MICROECONOMICS I.
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 week 9
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). Income elasticity of demand week 9 K®hegyi-Horn-Major
Applications and How does the demanded quantity react to the change in income? extensions of demand theory 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 is sensitive to the units of measurement. ∆I e.g.: income raises by 100 HUF, and then we consume 5 dkg=0,05 kg more butter. h i Then ∆x 0 05, if we use dkg ∆I = , Ft h i and ∆x 0 0005, if we use kg ∆I = , Ft 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) Income elasticity of demand (cont.) week 9 K®hegyi-Horn-Major De nition Applications and The income elasticity of demand ( ) is the proportionate change extensions of εx demand theory 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):
∆x/x ∆x I εx = ≡ ∆I /I ∆I x with continuous functions (elasticity at a point):
∂x I εx = ∂I x
Note The slope of the Engel curve is NOT income elasticity. Income elasticity of demand (cont.) week 9 K®hegyi-Horn-Major Statement Applications and An Engel curve with positive slope has income elasticity greater extensions of demand theory 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
De nition Necessity good: 1 > ε > 0 Luxury good: ε > 1 Income elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
Statement The weighted average of an individual's income elasticities equals 1, where the weights are the proportions of the budget spent on p1x1 pi xi pn xn each commodity. So if k1 ≡ I , ..., ki ≡ I , ..., kn ≡ I , then n X k1ε1 + . .. + ki εi + ... + knεn = ki εi = 1 i=1 Income elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory Proof 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:
∂x1 ∂x2 ∂xn p1 + p2 + . .. + pn = 1 ∂I ∂I ∂I Income elasticity of demand week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory Proof Let's expand all terms:
p1x1 I ∂x1 p2x2 I ∂x2 pnxn I ∂xn + + . .. + = 1 I x1 ∂I I x2 ∂I I xn ∂I
k1ε1 + . .. + ki εi + ... + knεn = 1 n X ki εi = 1 i=1 Income elasticity week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory 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. Income elasticity (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory 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 week 9 K®hegyi-Horn-Major
How sensitive is the demanded quantity on the change in price? Applications and extensions of We can de ne the price elasticity similarly to income elasticity: demand theory De nition 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:
∆x/x ∆x Px ηx = ≡ ∆Px /Px ∆Px x with continuous quantities:
∂x Px ηx = ∂Px x Price elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
Statement The price elasticity of a Gien good is positive, while it is negative for an ordinary good:
Ordinary good: ηx < 0
Gien good: ηx > 0 Price elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory 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. Price elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of Statement demand theory
If a consumer's demand for X is elastic, a reduction in price Px will ead to increased spending E ≡ Px x 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 (MEx ):
∂E ∂Px x ∂Px 1 MEx ≡ = Px + x = Px 1 + = Px 1 + ∂x ∂x Px ∂x ηx Cross elasticity of demand week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
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. Cross elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of De nition demand theory 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:
∆x/x ∆x Py ηxy = ≡ ∆Py /Py ∆Py x with continuous quantities:
∂x Py ηxy= ∂Py x Cross-price elasticity of demand week 9 We can de ne the relationship between two commodities with the K®hegyi-Horn-Major cross-price elasticity. If the are Applications and extensions of substitutes (butter-margarine), or demand theory complements (butter-bread ) commodities.
De nition X and Y commodities are
substitutes, if ηxy > 0
complements, if ηxy < 0
Note Cross-price elasticity can help in de ning 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 week 9 K®hegyi-Horn-Major
De nition Applications and extensions of n t The f (x) = f (x1, x2,..., xn), f : R → R function is a k h degree demand theory positive homogenous function if
k k f (tx) = f (tx1, tx2,..., txn) = t f (x1, x2,..., xn) = t f (x).
Statement Euler-theorem for (kth degree positive homogenous functions) Let n f (x) R → R be a dierentiable function. Let S open set such + that S ⊆ Df , x ∈ S ⇒ tx ∈ S, ∀t ∈ R . The f (x) function is a kth degree homogenous function on the S set i ∀x ∈ S
n X ∂f (x) xi = kf (x), ∂xi i=1
th where xi is the i coordinate of the x vector. Relations of elasticities week 9 K®hegyi-Horn-Major
Applications and extensions of Statement demand theory Demand functions are zero-degree positive homogenous functions.
Proof Tow variable case maximize: maximize:
U(x, y) → maxx,y U(x, y) → maxx,y subject to: px x + py y = I subject to: tpx x + tpy y = tI
The x(Px , Py , I ) and y(Px , Py , I ) demand functions are solutions to both maximizations, because there is no monetary illusion. Price elasticity matrix and income elasticity vector week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
η11 ········· η1n ε1 . . . . . .. . . . . . . . , εi . ηij . . . .. . . . . . ηn1 ········· ηnn εn Income elasticity of demand week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
Statement n X ηij + εi = 0; i = 1, ..., n j=1 Income elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory Proof Two variable case Let us use the Euler theorem for zero-degree positive homogenous functions:
∂x ∂x ∂x px + py + I = 0 ∂px ∂py ∂I
∂y ∂y ∂y px + py + I = 0 ∂px ∂py ∂I Price elasticity of demand week 9 K®hegyi-Horn-Major
Applications and extensions of Proof demand theory Let us divide the equations by x, and y variables:
px ∂x py ∂x I ∂x + + = 0 x ∂px x ∂py x ∂I
px ∂y py ∂y I ∂y + + = 0 y ∂px y ∂py y ∂I Using the de nition of elasticity:
ηx x + ηx y + εx = 0
ηy x + ηy y + εy = 0 Price elasticity of demand week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
Note The demand for a (normal) good is more price elastic the more close substitutes it has and the higher its income elasticity is. Price elasticity of demand (cont.) week 9 K®hegyi-Horn-Major De nition
Hicks compensated price elasticity: Applications and extensions of demand theory x H p ∗ ∂ i j ηij ≡ H ∂pj xi Hicks compensated price elasticity matrix:
∗ ∗ η11 ········· η1n . . . . .. . . . . . . ∗ . . ηij . . .. . . . . ∗ ∗ ηn1 ········· ηnn
Statement
∗ ηij = ηij − kj εi , (i, j = 1, ..., n) Price elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of Proof (Two variable case) demand theory Using the Slutsky theorem:
∂x M ∂x H ∂x M = − x H ∂px ∂px ∂e
∂x M ∂x H ∂x M = − y H ∂py ∂py ∂e ∂y M ∂y H ∂y M = − x H ∂px ∂px ∂e ∂y M ∂y H ∂y M = − y H ∂py ∂py ∂e Price elasticity of demand week 9 K®hegyi-Horn-Major
Applications and extensions of Proof (Two variable case) demand theory
Let us multiply the equations with px /x, py /x, px /y and py /y and use the duality relations
x M p x H p x M p I ∂ x ∂ x ∂ x M x M = H − M ∂px x ∂px x ∂I x I x M p x H p x M p I ∂ y ∂ y ∂ y M y M = H − M ∂py x ∂py x ∂I x I y M p y H p y M p I ∂ x ∂ x ∂ x M x M = M − M ∂px y ∂px y ∂I y I y M p y H p y M p I ∂ y ∂ y ∂ y M y M = M − M ∂py y ∂py y ∂e y I Price elasticity of demand week 9 K®hegyi-Horn-Major
Applications and extensions of De nition demand theory ∗ Net substitutes: ηij > 0 ∗ Net complements: ηij < 0
Statement ∗ For all cases ηii ≤ 0, (i = 1,..., n) For normal goods ηii ≤ 0, (i = 1,..., n)
Statement n X ∗ ηij = 0 j=1 Price elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of Proof demand theory ∗ Let's sum the equations ηij = ηij − kj εi derived from the Slutsky theorem by j. n n n X X ∗ X ηij = ηij − kj εi . j=1 j=1 j=1 Since Pn k 1, thus j=1 j =
n n X X ∗ ηij = ηij − εi . j=1 j=1 We have proved the right side to be zero. Price elasticity of demand week 9 K®hegyi-Horn-Major
Applications and extensions of Consequence demand theory All commodities has to have at least one net substitute.
Statement The Hicks compensated price elasticities are NOT symmetric.
∗ ∗ ηij 6= ηji
De nition The Hicks-Allen substitution elasticity:
∗ ηij pj xj σij ≡ , kj = kj I Price elasticity of demand (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of Statement demand theory The Hicks-Allen substitution elasticities are symmetric:
σij = σji
Proof ∗ H H ηij ∂xi pj I ∂xi I σij ≡ ≡ H ≡ H kj ∂pj xi pj xj ∂pj xi xj ∗ H H ηji ∂xj pi I ∂xj I σji ≡ ≡ H ≡ H ki ∂pi xj pi xi ∂pi xj xi Relations of elasticities week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
k1ε1 + ... + ki εi + ... + knεn = 1 Pn 0 i 1 n j=1 ηij + εi = ; = ,..., ∗ ηij = ηij − kj εi , i, j = 1,..., n Pn ∗ 0 j=1 ηij = σij = σji , i, j = 1,..., n Fitting demand curves week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
Elasticities of market demand: ∆X M , or ∂X M , where M is the total income of εX = ∆M X εX = ∂M X the society M Pn I , or the weighted average of their = i=1 i income.
∆X Px ∂X Px ηX = , or ηX = ∆Px X ∂Px X
∆X PY ∂X PY ηXY = , or ηXY ∆PY X = ∂PY X Fitting demand curves (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory Interpreting elasticities graphically week 9 K®hegyi-Horn-Major
Applications and Geometrical measure extensions of demand theory 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 FF 0, the demand elasticity at point T is identical with the elasticity at T along the tangent straight-line demand curve DD0 . Interpreting elasticities graphically (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory Ordinary demand curves week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
X = A + BPX , where A and B constant parameters (constant slope). Example
X = A + BPX + CI + DPY + EPZ + . .. Ordinary demand curves (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of b demand theory X = aPX , or log X = log a + b log PX (constant elasticity).
Example
b c d e X = aPX I PY PZ . ..
log X = log a + b log PX + c log I + d log PY + e log PZ + ...
ηX = b; εX = c; ηXY = d; ηXZ = e; ... Pl.: Demand for coee:
log C = 0, 16 log PC +0, 51 log Ib+0, 15 log Pt −0, 009 log T +constant Determinants of responsiveness of demand to week 9 price K®hegyi-Horn-Major
Applications and extensions of demand theory
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. Determinants of responsiveness of demand to week 9 price (cont.) K®hegyi-Horn-Major
Applications and extensions of demand theory Coupon rationing week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory Coupon rationing (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory Coupon rationing (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
Px x + Py ≤ I income constraint
px x + py y ≤ N point constraint Coupon rationing (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory Coupon rationing (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
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). Coupon rationing (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
Average weekly purchases by housekeeping families in cities (lb.) Income 1000 dollars 10002000 20003000 30004000 Over 4000 or less dollars dollars dollars dollars 1942 Cheese 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 Coupon rationing (cont.) week 9 K®hegyi-Horn-Major
Applications and extensions of demand theory
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