an increase in the price of one good – holding tastes, income, and the price of other goods...

An increase in the price of one good – holding tastes, income, and the price of other goods constant – causes a movement along the demand curve.

We use consumer theory to show how a consumer’s choice changes as the price changes, thereby tracing out the demand curve.

In the previous chapter, we used calculus to maximize utility subject to a budget constraint.

We solved for the optimal quantities that a consumer chooses as a functions of prices and income.

We solved for the consumer’s system of demand functions for these goods.

For example, Lisa chooses between pizzas, q1, and burgers, q2, so her demand functions are of the form

1 1 1 2

2 2 1 2

, ,

, ,

q q p p Y

q q p p Y

where p1 is the price of pizza, and p2 is the price of burger, and Y is her income.

We showed that if a consumer has a Cobb-Douglas utility function

such demand functions are given by

11 2a aU q q

1 21 2

and 1Y Y

q a q ap p

Thus the Cobb-Douglas utility function has the unusual property that the demand function for each good depends only on its own price and not the price of the other good.

Illustration:Consider Michael, who consumes only beers,

q1, and wines, q2. Let p1 be the price of beers, p2 be the price of wines, and Y is his income. We can derive his demand curve for beer by showing the quantities of beer he will purchase for some alternative prices of beer with price of wine and his income constant.

Suppose that p2 = 35 and Y = 420. Let us find q1 for p1 = 12, 6, and 4.

0

Wine (q2)

Beer (q1)

0

12.0

5.5

59

4.5

27 45

3.0

5927 45

4

12

6

p1

Beer (q1)

D1E3

E2

E1

e3e2

e1 I2I3

I1

Price-consumption curve

L1 (p1 =12) L2 (p1 =6)L3 (p1 =4)

p2 = 35 and Y = 420

An increase in an individual’s income, holding tastes and prices constant, causes a shift of the demand curve.

An increase in income causes a parallel shift of the budget constraint away from the origin, prompting a consumer to choose a new optimal bundle with more of some or all goods.

With price of beers fixed at p1 = 12, and price of wines fixed at p2 = 35, let us find q1 for incomes, Y = 420, 630, and 840.

Wine (q2)

Beer (q1)0

Beer (q1)0

p127

27

12

38 49

38 49

e13

7

5e2

e3

I1I2

I3

Income-consumption curve

L 2 (Y = 630)

L 3 (Y = 840)

L 1 (Y = 420)

D1

D2

D3

E3E2E1

p1 = 12 and p2 = 35

Wine (q2)

Beer (q1)027 38 49

e13

7

5e2

e3

I1I2

I3

Income-consumption curve

L 2 (Y = 630)

L 3 (Y = 840)

L 1 (Y = 420)

p1 = 12 and p2 = 35

840E3

E2

E1

630

420

Engel curve

Y

Beer (q1)0 27 38 49

Wine (q2)

Beer (q1)02719

e13.0

20.5

11.5e2

e3

I1

I2

I3

Income-consumption curve

L 2 (Y = 630)

L 3 (Y = 840)

L 1 (Y = 420)

if beer is an inferior good

840 E3

E2

E1

630

420

Engel curve

Y

Beer (q1)0 2710 19

10

p1 = 12 and p2 = 35

Holding tastes, other prices, and income constant, an increase in a price of a good has two effects on an individual’s demand.

1.Substitution Effect2. Income Effect

The change in the quantity of a good that a consumer demands when the good’s price rises, holding other prices and the consumer’s utility constant.

If utility is held constant as the price of the good increases, consumers substitute other, now relatively cheaper goods for that one.

The change in the quantity of a good a consumer demands because of a change in income, holding prices constant.

An increase in price reduces a consumer’s buying power, effectively reducing the consumer’s income or opportunity set and causing the consumer to buy less of at least some goods.

Total effect

Income

effect

q2

q10166 10

Substitution effect

e1

e2

e*

L1L2 L*

I1I2

The total effect from the price change is the sum of the substitution and income effects.

total effect = substitution effect + income effect

Because indifference curves are convex to the origin, the substitution effect is unambiguous.

Less of a good is consumed when its price rises.

A consumer always substitutes a less expensive good for a more expensive one, holding utility constant.

The substitution effect causes a movement along an indifference curve.

The income effect causes a shift to another indifference curve due to a change in the consumer’s opportunity set.

The direction of the income effect depends upon the type of good.

Since the good is normal, the income effect is negative with respect to price increase.

Thus, both the substitution and income effect go in the same direction, so the total effect of the price increase must be negative.

Total effect

Income

effect

q2

q101610

Substitution effect

e1

e2

e*

L1L2 L*

I1

I2

12

Assume q1 is an inferior good

If a good is inferior, the income effect goes in the opposite direction from the substitution effect.

For most inferior goods, the income effect is smaller than the substitution effect.

The total effect moves in the same direction as the substitution effect, but the total effect is smaller.

However, the income effect can more than offset the substitution effect in extreme cases.

A good is called a Giffen good if an increase in its price causes the quantity demanded to rise.

Thus, the demand curve for a Giffen good slopes upward.

It was named after Robert Giffen, a 19th century British economist who argued that poor people in Ireland increased their consumption of potatoes when the price rose because of a potato blight.

Total effect

Substitution effect

q2

q10168

Income

effect

e1

e2

e*

L1

L2

L*

I1

I2

14

Assume q1 is a Giffen good

We could derive a compensated demand curve, where we determine how the quantity demanded changes as the price rises, holding utility constant.

The change in the quantity demanded reflects only pure substitution effects when the price changes.

It is called the compensated demand curve because we would have to compensate an individual – give the individual extra income – as the price rises so as to hold the individual’s utility constant.

It is also called the Hicksian demand curve, after John Hicks, who introduced the idea.

The compensated demand function for q1 is

1 1 2, ,q H p p U where we hold utility constant at .U We cannot observe the compensated

demand curve directly because we do not observe utility levels.

Because the compensated demand curve reflects only substitution effects, the Law of Demand must hold: A price increase causes the compensated demand for a good to fall.

0

q2

q1

0

12.0

27 45

27 45

12

6

p1

Uncompensated Demand Curve (D)

E3

E1

E2

e3 e1

e2 I1

I2L2 (p1 =12) L1 (p1 =6)

p2 = 35 and Y = 420

41

41

Compensated Demand Curve (H)q1

L*

We can also derive the compensated demand curve by using the expenditure function.

1 2, ,E E p p U

Differentiating the expenditure function with respect to p1, we obtain the compensated demand function for q1.

1 2 11

, ,E

H p p U qp

Illustration:Find the compensated demand function for q1

given a Cobb-Douglas utility function below with a = 0.6.

The expenditure function for this Cobb-Douglas utility function is

11 2a aU q q

1

1 2

1

a ap p

E Ua a

Differentiating the expenditure function with respect to p1 we have the compensated demand function given as

Given that a = 0.6, then the expenditure function becomes

1

21

1 11

apE a

q Up a p

0.6 0.40.6 0.41 2

1 21.960.6 0.4p p

E U Up p

The compensated demand function for q1 is

0.4 0.4

2 21

1 1

0.61.18

0.4p p

q U Up p