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Lecture 3: Consumer Theory (cont’d) Topics 1.5 Properties of Demand Keywords: Relative prices, real income Substitution and income effects, Slutsky equation, Law of Demand mf620 Chaiyuth Punyasavatsut 1/2007 1 1.5.1 Relative Prices and real income Relative prices of good x in terms of good y = units of good y foregone per unit of good x acquired. px / py = units of y / a unit of x Example. px / py = 3 means that to get a unit of x, we have to sacrifice 3 units of y. In other words, price of x is 3 times higher than of y. mf620 Chaiyuth Punyasavatsut 1/2007 2 1.5.1 Relative Prices and real income Real income: number of units of goods we could get from our money income. Reflect our purchasing power over a single good. Nominal income / price of good x = units of x. Marshallian demand is HD 0 in prices and income. No money illusion mf620 Chaiyuth Punyasavatsut 1/2007 3 1.5.1 Relative Prices and real income Theorem 1.10: Homogeneity and Budget Balancedness “Given u is continuous, st. increasing, and st. quasiconcave, x(p,y) is HD zero in all prices and income, and it satisfies budget balancedness, px=y for all (p,y)”. Idea: as MU is positive, you will use up your money. For any (p,y), budget is binding. mf620 Chaiyuth Punyasavatsut 1/2007 4 1.5.1 Relative Prices and real income As for Homogeneity, we can use price of one good to be money unit. All things are compared to the price of this good. We call this good as numeraire. Say pn be numeraire; let t = 1/ pn x(p,y) = x(tp,ty) = x ( p1 /pn , ... , pn-1 /pn , y / pn) Demand depends on only n-1 relative prices, and real income. mf620 Chaiyuth Punyasavatsut 1/2007 5 1.5.2 Income and Substitution Effects How consumers respond to a relative price change. Suppose the price of good 1 becomes cheaper, we will buy more of good 1. Very likely, but could this be only an answer? Let talk about few concepts on mf620 Chaiyuth Punyasavatsut 1/2007 6 Effect of a Price Change Clothing (units per month) Assume: •I = $20 •PC = $2 •PF = $2, $1, $.50 10 A 6 U1 5 D B U3 4 Three separate indifference curves are tangent to each budget line. U2 4 12 20 Food (units per month) Effect of a Price Change Price of Food Individual Demand relates the quantity of a good that a consumer will buy to the price of that good. E $2.00 G $1.00 Demand Curve $.50 H 4 12 20 Food (units per month) Two Important Properties of Demand Curves 1) The level of utility that can be attained changes as we move along the curve. 2)At every point on the demand curve, the consumer is maximizing utility by satisfying the condition that the MRS of food for clothing equals the ratio of the prices of food and clothing. mf620 Chaiyuth Punyasavatsut 1/2007 9 Effects of Income Changes Clothing (units per month) Assume: Pf = $1 Pc = $2 I = $10, $20, $30 Income-Consumption Curve 7 D 5 U2 B 3 U3 An increase in income, with the prices fixed, causes consumers to alter their choice of market basket. U1 A 4 10 16 Food (units per month) Effects of Income Changes Price of food An increase in income, from $10 to $20 to $30, with the prices fixed, shifts the consumer’s demand curve to the right. E $1.00 G H D3 D2 D1 4 10 16 Food (units per month) Income Changes An increase in income shifts the budget line to the right, increasing consumption along the income-consumption curve. Simultaneously, the increase in income shifts the demand curve to the right. What else can shift the budget line? Changes in income distribution, population, prices of substitutes, preference, and price expectation mf620 Chaiyuth Punyasavatsut 1/2007 12 1.5.2 Income and Substitution Effects A fall in the price of a good has two effects: Substitution & Income Substitution Effect z Consumers will tend to buy more of the good that has become relatively cheaper, and less of the good that is now relatively more expensive. mf620 Chaiyuth Punyasavatsut 1/2007 13 1.5.2 Income and Substitution Effects A fall in the price of a good has two effects: Substitution & Income Income Effect zConsumers experience an increase in real purchasing power when the price of one good falls. mf620 Chaiyuth Punyasavatsut 1/2007 14 1.5.2 Income and Substitution Effects Substitution Effect The substitution effect is the change in an item’s consumption associated with a change in the price of the item, with the level of utility held constant. When the price of an item declines, the substitution effect always leads to an increase in the quantity of the item demanded. mf620 Chaiyuth Punyasavatsut 1/2007 15 1.5.2 Income and Substitution Effects Income Effect The income effect is the change in an item’s consumption brought about by the increase in purchasing power, with the price of the item held constant. When a person’s income increases, the quantity demanded for the product may increase or decrease. Even with inferior goods, the income effect is rarely Chaiyuth Punyasavatsut large enough tomf620 outweigh the substitution effect. 1/2007 16 Income and Substitution effects: Normal Good Clothing (units per month) R When the price of food falls, consumption increases by F1F2 as the consumer moves from A to B. The substitution effect,F1E, (from point A to D), changes the A relative prices but keeps real income (satisfaction) constant. C1 D B C2 U2 Substitution Effect O F1 Total Effect The income effect, EF2, ( from D to B) keeps relative prices constant but increases purchasing power. U1 E S F2 T Income Effect Food (units per month) Income and Substitution Effects: Inferior Good Clothing (units per month) R Since food is an inferior good, the income effect is negative. However, the substitution effect is larger than the income effect. A B U2 D Substitution Effect O F1 U1 E S Total Effect F2 Income Effect T Food (units per month) p1 Decomposition of a price change p10 p1 p11 X1(p1, p2o, yo) x1h(p1, p2o, uo) x1 SE IE 1.5.2 Income and Substitution effect Total price effect = Substitution effect + Income effect dx1 = ∂x1 + ∂x ∂ I dp1 ∂p1 | u=u* ∂I ∂p1 Since ∂ I /∂p1 = x from budget equation. That is, when price of good 1 increases by one Baht, you need more of x Baht to feel the same after the change. This reflects a loss of purchasing power, so we use –x to replace this term. mf620 Chaiyuth Punyasavatsut 1/2007 20 1.5.2 Income and Substitution effect The term –x represents the amount of income needed to keep u* at the original level. Minus means change in purchasing power is in the opposite direction to a price change. mf620 Chaiyuth Punyasavatsut 1/2007 21 1.5.2 Income and Substitution effect Theorem 1.11: Slutsky Equation “ Let x(p,y), u* maximum utility at (p,y), then ∂x i (p,y) = ∂ x ih (p,u*) - xj (p,y) ∂x i(p,y) ∂p j ∂p j ∂y mf620 Chaiyuth Punyasavatsut 1/2007 22 1.5.2 Income and Substitution effect Theorem 1.11: Slutsky Equation “ Let i=1 and j=1 ∂x 1 (p,y) = ∂ x1h (p,u*) - x 1 (p,y) ∂x1(p,y) ∂p 1 ∂p 1 ∂y mf620 Chaiyuth Punyasavatsut 1/2007 23 1.5.2 Income and Substitution effect Theorem 1.11 “ Let i=1 and j=2 ∂x1 (p,y) = ∂ x1h (p,u*) - x2 (p,y) ∂x1(p,y) ∂p2 ∂p2 ∂y TE mf620 Chaiyuth Punyasavatsut 1/2007 SE IE 24 Theorem 1.11 Slutsky Equation Proof: xih (p, u*) = xi (p, e(p, u*) ) Dif. w.r.t pj and apply the chain rule ∂xih (p, u*) = ∂xi (p,e(p, u*)) + ∂ xi (p,e(p,u*)) ∂e(p,u*) ∂pj ∂pj ∂y ∂pj Use the fact that u*=v(p,y), thus e(p,u*)= e(p, v(p,y))= y ............(1) And the last term is just the Hicksian demand mf620 Chaiyuth Punyasavatsut 1/2007 25 Theorem 1.11 Slutsky Equation ∂e (p, u*) = xjh (p,u*) = xj h (p,v(p,y)) ∂pj The last term is just xj (p,y) ..............(2). Substitute (1) and (2), we obtain ∂xih (p, u*) = ∂xi (p,y) + ∂ xi (p,y) xj (p,y) .....Q.E.D. ∂pj ∂pj ∂y mf620 Chaiyuth Punyasavatsut 1/2007 26 Theorem 1.12 Negative own-SE Let xih(p,u) be Hicksian demand for good i, then ∂xih (p, u*) ≤ 0, i = 1, ..., n. ∂p i Proof: From Shepard’s lemma, ∂e (p, u) = xih (p,u) => ∂2e (p, u) = ∂ xih (p,u) ∂pi ∂pi2 ∂pi Since e is a concave function of p, thus all of its second-order own partial derivatives are nonpositive. mf620 Chaiyuth Punyasavatsut 1/2007 27 Theorem 1.13 Law of Demand A decrease in the own price of a normal good will cause quantity demanded to increase. If an own price decrease causes a decrease in quantity demanded, the good must be inferior. Proof: normal good, look at the Slutsky equation, we know the SE is negative (THM 1.12) and the IE is also negative (since ∂x/ ∂y > 0) mf620 Chaiyuth Punyasavatsut 1/2007 28 Theorem 1.13 Law of Demand Proof: inferior good, look at the Slutsky equation, we know the SE is negative (THM 1.12) and the IE is now positive (since ∂x/ ∂y < 0 by inferiors), it is possible then IE dominates SE, thus the total price effect becomes positive. mf620 Chaiyuth Punyasavatsut 1/2007 29 Theorem 1.14 Symmetric Substitution Terms Let xh (p,u) , e(p,u) is c2, then ∂xih (p, u*) = ∂xjh (p, u*) i, j = 1, ..., n. ∂pj ∂pi Proof. ∂2e (p, u) = ∂ xih (p,u) ∂pi ∂pj ∂pj mf620 Chaiyuth Punyasavatsut 1/2007 30 Theorem 1.15 Negative Semidefinite Substitution Matrix Substitution Matrix contains all Hicksian Substitution terms. But this matrix is just the Hessian Matrix or secondorder partial derivative matrix of the expenditure function. ∂ xih (p,u) = ∂2e (p, u) ∂pj ∂pi ∂pj mf620 Chaiyuth Punyasavatsut 1/2007 31 Theorem 1.16 Symmetric and Negative Semidefinite Slutsky Matrix Replace elements in Substitution matrix using Slutsky equation This property and HD property of demand can be used to test the consumer theory. In empirical demand function, we can test if these properties hold. mf620 Chaiyuth Punyasavatsut 1/2007 32 1.5.3 Some Elasticity Relations Price Elasticity of Demand Recall: Price elasticity of demand measures the percentage change in the quantity demanded resulting from a one percent change in price. ΔQ/Q ΔQ / ΔP EP = = Q/P ΔP/P mf620 Chaiyuth Punyasavatsut 1/2007 33 1.5.3 Some Elasticity Relations Income Elasticity of Demand Recall: Income elasticity of demand measures the percentage change in the quantity demanded resulting from a one percent change in income. ηi = xi /xi y / y mf620 Chaiyuth Punyasavatsut 1/2007 34 1.5.3 Some Elasticity Relations 1. Budget constraint must hold with equality at any given prices and income. y = p1x1 (p,y) + p2x2 (p,y) 2. Engel Aggregation: Sum of income elasticities weighted by its budget share =1. s1 η1 + s2 η2 = 1 where s1=p1x1/y mf620 Chaiyuth Punyasavatsut 1/2007 35 1.5.3 Some Elasticity Relations 3. Cournot Aggregation: weighted sum of crossprice elasticities of good j = - budget share of good j. s1 ∈1j + s2 ∈2j = -sj , j =1, 2 mf620 Chaiyuth Punyasavatsut 1/2007 36 Summary Theorem 1.10 – 1.17 All are implications from UM behavior. 1.10 Homogeneity tells how demand responds to an equiproprotionate change in all prices and income. 1.10 Budget balancedness: income will be used up. 1.11 Slutsky gives us sign restrictions on systems of demands on how it responds to a price change. In addition, tell us how to separate the price effect into SE and IE. mf620 Chaiyuth Punyasavatsut 1/2007 37 Summary Theorem 1.10 – 1.17 1.12 Negative own SE effect (own price change) 1.13 Law of demand 1.14 Symmetric SE matrix (n x n) (Cross price change) 1.15 Negative Semidefinite SE matrix 1.16 Sym and neg semidef Slutsky matrix. (here we allow for all prices and income change on the system of Marshallian demands) mf620 Chaiyuth Punyasavatsut 1/2007 38 Summary Theorem 1.10 – 1.17 1.17 Engel Aggregation tells us how the quantities demanded across demand systems hang together, as a response to an income change and 1.18 to a single price change (Cournot aggregation). mf620 Chaiyuth Punyasavatsut 1/2007 39 Applications of demand theory Marshallian demand, although observable, is not good for welfare comparison, as v(p,y) is based on ordinal concept, so we cannot compare utility of Mr. A vs. of Mr. B. Hicksian demand, although unobservable, is good for welfare comparison. We can use e(p1, u0) – e(p0, u0) across persons and compared. mf620 Chaiyuth Punyasavatsut 1/2007 40 Applications of demand theory What you can do when you know demand for your product, and their responsiveness to price (price Elasticity). What are the determinants of price elasticity? SE: more substitutes=>more elastic IE: luxuries =>more elastic than necessities Budget share: more share =>more elastic (not always true.)Ex. salt is price inelastic because of no close substitutes and necessities, rather than having a tiny share of budget. mf620 Chaiyuth Punyasavatsut 1/2007 41 Applications of demand theory How about Budget share: more share =>more elastic. This is not always true. Ex. salt is price inelastic because of no close substitutes and necessities, rather than having a tiny share of budget. How about high-priced and low-prices goods: high price=>more elastic. Contradiction, we can have a demand curve with constant elasticity along price ranges. mf620 Chaiyuth Punyasavatsut 1/2007 42 Applications of demand theory We can use to analyze the effects of Gasoline price increase Tariff on Thai exports by foreign countries. Village-fund Unemployment benefits Food coupon Quota on imports or exports Producer taxes or consumer taxes (subsidies). mf620 Chaiyuth Punyasavatsut 1/2007 43