Problem Set II: budget set, convexity

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1 Problem Set II: budget set, convexity Paolo Crosetto Exercises ill be solved in class on January 25th, 2010 Recap: Walrasian Budget set, definition Definition 1 (Walrasian budget set). A Walrasian budget set is given by B p, = {x X : hen prices are p and ealth. p x }, Note that instead of ealth endoments ω can be used. It represents all consumption bundles that are affordable given prices and ealth. Definition 2 (Budget hyperplane). A budget hyperplane is the upper contour of B: H p, = {x X : p x = }. It represent all the consumption bundles that are just affordable (the consumer fully expends his ealth) given p and. Recap: Walrasian Budget set, graphics Recap: Assumptions on demand, definitions Definition 3 (Homogeneity of degree zero). x(p, ) is homogeneous of degree zero if x(αp, α) = x(p, ), p,, α > 0. the consumer is not affected by money illusion if prices and ealth both change in the same proportion, then the individuals consumption choice does not change. 1

2 Definition 4 (Walras la). A demand function x(p, ) satisfies Walras la (at the individual level) if p 0, > 0, p x =, x x(p, ). the consumer fully expends his ealth there is some good that is alays desirable (not all goods are bads) savings are alloed once they are seen as just another commodity Recap: convexity, sets Definition 5 (Convexity, sets). A set S is convex if for all x, y S, then αx + (1 α)y S, α (0, 1). Recap: convexity, functions Definition 6 (Convexity, functions). A function f (x) defined on an interval I is convex if, for all x, y I and for all 0 < α < 1: f (αx + (1 α)y) α f (x) + (1 α) f (y) 1. MWG 2.D.2: building consumption and budget sets A consumer consumes one consumption good x and hours of leisure h. The price of the consumption good is p, and the consumer can ork at a age rate of s = 1. What is the consumption set X? What is the consumer Walrasian Budget set? Write them don analytically and dra geometrically in R

3 Graphical Solution Analytical solution Analytical Solution Consumption set is X = {(x, h) R 2 + : h 24} Budget set is B p, = {(x, h) R 2 + : px + sh } Since e kno that = 24 and s = 1, budget set boils don to B p, = {(x, h) R 2 + : px + h 24} that defines a Budget line ith equation x = 24 h, ith slope 1 p p 2. MWG 2.D.4 (ith changes): convexity consumption and budget sets A consumer consumes one consumption good x and hours of leisure h. The price of the consumption good is p = 1. The consumer can ork at a age rate of s = 1 for 8 hours, and at age s > s for extra time; hoever, he can ork only up to 14 hours a day. Dra the budget set in R 2 + [Hint: it s very similar to the one on MWG] and derive an analytical expression for it; then sho both graphically and analytically that the budget set you dre and derived is not convex. Graphical solution Analytical solution X and budget sets Consumption set is X = {(x, h) R 2 + : 10 h 24} Wealth is given by hours times salary, p and s are at 1; { {(x, h) R 2 Budget set is B p, = + : x + h 24} for 16 h 24 {(x, h) R 2 + : x + s h s } for 10 h < 16 3

4 Convexity To sho that a set is not convex it is enough to find one exception to the rule. Definition of convexity: if a, b S, S is convex if and only if: αa + (1 α)b S, α (0, 1). Let s take to points that belong to B, namely y (24, 0) and x (10, 8 + 6s ) No let s take the midpoint, i.e. a convex combination ith α = 0.5 The midpoint z ill have coordinates (17, 4 + 3s ). The utmost reachable point on the budget line has coordinates (17, 7); since by hypothesis s > s, s > 1; hence 4 + 3s > 7 and z / B. 3. MWG 2.E.1. Suppose L = 3 and consider the demand function x(p, ) defined by: p x 1 (p, ) = 2 p 1 + p 2 + p 3 p 1 p x 2 (p, ) = 3 p 1 + p 2 + p 3 p 2 βp x 3 (p, ) = 1 p 1 + p 2 + p 3 p 3 Does this demand function satisfy homogeneity of degree zero and Walras la hen β = 1? What about hen β (0, 1)? Solution: Homogeneity of degree zero Homogeneity of degree zero means that multiplying all arguments of a function by a constant does not change the function. Formally, f (αa, αb) = f (a, b), α. So, e ill do just that: multiply all arguments by α. x 1 (αp, α) = This is so as α simplifies everyhere. αp 2 αp 1 + αp 2 + αp 3 α αp 1 = x 1 (p, ) the very same calculation can be carried out for the other to cases, ith similar results: Hence, x(p, ) is homogeneous of degree zero. Solution: Walras la (individual level) Walras la states that the consumer fully spends his ealth: x(p, ) satisfies the la if p 0, > 0, p x =, x x(p, ). To check, it is then enough to apply the definition, carrying on the vector product of p and a generic x. We ill hence calculate p x(p, ) = p 1 x 1 (p, ) + p 2 x 2 (p, ) + p 3 x 3 (p, ) Calculations sho this to be equal to p x(p, ) = βp 1 + p 2 + p 3 p 1 + p 2 + p 3 Which is equal to if β = 1, but it is NOT otherise. Hence x(p, ) satisfies Walras la if and only if β = 1. 4

5 Added Magic. MWG 2.E.4: demand, Engel functions Sho that if x(p, ) is homogeneous of degree one ith respect to, i.e. x(p, α) = αx(p, ) for all α > 0, and satisfies Walras la, then ε l (p, ) = 1 for every l. Interpret. Can you say something about D x(p, ) and the form of the Engel functions and curves in this case? Solution: elasticity equals 1, I Start ith the definition of elasticity: ε l (p, ) = x l(p, ), for all l x l (p, ) in hich the first term is the derivative of demand.r.t. ealth for good l i.e., for all goods, it is D x(p, ). Where can e get that from? Let s differentiate by α the definition x(p, α) = αx(p, ) By the chain rule, e kno that: x(p, α) = x(p, α) (α) Since x(p, α) = αx(p, ), the lhs of the equation becomes x(p, α) = x(p, ) = x(p, ) and the rhs becomes hence, summing up x(p, α) (α) = D α x(p, α) D α x(p, α) = x(p, ) D α x(p, α) = x(p, ) Solution: elasticity equals 1, II Evaluating at α = 1 e get D x(p, ) = x(p, ) for every good. hence, for the good l e have hich is hat e ere looking for. x l (p, ) = x l(p, ) By plugging this result into the definition of elasticity, e get as requested by the exercise. ε l (p, ) = x l(p, ) x l (p, ) = x l(p, ) x l (p, ) = 1 5

6 Considerations The result above tells us that an increase in income ill increase the consumption of all goods by the same amount. Using the homogeneity assumption e can deduce that x(p, ) is linear in. Then, x(p, ) defines a function of p only, x(p, 1). Then, the matrix of ealth effects D x(p, ) is a function of p only. The ealth expansion path is the locus of demanded bundles for a given set of prices hen e let the ealth vary: E p = {x(p, ) : > 0} Since linear ealth elasticity implies that ealth effects are only functions of prices, ealth just increases all quantities demanded proportionally; hence the ealth expansion paths are straight lines, rays through x(p, 1) this defines homothetic preferences. Homotheticity, graphics 6