Changing m Elasticity Examples Changing own price Changing other good’s price Examples Analyzing demand Intermediate Micro Lecture 6 Chapter 6 of Varian
Changing m Elasticity Examples Changing own price Changing other good’s price Examples
Analyzing demand
Intermediate Micro
Lecture 6
Chapter 6 of Varian
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
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)
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
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?
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 (↗)
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 (↗)
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
Changing m Elasticity Examples Changing own price Changing other good’s price Examples
↑ Income offer curveaxes: x1, x2
← Engel curvesaxes: xi ,m
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)
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
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
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
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
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
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
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
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
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!
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 =∞
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
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 =∞
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
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 (↘)
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 (↘)
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
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
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
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