Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest

MICROECONOMICS I.

Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics,

Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest

Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest

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

June 2010

ELTE Faculty of Social Sciences, Department of Economics

MICROECONOMICS I.

week 9

Consumption and demand, part 3

Gergely, K®hegyiDániel, HornKlára, Major

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).

Applications and extensions of demand theory

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 ∆xcould 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

dkg F t

i

and ^∆x_∆I = 0,0005, if we useh

kg F t

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)

Denition 1. The income elasticity of demand (εx) 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 1. The slope of the Engel curve is NOT income elasticity.

Statement 1. 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 2. 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 2. • Necessity good: 1> ε >0

• Luxury good: ε >1

Statement 3. 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, . . . , ki ≡^pi_I^xi, . . . , kn ≡^pn_I^xn, then

k₁ε₁+. . .+k_iε_i+. . .+k_nε_n=

n

X

i=1

k_iε_i= 1

Proof 1. Let us substitute the Marshall demand functions into the budget line:

p1x1(p1, p2, . . . , pn;I) +p2x2(p1, p2, . . . , pn;I) +. . .+ +. . .+pnxn(p1, p2, . . . , pn;I) =I

Let us dierentiate both sides by income:

p1

∂x1

∂I +p2

∂x2

∂I +. . .+pn

∂xn

∂I = 1 Income elasticity of demand

Proof 2. 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+. . .+kiεi+. . .+knεn= 1

n

X

i=1

kiεi= 1

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.

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 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 3. 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

∆Px/Px

≡ ∆x

∆Px

P_x x

• with continuous quantities:

ηx= ∂x

∂Px

Px

x

Statement 4. The price elasticity of a Gien good is positive, while it is negative for an ordinary good:

• Ordinary good: ηx<0

• Gien good: η_x>0

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.

Statement 5. If a consumer's demand for X is elastic, a reduction in pricePxwill ead to increased spending E ≡Pxxon commodity X. If demand is inelastic, a price reduction decreases Ex. If the demand elasticity is unitaryEx remains the same.

Proof 3. Marginal expenditure (M Ex):

M E_x≡ ∂E

∂x =P_x+x∂P_x

∂x =P_x

1 + x Px

∂P_x

∂x

=P_x

1 + 1 ηx

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.

Denition 4. 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:

ηxy=

∂x

∂Py

Py

x

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 5. X andY commodities are

• substitutes, if ηxy>0

• complements, if ηxy<0

Note 2. 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

Repeating the math

Denition 6. The f(x) = f(x1, x2, . . . , xn), 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 6. Euler-theorem for (k^th degree positive homogenous functions) Letf(x) Rⁿ →R be a dier- entiable function. Let S open set such that S⊆Df,x∈S⇒tx∈S,∀t∈R⁺. The f(x)function is ak^th degree homogenous function on theS set i ∀x∈S

n

X

i=1

xi

∂f(x)

∂xi

=kf(x),

wherexi is thei^th coordinate of thexvector.

Relations of elasticities

Statement 7. Demand functions are zero-degree positive homogenous functions.

Proof 4. Tow variable case

• maximize: U(x, y)→max_x,y

• subject to: pxx+pyy=I

• maximize: U(x, y)→maxx,y

• subject to: tpxx+tpyy=tI

Thex(Px, Py, I)andy(Px, Py, I)demand functions are solutions to both maximizations, because there is no monetary illusion.

Price elasticity matrix and income elasticity vector



















η₁₁ · · · η_1n

... ... ...

... ηij ...

... ... ...

ηn1 · · · ηnn

















 ,















 ε₁

...

ε_i ...

εn

















Income elasticity of demand

Statement 8. _n

X

j=1

ηij+εi= 0;i= 1, . . . , n

Proof 5. Two variable case Let us use the Euler theorem for zero-degree positive homogenous functions:

px

∂x

∂px

+py

∂x

∂py

+I∂x

∂I = 0

p_x ∂y

∂px

+p_y ∂y

∂py

+I∂y

∂I = 0 Price elasticity of demand

Proof 6. Let us divide the equations by x, andy variables:

p_x x

∂x

∂px

+p_y x

∂x

∂py

+I x

∂x

∂I = 0 px

y

∂y

∂px

+py

y

∂y

∂py

+I y

∂y

∂I = 0 Using the denition of elasticity:

ηxx+ηxy+εx= 0 ηyx+ηyy+εy = 0

Price elasticity of demand

Note 3. The demand for a (normal) good is more price elastic the more close substitutes it has and the higher its income elasticity is.

Denition 7. Hicks compensated price elasticity:

η^∗_ij≡ ∂x^H_i

∂pj

p_j x^H_i Hicks compensated price elasticity matrix:



















η₁₁^∗ · · · η_1n^∗

... ... ...

... η^∗_ij ...

... ... ...

η^∗_n1 · · · η_nn^∗



















Statement 9.

ηij =η^∗_ij−kjεi,(i, j= 1, . . . , n) Proof 7 (Two variable case). Using the Slutsky theorem:

∂x^M

∂px

= ∂x^H

∂px

−∂x^M

∂e x^H

∂x^M

∂p_y = ∂x^H

∂p_y −∂x^M

∂e y^H

∂y^M

∂px

= ∂y^H

∂px

−∂y^M

∂e x^H

∂y^M

∂p_y =∂y^H

∂p_y −∂y^M

∂e y^H

Price elasticity of demand

Proof 8 (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

∂py

p_y

x^M = ∂x^H

∂py

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

Price elasticity of demand

Denition 8. • Net substitutes: η_ij^∗ >0

• Net complements: η_ij^∗ <0

Statement 10. • For all casesη_ii^∗ ≤0,(i= 1, . . . , n)

• For normal goodsηii≤0,(i= 1, . . . , n)

Statement 11. _n

X

j=1

η^∗_ij= 0

Proof 9. Let's sum the equations ηij =η^∗_ij−kjεi derived from the Slutsky theorem byj.

n

X

j=1

ηij =

n

X

j=1

η_ij^∗ −

n

X

j=1

kjεi.

Since Pn

j=1k_j = 1, thus

n

X

j=1

ηij =

n

X

j=1

η_ij^∗ −εi.

We have proved the right side to be zero.

Price elasticity of demand

Consequence 1. All commodities has to have at least one net substitute.

Statement 12. The Hicks compensated price elasticities are NOT symmetric.

η^∗_ij6=η_ji^∗ Denition 9. The Hicks-Allen substitution elasticity:

σij ≡η^∗_ij kj

, kj =p_jx_j I

Statement 13. The Hicks-Allen substitution elasticities are symmetric:

σ_ij=σ_ji Proof 10.

σij ≡η^∗_ij

k_j ≡ ∂x^H_i

∂p_j pj

x^H_i I

p_jx_j ≡ ∂x^H_i

∂p_j I x^H_i xj

σ_ji≡η^∗_ji ki

≡ ∂x^H_j

∂pi

p_i x^H_j

I pixi

≡ ∂x^H_j

∂pi

I x^H_j xi

Relations of elasticities

• k1ε1+. . .+kiεi+. . .+knεn= 1

• Pn

j=1ηij+εi= 0;i= 1, . . . , n

• ηij=η^∗_ij−kjεi, i, j= 1, . . . , n

• Pn

j=1η_ij^∗ = 0

• σij =σji, i, j= 1, . . . , n

Fitting demand curves

Elasticities of market demand:

• ε_X =_∆M^∆XM_X, orε_X= _∂M^∂XM_X, whereM is the total income of the societyM =Pn

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

• η_X= _∆P^∆X

x

P_x

X, orη_X = _∂P^∂X

x

P_x X

• ηXY =_∆P^∆X

Y

PY

X , orηXY= ∂X

∂PY

PY

X

Interpreting elasticities graphically Geometrical measure of elasticity

Elasticity at a point T along a linear demand curveDD⁰is 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 asF F⁰, the demand elasticity at point T is identical with the elasticity at T along the tangent straight-line demand curveDD⁰ .

Ordinary demand curves

• X =A+BP_X, where AandB constant parameters (constant slope).

Example 1.

X=A+BP_X+CI+DP_Y +EP_Z+. . .

• X =aP_X^b, orlogX = loga+blogP_X (constant elasticity).

Example 2.

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

logX= loga+blogP_X+clogI+dlogP_Y +elogP_Z+. . . ηX =b;εX =c;ηXY =d;ηXZ=e;. . .

Pl.: Demand for coee:

logC= 0,16 logPC+ 0,51 logIb+ 0,15 logPt−0,009 logT+constant 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.

Coupon rationing

• P_xx+P_y≤I income constraint

• pxx+pyy≤N point constraint

Binding constraints

The point constraint limits consumption in the rangeGM (the solid portion of the lineGH), and the income constraint in the rangeM L(the solid portion of KL).

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

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