Income and Substitution Effects Engel Curves and the Slutsky Equation

Income and Substitution Effects

Engel Curves and the Slutsky Equation

Demand and income

• If your income is initially X1, you buy A1 apples

• When your income rises to X2, you buy A2 apples.

• To make the obvious point, demand is a function of income

X

A1

I1

X1

A2

X2

I2

How demand rises with income

• Lets plot the combinations of apples and income (X) from the previous graph.

A1

A2

X1 X2

How demand rises with income

• Lets plot the combinations of apples and income (X) from the previous graph.

• Connecting all possible points, we get the Engel curve, giving demand as a function of income.

A1

A2

X1 X2

The Shape of the Engel Curve

• The shape of the Engel Curve gives us the income elasticity of demand for the good

• If the Engel Curve is a straight line, the income elasticity is 1.0

A

X

The Shape of the Engel Curve

• The shape of the Engel Curve gives us the income elasticity of demand for the good

• If the Engel Curve has increasing slope the elasticity is greater than 1.0

A

X

The Shape of the Engel Curve

• The shape of the Engel Curve gives us the income elasticity of demand for the good

• If the Engel Curve has decreasing slope the elasticity is less than 1.0

A

X

The Shape of the Engel Curve

• Of course the Engel Curve need not be so well behaved

• This Engel Curve corresponds to a good that is both inferior and superior, depending on income

A

X

Income and Substitution Effects

• We know that both price and income influence demand.

Income and Substitution Effects

• We know that both price and income influence demand.

• A price change means an income change.

Income and Substitution Effects

• We know that both price and income influence demand.

• A price change, means an income change.– You are purchasing 10

apples at $1 each.– If the price falls to 50¢,

you effectively get $5 more income

Income and Substitution Effects

• Let’s draw the indifference curves between money and apples.

$

A

Yo

Yo/pA

I1

Income and Substitution Effects

• Let’s draw the indifference curves between money and apples.

• Your income is Yo; Apples initially cost pa

$

A

Yo

Yo/pA

I1

Income and Substitution Effects

• Let’s draw the indifference curves between money and apples.

• Your income is Yo; Apples initially cost pa

• You are are on indifference curve I1.

$

A

Yo

Yo/pA

I1

Income and Substitution Effects

• Suppose the price of apples drops to p*a

$

A

Yo

Yo/pA

Yo/p*A

I1

I2

Income and Substitution Effects

• Suppose the price of apples drops to p*a

• The budget line rotates out and you move to indifference curve I2.

$

A

Yo

Yo/pA

Yo/p*A

I1

I2

Income and Substitution Effects

• Suppose the price of apples drops to p*a

• The budget line rotates out and you move to indifference curve I2.

• Two things have occurred: a price cut and an increase in income.

$

A

Yo

Yo/pA

Yo/p*A

I1

I2

Income and Substitution Effects

• The substitution effect $

A

Yo

Yo/pA

Yo/p*A

I1

I2

Income and Substitution Effects

• The substitution effect– To isolate the effect of

the lower price, imagine a budget line like the red line, reflecting the lower price but tangent to the old indifference curve.

$

A

Yo

Yo/pA

Yo/p*A

I1

I2

Income and Substitution Effects

• The substitution effect– To isolate the effect of

the lower price, imagine a budget line like the red line, reflecting the lower price but tangent to the old indifference curve.

– The move to the red point on I1 shows the substitution effect.

$

A

Yo

Yo/pA

Yo/p*A

I1

I2

Income and Substitution Effects

• The substitution effect is always negative– Diminishing MRS

guarantees it

$

A

Yo

Yo/pA

Yo/p*A

I1

I2

Income and Substitution Effects

• The substitution effect is always negative

• The income effect– Of course, income has

gone up as well, and the movement from the red point to the green point reflects that.

$

A

Yo

Yo/pA

Yo/p*A

I1

I2

Income and Substitution Effects

• We effectively break the price change down into its two components.– The substitution effect

– The income effect.

$

A

Yo

Yo/pA

Yo/p*A

I1

I2

Income and Substitution Effects

• We effectively break the price change down into its two components.– The substitution effect

– The income effect.

• While the substitution effect is always negative, the income effect may or not be positive

$

A

Yo

Yo/pA

Yo/p*A

I1

I2

A Summary Table

Change in Quantity Demanded for Normal and Inferior Goods

Normal Good Inferior Good

Price Increase

Price Decrease

Price Increase

Price Decrease

Change in quantity demanded due to substitution effect (holding utility constant)

(-) (+) (-) (+)

Change due to income effect (holding prices constant)

(-) (+) (+) (-)

Combined Effect (-) (+) ? ?

The Slutsky Equation

• These effects are often summarized in the Slutsky equation

I

QQ

P

Q

P

Q

tConsU tan

The Slutsky Equation

• These effects are often summarized in the Slutsky equation

• The substitution effect shows the change in demand from a movement along the indifference curve.

I

QQ

P

Q

P

Q

tConsU tan

The Slutsky Equation

• These effects are often summarized in the Slutsky equation

• The income effect shows the change in demand from the effective increase in income.

I

QQ

P

Q

P

Q

tConsU tan

An Application

Apples Cost 50¢ Income of $100 50 Demanded

An Application

Apples Cost 50¢ Income of $100 50 Demanded

Apples Cost 45¢ Income of $100 53 Demanded

An Application

Apples Cost 50¢ Income of $100 50 Demanded

Apples Cost 45¢ Income of $100 53 Demanded

Apples Cost 50¢ Income of $101 51 Demanded

An Application

Apples Cost 50¢ Income of $100 50 Demanded

Apples Cost 45¢ Income of $100 53 Demanded

Apples Cost 50¢ Income of $101 51 Demanded

I

QQ

P

Q

P

Q

tConsU tan

An Application

(Q/P) = 3/(-0.05) = - 60

Apples Cost 50¢ Income of $100 50 Demanded

Apples Cost 45¢ Income of $100 53 Demanded

Apples Cost 50¢ Income of $101 51 Demanded

I

QQ

P

Q

P

Q

tConsU tan

An Application

(Q/P) = - 60Q(Q/I) = 50 (1) = 50

Apples Cost 50¢ Income of $100 50 Demanded

Apples Cost 45¢ Income of $100 53 Demanded

Apples Cost 50¢ Income of $101 51 Demanded

I

QQ

P

Q

P

Q

tConsU tan

An Application

(Q/P) = - 60Q(Q/I) = 50

-60 (Q/P)U=Constant –50

Apples Cost 50¢ Income of $100 50 Demanded

Apples Cost 45¢ Income of $100 53 Demanded

Apples Cost 50¢ Income of $101 51 Demanded

I

QQ

P

Q

P

Q

tConsU tan

An Application

-60 (Q/P)U=Constant –50

(Q/P)U=Constant = -10

Apples Cost 50¢ Income of $100 50 Demanded

Apples Cost 45¢ Income of $100 53 Demanded

Apples Cost 50¢ Income of $101 51 Demanded

I

QQ

P

Q

P

Q

tConsU tan

A Caution

• The version of the Slutsky equation we use is only an approximation.

A Caution

• The version of the Slutsky equation we use is only an approximation.

• We are assuming discrete changes in price and income; the correct equation assumes infinitesimal changes.

Why spend time on this topic?

• Giffin Goods

Why spend time on this topic?

• Giffin Goods

• The Demand for Leisure

Why spend time on this topic?

• Giffin Goods

• The Demand for Leisure – As wage rates increase, the cost of an hour of

leisure increases– Demand goes up because the income effect

dominates the substitution effect.

Why spend time on this topic?

• Giffin Goods

• The Demand for Leisure

• Different Slopes.

Why spend time on this topic?

• Giffin Goods

• The Demand for Leisure

• Different Slopes– Changes in the price of one brand versus

changes in the prices of all brands.

.

Why spend time on this topic?

• Giffin Goods

• The Demand for Leisure

• Different Slopes– Changes in the price of one brand versus

changes in the prices of all brands.– Heavily purchased goods versus lightly

purchased goods.

A Final Point

I

QQ

P

Q

P

Q

tConsU tan

A Final Point

• The slope of the Marshallian, or uncompensated demand function

I

QQ

P

Q

P

Q

tConsU tan

A Final Point

• The slope of the Marshallian, or uncompensated demand function

• The slope of the Hicksian, or compensated demand function.

I

QQ

P

Q

P

Q

tConsU tan

©2003 Charles W. Upton