• Find the IEP and Engel Curve for a consumer with
• Recap: last lecture we covered: – Income Expansion Paths and Engel curves – Inferior and Normal Goods – Necessities and Luxuries – “Marshallian” Demand Curves
Spring 2001
Econ 11-Lecture 6
Cobb Douglass Utility Function : U ( x, y ) = xα y1−α Budget Constraint : px x + p y y = I
• To find the solution: – Solve for the Marshallian demand curves. This will automatically give you the Engel Curve – Solve each demand curve for income – Set these equations equal to each other to derive the IEP. 1
Spring 2001
Solved Example
• Find the Marshallian demand curves:
L = xα y1−α − λ ( p x x + p y y − I )
x( px , p y , I ) = α
• Calculate the first order conditions:
α
− λp x = 0
Spring 2001
éxù ∂L = α ê ú − λp y = 0 ∂y ë yû
Econ 11-Lecture 6
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Solved Example (III) px x* α
I=
y ( p x , p y , I ) = (1 − α )
I py
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What happens to demand when price changes?
• Solve each demand curve for income: I=
I px
• These demand curves are the same as the Engel curves, since they show how the optimal levels of x and y change with income. • Note that for Cobb-Douglass utility, Engel curves are linear in income.
∂L = px x + p y y − I = 0 ∂λ 1−α
2
Solved Example (II)
• Set up the Lagrangian:
∂L é yù =αê ú ∂x ëxû
Econ 11-Lecture 6
x2
p y y* 1−α
• Setting these equations equal to each other gives the income expansion path: py y* 1−α
=
(1 − α ) p x x* px x* Þ y* = α αp y
slope = −
slope = −
• For Cobb-Douglass utility, the IEP is linear.
* 1
p p2
p10 p2
x1 Spring 2001
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Professor Jay Bhattacharya
Spring 2001
Marshallian Demand Curve
What Causes the Change in Demand?
p1
• 2 reasons why demand for x1 changes
p1*
– it is more expensive relative to x2 – consumer effectively has less income
• We label these 2 effects as:
p10
– the ‘substitution effect’ • (“Hicks” substitution effect)
– the ‘income effect’ x1 Spring 2001
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The “Hicks” Substitution Effect
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The “Hicks” Income Effect
x2
x2 The budget constraint tilts along the original utility curve until its slope reflects the new relative prices.
Then the tilted budget constraint shifts back to reflect the new budget constraint.
x1 Spring 2001
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• The substitution effect must be negative • The income effect can be positive or negative. Why?
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“Hicks” vs. “Slutsky” • The “Hicks” substitution effect holds utility constant
– some goods are inferior
– rotate along the indifference curve
• The “Slutsky” substitution effect holds purchasing power constant – rotate around the original consumption bundle
• It’s easier to derive the Slutsky equation and the size of the income effect from the latter x1 inferior Spring 2001
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x1 normal Econ 11-Lecture 6
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Professor Jay Bhattacharya
Spring 2001
The “Slutsky” Substitution Effect x2
The “Slutsky” Income Effect x2
x1 Spring 2001
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x1 13
~
~
(+ or -) Income Effect Ι
∆Χ 1 = ∆Χ 1 + ∆Χ 1
0
– The size of the substitution effect depends upon how much of a change is needed to get to the point where the MRS is equal to the slope of the new budget constraint.
( -) Substitution Effect
• This distance depends upon the curvature of the indifference curve.
S
– If the indifference curve is flat, the substitution effect will be large. – If the indifference curve is “very convex,” the substitution effect will be small.
The “Slutsky” Equation
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How large is the income effect?
How large is the substitution effect? Flatter indifference curve means larger substitution effect.
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• The substitution effect represents a movement along an indifference curve.
= (Χ1 − Χ1 ) + (Χ1 − Χ1 ) *
Econ 11-Lecture 6
How large is the substitution effect?
Change in Total Demand ∆Χ 1 = Χ1∗ − Χ10
Spring 2001
Steeper indifference curve means smaller substitution effect.
• Intuition: The income effect will be larger, the more x1 originally purchased. • How to see this: Think about how much extra income is needed to get back the original bundle of goods when one price increases.
• “Uncompensated” Change is the total change resulting from a price change. – Marshallian Demand – What we observe • We separate the uncompensated change into 2 effects – substitution effect – income effect • “Compensated” demand is the change holding utility constant, i.e., the substitution effect.
New
Income Needed
This quantity is larger if x10 is larger. Thus:
−∆I = ∆p1 x10 Spring 2001
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Spring 2001
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Derivation of Slutsky’s Equation
Slutsky’s Equation
∆x1 = ∆x1S + ∆x1I
• An algebraic decomposition of the total change in demand into income and substitution effects • What do the relative size and sign of the two effects imply for the change in demand?
∆x1I = =
∆x1I ∆I ∆I
∆x1 • (−∆p1 x10 ) ∆I
∆x1 = ∆x1 − S
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Spring 2001
Slutsky’s Equation (in terms of rates of change) ∆x1 = ∆x1 − S
Econ 11--Lecture 6
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• The rate of change in demand as price changes (holding income fixed) is equal to – the rate of change in demand as prices change, adjusting income; and – the rate of change in demand as income changes holding prices fixed
∆x1 ∆x ∆x 0 = − x1 ∆p1 ∆p1 ∆I
Spring 2001
Econ 11-Lecture 6
Interpretation
∆x1I 0 ∆p1 x1 ∆I
S
∆x1I 0 ∆p1 x1 ∆I
I 1
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Professor Jay Bhattacharya
Spring 2001
Law of Demand and Giffen Goods
Example of a Giffen Good x2
• The change in demand can be positive or negative since the income effect can be positive or negative.
• Case I: ‘Law’ of Demand – Occurs if:
p1
dx1 income effect
• Case II: ‘Giffen Good’ – Occurs if: • x1 is inferior, and • income effect > substitution effect
dx1 >0 dp x1
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Spring 2001
(Own) Price Elasticity of Demand
= x1
• Fact: the price elasticity of demand tells us how the total expenditure on a good changes with price • Let T = total expenditure on x1 = x 1 p 1 • How total expenditure changes with price = dT d p1 Econ 11-Lecture 6
