+ All Categories
Home > Documents > Intermediate Micro Lecture 6 - Georgetown...

Intermediate Micro Lecture 6 - Georgetown...

Date post: 06-Mar-2018
Category:
Upload: vohanh
View: 215 times
Download: 0 times
Share this document with a friend
29
Changing m Elasticity Examples Changing own price Changing other good’s price Examples Analyzing demand Intermediate Micro Lecture 6 Chapter 6 of Varian
Transcript
Page 1: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Analyzing demand

Intermediate Micro

Lecture 6

Chapter 6 of Varian

Page 2: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Analyzing demand

Can model utility function and decisions

I Even for p,m we don’t observeI Can use demand functions to model comparative statics

I Comparative statics: Studying the effect on the equilibriumoutcome due to a change in parameters.

I How does demand change as income increases?I What are the effects on demand of a change in prices?I Categorize goods based on comparative statics

Page 3: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Review: Demand function

Start with basic consumer decision problem:

maxx1,x2u(x1, x2)

s.t.p1x1 + p2x2 = m

Leave p1, p2,m as parameters, and obtain demand functions

x1(p1, p2,m)

x2(p1, p2,m)

Page 4: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Changing m

x1(p1, p2,m)

I Consider effect of ↑ mI Easiest measure: dx1

dm (= ddmx1(p1, p2,m))

I Normal good: x1 is a normal good (at (p1, p2,m)) if dx1dm > 0

I Inferior good: x1 is an inferior good (at (p1, p2,m)) if dx1dm < 0

Page 5: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Normal vs Inferior

x1 is normalx2 is normal

x1 is inferiorx2 is normal

Can both goods be inferior?

Page 6: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Income offer curve

I Income offer curve: Agraph of all optimalbundles for a givenp1, p2, for all values of m

I m varies

I p1, p2 stay constant

I Plug various values of minto demand functions,plot results

I If both goods are normal,income offer curve isupward sloping (↗)

Page 7: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Income offer curve

I Income offer curve: Agraph of all optimalbundles for a givenp1, p2, for all values of m

I m varies

I p1, p2 stay constant

I Plug various values of minto demand functions,plot results

I If both goods are normal,income offer curve isupward sloping (↗)

Page 8: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Engel curve

I Engel curve: A graph of the demand for one of the goods, forall values of m, holding constant p1, p2

I m varies

I p1, p2 stay constant

I Plug various values of m into demand function, plot results

I If the good is normal, Engel curve is upward sloping (↗)

I The Engel curve never slopes downward

Page 9: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

↑ Income offer curveaxes: x1, x2

← Engel curvesaxes: xi ,m

Page 10: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Elasticity - Not in book!

I Income elasticity of demand: εxi ,m = dxidm ∗

mxi

I This formula is called point (income) elasticityI Percent change in xi relative to the percent change in m

I Non-calculus formula: ∆x/x∆m/m

I Called arc (income) elasticity

I Formula for (instantaneous) percent growth of y due to z :ddz ln(y(z))

I εxi ,m =ddm

ln(x1(p1,p2,m))ddm

ln(m)

Page 11: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Why use elasticity - Not in book!

I dxidm : change in xi due to increase in m

I dxidm > 0: increasing in m

I dxidm < 0: decreasing in m

I Scale?

I dxidm ∗

mxi

(income elasticity of demand): Same sign as dxidm

I Note that elasticity (and slope!) can vary with m

Page 12: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Elasticity-based definitions- Not in book!

I Unit elasticity: whenεxi ,m = 1

I xi grows at same rate asm

I Any ray through theorigin has unit elasticity

Engel curve with unit elasticity

Page 13: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Homothetic preferences

I Homothetic preferences: A set of preferences with theproperty that, if (x1, x2) ∼ (y1, y2), then(tx1, tx2) ∼ (ty1, ty2),∀t ≥ 0

I Equivalent properties:I Income offer curves are straight lines through the origin, for

any (p1, p2)I Engel curves are straight lines through the origin, for any

(p1, p2)I εxi ,m for any (p1, p2,m), for any good i

Page 14: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Elasticity-based definitions- Not in book!

I Luxury good: xi forwhich εxi ,m > 1

I xi grows at faster ratethan m

I To identify on Engelcurve

1. Draw ray from originto point

2. If curve crosses rayfrom left to right,good is luxury at this(p1, p2,m) Engel curve for Ikea furniture

Page 15: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Elasticity-based definitions- Not in book!

I Necessary good: xi forwhich εxi ,m < 1

I xi grows at slower ratethan m

I To identify on Engelcurve

1. Draw ray from originto point

2. If curve crosses rayfrom right to left,good is ncessary atthis (p1, p2,m) Engel curve for Ikea furniture

Page 16: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Perfect substitutes

u(x1, x2) = 2x1 + 3x2

m = x1 + 2x2

x1(1, 2,m) = m, x2(1, 2,m) = 0

Income offer curve Engel curve for x1

Page 17: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Cobb Douglas

u(x1, x2) = x0.41 x0.6

2

m = 0.5x1 + 1.5x2

x1(0.5, 1.5,m) = 0.8m, x2(0.5, 1.5,m) = 0.4m

Income offer curve Engel curve for x2

Page 18: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Quasilinear

u(x1, x2) = ln(x1) + 0.25x2

m = x1 + x2

x1(1, 1,m) =

{m if m < 44 if m ≥ 4

}, x2(1, 1,m) =

{0 if m < 4m − 4 if m ≥ 4

}

Income offer curve Engel curve for x1

Page 19: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Changing p1: effect on x1

x1(p1, p2,m)

I Consider effect of ↑ pi on xiI Derivative: dxi

dpi

I dxidpi

< 0 for all known goods

I Giffin good: A good for which dxidpi

> 0I No documented examples!

Page 20: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Own-price elasticity - Not in book!

I Own-price elasticity of demand: εxi ,pi = − dxidpi∗ pi

xiI % ↓ in xi relative to % ↑ in p1

εxi ,pi Description Interpretation

εxi ,pi = 0 Perfectlyinelastic

xi does notchange when pidoes

0 < εxi ,pi < 1 Inelastic xi changes lessthan pi does

εxi ,pi = 1 Unit elastic xi changes bysame % pi does

1 < εxi ,pi <∞ Elastic xi changes morethan pi does

εxi ,pi =∞ Perfectlyelastic

↑ pi ⇒ xi = 0,↓ pi ⇒ xi =∞

Page 21: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Inverse demand

x1(p1, p2,m)

I Take m, p2 as fixed

I Rewrite demand function as x1(p1)I Can find inverse demand function: p1(x1)

I Gives p1 that causes x1 to be optimalI Only exists if each value x1 optimal only for one p1

Page 22: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Implications of own-price elasticity - Not in book!

I Expenditure on good 1 = p1x1I If εxi ,pi > (=, <)1

I ↑ p1 causes ↓ ( no change, ↑) in p1x1

εxi ,pi Description Interpretation

εxi ,pi = 0 Perfectlyinelastic

xi does notchange when pidoes

0 < εxi ,pi < 1 Inelastic xi changes lessthan pi does

εxi ,pi = 1 Unit elastic xi changes bysame % pi does

1 < εxi ,pi <∞ Elastic xi changes morethan pi does

εxi ,pi =∞ Perfectlyelastic

↑ pi ⇒ xi = 0,↓ pi ⇒ xi =∞

Page 23: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Changing p1: effect on x2 - Not in book!

x2 =m

p2− p1x1

p2

I Suppose ↑ p1

I dx2dp1

> (=, <)0 when εxi ,pi > (=, <)1

I Complements: Two goods for which dx2dp1

< 0

I Substitutes: Two goods for which dx2dp1

> 0

Page 24: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Price offer curve

I Price offer curve: Agraph of all optimalbundles for a givenm, p2, for all values of p1

I p1 varies, m, p2 constant

I Plug values of p1 intodemand functions, plot

I Complements: POCupward sloping (↗)

I Substitutes: POCdownward sloping (↘)

Page 25: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Price offer curve

I Price offer curve: Agraph of all optimalbundles for a givenm, p2, for all values of p1

I p1 varies, m, p2 constant

I Plug values of p1 intodemand functions, plot

I Complements: POCupward sloping (↗)

I Substitutes: POCdownward sloping (↘)

Page 26: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Demand curve

I Demand curve: A graphof the demand for goodi , for all values of pi ,holding constantm, pnot i

I Non-Giffin goods:downward-sloping or flat

Page 27: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Perfect substitutesu(x1, x2) = x1 + x2

10 = p1x1 + x2

x1(p1, 1, 10) =

10p1

if p1 < 1

[0, 10] if p1 = 10 if p1 > 1

, x2(p1, 1, 10) =

0 if p1 < 110− x1 if p1 = 110 if p1 > 1

Price offer curve Demand curve for x1

Page 28: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Cobb-Douglas

u(x1, x2) = x0.751 x0.25

2

40 = 2x1 + p2x2

x1(2, p2, 40) = 15, x2(2, p2, 40) = 10p2

Price offer curve Demand curve for x2

Page 29: Intermediate Micro Lecture 6 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/6Demand.pdf · Intermediate Micro Lecture 6 ... Price o er curve I Price o er curve:

Changing m Elasticity Examples Changing own price Changing other good’s price Examples

Quasilinear

u(x1, x2) = ln(x1) + x2

10 = p1x1 + x2

x1(p1, 1, 10) = 1p1, x2(p1, 1, 10) = 9

Price offer curve Demand curve for x1


Recommended