consumer Demand’s Before Price Increase

Consumer solves:

Optimality condition:

2*

21*

1

22

1

1,

200

..

10max21

xpxp

ts

xxxx

*

*

*

2

1 12

1

1

5p

p

p

x

Find Consumer Demand’s Before Price Increase

Since:

Demand for gasoline is:

Demand for composite good:

251

25 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 2

1

1**

** p

x

2$**1 p

5.1874

252200**

2 x

Compute Change in Consumer’s Surplus

1x

Inverse demand function:

Consumer surplus:

1p

2

1

1

15

xp

254

25

2

1.2/25

25**

*

CS

CS

.2/25*** CSCS

Compute Compensating Variation

Government pays amount CV such that:

Plug in numbers:

CVuu 5.187,

4

25175,25

CV

5.187

4

25101752510

2

1

2

1

Compute Compensating Variation

Plug in numbers:

Get:

EV

5.187

4

25101752510

2

1

2

1

2

25EV

Compute Equivalent Variation

Government pays amount EV such that:

Plug in numbers:

EVuu

175,255.187,

4

25

EV

17525105.187

4

2510 2

12

1

Compute Equivalent Variation

Plug in numbers:

Get:

EV

17525105.187

4

2510 2

12

1

2

25EV

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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