Income elasticity of demand

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MICROECONOMICS I.

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ELTE Faculty of Social Sciences, Department of Economics

Microeconomics I.

week 9

CONSUMPTION AND DEMAND, PART 3 Authors:

Gergely K®hegyi, Dániel Horn, Klára Major Supervised by

Gergely K®hegyi

June 2010

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K®hegyi-Horn-Major

Applications and extensions of demand theory

The course was prepaerd by Gergely K®hegyi, using Jack Hirshleifer, Amihai Glazer and David Hirshleifer (2009) Mikroökonómia. Budapest: Osiris Kiadó, ELTECON-books (henceforth HGH), and Gábor Kertesi (ed.) (2004) Mikroökonómia el®adásvázlatok.

http://econ.core.hu/ kertesi/kertesimikro/ (henceforth KG).

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K®hegyi-Horn-Major

Income elasticity of demand

How does the demanded quantity react to the change in income?

For any good X the change in consumption∆I due to a change in income∆x could be measured by the ratio ^∆_∆^x_I. (This ratio is the slope of the Engel curve over the relevant range)

Problem: ^∆_∆^x_I is sensitive to the units of measurement.

e.g.: income raises by 100 HUF, and then we consume 5 dkg=0,05 kg more butter.

Then ^∆_∆^x_I =0,05, if we useh

dkgFt

i

and ^∆_∆I^x =0,0005, if we useh

kgFt

i

This can cause trouble, especially if we want to compare the income sensitivity of dierent goods. (e.g. pieces of

watermelon and apple, or grams of coee and bags of tea)

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K®hegyi-Horn-Major

Income elasticity of demand (cont.)

Denition

The income elasticity of demand (ε_x) is the proportionate change in the quantity purchased divided by the proportionate change in income. In other words, it shows how much (%) demanded quantity changes if income changes by 1%.

with discrete quantity (elasticity over a range or arc):

ε_x =∆x/x

∆I/I ≡∆x

∆I I x with continuous functions (elasticity at a point):

ε_x =∂x

∂I I x

Note

The slope of the Engel curve is NOT income elasticity.

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K®hegyi-Horn-Major

Income elasticity of demand (cont.)

Statement

An Engel curve with positive slope has income elasticity greater than, equal to, or less than 1 depending upon whether the slope along the Engel curve is greater than, equal to, or less than the slope of a ray drawn from the origin to the curve.

Statement

If the income elasticity of a good is positive, it is a normal good, if it is negative, then it is an inferior good.

Normal good: ε >0 Inferior good: ε <0

Denition

Necessity good: 1> ε >0 Luxury good: ε >1

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K®hegyi-Horn-Major

Income elasticity of demand (cont.)

Statement

The weighted average of an individual's income elasticities equals 1, where the weights are the proportions of the budget spent on each commodity. So if k₁≡^p¹_I^x¹, . . . ,k_i ≡^pⁱ_I^xⁱ, . . . ,k_n≡^pⁿ_I^xⁿ, then

k₁ε₁+. . .+k_iε_i+. . .+k_nε_n=

n

X

i=1

k_iε_i =1

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K®hegyi-Horn-Major

Income elasticity of demand (cont.)

Proof

Let us substitute the Marshall demand functions into the budget line:

p₁x₁(p₁,p₂, . . . ,p_n;I) +p₂x₂(p₁,p₂, . . . ,p_n;I) +. . .+ +. . .+p_nx_n(p₁,p₂, . . . ,p_n;I) =I

Let us dierentiate both sides by income:

p1∂x1

∂I +p2∂x2

∂I +. . .+pn∂xn

∂I =1

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K®hegyi-Horn-Major

Income elasticity of demand

Proof

Let's expand all terms:

p₁x₁ I

I x1

∂x₁

∂I +p₂x₂ I

I x2

∂x₂

∂I +. . .+p_nx_n I

I xn

∂x_n

∂I =1 k₁ε₁+. . .+k_iε_i+. . .+k_nε_n=1

n

X

i=1

kiε_i =1

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K®hegyi-Horn-Major

Income elasticity

Unitary income elasticity

The straight line Engel curve ADB has income

elasticity 1, because the slope along the curve is the same as the slope of a ray from the origin to any point on the curve.

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Income elasticity (cont.)

Eect of income on expenditures (income elasticities)

Lowest Highest

income income

Category group group

Food 0,63 0,84

Housing 1,22 1,80

Household operation 0,66 0,85

Clothing 1,29 0,98

Transportation 1,50 0,90

Tobacco and alcohol 2,00 0,85

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K®hegyi-Horn-Major

Price elasticity of demand

How sensitive is the demanded quantity on the change in price?

We can dene the price elasticity similarly to income elasticity:

Denition

The price elasticity of demand is the proportionate change in quantity purchased divided by the proportionate change in price.

In other words, it shows how much (%) demanded quantity changes if price changes by 1%.

with discrete quantities:

η_x = ∆x/x

∆P_x/P_x ≡ ∆x

∆P_x P_x

x with continuous quantities:

η_x = ∂x

∂P_x P_x

x

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Price elasticity of demand (cont.)

Statement

The price elasticity of a Gien good is positive, while it is negative for an ordinary good:

Ordinary good: η_x <0 Gien good: η_x >0

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Price elasticity of demand (cont.)

market price elasticity of demand

The four demand curves represent dierent responses of quantity purchased to changes in price. Since demand curves are conventionally drawn with price on the vertical axis, a greater response is represented by a atter demand curve.

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K®hegyi-Horn-Major

Price elasticity of demand (cont.)

Statement

If a consumer's demand for X is elastic, a reduction in price P_x will ead to increased spending E ≡Pxx on commodity X. If demand is inelastic, a price reduction decreases Ex. If the demand elasticity is unitary Ex remains the same.

Proof

Marginal expenditure (ME_x):

MEx ≡ ∂E

∂x =Px+x∂Px

∂x =Px

1+ x

P_x

∂Px

∂x

=Px

1+ 1

η_x

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Cross elasticity of demand

the demanded quantity of butter depends not only on the price of the butter, but also from the price of other related commodities such as e.g. bread or cheese.

Now the fact that elasticity is not sensitive to unit measures comes very handy.

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K®hegyi-Horn-Major

Cross elasticity of demand (cont.)

Denition

The cross-price elasticity of demand is the proportionate change in quantity purchased divided by the proportionate change in price of another good. In other words, it shows how much (%) demanded quantity changes if price of another good changes by 1%.

with discrete quantities:

η_xy = ∆x/x

∆Py/Py ≡ ∆x

∆Py

P_y x with continuous quantities:

η_xy=

∂x

∂P_y Py

x

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K®hegyi-Horn-Major

Cross-price elasticity of demand

We can dene the relationship between two commodities with the cross-price elasticity. If the are

substitutes (butter-margarine), or

complements (butter-bread ) commodities.

Denition

X and Y commodities are substitutes, ifη_xy>0 complements, ifη_xy <0

Note

Cross-price elasticity can help in dening the relevant market

Demand elasticities of two pharmaceuticals

Brand 1 Generic 1 Brand 3 Generic 3

Brand 1 −0,38 1,01 −0,20 −0,21

Generic 1 0,79 −1,04 −0,09 −0,10

Brand 3 0,52 0,53 −1,93 1,12

Generic 3 0,21 0,23 2,00 −2,87

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Repeating the math

Denition

The f(x) =f(x₁,x₂, . . . ,x_n),f :Rⁿ →R function is a k^th degree positive homogenous function if

f(tx) =f(tx1,tx2, . . . ,txn) =t^kf(x1,x2, . . . ,xn) =t^kf(x).

Statement

Euler-theorem for (k^th degree positive homogenous functions) Let f(x)Rⁿ→Rbe a dierentiable function. Let S open set such that S ⊆Df, x∈S ⇒tx∈S,∀t ∈R⁺. The f(x)function is a k^th degree homogenous function on the S set i∀x∈S

n

X

i=1

xi∂f(x)

∂x_i =kf(x), where xi is the i^th coordinate of the x vector.

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Relations of elasticities

Statement

Demand functions are zero-degree positive homogenous functions.

Proof

Tow variable case maximize:

U(x,y)→max_x,y

subject to: pxx+pyy=I

maximize:

U(x,y)→max_x,y

subject to: tpxx+tpyy =tI

The x(P_x,P_y,I)and y(P_x,P_y,I)demand functions are solutions to both maximizations, because there is no monetary illusion.

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K®hegyi-Horn-Major

Price elasticity matrix and income elasticity vector







η₁₁ · · · η_1n

... ... ...

... η_ij ...

... ... ...

η_n1 · · · η_nn





 ,





 ε₁

...

ε_i ...

ε_n







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K®hegyi-Horn-Major

Income elasticity of demand

Statement

n

X

j=1

η_ij+ε_i=0;i=1, . . . ,n

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K®hegyi-Horn-Major

Income elasticity of demand (cont.)

Proof

Two variable case Let us use the Euler theorem for zero-degree positive homogenous functions:

px ∂x

∂p_x +py ∂x

∂p_y +I∂x

∂I =0 p_x ∂y

∂px +p_y ∂y

∂py +I∂y

∂I =0

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K®hegyi-Horn-Major

Price elasticity of demand

Proof

Let us divide the equations by x, and y variables:

p_x x

∂x

∂p_x +p_y x

∂x

∂p_y +I x

∂x

∂I =0 p_x

y

∂y

∂px +p_y y

∂y

∂py + I y

∂y

∂I =0 Using the denition of elasticity:

η_xx+η_xy+ε_x =0 η_yx+η_yy+ε_y =0

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K®hegyi-Horn-Major

Price elasticity of demand

Note

The demand for a (normal) good is more price elastic the more close substitutes it has and the higher its income elasticity is.

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K®hegyi-Horn-Major

Price elasticity of demand (cont.)

Denition

Hicks compensated price elasticity:

η^∗_ij≡ ∂x_i^H

∂pj

p_j x_i^H Hicks compensated price elasticity matrix:







η₁₁^∗ · · · η_1n^∗

... ... ...

... η_ij^∗ ...

... ... ...

η^∗_n1 · · · η_nn^∗







Statement

η_ij =η_ij^∗−kjε_i,(i,j=1, . . . ,n)

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K®hegyi-Horn-Major

Price elasticity of demand (cont.)

Proof (Two variable case)

Using the Slutsky theorem:

∂x^M

∂p_x = ∂x^H

∂p_x −∂x^M

∂e x^H

∂x^M

∂py = ∂x^H

∂py −∂x^M

∂e y^H

∂y^M

∂p_x = ∂y^H

∂p_x −∂y^M

∂e x^H

∂y^M

∂py = ∂y^H

∂py −∂y^M

∂e y^H

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K®hegyi-Horn-Major

Price elasticity of demand

Proof (Two variable case)

Let us multiply the equations with px/x,py/x,px/y and py/y and use the duality relations

∂x^M

∂p_x px

x^M =∂x^H

∂p_x px

x^H −∂x^M

∂I x^M px

x^M I I

∂x^M

∂p_y p_y

x^M =∂x^H

∂p_y p_y

x^H −∂x^M

∂I y^M p_y x^M

I I

∂y^M

∂p_x px

y^M =∂y^H

∂p_x px

y^M −∂y^M

∂I x^M px

y^M I I

∂y^M

∂py

p_y

y^M =∂y^H

∂py

p_y

y^M −∂y^M

∂e y^M p_y y^M

I I

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K®hegyi-Horn-Major

Price elasticity of demand

Denition

Net substitutes: η^∗_ij>0 Net complements: η_ij^∗<0

Statement

For all cases η_ii^∗≤0,(i=1, . . . ,n) For normal goods η_ii≤0,(i=1, . . . ,n)

Statement

n

X

j=1

η^∗_ij =0

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K®hegyi-Horn-Major

Price elasticity of demand (cont.)

Proof

Let's sum the equationsη_ij =η^∗_ij−k_jε_i derived from the Slutsky theorem by j.

n

X

j=1

η_ij =

n

X

j=1

η_ij^∗−

n

X

j=1

kjε_i. SincePn

j=1kj =1, thus

n

X

j=1

η_ij =

n

X

j=1

η_ij^∗−ε_i.

We have proved the right side to be zero.

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K®hegyi-Horn-Major

Price elasticity of demand

Consequence

All commodities has to have at least one net substitute.

Statement

The Hicks compensated price elasticities are NOT symmetric.

η_ij^∗6=η^∗_ji

Denition

The Hicks-Allen substitution elasticity:

σ_ij≡ η_ij^∗

k_j,k_j =pjxj

I

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K®hegyi-Horn-Major

Price elasticity of demand (cont.)

Statement

The Hicks-Allen substitution elasticities are symmetric:

σ_ij =σ_ji

Proof

σ_ij ≡ η_ij^∗ kj ≡ ∂x_i^H

∂pj

p_j x_i^H

I

pjxj ≡ ∂x_i^H

∂pj

I x_i^Hx_j σ_ji≡ η_ji^∗

k_i ≡∂x_j^H

∂p_i pi

x_j^H I

p_ix_i ≡ ∂x_j^H

∂p_i I x_j^Hxi

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K®hegyi-Horn-Major

Relations of elasticities

k₁ε₁+. . .+k_iε_i+. . .+k_nε_n=1 P_n

j=1η_ij+ε_i =0;i =1, . . . ,n η_ij=η_ij^∗−k_jε_i,i,j =1, . . . ,n Pn

j=1η_ij^∗=0

σ_ij=σ_ji,i,j=1, . . . ,n

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K®hegyi-Horn-Major

Fitting demand curves

Elasticities of market demand:

ε_X = _∆^∆_M^X^M_X, orε_X = _∂^∂_M^X^M_X, where M is the total income of the society M =Pn

i=1I_i, or the weighted average of their income.

η_X =_∆^∆_P^X

x

Px

X, orη_X =_∂^∂_P^X

x

Px

X

η_XY =_∆^∆_P^X

Y

PY

X , orη_XY= ∂X

∂PY

PY

X

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K®hegyi-Horn-Major

Fitting demand curves (cont.)

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Interpreting elasticities graphically

Geometrical measure of elasticity

Elasticity at a point T along a linear demand curve DD⁰ is the slope of a ray OT from the origin to point T divided by the slope of the demand curve. For a non-linear demand curve such as FF⁰, the demand elasticity at point T is identical with the elasticity at T along the tangent straight-line demand curve DD⁰ .

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Interpreting elasticities graphically (cont.)

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Ordinary demand curves

X =A+BPX, where A and B constant parameters (constant slope).

Example

X =A+BPX+CI+DPY+EPZ+. . .

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Ordinary demand curves (cont.)

X =aP_X^b, or log X =log a+b log PX (constant elasticity).

Example

X =aP_X^bI^cP_Y^dP_Z^e. . .

log X =log a+b log P_X+c log I+d log P_Y +e log P_Z+. . . η_X =b;ε_X =c;η_XY =d;η_XZ =e;. . .

Pl.: Demand for coee:

log C =0,16 log PC+0,51 log Ib+0,15 log Pt−0,009 log T+constant

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Determinants of responsiveness of demand to price

Availability of substitutes. Demand for a commodity will be more elastic the more numerous and the closer the available substitutes.

Luxuries versus necessities: Demand for a luxury tends to be more elastic than demand for a necessity.

High-priced versus low-priced goods.

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Determinants of responsiveness of demand to

price (cont.)

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Coupon rationing

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Coupon rationing (cont.)

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K®hegyi-Horn-Major

Coupon rationing (cont.)

P_xx+P_y ≤I income constraint pxx+pyy ≤N point constraint

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K®hegyi-Horn-Major

Coupon rationing (cont.)

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K®hegyi-Horn-Major

Coupon rationing (cont.)

Binding constraints

The point constraint limits consumption in the range GM (the solid portion of the line GH), and the income constraint in the range ML (the solid portion of KL).

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K®hegyi-Horn-Major

Coupon rationing (cont.)

Average weekly purchases by housekeeping families in cities (lb.) Income 1000 dollars 10002000 20003000 30004000 Over 4000

or less dollars dollars dollars dollars

1942Cheese 0,26 0,57 0,64 0,81 1,03

Canned sh 0,21 0,36 0,56 0,44 0,37

1944 0,24 0,33 0,44 0,49 0,52

Cheese

Canned sh 0,06 0,12 0,17 0,22 0,28

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K®hegyi-Horn-Major

Coupon rationing (cont.)

Time as income constraint:

Average values of variables

Chevron Non-Chevron

Gallons purchased 11,6 8,8

% weekend customers 31,2 26,2

% with passengers 7,3 18,0

% employed full-time 67,9 83,6

% housewives 5,5 3,3