Recitation 2-09/01/2017 (Solution)

Recitation 2-09/01/2017 (Solution) 1. Checking properties of the Cobb-Douglas utility function. Consider the utility function u(x) Y n i1 x i i ; where x denotes a vector of n di erent goods x 2 R n +, and 1 > i > 0. Check if this utility function satis es: additivity, homogeneity of degree k, and homotheticity. First, note that this utility function is just a generalization of the Cobb-Douglas utility function to n goods. Indeed, for n 2 goods u(x) Y n i1 x i i x 1 2, where 1 ; 2 > 0 Additivity. The utility function is not additive, since the marginal utility of additional amounts of good k, u k (x), is u k (x) Y k n x k i1 x i i and therefore, depends on the amounts of other goods consumed, i.e., @u k(x) @x j 6 0. [For a utility function to be additive, we should have obtained that the marginal utility of good k, u k (x) is independent on the amount of other goods, i.e., @u k(x) @x j 0 for all j 6 k.] This can also be con rmed for the case of n 2 goods, u(x) x 1 2, where the marginal utility of good 1 is u 1 (x) 1 x 1 1 2 1 x 1 2 x 1 which depends on both x 1 and x 2. For this utility function to be additive, we should have obtained that u 1 (x) does not depend on x 2. A similar argument applies to u 2 (x). Homogeneity. Let us now check its degree of homogeneity. Simultaneously increasing all of its arguments by a common factor, we obtain u(x) Y n i1 (x i) i np i1 i Y n i1 x i i np i i1 u(x) P Therefore, utility function u(x) is homogeneous of degree n i. In the case of a Cobb-Douglas utility function for only two goods, u(x) x 1 2, we have that 1 i1

the degree of homogeneity is n2 P i 1 + 2. Intuitively, when 1 + 2 > 1(< 1), i1 a common increase in the amounts of all goods produces a more-than-proportional (less-than-proportional, respectively) increase in the consumer s utility level. If 1 + 2 1, then a common increase,, in the consumption of all goods, generates a proportional increase in the utility level of this individual, i.e., his utility increases by exactly. np i > 1 or < 1. i1 A similar argument applies to n goods, where either Homotheticity. Finally, the utility function satis es homotheticity since it is homogeneous, and thus we can apply a monotonic transformation g() on u(x) that produces a homothetic function v(x) g (u(x)). Nonetheless, let us check for homotheticity as a practice. In particular, let us rst nd the marginal rate of substitution between any two goods k 6 l MRS l;k k l x l x k As a consequence, the MRS l;k between two goods k 6 l only depends on the ratio of these two goods that the consumer enjoys, but is independent on any other good h. Therefore @MRS l;k @x h 6 0, for any third good h 6 k 6 l, implying that the MRS l;k (the slope of the consumer s indi erence curves) only depends on the proportion of good l and k that the individual consumes. Graphically, this implies that the slope of the indi erence curve coincides along any ray from the origin (since rays from the origin maintain the ratio between x l and x k constant). 2. Testing more properties of the Cobb-Douglas utility function. Check if the Cobb-Douglas utility function u(x 1 ; x 2 ) x, where ; > 0, satis es the following properties: (a) local non-satiation; (b) decreasing marginal utility for both goods 1 and 2; (c) quasiconcavity; and (d) homotheticity. (a) Local non-satiation (LNS). When working with a di erentiable utility function we can check LNS by directly checking for monotonicity (since monotonicity implies LNS). In order to test for monotonicity, we just need to con rm that the marginal utility from additional amounts of goods 1 and 2 are non-negative, @u(x 1 ; x 2 ) @x 1 x 1 x x 1 > 0 if and only if > 0 @u(x 1 ; x 2 ) @x 2 x 1 x 1 2 x x 2 > 0 if and only if > 0 2

In fact, since the marginal utility of increasing either good is strictly positive, the Cobb-Douglas utility function not only satis es monotonicity, but also strong monotonicity. (b) Decreasing marginal utility. We need to show that the marginal utilities we found above are nonincreasing. That is, @ 2 u(x 1 ; x 2 ) @x 2 1 ( 1)x 2 0 if and only if 1 @ 2 u(x 1 ; x 2 ) @x 2 2 ( 1)x 2 0 if and only if 1 Hence, while additional units of good 1 or good 2 increase this individual s utility, they do it at a decreasing rate. (c) Quasiconcavity. Let us rst simplify the expression of the utility function by applying a monotonic transformation on u(x 1 ; x 2 ), since any monotonic transformation of a utility function maintains the same preference ordering. In this case, we apply z 1 ln u() ln x 1 + ln x 2 We now need to nd the bordered Hessian matrix, and then nd its determinant. If this determinant is greater than (or equal to) zero, then this utility function is quasiconcave; otherwise it is quasiconvex. 1 The bordered Hessian matrix is 0 z 1 z 2 z 1 z 11 z 12 z 2 z 21 z 22 0 x 1 x 1 x 2 0 x 2 1 x 2 0 x 2 2 and the determinant of this matrix is 0 + 0 + x 1 x 1 x 2 2 x 2 0 x 2 1 x 2 0 + 2 + 2 x 2 1x 2 2 x 2 1x 2 2 ( + ), x 2 1x 2 2 which is positive for all x 1 ; x 2 2 R +, ultimately implying that the Cobb-Douglas utility function u() is quasiconcave. (d) Homothetic preferences. We know that the Cobb-Douglas utility function is homegeneous, and that all homogeneous functions are homothetic. Hence, the Cobb- 1 See Simon and Blume s Mathematics for Economists for references about the bordered Hessian matrix

Douglas utility function must be homothetic. For a more algebraic proof, however, let us rst nd the marginal rate of substitution MRS 1;2 x 1 : x 1 x 1 2 Scaling up the amounts of all goods by a common factor t, we obtain MRS 1;2 (tx 1) 1 (tx 2 ) (tx 1 ) (tx 2 ) 1 t 1+ t + 1 x 1 x 1 x 1 2 x 1 x 1 x 1 2 which shows that the MRS 1;2 does not change when we scale up all goods by a common factor t, i.e., if we depict a ray from the origin (where the ratio between x 1 and x 2 is constant), indi erence curves would have the same slope at the point where they are crossed by the ray. A few remarks on Homothetic preferences. When preferences are homothetic, the MRS between the two goods is just a function of the consumption ratio x between the goods, 1 x 2, but it does not depend on the absolute amounts consumed. As a consequence, if we double the amount of both goods, the MRS (i.e., the willingness of the individual to substitute one good for another) does not change; as depicted in gure 2.1. Figure 2.1. Homothetic preferences. Recall that this type of preferences induce wealth expansion paths that are straight lines from the origin, i.e., if we double the wealth level of the individual, then his wealth expansion path (the line connecting his demanded bundles for the initial and the new wealth level) are straight lines. A corollary of this property is that the demand function obtained from homothetic preferences must have an income-elasticity equal to 1, i.e., when the consumer s 4

income increases by 1%, the amount he purchases of any good k must increase by 1% as well. Examples of preference relations that are homothetic: Cobb-Douglas (as in the previous example), preferences over goods that are considered perfect substitutes, preferences over goods that are considered perfect complements, and CES preferences. homothetic. In contrast, quasilinear preference relations are not. [Finding Walrasian demands-i.] Determine the Walrasian demand x(p; w) (x 1 (p; w); x 2 (p; w)) and the indirect utility function v(p; w) for each of the following utility functions in R 2 +. Brie y describe the indi erence curves of each utility function and nd the marginal rate of substitution, MRS 1;2 (x). Please consider the following two points in your analysis. Existence: First, in all three cases the budget set is compact (it is closed, since the bundles in the frontier are available for the consumer, and bounded). Additionally, all utility functions are continuous. Therefore, we can apply Weierstrass theorem to conclude that each of the utility maximization problems (UMPs) we consider has at least one solution. Binding constraints: We know that, if preferences are locally non-satiated, then the budget constraint will be binding, i.e., the consumer will be exhausting all his wealth. We can easily check that these utility functions are increasing in both x 1 and x 2, which implies monotonicity and, in turn, entails local non-satiation. Hence, we can assume thereafter that the budget constraint is binding. (a) Cobb-Douglas utility function, u(x) x 1x 4 2. 1. This utility function is a Cobb-Douglas utility function, with smooth indifference curves that are bowed-in towards the origin. Regarding the marginal rate of substitution between goods x 1 and x 2, MRS x1 ;x 2, we have MRS x1 ;x 2 @u(x) @x 1 @u(x) @x 2 x2 1x 4 2 4x 1x 2 x 2 4x 1 2. The UMP is given by subject to max x 1 ;x 2 u(x) x 1x 4 2 x 1 + x 2 w 5

x 1 ; x 2 0 As mentioned above, the budget constraint will be binding. Furthermore, since the utility from consuming zero amounts of either of the goods is zero, i.e., u (0; ) u (; 0) 0, and the consumer s wealth is strictly positive, w > 0, then it can never be optimal to consume zero amounts of either of the goods. Therefore, we do not need to worry about the nonnegativity constraints x 1 ; x 2 0, i.e., there are no corner solutions. The Lagrangian of this UMP is then L(x 1 ; x 2 ; ) x 1x 4 2 [ x 1 + x 2 w] The rst order conditions are: @L @x 1 x 2 1x 4 2 0 @L @x 2 4x 1x 2 0 Solving for on both rst order conditions, we obtain x 2 1x 4 2 4x 1x 2 () x 2 4x 1 This is the well-known equal bang for the buck condition across goods at utility maximizing bundles. (Intuitively, the consumer adjusts his consumption of goods 1 and 2 until the point in which the marginal utility per dollar on good 1 coincides with that of good 2.) Using now the budget constraint (which is binding), we have x 1 + x 2 w () x 1 w x 2 and substituting this expression of x 1 into the above equality, yields the Walrasian demand for good 2 x 2 4 w x 2 () x 2 4 w 7 and similarly solving for x 1, we obtain the Walrasian demand for good 1, w x 1 4x 1 () x 1 7 w 6

Hence, the Walrasian demand function is w x(p; w) ; 4 w 7 7 And the indirect utility function v(p; w) is given by plugging the Walrasian demand of each good into the consumer s utility function, which provides us with his utility level in equilibrium, as follows: w 4 w v(p; w) 7 7 4 4 4 7 7 w 7 p 1p 4 2 (b) Preferences for substitutes (linear utility function), u(x) x 1 + 4x 2. In order to draw indi erence curves for this utility function, just consider some xed utility level, e.g., u 10, and then solve for x 2, obtaining x 2 u x 4 4 1. Note that the resulting expressions are functions of x 1 only, and importantly, they are linear in x 1 ; as depicted in gure 2.1. Intuitively, this indicates that both goods can be substituted at the same rate, regardless of the amount of goods the consumer owns of every good (goods are perfect substitutes). The MRS x1 ;x 2 con rms this intuition, since it is constant for any amount of x 1 and x 2, MRS x1 ;x 2 @u(x) @x 1 @u(x) @x 2 4 Figure 2.1. Indi erence curves of u(x) x 1 + 4x 2. The MRS x1 ;x 2 condition in this case entails 4, or 4, which represents in the left (right) the marginal utility per dollar spent on good 1 (good 2, respectively), i.e., the bang for the buck on each good. When > 4, the consumer seeks to purchase good 1 alone, giving rise to a corner solution with x 1 (p; w) w 7 and x 2 (p; w) 0 as Walrasian demands.

Similarly, when < 4 a corner solution emerges with only good 2 being consumed, i.e., x 1 (p; w) 0 and x 2 (p; w) w. Finally, when 4, a continuum of equilibria arise as the consumer is indi erent between dedicating more money into good 1 or good 2; that is, all (x 1 ; x 2 )-pairs on the budget line x 1 + x 2 w are utility-maximizing bundles. For completeness, we next show that we can obtain the same solutions if we were to set up the consumer s UMP, his associated Lagrangian, and take Kuhn-Tucker conditions. The UMP in this case is max x 1 ;x 2 u(x) x 1 + 4x 2 subject to x 1 + x 2 w x 1 ; x 2 0 As mentioned above, the budget constraint will be binding. The nonnegativity constraints, however, will not necessarily be binding, implying that in certain cases the consumer might choose to select zero amounts of some good. Therefore, we face a maximization problem with inequality constraints, x 1 ; x 2 > 0, and hence must use Kuhn-Tucker conditions. First, we set up the Kuhn-Tucker style Lagrangian of this UMP L(x 1 ; x 2 ; 1 ; 2 ; ) x 1 + 4x 2 1 [ x 1 + x 2 w] + 2 x 1 + x 2 The Kuhn-Tucker conditions are: @L @x 1 1 + 2 0 @L @x 2 4 1 + 0 x 1 + x 2 w x 1 ; x 2 0 1 [ x 1 + x 2 w] 0, 2 x 1 0, and 1 x 2 0 While we include all Kuhn-Tucker conditions for this type of maximization problems, some of them can be eliminated, since we know that the budget constraint is binding, i.e., x 1 + x 2 w. Additionally, solving for 1 in 8

the rst two expressions, we obtain + 2 4 + ((1)) Now we are ready to consider the solutions that can arise in the four possible cases in which the nonnegativity constraints can be met. These cases are 1. 2 0 and 0 i.e., x 1 > 0 and x 2 > 0 2. 2 0 and 6 0 i.e., x 1 > 0 and x 2 0. 2 6 0 and 0 i.e., x 1 0 and x 2 > 0 4. 2 6 0 and 6 0 i.e., x 1 0 and x 2 0 CASE 1. Interior solution, x 1 > 0 and x 2 > 0, i.e., 2 0 and 0. This implies that equation (1) becomes 4 () 4 Hence, we can only have an interior solution when the price ratio is exactly 4. In such a case, the budget line totally overlaps a indi erence curve with the same slope, (as depicted in gure 2.2), and the consumer can choose any 4 consumption bundle on the budget line. In particular, any bundle (x 1 ; x 2 ) satisfying x 1 + x 2 w is optimal as long as the price ratio is exactly 4. Figure 2.2. Case 1: Interior solutions CASE 2. Lower corner solution, x 1 > 0 and x 2 0, i.e., 2 0 and > 0. Since wealth is fully spent on good 1, we have that x 1 w. In order to determine when this solution is optimal, we can use expression (1), and the 9

fact that 2 0, to obtain 4 + () 4 and since > 0 we nd that this solution is optimal when 4 > 0 () < 4 Graphically, this happens when the linear indi erence curves are steeper than the budget line, i.e., MRS >, as depicted in gure 2.. Figure 2.. Case 2: Corner solution-i CASE. Upper corner solution, x 1 0 and x 2 > 0, i.e., 2 > 0 and 0. Since wealth is fully spent on good 2, we have that x 2 w. In order to determine when this solution is optimal, we can use expression (1), and the fact that 0, which yields + 2 4 () 2 4 and since 2 > 0 we nd that this solution becomes optimal when 2 4 > 0 () > 4 Graphically, this occurs when the linear indi erence curves are atter than 10

the budget line, i.e., MRS <, as illustrated gure 2.4. Figure 2.4. Case : Corner solution-ii CASE 4. Corner solution, x 1 0 and x 2 0, i.e., 2 > 0 and > 0. This solution would not exhaust this individual s wealth, i.e., it would imply x 1 + x 2 < w. Indeed, since the utility function is monotone, the budget constraint should be binding (the consumer should spend his entire wealth). Hence, case 4 cannot arise under any positive price-wealth pairs (p; w). SUMMARY. We can now summarize the Walrasian demand correspondence 8 0; >< wp2 if > 4 x(p; w) any (x 1 ; x 2 ) 2 R 2 + s.t. x 1 + x 2 w if 4 >: ; w ; 0 if < 4 Plugging this Walrasian demand into the utility function, yields the indirect utility function v(p; w) ( 4 w if 4 w if < 4 4. [WARP for undergrads-i] The budget of a given student is entirely spent on pizza and video games during the months of September and October, and then the student needs to borrow money from his parents for the rest of the semester! Sept Oct $ $8 $4 $6 x 1 4 x 2 4 11

Let x 0 (x 0 1; x 0 2) denote his consumption of goods 1 (pizza) and 2 (video games) during September, and let x 1 (x 1 1; x 1 2) be his consumption during October. Similarly, let p 0 (p 0 1; p 0 2) be the vector of prices for goods 1 and 2 during September, and ( 1; 2) be this vector during October. (a) Determine if his consumption bundle during October, x 1, is a ordable at September prices p 0, i.e., check if p 0 x 1 < p 0 x 0. His consumption bundle in September, under September prices, costs p 0 x 0 (4 ) + ( 4) 24 and if the student were to buy his October consumption bundle x 1 under September prices, p 0, he would have to pay p 0 x 1 ( ) + (4 4) 25 Hence, the October bundle x 1 is not a ordable at September prices p 0. (b) Determine if his consumption bundle during September, x 0, is a ordable at October prices, i.e., check if x 0 < x 1. His consumption bundle in October, under October prices, costs x 1 (8 ) + (6 4) 48 and if the student were to buy his September consumption bundle x 0 under the new October prices,, he would have to pay x 0 (8 4) + (6 ) 50 Hence, the September bundle x 0 is not a ordable at October prices. That is, the student cannot keep consuming his initial consumption bundle x 0 after the prices change. (c) Based on your answers on parts (a) and (b), do his preferences violate the weak axiom of revealed preference (WARP)? Figure 2.10 depicts the budget line in September: facing a wealth level of w 0 $24 (as found in the previous part of the exercise), budget line B p 0 ;w 0 is x 1 + 4x 2 24, or x 2 6 4 x 1. Similarly, in October, wealth is w 1 $48 (as found in the previous part of the exercise), thus yielding a budget line 8x 1 + 6x 2 48, or x 2 8 4 x 1. As a remark, note that both budget lines 12

cross at 6 x 4 4 1 8 x 1, or x 1 :42, which implies x 2 :42, thus entailing that the crossing point occurs at the 45 0 line. Figure 2.10. September and October budget lines. Figure 2.10 also locates the Walrasian bundle that the consumer selects in September, x 0 (4; ), and in October, x 1 (; 4). Let us now check the WARP. Recall that, in order to check for WARP, we would need that his October consumption bundle, x 1, to be a ordable under September prices, i.e., p 0 x 1 < w p 0 w 0 ; and that his September consumer bundle, x 0, to be una ordable under October prices, i.e., x 0 > w w 1. In particular, for WARP to be violated, we need that the premise holds, p 0 x 1 < w, while the conclusion does not, x 0 < w. However, in this exercise, the premise does not hold, since his October bundle, x 1, is una ordable under September prices, p 0 x 1 > w (graphically, bundle x 1 lies strictly above the budget line for September, B p 0 ;w0, in gure 2.10), thus implying that we cannot show a violation of WARP. 1