INCOME and SUBSTITUTION EFFECTS Two Demand Functions
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INCOME AND SUBSTITUTION EFFECTS [See Chapter 5 and 6] 1 Two Demand Functions m • Marshallian demand xi(p1,…, pn, ) describes how consumption varies with prices and income. – Obtained by maximizing utility subject to the budget constraint. u • Hicksian demand hi(p1,…, pn, ) describes how consumption varies with prices and utility. – Obtained by minimizing expenditure subject to the utility constraint. 2 CHANGES IN INCOME 3 1 Changes in Income • An increase in income shifts the budget constraint out in a parallel fashion • Since p1/p2 does not change, the optimal MRS will stay constant as the worker moves to higher levels of utility. 4 Increase in Income • If both x1 and x2 increase as income rises, x1 and x2 are normal goods Quantity of x2 As income rises, the individual chooses to consume more x1 and x2 C B A U3 U2 U1 Quantity of x1 5 Increase in Income • If x1 decreases as income rises, x1 is an inferior good As income rises, the individual chooses to consume less x1 and more x 2 Quantity of x 2 Note that the indifference C curves do not have to be “oddly” shaped. The B U3 preferences are convex U2 A U1 Quantity of x1 6 2 Changes in Income • The change in consumption caused by a change in income from m to m’ can be computed using the Marshallian demands: ∆ = − x1 x1( p1, p2 ,m )' x1( p1, p2 ,m) m ∂ ∂m ≥ • If x1(p1,p2, ) is increasing in m, i.e. x1/ 0, then good 1 is normal. m ∂ ∂m • If x1(p1,p2, ) is decreasing in m, i.e. x1/ < 0, then good 1 is inferior. 7 Engel Curves • The Engel Curve plots demand for x i against income, m. • 8 OWN PRICE EFFECTS 9 3 Changes in a Good’s Price • A change in the price of a good alters the slope of the budget constraint • When the price changes, two effects come into play – substitution effect – income effect • We separate these effects using the Slutsky equation. 10 Changes in a Good’s Price Suppose the consumer is maximizing Quantity of x 2 utility at point A. If p 1 falls, the consumer will maximize utility at point B. B A U2 U1 Quantity of x1 Total increase in x1 11 Demand Curves • The Demand Curve plots demand for x i against p i, holding income and other prices constant. 12 4 Changes in a Good’s Price • The total change in x1 caused by a change in its price from p1 to p1' can be computed using Marshallian demand: ∆ = − x1 x1( p1 ,' p2 ,m) x1( p1, p2 ,m) 13 Two Effects • Suppose p 1 falls. 1. Substitution Effect – The relative price of good 1 falls. – Fixing utility, buy more x 1 (and less x 2). 2. Income Effect – Purchasing power also increases. – Agent can achieve higher utility. – Will buy more/less of x 1 if normal/inferior. 14 Substitution Effect Quantity of x2 Let’s forget that with a fall in price we can move to a higher indifference curve. The substitution effect is the movement from point A to point C A C The individual substitutes good x1 for good x2 because it is now U 1 relatively cheaper Quantity of x1 Substitution effect 15 5 Substitution Effect • The substitution effect caused by a change in price from p1 to p1' can be computed using the Hicksian demand function: = − Sub. Effect h1( p1 ,' p2 ,U ) h1( p1, p2 ,U ) 16 Income Effect Now let’s keep the relative prices constant at the Quantity of x2 new level. We want to determine the change in consumption due to the shift to a higher curve The income effect is the movement from point C to point B B If x is a normal good, A C 1 the individual will buy U2 more because “real” U1 income increased Quantity of x Income effect 1 17 Income Effect • The income effect caused by a change in price from p1 to p1' is the difference between the total change and the substitution effect: Income Effect = − − − [x1 ( p1 ,' p2 , m) x1 ( p1 , p2 , m)] [h1 ( p1 ,' p2 ,U ) h1 ( p1 , p2 ,U )] 18 6 Increase in a Good 1’s Price Quantity of x2 An increase in the price of good x1 means that the budget constraint gets steeper The substitution effect is the C movement from point A to point C A The income effect is the B movement from point C U1 to point B U2 Quantity of x1 Substitution effect Income effect 19 Hicksian & Marshallian Demand • Marshallian demand – Fix prices (p 1,p 2) and income m. – Induces utility u = v(p 1,p 2,m) – When we vary p 1 we can trace out Marshallian demand for good 1 • Hicksian demand (or compensated demand) – Fix prices (p 1,p 2) and utility u – By construction, h 1(p 1,p 2,u)= x 1(p 1,p 2,m) – When we vary p 1 we can trace out Hicksian demand for good 1. 20 Hicksian & Marshallian Demand • For a normal good, the Hicksian demand curve is less responsive to price changes than is the uncompensated demand curve – the uncompensated demand curve reflects both income and substitution effects – the compensated demand curve reflects only substitution effects 21 7 Hicksian & Marshallian Demand p1 At p1 the curves intersect because the individual’s income is just sufficient to attain the given utility level U p1 x1 h1 x1 Quantity of x1 22 Hicksian & Marshallian Demand At prices above p , income p 1 1 compensation is positive because the individual needs some help to remain on U p1’ p1 x1 h1 x ’ x'’ 1 1 Quantity of x1 23 Hicksian & Marshallian Demand p1 At prices below px, income compensation is negative to prevent an increase in utility from a lower price p1 p’ 1 x1 h1 x’ x’’ 1 1 Quantity of x1 24 8 Normal Goods • Picture shows price rise. • SE and IE go in same direction 25 Inferior Good • Picture shows price rise. • SE and IE go in opposite directions. 26 Inferior Good (Giffen Good) • Picture shows price rise • IE opposite to SE, and bigger than SE 27 9 SLUTSKY EQUATION 28 Slutsky Equation • Suppose p 1 increase by ∆p1. 1. Substitution Effect. – Holding utility constant, relative prices change. ∂h – Increases demand for x by 1 ∆p 1 ∂ 1 p1 2. Income Effect – Agent’s income falls by x* 1×∆p1. ∂x* – Reduces demand by x* 1 ∆p 1 ∂m 1 29 Slutsky Equation • Fix prices (p 1,p 2) and income m. • Let u = v(p 1,p 2,m). • Then ∂ ∂ ∂ x*( p , p ,m) = h ( p , p ,u) − x*( p , p ,m)⋅ x*( p , p ,m) ∂ 1 1 2 ∂ 1 1 2 1 1 2 ∂ 1 1 2 p1 p1 m • SE always negative since h 1 decreasing in p 1. • IE depends on whether x 1 normal/inferior. 30 10 Example: u(x 1,x 2)=x 1x2 • From UMP 2 * = m = m x1 ( p1, p2 , m) and v ( p1, p2 ,m) 2 p1 4 p1 p2 • From EMP 1/2 p = 2 = 2/1 h1( p1, p2 ,u) u and (e p1, p2 ,u) (2 up1 p2 ) p1 • LHS of Slutsky: ∂ 1 x* ( p , p ,m) = − mp −2 ∂ 1 1 2 1 p1 2 • RHS of Slutsky: ∂ ∂ 1 1 1 1 h − x* ⋅ x* = − u 2/1 p − 2/3 p 2/1 − mp −2 = − mp −2 − mp −2 ∂ 1 1 ∂m 1 1 2 1 1 1 p1 2 4 4 4 31 CROSS PRICE EFFECTS 32 Changes in a Good’s Price • The total change in x2 caused by a change in the price from p1 to p1' can be computed using the Marshallian demand function: ∆ = * − * x2 x2 ( p1 ,' p2 ,m) x2 ( p1, p2 ,m) 33 11 Substitutes and Complements • Let’s start with the two-good case • Two goods are substitutes if one good may replace the other in use – examples: tea & coffee, butter & margarine • Two goods are complements if they are used together – examples: coffee & cream, fish & chips 34 Gross Subs/Comps • Goods 1 and 2 are gross substitutes if ∂x* ∂x* 1 > 0 and 2 > 0 ∂ ∂ p2 p1 • They are gross complements if ∂x* ∂x* 1 < 0 and 2 < 0 ∂ ∂ p2 p1 35 Gross Complements Quantity of x 2 When the price of x2 falls, the substitution effect may be so small that the consumer purchases more x1 and more x2 In this case, we call x and x x’2 1 2 gross complements x2 U1 ∂ ∂ U0 x1/ p2 < 0 x1 x’1 Quantity of x1 36 12 Gross Substitutes Quantity of x 2 When the price of x2 falls, the substitution effect may be so large that the consumer purchases less x1 and more x2 x’ 2 In this case, we call x1 and x2 gross substitutes x2 U1 ∂ ∂ x1/ p2 > 0 U0 x’ 1 x1 Quantity of x1 37 Gross Substitutes: Asymmetry • Partial derivatives may have opposite signs: – Let x 1=foreign flights and x 2=domestic flights. – An increase in p 1 may increase x 2 (sub effect) – An increase in p 2 may reduce x 1 (inc effect) • Quasilinear Example: U(x,y) = ln x + y – From the UMP, demands are m x1 = p2 /p1 and x 2 = ( – p2)/p 2 – We therefore have ∂x /∂p > 0 and ∂x /∂p = 0 1 2 2 1 38 Net Subs/Comps • Goods 1 and 2 are net substitutes if ∂h ∂h 1 > 0 and 2 > 0 ∂ ∂ p2 p1 • They are net complements if ∂h ∂h 1 < 0 and 2 < 0 ∂ ∂ p2 p1 • Partial derivatives cannot have opposite signs – Follows from Shepard’s Lemma (see EMP notes) • Two goods are always net substitutes.