EconS 527 Homework 3 Answer Key

Transcription

1 EconS 57 Homework 3 Answer Key. Consider a consumer with Cobb-Douglas utility function u(, ) = α α over two goods, a) Find the Walrasian demand function for goods and. The Lagrangian of this UMP is then L(, ; λ) = α α λ[p + p ω] The first order conditions are L = α α α λp = 0 L = ( α) α ( α) λp = 0 Solving for λ on both first order conditions, we obtain which solving for yields α α α = ( α) α p = ( α)p αp Using now the budget constraint (which is binding), we have p + p = ω = ω p p p and plugging this epression of into the above equality, yields the Walrasian demand for good = ( α)p ( ω p ) αp p p = α ω p and similarly solving for, we obtain the Walrasian demand for good, = ω p ( p Hence, the Walrasian demand function is ( α) ω) p p p α = α p ω (p, ω) = ( α ω, α ω) p p which, as usual for the Cobb-Douglas utility function, represents that the consumer spends a fraction of his wealth on each good, i.e. α for good and α for good. p p b) What restrictions on α would ensure that the consumer demands positive amounts of the goods (interior solution)?

2 In order to have a positive demand for the first good (p, ω) 0, we just need α 0, since ω > 0 by definition. On the other hand, a positive demand for the second good, (p, ω) 0 is satisfied if and only if α 0, that is, if α. Combining both conditions, we can guarantee that the consumer demands non-negative amounts of both goods when α [0, ].. Let denote the number of phone calls, and y denote spending on other goods. The epression of the budge line under Plan A, BL A is y = 60, or y = , as depicted in the solid line of the following figure that originates at y = 60 and which crosses the horizontal ais at = 0. Under Plan B, Tyler s budget line, BL B, is 0. + y = 40, or y = 40 0., as illustrated in the figure by the dashed line that originates at y = 40 and crosses the horizontal ais at = 00. These two budge lines intersect each other at (40 0.) = 60, i.e., = Hence, y = = 40 ( ) = 6.67 Therefore, BL A and BL B intersect at bundle (66.67, 6.67)

3 b. According to WARP, if the consumption bundle under new prices and wealth was affordable under the original prices and wealth, p (p,, w, ) w, then the bundle the decision makes selected under the old prices and wealth cannot be affordable under the new prices and wealth, i.e., p, (p, w) w,. In this contet, where the consumer moves from facing budget line BL B to BL A, WARP states that, if the consumption bundle under BL B, (p,, w, ), is affordable under BL A, it must lie on segment KJ in the above figure, i.e., this is equivalent to the premise of WARP, p (p,, w, ) w. Hence, the bundle selected when facing budget line BL A, (p, w), must be unaffordable under BLB; that is, (p, w) must lie on segment LJ of budget line BL A. Notice that bundles in segment JM are instead affordable under BL B, thus violating WARP. 3. a) Increasing prices and wealth by a common factor λ, we obtain (λp, λω) = λp λ = p = (p, ω) (λp, λω) = λp λ = p = (p, ω) 3 (λp, λω) = λω λ = ω = 3 (p, ω) That is, increasing both prices and wealth by the same factor λ does not change this consumer s demand. Intuitively, if we double the price of all change goods but also double his income, the individual s demand is unaffected. b) Recall that Walras Law states that for a strictly positive price vector (p 0) and a positive wealth 3 level (ω > 0), p = ω, or alternatively, i = ω. Hence, in this contet, 3 i= i = p (p, ω) + p (p, ω) + 3 (p, ω) i= and further rearranging, we obtain 3 = p p + p ( p ) + ω i = p p p p + ω = ω i= Therefore, Walras Law is satisfied, confirming that the individual spends all his income on goods, and 3. c) Let us use a countereample, ω = p = (,,) which yields a demand of (p, ω) = (,,)

4 ω = p = (,,) which yields a demand of (p, ω ) = (,, ) We know that WARP is satisfied if for any pair of prices and wealth (p, ω) and (p, ω ) p (p, ω ) ω and (p, ω ) (p, ω) then p (p, ω) > ω In our eample, the bundle that the consumer selects at the final price wealth pair is affordable under initial prices and wealth, p (p, ω ) = [,,] [ ] = + = ω (since ω = ) However, the consumption bundle at initial prices and wealth, (p, ω), is affordable under final prices and wealth. In particular, p (p, ω) = [,,] [ ] = + = Hence, since ω =, bundle (p, ω) is eactly affordable at final prices and wealth, implying that the conclusion of WARP, p (p, ω) > ω is not satisfied. Therefore, WARP is violated. d) Let us first recall the Slutsky matri: where every component s ik is defined as s s s N s s s N S(p, ω) = [ ] s N s N s NN s ik = i(p,ω) + i(p,ω) p k ω k(p, ω) (Slutsky equation) Hence, Slutsky equation informs about what is the change in the demand for good i after the price of good k varies, once the consumer s wealth is appropriately compensated. Let us now find each of the components of the Slutsky matri for this particular eercise. s = (p, ω) p s = (p, ω) p s 3 = (p, ω) s = (p, ω) p + (p, ω) ω (p, ω) = = 0 + (p, ω) ω (p, ω) = + 0 = + (p, ω) ω 3 (p, ω) = p p + 0 = p 3 + (p, ω) ω (p, ω) = + 0 =

5 s = (p, ω) p s 3 = (p, ω) s 3 = 3(p, ω) p s 3 = 3(p, ω) p s 33 = 3(p, ω) Therefore, the Slutsky matri is + (p, ω) ω (p, ω) = = 0 + (p, ω) ω 3 (p, ω) = p p + 0 = p 3 + 3(p, ω) ω (p, ω) = 0 + p = p p 3 + 3(p, ω) ω (p, ω) = 0 + ( p ) = p p 3 + 3(p, ω) ω 3 (p, ω) = ω p + ω = 0 3 S(p, ω) = 0 p 0 p p 3 p p 0 [ ] 4. a) The consumer solves a UMP given by ma u() subject to p ω Using the shortcut MRS i,j = p j, we obtain interior solutions α i i = p j, or α i p j = i, which together with budget constraint yields a Walrasian demand of i (p, ω) = α iω for every good i () In addition, we can obtain the Lagrange multiplier, λ, from the first-order condition which, combined with () yields and solving for λ we obtain 0 u i = λ, or α i i = λ α i α i ω = λ

6 λ(p, ω) = ω Hence, the marginal value of relaing the constraint (i.e., the shadow price of wealth) is ω. b) The indirect utility function is Hence, the marginal utility of wealth is L v(p, ω) = α i ln ( α iω ) L i= v(p, ω) α i = α ω i α i ω = ω α j = ω i= = i L Interestingly, this result is generalizable to settings in which, given the separable nature of the utility function, the consumer focuses on a subset of goods {,,, L } where L < L, {L +,, L }, etc. and solves a separated UMP for each of these subsets of goods, i.e., one UMP for goods {,,, L }, another UMP for goods {L +,, L }, etc.. The consumer s solution to these separated UMPs must coincide with that in part (a), where the consumer simultaneously considers all L goods.