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Demand Functions Professor Jay Bhattacharya Spring 2001 Demand Part I Demand Functions • Recap: The Consumer’s Optimization Problem • The budget constraint and the tangency condition •AMarshalliandemand function relates the determine the amount of each good the consumer quantity demanded of a good to prices and income will purchase. • Demand depends on all prices X2 • Preferences and constraints together determine the The consumer’s choice of (X ,X ) 1 2 shape of demand (i.e. demand for X1 and X2) depends upon: •prices(p , p ) = 1 2 X1 f ( p1, p2 , I) •income(I) • preferences—U(X1,X2) = X 2 g( p1, p2 , I) X1 Spring 2001 Econ 11--Lecture 5 1 Spring 2001 Econ 11--Lecture 5 2 Comparative Statics Zero Degree Homogeneity of • What happens to demand when prices or income Demand changes? • A function f(x1,x2,…xn) is homogenous of –e.g., if prices f ()()tx ,tx ,...tx = t k f x , x ,...x I double and degree k if 1 n n 1 n n p 2 income doubles, • Marshallian demand functions are what happens to homogenous of degree zero. This fact is demand? I consistent with the absence of “money 2 p 2 illusion.” = Ι X 1 ( p1, p2 , I ) X 1 (2 p1,2 p2 ,2 ) I I 2 p1 p1 Spring 2001 Econ 11--Lecture 5 3 Spring 2001 Econ 11--Lecture 5 4 What happens to demand when income Income Expansion Path changes? (Income-Offer Curve) x • Budget constraint shifts in/out. Slope of budget 2 constraint does not change. I Prices are fixed 2 along the income x p2 2 expansion path I1 Increasing income p2 I0 p2 I0 I1 I2 x p1 p1 p1 1 Spring 2001 Econ 11--Lecture 5x1 5 Spring 2001 Econ 11--Lecture 5 6 Econ 11--Lecture 5 1 Professor Jay Bhattacharya Spring 2001 Engel Curves • But what if the IEP or Engel curve looks like this? An increase in income leads to • Engel curve relates income to quantity more x but less x . demanded. 2 1 A “Normal” Good when income x2 – x1 is an “inferior” good. x 1 rises, the consumer buys more of x1 x1 IEPx1 Engel Curve Income Income Spring 2001 Econ 11--Lecture 5 7 Spring 2001 Econ 11--Lecture 5 8 Normal and Inferior Goods • Normal Good:Demand for a good x Examples increases with income – This implies that the slope of the Engel curve is Normal Goods Inferior Goods positive. ∂X Beef Potatoes > 0. ∂Ι Cars Bus tickets • Inferior Good:Demand for a good x Haircuts at a salon Haircuts by your mother decreases with income – This implies that the slope of the Engel curve is negative. ∂X < 0. ∂Ι Spring 2001 Econ 11--Lecture 5 9 Spring 2001 Econ 11--Lecture 5 10 All Goods Can’t Be Inferior All Goods Can’t Be Inferior • “Proof” #1: If income expands, the IEP cannot • Proof #2: use budget constraint. point toward the origin. + = Ι x x p1x1 p2 x2 2 x2 2 ∂x ∂x IEP p 1 + p 2 = 1 IEP 1 ∂Ι 2 ∂Ι IEP ∂x ∂x Both1 and2 can’t be negative. ∂Ι ∂Ι x x 1 x1 1 Both Normal x1 Inferior x1 Normal x2 Normal x2 Inferior Thus, both x1 and x2 can’t be inferior goods. Spring 2001 Econ 11--Lecture 5 11 Spring 2001 Econ 11--Lecture 5 12 Econ 11--Lecture 5 2 Professor Jay Bhattacharya Spring 2001 A Good Can’t be Inferior at all Income Levels Consider 2 Engel Curves x • Why not? Start with zero income. As income increases, if you x ever consume that good, it is normal (at that income level). Demand • In order for a consumer to purchase less of a good as income increases increases, he must once have consumed some of it. Demand “faster” than increases income. x “slower” than 1 Engel Curve income. I I inferior normal % change in x < % change in I % change in x > % change in I Necessity Luxury Income Spring 2001 Econ 11--Lecture 5 13 Spring 2001 Econ 11--Lecture 5 14 Elasticity Elasticity • The elasticity of y with respect to x is • Elasticity of y with respect to x is defined as defined as the percentage change in y the percentage change in y induced by a induced by a small percentage change in x. small percentage change in X. • Why do we need this concept? It is unit dy x ∆y ∆x free. ε = ⋅ ≈ y,x / – e.g. How much coffee are you willing to trade dx y y x for a bagel? Determined by the slope of indifference curve = MRS. ε = d ln y dy Y ,X since d ln y = – The value of the MRS depends upon the “units” d ln x y of coffee we are using. Spring 2001 Econ 11--Lecture 5 15 Spring 2001 Econ 11--Lecture 5 16 Income elasticity of demand • Engel curves for luxuries have a convex • Elasticity of a good, x, with respect to shape: income. x dx x* slope = dx Ι slope = dI ()x* ,I * ⋅ ≡ e I * dΙ x x,I x* dx x* Definitions > * dI ()x* ,I * I e <0 Inferior e <1 Necessity x,I x,I I * dx Þ ≡ e >1 e >0 Normal e >1 Luxury * I ,x x,I x,I x dI ()x* ,I * I* I Spring 2001 Econ 11--Lecture 5 17 Spring 2001 Econ 11--Lecture 5 18 Econ 11--Lecture 5 3 Professor Jay Bhattacharya Spring 2001 •Ifx is a necessity, then as income increases, Engel’s Law the share of income spent on x decreases: – Define the share of income spent on x as Sx •Engel’sLaw: “Food is a necessity” xp S = x • Expenditure on Food / Income x Ι • 1935-1939 35.4% – I will prove that if x is a necessity: • 1952 32.2% dS • 1963 25.2% x < 0 dΙ • 1998 19% Spring 2001 Econ 11--Lecture 5 19 Spring 2001 Econ 11--Lecture 5 20 log S = log x + log p − log Ι I dS I dx x x x = −1 Totally differentiate this log “share” equation: S x dI x dI = + − ∆ Ι d log S x d log x d log px log I dS Þ x = e −1 Hold prices constant, i.e., set = I ,x d log px 0 S x dI Þ d log S = d log x − d log Ι < x but for necessities, eI ,x 1 d log S x d log x Þ = −1 I dS x dS x d log Ι d log Ι Þ < 0 Þ < 0 so the data confirm S x dI dI Engel’s Law I dS I dx Þ x = −1 S x dI x dI Spring 2001 Econ 11--Lecture 5 21 Spring 2001 Econ 11--Lecture 5 22 • The expenditure weighted sum of income • Proof: Start with the budget constraint elasticities is equal to 1. p x + p x = Ι • Thus, all goods cannot be necessities. Nor 1 1 2 2 dx dx can all goods be luxuries. Þ p 1 + p 2 =1 1 dΙ 2 dΙ S e + S e = 1 1 I ,1 2 I ,2 x Ι dx x Ι dx Þ 1 p 1 + 2 p 2 = 1 Ι x 1 dΙ Ι x 2 dΙ where 1 2 æ x p öæ Ι dx ö æ x p öæ Ι dx ö x p x p Þ ç 1 1 ÷ç 1 ÷ + ç 2 2 ÷ç 2 ÷ = 1 S = 1 1 S = 2 1 è Ι øè x dΙ ø è Ι øè x dΙ ø 1 Ι 2 Ι 1 2 Þ + = S1eI ,1 S2eI ,2 1 Spring 2001 Econ 11--Lecture 5 23 Spring 2001 Econ 11--Lecture 5 24 Econ 11--Lecture 5 4 Professor Jay Bhattacharya Spring 2001 What happens to demand when price “Marshallian” Demand Curve (Demand Curve) changes? • In the graph, we hold constant income and x2 “Price-Consumption Curve” the prices of all other goods. or “Price-Offer” Curve p1 p1 p′ slope = − 1 ′ p p1 p 2 slope = − 1 p2 x1 x1 Spring 2001 Econ 11--Lecture 5 25 Spring 2001 Econ 11--Lecture 5 26 The Law of Demand • The ‘Marshallian” demand curve slopes downward (usually). – The “weak” law of demand. – It is theoretically possible for the Marshallian demand curve to slope upward. • The “Marshallian demand curve is the demand curve that we most often use. Thus, we often just call it the “demand curve.” Spring 2001 Econ 11--Lecture 5 27 Econ 11--Lecture 5 5.
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