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Changing m Elasticity Examples Changing own price Changing other goods price Examples

Analyzing demand

Intermediate Micro

Lecture 6

Chapter 6 of Varian

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Analyzing demand

Can model utility function and decisions

I Even for p,m we dont 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

Changing m Elasticity Examples Changing own price Changing other goods 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)

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Changing m

x1(p1, p2,m)

I Consider effect of mI Easiest measure: dx1dm (=

ddmx1(p1, p2,m))

I Normal good: x1 is a normal good (at (p1, p2,m)) ifdx1dm > 0

I Inferior good: x1 is an inferior good (at (p1, p2,m)) ifdx1dm < 0

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Normal vs Inferior

x1 is normalx2 is normal

x1 is inferiorx2 is normal

Can both goods be inferior?

Changing m Elasticity Examples Changing own price Changing other goods 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 ()

Changing m Elasticity Examples Changing own price Changing other goods 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 ()

Changing m Elasticity Examples Changing own price Changing other goods 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

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Income offer curveaxes: x1, x2

Engel curvesaxes: xi ,m

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Elasticity - Not in book!

I Income elasticity of demand: xi ,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/xm/mI Called arc (income) elasticity

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

I xi ,m =ddm

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

ln(m)

Changing m Elasticity Examples Changing own price Changing other goods 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 dxidmI Note that elasticity (and slope!) can vary with m

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Elasticity-based definitions- Not in book!

I Unit elasticity: whenxi ,m = 1

I xi grows at same rate asm

I Any ray through theorigin has unit elasticity

Engel curve with unit elasticity

Changing m Elasticity Examples Changing own price Changing other goods 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 xi ,m for any (p1, p2,m), for any good i

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Elasticity-based definitions- Not in book!

I Luxury good: xi forwhich xi ,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

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Elasticity-based definitions- Not in book!

I Necessary good: xi forwhich xi ,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

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Perfect substitutes

u(x1, x2) = 2x1 + 3x2m = x1 + 2x2

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

Income offer curve Engel curve for x1

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Cobb Douglas

u(x1, x2) = x0.41 x

0.62

m = 0.5x1 + 1.5x2x1(0.5, 1.5,m) = 0.8m, x2(0.5, 1.5,m) = 0.4m

Income offer curve Engel curve for x2

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Quasilinear

u(x1, x2) = ln(x1) + 0.25x2m = 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

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Changing p1: effect on x1

x1(p1, p2,m)

I Consider effect of pi on xiI Derivative: dxidpiI dxi

dpi< 0 for all known goods

I Giffin good: A good for which dxidpi > 0I No documented examples!

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Own-price elasticity - Not in book!

I Own-price elasticity of demand: xi ,pi = dxidpi pixi

I % in xi relative to % in p1

xi ,pi Description Interpretation

xi ,pi = 0 Perfectlyinelastic

xi does notchange when pidoes

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

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

1 < xi ,pi

Changing m Elasticity Examples Changing own price Changing other goods 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

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Implications of own-price elasticity - Not in book!

I Expenditure on good 1 = p1x1I If xi ,pi > (=,

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Changing p1: effect on x2 - Not in book!

x2 =m

p2 p1x1

p2

I Suppose p1I dx2

dp1> (=, (=, 0

Changing m Elasticity Examples Changing own price Changing other goods 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 ()

Changing m Elasticity Examples Changing own price Changing other goods 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 ()

Changing m Elasticity Examples Changing own price Changing other goods 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

Changing m Elasticity Examples Changing own price Changing other goods 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

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Cobb-Douglas

u(x1, x2) = x0.751 x

0.252

40 = 2x1 + p2x2x1(2, p2, 40) = 15, x2(2, p2, 40) =

10p2

Price offer curve Demand curve for x2

Changing m Elasticity Examples Changing own price Changing other goods price Examples

Quasilinear

u(x1, x2) = ln(x1) + x210 = p1x1 + x2

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

Price offer curve Demand curve for x1

Changing mElasticityExamplesChanging own priceChanging other good's priceExamples

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