i) This is simply an application of Berge s Maximum Theorem, but it is actually not too difficult to prove the result directly.

Transcription

1 Bocconi University PhD in Economics - Microeconomics I Prof. M. Messner Problem Set 3 - Solution Problem 1: i) This is simply an application of Berge s Maximum Theorem, but it is actually not too difficult to prove the result directly. So assume that x(p, w) is a function (i.e. not a correspondence). We have to show that if q n = (p n, w n ) q = (p, w) then x n = x(q n ) x(q). For later reference notice that since {q n } converges we know that for any t (0, 1) there must be a N such that B(q n ) B(tp, (1+t)w) for all N > n. Thus, we may see {x n } as a sequence in a compact set. We will proceed in two steps. In a first step we will show that for each y B(q) there is a sequence {y n } such that y n B(q n ) and y n y. In a second step we will use this fact to show the desired result. So fix y B(q) and take any point ŷ such that pŷ < w (i.e. ŷ is in the interior of the set B(q)). Then p n ŷ < w n for n large enough. Now define (for large enough n) y n = t n y + (1 t n )ŷ where t n is the maximal number in [0, 1] such that y n B(q n ) (since ŷ is in the interior of B(q n ) for n large enough, such a t n must exist). We want to show that y n y. Clearly, this can be the case only if t n 1. Now since {t n } is a sequence in the compact set [0, 1] it must have a cluster point t (i.e. there must be a subsequence {t l } such that t l t). We will argue now that there cannot be any cluster point t < 1. If this were the case then we would have for l large enough p l y l = w l and thus lim p l (t l y + (1 t l )ŷ) w l = p(ty + (1 t)ŷ) w = 0. l But since pŷ < w this would imply py > w, which contradicts our starting assumption that y B(q). Hence, the only cluster point of our sequence is t = 1 and thus t n 1. We know now that for an arbitrarily chosen y B(q) there exists a sequence {y n } such that y n B(q n ) and y n y. Then by definition of the demand function we have that x n y n. Since the sequence {x n } is a sequence of elements of a compact set it must have a cluster point x (i.e. there must be a subsequence {x n k } such that x n k x ). p n k x n k w n k implies px w or equivalently x B(q). Since x n k y n k it follows by continuity of that x y. Since y was chosen arbitrarily it follows that we must have x y for all y B(q). Or put differently, any cluster point of {x n } must be at least as good as any y B(q). But since x(q) is by assumption the only such point in B(q) it follows that x n x(q). ii) First of all observe that the second utility function is simply a strictly monotonic transformation of the first one (ũ (x) = ln [u (x)]). Therefore both utility functions represent the same preferences and thus must give rise to identical demand functions. In order to calculate the demand for the Cobb-Douglas utility function, notice first that this function is strictly monotonic everywhere in in the interior of X = R L +. Moreover, for 1

2 any point x in the interior of the consumption set we have that u(x) > u(y) for all vectors y which have at least one component which is equal to zero. These two observations imply that if x solves the consumer s problem for the pair (p, w), then px = w and x l > 0 for all l. We also know that any solution must satisfy the Kuhn-Tucker necessary conditions. Using the previous observations and the fact that u(x)/ x l > 0 for all x in the interior of X, the K-T conditions reduce for some λ > 0. u(x ) = λp px = w It is straightforward to show that this set of equations has a unique solution (which therefore must be the unique solution of the maximization problem) which is given by x l (p, w) = α l I n α. n p l Notice that the demand for good l does not depend on the prices of any of the other goods. Moreover, the share of income which a consumer with Cobb-Douglas preferences spends on good l is a constant (and thus independent of both prices and income; p l x l (p, w)/i = (α l / n α n)). Problem 2: i) We have to show that linear homogeneity of u implies linear homogeneity of the demand in w. The statement of the problem contains assumptions regarding differentiability of u. We will prove the statement without using these assumptions. We just use strict quasi-concavity which implies that x(p, w) is a function. So let x = x (p, w). Notice that since px w it also follows that for any λ > 0, λx is affordable at (p, λw), that is p (λx) λw. Next, observe that if y B(p, λw), then y/λ B(p, w) (py λw p(y/λ) w). Since x is chosen at (p, w) it follows that u ( y λ) < u (x) for all y B(p, λw). By linear homogeneity we have ( y u u (x) u (y) < λu (x) u (y) < u (λx). λ) This means that λx = x (p, λw). As for the indirect utility function simply observe that v (p, λw) = u (x (p, λw)) = u (λx (p, w)) = λu (x (p, w)) = λv (p, w). In the preceding proof we did not at all use differentiability of the utility function. We will now also provide a proof which relies on differentiability arguments. In our argument we will refer to the Kuhn-Tucker (K-T) conditions. We know that if u is differentiable then x = x(p, w) implies that there is some α 0 such that u(x ) αp, x [ u(x ) αp] = 0 px w 0, α(px w) = 0. 2

3 Since u is strictly quasi-concave it follows that there cannot exist any x 0 such that u(x) = 0 = u(0) (why? if there were such an x then for any α (0, 1) we would have by strict quasi-concavity u(αx) > u(x) = u(0) = 0; by linear homogeinity instead we would have (1/α)u(αx) = u(x) > 0, a contradiction). This also means that we either have, u(x) > u(0) = 0 for all x 0 or u(0) > u(x) for all x 0 (if there were two non-zero bundles x and x such that u(x) > 0 and u(x ) < 0 then by continuity - which is implied by differentiability - there would have to exist some convex combination of x and x, x say, such that u(x ) = 0, which, as we have already seen, is impossible). In the second case 0 is the unique optimal solution for all (p, w) and hence we have 0 = x(p, w) = λx(p, w) = x(p, λw) = 0. In the first case instead we must have (by linear homogeneity) that the unique optimal solution lies on the budget line (we know that the optimal solution x satisfies u(x ) > 0; suppose px < w, then there exists some λ > 1 such that p(λx ) = w and u(λx ) = λu(x ) > u(x ), which contradicts optimality of x and thus we must have px = w). Moreover, in that case we must have that that at each x such that u(x) > 0 there must be some l such that u(x)/ x l > 0 (linear homogeneity implies u(x) = l x l( u(x)/ x l )). This implies that the K-T conditions can be satisfied only for a strictly positive Lagrange multiplier, i.e. α > 0. Next observe that by linear homogeneity we know that for all λ > 0 and all x R L + the equation u(λx) = λu(x) holds. Thus also the derivatives wrt x l of both sides of this equation must coincide: λ u(λx) x l = λ u(x) x l. Since λ > 0 this means that the partial derivatives of u are homogeneous of degree zero. But then if (x, α) satisfies the K-T conditions for the price-wealth pair (p, w), then (λx, α) must satisfy the K-T conditions for the price-wealth pair (p, λw): The conditions u(x )/ x l αp l, x l ( u(x )/ x l αp l ) = 0 and px = w are equivalent to the conditions u(λx )/ x l αp l, λx l ( u(λx )/ x l αp l ) = 0 and λpx = λw. Finally, the result that λx(p, w) = x(p, λw) follows by observing that the K-T conditions are not only necessary but also sufficient for an optimum given that the utility function is (strictly) quasi-concave and strictly increasing in some dimension. ii) In order to show that the utility function is linear homogeneous we first argue that the demand function must be linear homogeneous in in w. This is most easily done under the assumption that the indirect utility function is differentiable. Notice that λv(p, w) = v(p, λw) implies v(p, w) v(p, λw) λ = p l p l But then by Roy s Identity it follows that and v(p, w) w x l (p, λw) = v(p, λw)/ p l v(p, λw)/ w = λ v(p, w)/ p l v(p, w)/ w = v(p, λw). w = λx(p, w). 3

4 Next, observe that combining v (p, λw) = u (x (p, λw)) = u (λx (p, w)) with v (p, λw) = λv (p, w) = λu (x (p, w)) implies u (λx (p, w)) = λu (x (p, w)) That is, the condition u(λx) = λu(x) holds at least whenever x is an optimal choice for some (p, w). Finally observe that since u is strictly quasi-concave it follows that for every x there exists some (p, w) such that x = x(p, w). Problem 3: i) Consider two price-wealth pairs (p, w) and (p, w ) such that p 1 = 1, and assume that the respective optimal choises are x and x (remember that by strict convexity of preferences the optimal choices must be unique). We have to show that x = x = (x 1 + w, x 2,..., x L ), where w = w w (i.e. the extra income is entirely spent on commodity 1). We will do so by contraposition. So assume that x x. Then, x x and thus by quasi-linearty ˆx = x ( w, 0,..., 0) x ( w, 0,..., 0) = x. Now since pˆx = p(x ( w, 0,..., 0)) = px w w w = w ˆx it also follows that ˆx belongs to the budget B(p, w). But ˆx B(p, w) and ˆx x contradicts the assumption that x is the unique optimal choice from B(p, w) and so we can conclude that we must have x = x. Remark: Remember that quasi-linearity in commodity one of the preference implies that it is representable by a utility function of the form u(x) = x 1 + η(x 1 ). If we assume that η is differentiable then the above result (x 1 (p, w) is independent of w) may also be shown by using the FOC of the UMP. In order to see this observe that since u is linear in x 1 it follows that x(p, w) must satisfy Walras Law (the budget must always be exhausted). Moreover, linearity of u in x 1 also implies that the constraint optimization problem max (x 1,x 1) R R L 1 + x 1 + η (x 1 ) s.t. x 1 + p 1 x 1 = w is equivalent to (using the budget equation in order to eliminate x 1 from the objective function) The FOC for this problem max (x 1) R L 1 + w p 1 x 1 + η (x 1 ). η xl (x 1 ) p l 0 and x l (η xl (x 1 ) p l ) = 0, l = 2,..., L are clearly independent of w and so we can conclude that the demand x 1 (p, w) (which must satisfy the FOC) must be independent of w. 4

5 ii) We have argued before that the demand x(p, w) must satisfy Walras Law. Thus, if x 1 (p) is the (wealth independent) demand for the commodities l 2 we have that x 1 (p, w) = w p 1 x 1 (p). This implies that the indirect utility at (p, w) is given by v (w, p) = w p 1 x 1 (p) + η (x 1 (p)). Thus, if we define φ(p) = η(x 1 (p)) p 1 x 1 (p) we have v (w, p) = w + φ(p). iii) First of all, observe that strict quasi-concavity of the utility function u implies that the function η must be strictly quasi-concave too. In order to see this take x 2 and x 2 such that x 2 x 2 and η(x 2 ) η(x 2). We have to show that η(x 2 (λ)) > η(x 2 ) for all λ (0, 1), where of course x 2 (λ) = λx 2 + (1 λ)x 2. For any x 1 we have x 1 + η(x 2) x 1 + η(x 2 ). Thus by strict quasi-concavity of u we have λx 1 + (1 λ)x 1 + η(x 2 (λ)) = x 1 + η(x 2 (λ)) > x 1 + η(x 2 ) for all λ (0, 1) and so we are done. Now consider the UMP(p, w) max x 1 + η (x 2 ) s.t. x 1 + p 2 x 2 = w. (x 1,x 2) R 2 + This problem is equivalent to the problem UMP (p, w) given by max w p 2 x 2 + η (x 2 ) s.t. p 2 x 2 w. x 2 R + The non-negativity constraint on x 1 in UMP(p, w) is irrelevant if and only if in UMP (p, w) the constraint p 2 x 2 w is not strictly binding. That is, if and only if the relaxed problem UMP (p, w) defined as max w p 2 x 2 + η (x 2 ) x 2 R + is solved by a point ˆx 2 w/p 2. Since the objective function of this problem is strictly quasi-concave (we know that η(x 2 ) is strictly quasiconcave; you can easily verify that adding the linear term, p 2 x 2, does not affect strict quasi-concavity) we can conclude that ˆx 2 w/p 2 if and only if the objective function is non-increasing at w/p 2, i.e. iff x 2 > w/p 2 implies η(x 2 ) η(w/p 2 ) p 2 x 2 w. If the function η is differentiable then this condition can be expressed as η (w/p 2 ) p 2. Problem 4: In class we have discussed necessary and sufficient conditions for rationalizability. For the first two demand functions given in the text we will not verify these conditions in detail, because it is rather straightforward to directly guess preferences by which they are generated. In the case of the third demand function we will show instead that it violates WARP, which implies that it cannot be the outcome of preference maximization. 5

6 i) Inspecting the demand function x (p, w) = ( 2w 2p 1 + p 2, w 2p 1 + p 2 immediately reveals that the consumer likes to consume the two goods in the fixed proportion 2 : 1. But that simply means that he considers the two goods perfect complements and so we can conclude that his demand can be rationalized by the preference,, defined by x y if min{x 1, 2x 2 } min{y 1, 2y 2 }. ) ii) The demand function x (p, w) = ( ) w p 1, 0 ( ) 1, w p1 p 2 if w p 1 1 else may be obtained by maximizing the preference represented by the utility function u defined as { x1 if x 1 < 1 u(x) = 1 + x 2 if x 1 1. In order to see this just observe that as long as prices and wealth are such that B(p, w) does not contain any x with x 1 1, it is optimal for the consumer to spend his entire wealth on good 1. If instead the consumer can afford a bundle x such that x 1 1 then it is optimal for him to acquire exactly one unit of good 1 and to spend the rest of the income on good 2. iii) If x (p, w) is such that it satisfy the two equations p 2 x 1 (p, w) = p 1 x 2 (p, w) and p 1 x 1 (p, w) + p 2 x 2 (p, w) = w then we must have that p i x i (p, w) = p w. p2 2 This demand function clearly satisfies homogeneity of degree zero. Now consider the two price-wealth pairs (p, w) = (2, 1, 5) and (p, w ) = (1, 2, 5) and observe that x(p, w) = (2, 1) (1, 2) = x(p, w ). Since p x(p, w) = px(p, w ) = = 4 < 5 it follows that x(p, w, ) B(p, w ) and x(p, w ) B(p, w). This of course means that the demand function does not satisfy the WARP and so it violates one of the necessary conditions for rationalizability. Problem 5: i) Consider the EMP min px x R L + s.t. u (x) u 6

7 where p R L ++. Take an arbitrary x Y = {x : u(x) u} (given that the feasible set is non-empty there is always such an element) and consider the set Y = {x : u(x) u} {x : px px }. This set is non-empty (at least x belongs to it). Moreover, since both Y and {x : px px } are closed and {x : px px } is bounded, it follows that Y must be closed and bounded (and thus compact). But then, given that the expenditure is linear (and thus continuous) in x the problem min px s.t. x Y x R L + must have a solution. Denote this solution by x and observe that for all x Y Y we must have px px px. Thus x must also solve the original EMP. ii) Consider the solution set h(p, u) for the EMP(p, u). If h(p, u) is a singleton there is nothing to show. So assume that x, x h(p, u) and x x. We have to show that for all α [0, 1] we have x(α) = αx + (1 α)x h(p, u). First, observe that if is convex, then the utility function u( ) must be quasi-concave. But quasi-concavity of the utility function u( ) implies that x(α) must belong to the feasible set of EMP(p, u). Moreover, px(α) = α(px)+(1 α)px = αe(p, u)+(1 α)e(p, u) = e(p, u). Thus, for all α [0, 1] the bundle x(α) delivers the minimal expenditure and so we can conclude that x(α) h(p, u). If is strictly convex, then u( ) is strictly quasi-concave. Now assume - by contraposition - that h(p, u) is not a singleton. In particular, let again x, x h(p, u) and x(α) = αx+(1 α)x. Then, by strict quasi-concavity we have u(x(α)) > min{u(x), u(x )} u for all α (0, 1). If x(α) yields a strictly larger utility then by continuity there is a t < 1 such that u(tx(α)) u. But if tx(α) is feasible and tpx(α) = t(αpx + (1 α)px < px = px we obtain a contradiction with the assumption that x and x are optimal. So we can conclude that under the assumption of strictly convex preferences h(p, u) must be a singleton. iii) We have to show that if u(λx) = λu(x) for all x and all λ > 0 then x h(p, u) implies λx h(p, λu). So assume that x h(p, u). Then x Y (u) = {x : u(x ) u}. By linear homogeneity of the utility function we have that u(λx) = λu(x) λu. Thus, λx is in the feasible set of EMP(p, λu). Now suppose (by contraposition) that λx does not solve EMP(p, λu), i.e. assume that there is some x Y (λu) = {x : u(x) λu} such that px < pλx. Then since u(x ) λu we have (1/λ)u(x ) u, or equivalently, u(x /λ) u. Hence x /λ is in the feasible set of the EMP(p, u). But since px < pλx we have px /λ < px which contradicts our assumption that x is a solution of EMP(p, u). We can therefore conclude that if x h(p, u) then λx h(p, λu). Linear homogeneity of e(p, u) in u follows immediately from linear homogeneity of h(p, u) in u: If x h(p, u) then λx h(p, λu) and thus λe(p, u) = λpx = p(λx) = e(p, λu). Problem 6: Consider the situation depicted in the following figure. Under budget B the two individuals choose the bundles x 1 and x 2 respectively. The average of these two bundles is represented by the point x. Under budget B, which is obtained by a rotation of the original budget line around x the two chosen bundles are y 1 and y 2, respectively. Notice that we have 7

8 by construction that x B. Moreover observe that consumer two reduces his consumption of good 2 as we move from budget B to B, while consumer one keeps his consumption of good two constant. Thus, the average of the bundles chosen under B must lie somewhere on the new budget line below the level x 2. This in turn implies that ȳ B. But then x and ȳ could never be rationalized as the choices of a preference maximizing consumer since these choices violate the WARP. Now the question is, whether we could rationalize the choices of the two individuals. First of all observe that the choices of the two consumers do not violate the WARP. Consider now consumer 1 who varies only his consumption of good 1. Such a behavior is easily explained as the outcome of the maximization of a preference relation which is quasi-linear in good 1. The behavior of consumer two may be explained as the choices of an individual for whom good two is a strongly inferior good. x 2 B x 2 y 2 x ȳ x 1 y 1 B x 1 8