Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter 2)

Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter 2) Tsun-Feng Chiang *School of Economics, Henan University, Kaifeng, China November 15, 2015 Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 1 / 29

2.1 Duality: A Closer Look The solutions to utility maximization problems and expenditure minimization problems are, in sense, the same. In this section, we shall explore further the connections among direct utility, indirect utility (from the maximization problem) and expenditure functions (from the minimization problem). The first duality is that every function of prices and utility that has all the properties of an expenditure function is in fact an expenditure function, i.e. there is a well-behaved utility function that generates it. Consider any function of prices and utility, E(p, u), that may or may not be an expenditure function. Suppose this function satisfies the expenditure function properties 1 to 7 of Theorem 1.7, so that it is continuous, strictly increasing, and unbounded above in u, as well as increasing in, homogeneous of degree one, concave, and differentiable in p. To show E is exactly an expenditure function, it must be shown there must exist a utility function on R n + whose expenditure function is precisely E. Indeed we shall give an explict procedure for constructing Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 2 / 29

this utility function. To see how the construction works, choose (p 0, u 0 ) R n ++ R +, and evaluate E to obtain a number. Use this number to construct the "half-space" in the consumption set, A(p 0, u 0 ) {x R n + p 0 x E(p 0, u 0 )}, illustrated in Figure 2.1(a) (see the next slide). Now choose different prices p 1, keep u 0 fixed, and construct another set, A(p 1, u 0 ) {x R n + p 1 x E(p 1, u 0 )}. Imagine proceeding like this for all prices p 0 and forming the infinite intersection A(u 0 ) p 0 A(p, u 0 ) = {x R n + p x E(p, u 0 ) for all p 0}.(2.1) The shaded area in Figure 2.1(b) illustrated the infinite intersection. It is easy to imagine that as more and more prices are considered and more sets are added to the intersection, the shaded area will be more closely resemble a superior set for some quasiconcave real-valued function. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 3 / 29

Figure 2.1. (a) The closed half-space A(p 0, u 0 ). (b) The intersection of a finite collection of the sets A(p 0, u 0 ). Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 4 / 29

The following theorem says the the function constructed by this way is increasing, unbounded above and quasiconcave, the properties like a direct utility fuunction. Theorem 2.1 Constructing a Utility Function from an Expenditure Function Let E : R n ++ R + R + satisfy properties 1 through 7 of an expenditure function given in Theorem 1.7. Let A(u) be as in (2.1). Then the function u : R n + R + given by u(x) max{u 0 x A(u)} is increasing, unbounded above, and quasiconcave. (This function, u(x), represents convex, monotonic prefernces.) If E(p, u) is really an expenditure function generated by some utility function u(x). From the definition of an expenditure function, p x E(p, u(x)) for all prices p 0. Because E is strictly increasing in u, u(x) is the largest value of u such that p x E(p, u) for all prices p 0. That is, u(x) is the largest value of u such that x A(u). Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 5 / 29

Theorem 2.1 tells us we can begin with an expenditure function and use it to construct a direct utility function representing some convex, monotonic preferences.the following theorem says a function satisfying Theorem 1.7 is indeed an expenditure function. Theorem 2.2 The Expenditure Function of Derived Utility, u, Is E Let E(p, u), defined on R n ++ R n +, satisfy properties 1 to 7 of an expenditure function given in Theorem 1.7 and let u(x) be derived from E as in Theorem 2.1. Then for all nonnegative prices and utility, E(p, u) = min x p x s.t. u(x) u. That is, E(p, u) is the expenditure function generated by derived utility u(x). Once we obtain the utitlity through an expenditure function, if the underlying preferences are continuous and strictly increasing, we can invert the function in u, obtain the associated indirect utility function, apply Roy s identity, and derive the system of Marshallian demands as well. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 6 / 29

What if the utiltiy function is not strictly increasing and not quasiconcave? What would be the properties of derived utility function constructed from an expenditure function? Suppose only that u(x) is continuous. Let e(p, u) be the expenditure function generated by u(x). The continuity of u(x) is enough to guarantee that e(p, u) is well-defined and continuous. Consider the utility function, call it w(x), generated by e( ) in the now familiar way, that is, w(x) max{u 0 p x e(p, u) p 0} By Theorem 2.1, w(x) is increasing and quasiconcave. Thus, regardless of whether or not u(x) is quasiconcave or increasing, w(x) will be both increasing and quasiconcave. Clearly, u(x) and w(x) need not coincide. By the definition of e(p, u), that is, p x e(p, u), and the defintion of w(x), we can see w(x) u(x). For any number u 0, the level-u superior set for u(x), say S(u), will be contained in the level-u superior set for w(x), say T (u). Moreover, because w(x) is quasiconcave, T (u) is convex. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 7 / 29

Figure 2.2. Duality between expenditure and utility. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 8 / 29

In Figure 2.2 (a), u(x) is quasiconcave and increasing, then the boundary of S(u) yields the negatively sloped, convex indifference curve u(x) = u. Note that each point on the boundary is the expenditure-minimizing bundle to achieve utility u at some price vector p 0. For any number u, by the definition of the expenditure function it must be that u(x) u, which implies u(x) u w(x). But it had been said w(x) u(x). Therefore, w(x) = u(x). The case depicted in Figure 2.2 (b) is more interesting. There, u(x) is neither increasing nor quasiconcave. Note that some bundles on the indifference curve never minimize the expenditure required to obtain utility level u regardless of the price vector. The thick lines in Figure 2.2(c) show those bundles that do minimize expenditure at some positive price vector. For those bundle x on the thick linesegments in Figure 2.2(c), we therefore have as before that w(x) = u(x) = u. But because w(x) is quasiconcave and increasing, the w(x) = u indifference curve must be as depicted in Figure 2.2(d). Thus, w(x) differs from u(x) only as much as is required to become strictly increasing and quasiconcave. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 9 / 29

The implication for this case is that any observable demand behavior that can be generated by a nonincreasing, nonquasiconcave utility function, like u(x), can also be generated by an increasing, quasiconcave utility function, like w(x). It is in the sense that the assumptions of monotonicity and convexity of preferences have no observable implications for our theory of consumer demand. Because the expenditure function and indirect utility are closely related (i.e. are inverses of each other), the duality between the expenditure function and the direct utility function implies there exists a duality between the indirect utility function and direct utility function. Suppose that u(x) generates the indirect utility function v(p, y). Then by definition of the indirect utility function (see (1.12)), for every x R n +, v(p, p x) u(x) holds for every p 0. In addition, there will be be some price vector for which the equality holds. Evidently, we may write u(x) = min p R n ++ v(p, p x). (2.2) (2.2) provides a mean for recovering the utility function u(x) from knowledge of only the indirect utility function it generates. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 10 / 29

Theorem 2.3 Duality Between Direct and Indirect Utility Suppose that u(x) is quasiconcave and differentiable on R n ++ with strictly positive partial derivative there. Then for all x R n ++, v(p, p x), the indirect utility function generated by u(x), achieves a minimum in p on R n ++, and u(x) = min p R n ++ v(p, p x). (T.1) Beacuse v(p, y) is homogeneous of degree zero in (p, y), we have v(p, p x) = v(p/(p x), 1) whenever p x > 0. Consequently, let p x = 1 we can rewrite (T.1) as u(x) = min p R n ++ v(p, 1). s.t. p x = 1 (T.1 ) Whether we use (T.1) or (T.1 ) to recover u(x) from v(p, y) does not matter. Simply choose that which is more convenient. One disadvantage of (T.1) is that it always possess multiple solutions becuase of the homogeneity of v (i.e., if p solves (T.1), then so does tp for all t > 0). Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 11 / 29

Example 2.1 Convert the indirect utility function v(p, y) = y(p r 1 + pr 2 ) 1/r to the direct utility function. We use (T.1 ) by setting y = 1, which yields v(p, 1) = (p r 1 + pr 2 ) 1/r. The direct utility function therefore will be the minimum-value function u(x 1, x 2 ) = min p1,p 2 (p r 1 + pr 2 ) 1/r s.t. p 1 x 1 + p 2 x 2 = 1. The first order conditions for the Lagrangian require that the optimal p 1 and p 2 satisfy (p r (p r 1 + p r 1 + p r Solve the system of equations to obtain 2 )( 1/r) 1 (p1 )r 1 λ x 1 = 0, 2 )( 1/r) 1 (p2 )r 1 λ x 2 = 0, 1 p1 x 1 p2 x 2 = 0. p 1 = x 1/(r 1) 1 x r/(r 1) 1 +x r/(r 1) 2, p2 = x 1/(r 1) 2 x r/(r 1) 1 +x r/(r 1) 2 Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 12 / 29

Example 2.1 (Continued) Substituting these into the objective function and forming u(x 1, x 2 ), we obtain [ r/(r 1) x 1 + x r/(r 1) ] 1/r 2 u(x 1, x 2 ) = (x r/(r 1) 1 + x r/(r 1) = 2 ) r Defining ρ r/(r 1) yields = (x r/(r 1) 1 + x r/(r 1) 2 ) (r 1)/r. u(x 1, x 2 ) = (x ρ 1 + x ρ 2 )1/ρ [ (x r/(r 1) 1 + x r/(r 1) 2 ) 1 r ] 1/r This is the CES direct utility function we started with in Example 1.2, as it should be. The last duality result we take up concerns the consumer s inverse demand function. The following theorem shows how to obtain the inverse demand from the direct utility function without solving the maximization problem. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 13 / 29

Theorem 2.4 (Hoteling, Wold) Duality and the System of Inverse Demands Let u(x) be the consumer s direct utility function and it differentiable. Then the inverse demand function for good i associated with income y = 1 is given by u(x)/ x i p i (x) = n j=1 x j( u(x)/ x j ) Example 2.2 Let s take the case of the CES utility function once again. If u(x 1, x 2 ) = (x ρ 1 + x ρ 2 )1/ρ, then u(x)/ x j = (x ρ 1 + x ρ 2 )(1/ρ) 1 x ρ 1 j multiplying by x j, summing over j = 1, 2, forming the required ratios, and invoking Theorem 2.4 gives the following system of inverse functions when income y = 1: p 1 = x ρ 1 1 (x ρ 1 + x ρ 2 ) 1 ; p 2 = x ρ 1 2 (x ρ 1 + x ρ 2 ) 1 Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 14 / 29

2.2 Integrability In Chapter 1.5, we showed that a utility maximizing consumer s demand function must satisfy homogeneity of degree zero in prices and income, budget balancedness, symmetric, and negative semidefiniteness, along with Cournot and Engel aggregation. But from Theorem 1,17 the two aggregation results are derived from budget balancedness so they are redundant. There is another redundancy as well. Of the remaining four conditions, only budget balancedness, symmetry, and negative semidefiniteness are truly independent: Homogeneity of degree zero is implied by budget balancedness and symmetry. Theorem 2.5 Budget Balancedness and Symmetry Imply Homogeneity If x(p, y) satisfies budget balancedness and its Slutsky matrix is symmetric, then it is homogeneous of degree zero in p and y. Proof: When budget balancedness holds, Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 15 / 29

Theorem 2.5 (Continued) y = n i=1 p ix i (p, y) Take the first derivatives with respect to prices and income to obtain for, i = 1,, n, n j=1 p j x j (p, y) p i = x i (p, y), (P.1) n j=1 p j x j (p, y) y = 1. (P.2) Fix p and y, then let f i (t) = x i (tp, ty) for all t > 0. Differentiating f i with respect to t gives f i (t) = n j=1 x i (tp, ty) p j p j + x i(tp, ty) y y (P.3) Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 16 / 29

Theorem 2.5 (Continued) Now by budget balancedness, tp x(tp, ty) = ty, so that dividing by t > 0, we may write: y = n j=1 p jx j (tp, ty) (P.4) Substituting from (P.4) for y in (P.3) and rearranging yields f i (t) = n j=1 p j [ xi (tp, ty) p j + x ] i(tp, ty) x j (tp, ty) y The term in bracket is the ij th element of the Slutsky matrix. When symmetry of the Slutsky matrix holds, the ij th element is equal to the ji th element. Consequently we may interchange i and j within those brackets and maintain equality. Therefore, f i (t) = n j=1 p j [ xj (tp, ty) p i + x ] j(tp, ty) x i (tp, ty) y Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 17 / 29

Theorem 2.5 (Continued) n = x j (tp, ty) n p j + x i (tp, ty) p i j=1 j=1 = 1 n x j (tp, ty) tp j + x i (tp, ty) 1 t p i t j=1 x j (tp, ty) p j y n j=1 tp j x j (tp, ty) y = 1 t [ x i(tp, ty)](by (P.1)) + x i (tp, ty) 1 [1](by (P.2)) t = 0. Since the demand f i (t) = x i (tp, ty) does not change in t, it is homogeneous of degree zero. In summary, for a utility-maximizer s system, the demand function must satisfies the three properties, Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 18 / 29

Budget Balancedness: p x(p, y) = y Negative Semidefiniteness: The associated Slutsky matrix s(p, y) must be negative semidefinite. Symmetry: s(p, y) must be symmetric. On the other hand, it can be proved that demand behavior is consistent with the theory of utility maximization if and only if it satisfies budget balancedness, negative semidefiniteness, and symmetry. This impressive result warrants a formal statement. Theorem 2.6 Integrability Theorem A continuously differentiable function x : R n+1 ++ Rn ++ is the demand function generated by some increasing, quasiconcave utility function if (and only if, when utility is continuous, strictly increasing, and strictly quasiconcave) it satisfies budget balancedness, symmetry, and negative semidefiniteness. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 19 / 29

In practice, this theorem allows us to recover the utility function given the demand functions satisfying the three properties. Example 2.3 Suppose there are three goods and that a consumer s demand behavior is summarized by the functions x i (p 1, p 2, p 3, y) = α iy p i, i = 1, 2, 3, where α i > 0, and α 1 + α 2 + α 3 = 1. It is straightforward to check that the demand functions satisfies budget balancedness, and the Slutsky equation is symmetric and negative semidefinite. Consequently, by Theorem 2.6, x(p, y) must be utility-generated. The first step is to derive the expenditure function. Remember the Shephard d lemma e(p 1, p 2, p 3, u) p i = x h i (p 1, p 2, p 3, u) Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 20 / 29

Example 2.3 (Continued) By duality between Hicksian demand functions and Marshallian demand function, x h i (p 1, p 2, p 3, u) = x i (p 1, p 2, p 3, e(p 1, p 2, p 3, u)) = α ie(p 1, p 2, p 3, u) p i Therefore, e(p 1, p 2, p 3, u) p i = α ie(p 1, p 2, p 3, u) p i, i = 1, 2, 3 This is a partial differential equation. Move e(p, u) to the left hand side, p i to the right hand side, and take the integral on the both side e(p, u) e(p, u) = α pi i, i = 1, 2, 3 p i You would have no trouble deducing that Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 21 / 29

Example 2.3 (Continued) ln(e(p, u)) = α i ln(p i ) + constant, i = 1, 2, 3 The only additional element to keep in mind is that when partially differentiating with respect to, say, p i, all the other variables, p 1, p 3, and u, are treated as constants. With this in mind, it is easy to see the three equations above imply the following three: ln(e(p, u)) = α 1 ln(p 1 ) + c 1 (p 2, p 3, u), ln(e(p, u)) = α 2 ln(p 2 ) + c 2 (p 1, p 3, u), ln(e(p, u)) = α 3 ln(p 3 ) + c 3 (p 1, p 2, u), where the c i ( ) functions are the constant terms. Because the three equations should hold simultaneously, the only possibility is ln(e(p, u)) = α 1 ln(p 1 ) + α 2 ln(p 2 ) + α 3 ln(p 3 ) + c(u), where c( ) is some function of u. But this means that Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 22 / 29

Example 2.3 (Continued) e(p, u) = c(u) p α 1 1 pα 2 2 pα 3 3 Because e(p, u) is strictly increasing in u, we can choose any c(u) that is strictly increasing. It does not matter what strictly increasing functions we choose because any monotonic transform can keep the order of utility. We may choose c(u) = u, so that e(p, u) = up α 1 1 pα 2 2 pα 3 3 We can then know the utility function. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 23 / 29

2.3 Revealed Preference We had discussed about the properties of demand functions derived from the consumer s preference or utility. The problem is that preference is not observable so we cannot directly examine the demand functions. The theory of revealed preference says that virtually every prediction ordinary consumer theory makes for a consumer s observable market can also be derived from a few simple and sensible assumption about the consumer s observable choices themselves, rather than about his unobservable preferences. The basic idea is simple: If the consumer buys one bundle instead of another affordable bundle, then the first bundle is considered to be revealed preferred to the second. Thus his tastes had been revealed by his choices. Instead of laying down axioms on a person s preferences as well we did before, we instead make assumptions about the consistency of the choices that are made. Formally, Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 24 / 29

Definition 2.1 Weak Axiom of Revealed Preference (WARP) A consumer s choice behavior satisfies WARP if for every distinct pair of bundles x 0, x 1 with x 0 chosen at prices p 0 and x 1 chosen at prices p 1, p 0 x 1 p 0 x 0 = p 1 x 0 > p 1 x 1 In other words, WARP holds if whenever x 0 is revealed preferred to x 1, x 1 is never revealed preferred to x 0. To better understand the implications of this definition, look at Figure 2.3.(see the next slide). In both graphs, the consumer facing p 0 chooses x 0, and facing p 1 chooses x 1. In (a), when the price is at p 0, the consumer chooses x 0 although x 1 is affordable to him (x 1 is inside the budget constraint). When the prices change from p 0 to p 1, he chooses x 1 because x 0 becomes not affordable to him. Therefore, his behavior does not violate WARP. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 25 / 29

In (b), at price p 0, both x 0 and x 1 are affordable to him. His choice of x 0 means he prefers x 0 to x 1. When the prices change from p 0 to p 1, he chooses x 1 instead of x 0 although x 0 is affordable to him. Figure 2.3. The Weak Axiom of Revealed Preference (WARP) Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 26 / 29

A Numerical Example: WARP In the table above, first bundle is chosen at first prices, second bundle at second prices and third bundle at third prices. So the diagonal corresponds to actual choices. From the first row, because we see the consumer chooses (10, 1) although (5, 5) and (5, 4) are affordable. So we know (10, 1) (5, 5) and (10, 1) (5, 4). From the second row, we know (5, 5) (5, 4). From the third row, we know (5, 5) (10, 1) which contradicts (10, 1) (5, 5). His behavior violates WARP. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 27 / 29

Now suppose a consumer s choice behavior satisfies WARP. Let x(p, y) denote the choice function (note well this is not a demand function because we have not mentioned utility or utility maximization), the quantities the consumer chooses facing prices p and income y. For p 0, we assume the choice function x(p, y) satisfies budget balancedness, i.e. p x(p, y) = y. The first consequence of WARP and budget balancedness is that the choice function x(p, y) must be homogeneous of degree zero in (p, y). Next consequence of WARP is the negative semidefiniteness of the Slutsky equation. Then it can also be proved the Slutsky equation is symmetric by budget balancedness and homogeneity of degree zero. However, symmetry only applies in the case of two goods. When there are only two goods, the choice function x(p, y) satisfies both negative definiteness and symmetry given the assumptions of WARP and budget balancedness. Then the choice function is actually a demand function because we would then be able to construct a utility function generating it. On the other hand, we also can use the choice function to recover the utility function using integrability. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 28 / 29

For more than two goods, WARP and budget balancedness does not necessarily imply symmetry of the Slutsky matrix because of the lack of transitivity. To have a choice function to be equivalent to the demand functions, we need a stronger axiom than WARP imposed on the consumer s behavior, that is Strong Axiom of Revealed Preference (SARP). Definition: Strong Axiom of Revealed Preference (Mas-Colell et. al, 1995) The choice function x(p, y) satisfies the strong axiom of revealed preference if for any list, (p 1, y 1 ), (p 2, y 2 ),, (p N, y N ) with x n+1 (p n+1, y n+1 ) x n (p n, y n ) for all n N 1, we have p N x 1 (p 1, y 1 ) > y N whenever p n x n+1 (p n+1, y n+1 ) y n for all n N 1. In words, if x 1 (p 1, y 1 ) is directly or indirectly preferred to x N (p N, y N ), then x N (p N, y N ) cannot be revealed preferred to x 1 (p 1, y 1 ). Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter November 2) 15, 2015 29 / 29