u(x) s.t. px w x 0 Denote the solution to this problem by ˆx(p, x). In order to obtain ˆx we may simply solve the standard problem max x 0

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1 Bocconi University PhD in Economics - Microeconomics I Prof M Messner Probem Set 4 - Soution Probem : If an individua has an endowment instead of a monetary income his weath depends on price eves In particuar, in this case we have wp) = p x The consumer probem then becomes max x ux) st px p x Denote the soution to this probem by ˆxp, x) In order to obtain ˆx we may simpy sove the standard probem max x ux) st px w which deivers the standard Warasian demand xp, w) and then substitute w with p x, ie ˆxp, x) = xp, p x) i) Now assume that the demand is a differentiabe function Then ˆx p, x) p k = x p, p x) p k + x p, p x) p) p k Obviousy, we have that p)/ p k = x k Sutsky equation we have Moreover, remember that by the standard x p, p x) p k = p, u) p k Using these two observations we obtain x p, p x) x k p, p x) ˆx p, x) = p, u) x p, p x) x k p, p x) x k ) p k p k Notice that in this mode the sign of the income effect of a change of price p k on the demand for good not ony depends on whether good is norma or inferior but aso on whether or not the quantity consumed of good k exceeds the endowment quantity of that good ii) Let ˆvp, x) = uˆxp, x)) be the indirect utiity function for this mode and notice that ˆvp, x) = vp, p x) Therefore By Roy s Identity we have Using this we obtain ˆvp, x) p ˆvp, x) p = vp, p x) p = vp, p x) p = + vp, p x) x vp, p x) x p, p x) vp, p x) x x p, p x)) Since vp, w)/ > at east under assumptions which guarantee that the consumer aways wants to exhaust his budget) it foows that the consumer must be worse off as the price of good decreases

2 The intuition for this resut is straightforward: If the consumer is seing part of his endowment of good in order to finance a higher consumption of the other good, then a decrease of the price of good means that the vaue of the good which he is trying to se decreases Thus a decrease of p in such a situation has the same effect as a decrease of the income in the standard mode Probem : We know that for a p and u we have e p, u) p = x p, e p, u)) Thus both sides of this equation must vary in the same way as u changes This impies that e p, u) p u = x p, e p, u)) ep, u) u If the preference reation from which the expenditure function is derived satisfies oca nonsatiation then ep, u)/ u > Thus, x p, ep, u))/) > if and ony if e p, u) / p u > Probem 3: i) We know that if a utiity function u ) represents the preference then aso any stricty positive monotonic transformation of u ) is a utiity representation for Denote the indirect utiity function corresponding to u ) by v, ) Under conditions which impy Waras Law the expenditure function corresponding to the utiity function u )), e, ), is stricty increasing in its second argument Hence, for any fixed p the function ũp; x) = ep, ux)) represents the preference Consequenty, for any fixed p, the function fp; p, w) = ũp; x p, w)) is an indirect utiity function Now simpy notice that since fp; p, w) = ep, v p, w) it foows that f measures the indirect utiity in monetary units Given that e p, ux p, w)) = w it foows that f p; p, w) = w Moreover, by Shepard s Lemma we must have that for any p and any pair p, w) ep, v p, w))/ p = p, v p, w) = x p, ep, v p, w))) The differentia equation given in the text has exacty this form except for the fact that there the price of good two is normaized to : e p, v p, w)) p = h p, v p, w)) = x p, ep, v p, w))) ) Morever, for p = p we shoud have e p, v p, w)) must be equa to w ii) Soving the differentia equation for a given pair p, w) is rather straightforward If we denote the partia derivative of the function f wrt its first argument by f p we can write f p p, p, w) fp, p, w) = α p The eft hand side of this expression is simpy the derivative of n f with respect to p The right hand side instead is obtain as derivative of n p α Thus we must have that or equivaenty, n fp, p, w) = n p α + c, fp, p, w) = cp α

3 where c = e c is some constant which depends on the parameters p and w and is determined by the initia condition: We thus have f p, p, w) = c n p α = w c = w p α fp, p, w) = p α We finay, have to argue that if we fix p at some arbitrary eve, then the function g p, w) = fp, p, w) satisfies a properties of a normaized) indirect utiity function In order to do so it is convenient to undo the normaization Suppose g p, w) = A w/ p α where A = p α ) has been obtained through normaization from the function v p, p, w) Then, we must have v p, p, w) = v p p,, ) w p p = g, p w p α ) w = A p w p α p α) We now verify that v indeed satisfies a properties of an indirect utiity function By construction v is homogeneous of degree zero Moreover, v is continuous, stricty increasing in w and stricty decreasing in p i So it ony remains to be shown that v is quasi-convex and that it satisfies Roy s Identity Quasi-convexitiy: Let vp, p, w) vˆp, ˆp, ŵ) and p i λ) = λp i + λ)ˆp i, wλ) = λw + λ)ŵ We have to show that This condition is equivaent to vp λ), p λ), wλ)) vp, p, w) vp λ), p λ), wλ)) vp, p, w) = λw + λ)ŵ w Observe that vp, p, w) vˆp, ˆp, ŵ) impies that ŵ w ˆpα ˆp α p α p α p α p α p λ) α p λ) α Thus the hs of the previous inequaity is no arger than λ + λ) ˆpα ˆp α ) p α p α p α p α p λ) α p λ) α and so it is sufficient to show that this expression is smaer than, or equivaenty, λp α p α + λ)ˆp α ˆp α p λ) α p λ) α Since Cobb-Dougas functions whose exponents sum to are concave this condition must hod true and so we are done Roy s Identity: We have vp, p, w)/ p Aαwp α vp, p, w)/ = vp, p, w)/ p vp, p, w)/ Thus, Roy s Identity is satisfied p α) Ap α p α) α)wp α = A p α) Ap α p α) = αw p = x p, p, w) = α)w p = x p, p, w) 3

4 Probem 4: i) Consider the set UCS x) = { y R L ++ : u y) u x) } Cosedness: We have to show that every converging sequence in UCSx) converges to a point which beongs to UCSx) So take {y n } UCS x) such that y n y Since {y n } converges to y it foows by continuity of u that {uy n )} must converge to uy) Moreover, since y n UCS x) we know that u y n ) u x) for a n Hence, aso the imit of the sequence {uy n )} cannot be stricty smaer than ux) Therefore, uy) ux) which in turn means that y UCSx) Convexity: Take y, z beonging to UCS x) and λ [, ] Then by quasi-concavity of the utiity function we know that u λy + λ) z) min {u y), u z)} Since u y) u x) and u z) u x) it foows that u λy + λ) z) u x) and so λy + λ)z UCS x) ii)+iii) In order to show that there exists a p x such that p x x p x y for a y UCSx), we can rey on the Supporting Hyperpane Theorem SHT) The SHT see the appendix of the Mas-Coe et a) tes us that for any point x of a cosed and convex set C, such that x does not beong to the interior of C, there exists a p x such that p x x p x y for a y C We know aready that for any x the upper contour set UCSx) is both cosed and convex In order to appy the SHT we thus ony have to argue that x is not an interior point of U CSx) But this is straightforward: Since the utiity function is stricty monotonic we know that for a t, ) utx) < ux) Hence any neighborhood of x contains an eement of the compement of UCSx) and thus x does not beong to the interior of UCSx) Next we show that p x Suppose - by contraposition - that there is some such that p x, By strong monotonicity we know that for any ε > the pointx = x+εe beongs to the interior of UCSx) strong monotonicity impies that ux ) > ux) and continuity of u impies that there must be some ba around x such that a eements in that ba give a stricty higher utiity than x) Since x and x differ ony in the -th component and p x, we have p x x p x x Moreover, given that x is an interior point of UCSx) and u is continuous, there must be some t, ) such that utx ) ux) But since p x tx ) < p x x p x x this means that UCSx) contains a point which impies a stricty smaer expenditure at p x than x does Of course, this contradicts the assumption that p x x p x y for a y UCSx) and so we can concude that p x cannot have any component which is not stricty positive iv) Fix x and notice that vp, px) is defined as v p, px) = max u z) z Bp,px) Since x Bp, px) for a p it foows that vp, px) ux) for a p V = {v : v = vp, px) for some p } is bounded beow by ux) Thus the set 4

5 On the other hand we have seen in the preceding parts of this probem that there must exist a p x such that uy) ux) impies p x y p x x But then vp x, p x x) = max z Bpx,p xx) uz) = ux) and so ux) V But if ux) is a ower bound of V which is contained in V then ux) must be the minima eement of V, or equivaenty ux) = min vp, px) p v) The indirect utiity function is homogeneous of degree zero in prices and weath Thus, for each p we have that vp, px) = vp/px, ) Of course, p/px)x = and hence p/px beongs to the feasibe set of the probem min vq, ) ) q {q :q x=} Consequenty the vaue of Probem cannot be arger than the vaue of the probem min vp, px) 3) p Conversey, fix q {q : qx = } and observe that for any p we have that vq, ) = vpxq, px) = vp, px) Thus, the vaue of Probem 3) can be no arger than the vaue of Probem ) vi) According to part v) of this probem the utiity function we are ooking for is the vaue function for the foowing cass of minimization probems one for each x ): min p Of course this probem is equivaent to p α p β s t p x + p x = min p,/x ) ) β p x p α x We know that this probem has a soution Given that the set of admissibe choices is an open interva it foows that a soutions of the probem must satisfy the first order condition or equivaenty, αp α p x x ) β βp α p x x ) β x x = α p x ) βp x = Since this equation admits ony the soution p = α α + β it foows that this must be the soution of our minimization probem By pugging this expression into the objective function we obtain the vaue x ux) = Ax γ xδ, 5

6 where A = αβ)/α+β) >, γ = α > and δ = β > Hence, we can concude that the indirect utiity function v has been derived from a utiity function of the Cobb-Dougas type Probem 5: i) In order to show that Ux) ux) for a x we have to argue that for any x the set {u : x V u)} contains the vaue ux) Of course we must have ux) {u : x V u)} if x V ux)) We wi show now that x V ux)) must indeed hod First observe that for any p, u) the set W p, u) = {x : px ep, u)} contains the set Y u) = {x : ux ) u} since ep, u) = min{px : x Y u)} it foows that px ep, u) for a x Y u) and thus Y u) W u, p)) This aso impies that Y u) V u) = p W u, p) Now take u = ux) Since x Y ux)) the previous observations impy x V ux)) Next we wi have to argue that {u : x V u)} does not contain any û > ux) if x = hp, u ) for some p, u ) We wi do so by contraposition So assume that {u : x V u)} contains an eement û such that û > ux) and x = hp, u ) for some p, u ) The first assumption means that x V û) or equivaenty x W û, p) for a p Thus px ep, û) for a p and so in particuar p x ep, û) We aso know that p x = ep, u ) Now remember that if the utiity function u ) is stricty monotonic and continuous then we must have u = ux) < û Aso reca that strict monotonicity of the utiity function impies that the expenditure function must be stricty increasing in u Combining these observations we obtain p x = ep, u ) < ep, û) p x, which is obviousy nonsense and so we can concude that we must have Ux) = ux) whenever there is a pair p, u ) such that x soves EMPp, u ) ii) Without oss of generaity assume that u x) u x ) Then consider the set V ux )) = {y : py ep, ux )) for a p} This set contains both x and x both beong to Y ux )) Moreover, since this set is convex it must aso contain x α x α V ux )) impies that ux ) {u : x α V u)} and so Ux α ) ux ) Since ux ) > ux α ) it foows therefore that Ux α ) > ux α ) iii) By part i) it is sufficient to show that quasi-concavity of u ) impies that for every x there is a pair p x, u x ) such that x soves EMPp x, u x ) If u ) is quasi-concave then for any x the set UCSx) = {x : ux ) ux)} is convex Moreover, by continuity and strict monotonicity it foows that U CSx) is cosed and that x is not an interior point of this set In Probem 4 we have shown that there must exist a vector p x such that p x x p x y for a y in UCSx) But this means that x must be a soution for EMPp, ux)) and so we are done iv) Consider the set V u) = { x R L + : p x + p x up } p for a p p + p 6

7 and observe that the condition is equivaent to which in turn is equivaent to p x + p x up p p + p for a p p p x + x up /p ) p /p + x sup{ uq q> + q qx } for a p The function fq, u, x ) = uq)/ + q) qx is continuous and stricty concave in q Moreover, observe that since f q q, u, x ) = u/ + q) x it foows that f is stricty decreasing in q whenever x u and hence in this case we have sup q> {fq, u, x } = im q fq, u, x ) = Next, notice that if = x < u then f q is positive for a q and thus sup q> fq, u, ) = im q fq, u, ) = u Finay, consider the case < x < u In this situation we have that f q is positive for q sufficienty cose to and negative for sufficienty arge q Given strict concavity, this means that in these cases the probem max q> fq, u, x ) has a soution which is characterized by the first order condition u + q) x = Soving this equation yieds ˆqu, x ) = u/x Using this we obtain sup q> fq, u, x ) = fˆqu, x ), u, x ) = u + x + u x Taken together the preceding observations impy that V u) = {x R L + : either x u or [ x < u and x u + x u x ]} The fina step of our probem is to cacuate the function U defined by Ux) = sup{u : x V u)} From the definition of V it foows immediatey that x V u) for a u x Moreover, for u > x we have that x V u) if and ony if x u + x u x The expression on the rhs of this inequaity is increasing in u if u > x Thus, the maxima u which satisfies this condition is simpy the one which soves the equation x = u + x u x which is more convenienty written in the form u 4x u x x ) = Soving this equation yieds Ux) = x + x ) x +x x = x x ) x + x + x ) x x ) = x + x ) Notice that this utiity function is of the CES type α = β =, ρ = /) Probem 6: 7

8 i) In cass we have seen that if p k = p k = p k for a k and p p, then CV p, p, w ) = EV p, p, w ) = p p p, p, u ) dp p, p, u ) dp Thus, CV p, p, w ) > EV p, p, w ) if it is the case that, p, u ) >, p, u ) exacty when p > p Observe that since hp, u) = xp, ep, u)) for a p and a u we must have that p, u) u = x p, ep, u)) ep, u) u Since, the expenditure function is stricty increasing in u at east if preferences are ocay non-satiated) it therefore foows that the Hicksian demand for good is decreasing in u at p, u) if and ony if good is an inferior good at p, w) = p, ep, u)) Now observe that p > p impies u > u Thus, if good is inferior at a p, w) = p, ep, u)) for a p such that p = p and p [p, p ] and u [u, u ] we have that p, p, u ) > p, p, u ) for a p [p, p ] To the case p < p a perfecty anaogous argument appies ii) Consider again the case p > p and assume first that good is inferior for a reevant price and weateves CS p, p, w ) = EV p, p, w ) = CV p, p, w ) = p p p x p, p, w) dp p, p, u ) dp p, p, u ) dp Since good is inferior we have x p, p, w) p > hp, p, vp, p, w)) p Combining this with x p, w) = p, vp, w)) and x p, w ) = p, vp, w)) deivers p, p, u ) > x p, p, w) > p, p, u ) for a p [p, p ] Hence, in this case we have CV p, p, w ) > CS p, p, w ) > EV p, p, w ) If good is a norma good at a reevant price and weath pairs, then the ordering of the three measures is inverted iii) If preferences are quasi-inear in good then the demand of a goods k, k > is independent of weath Therefore if we have p, p, u ) = x p, p, w) = p, p, u ) and the previous three integras are a equa 8