A utility function can generate Giffen behavior

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1 A utility function can generate Giffen behavior Junko Doi Department of Economics, Kansai University Kazumichi Iwasa GSE, Kobe University December, 006 Koji Shimomura RIEB, Kobe University Abstract We present a specific utility function which generates Giffen behavior. The derived demand function of each good is not only continuous in its price and income but also partly increasing in its price and decreasing in income. Moreover, we show that Giffen behavior is compatible with any level of utility and an arbitrarily low share of income spent on the inferior good; thus the standard margarine-butter paradigm may not adequately explain Giffen s paradox. Acknowledgement Comments from Murray C. Kemp, Binh Tran- Nam, Masao Oda, Ngo Van Long, and Chiaki Hara have greatly improved the paper. We have also benefitted from the discussion with Koichi Hamada, Satya Das, Takashi Kamihigashi, Tomoyuki Kamo, Takeshi Nakatani, Kazuo Nishimura, Ken-Ichi Shimomura, and Stephen J. Turnovsky. Introduction Giffen s paradox is usually construed as a possible economic behavior acted by households with low real wealth levels. For a recent example, a standard textbook of microeconomic theory gives an intuitive explanation of Giffen behavior as follows: Low-quality goods may well be Giffen goods for consumers with low wealth levels. For example, imagine that a poor consumer initially is fulfilling much of his dietary requirements with potatoes because they are a low-cost way to avoid hunger. If the price of potatoes falls, he can then afford to buy other, more desirable foods Corresponding author:rieb, Kobe University, - Rokkodai-cho, Nada-ku, Kobe, , Japan:Tel: :Fax: ; simomura@rieb.kobe-u.ac.jp

2 that also keep him from being hungry. His consumption of potatoes may well fall as a result. Note that the mechanism that leads to potatoes being a Giffen good in this story involves a wealth consideration: When the price of potatoes falls, the consumer is effectively wealthier (he can afford to purchase more generally), and so he buys fewer potatoes...(mas-colell, Whinston, and Green, 995, p.6). The purpose of this paper is to give a theoretical example which clearly shows the existence of Giffen behavior to be distinguished from the standard interpretation. Specifically, we show that there exists a continuous, non-decreasing, quasi-concave utility function u = u(x, x ) which maps the two-dimensional positive orthant R++ onto the set of real numbers R such that it has the following properties. Property I For any given positive income I and prices of good and good, p ( p) and p ( ), the textbook utility maximization problem has a unique, positive and interior optimal consumption bundle; continuous demand functions, say x (p, I) and x (p, I), exist for any positive pair (p, I). That is, those functions map the set of price-income pairs which is equal to R ++ into the set of positive real numbers, R ++. Property II The set of price-income pairs, R ++, is decomposed into three mutually exclusive subsets, g, i, and n, i.e., R ++ = g i n and s t =, s, t = g, i, n, s t, as follows. (i) Good is a Giffen good [x (p, I) is increasing in p and decreasing in I over the interior of g ]; (ii) Good is an inferior good with negatively sloped demand curve [x (p, I) is decreasing in both p and I over the interior of i ]; (iii) Good is a normal good [x (p, I) is decreasing in p and increasing in I over the interior of n ]. Property III Define a set of positive pair (p, I) Ω(u) {(p, I) : u = u(x (p, I), x (p, I))} For any utility level u there are three nonempty subsets Ω g (u), Ω i (u), and Ω n (u) such that (i) x (p, I) is increasing in p and decreasing in I over the interior of Ω g (u); good is a Giffen good; (ii) x (p, I) is decreasing in both p and I over the interior of Ω i (u); good is an inferior good with negatively sloped demand curve; (iii) x (p, I) is decreasing in p and increasing in I over the interior of Ω n (u); good is a normal good. Property IV Denote a share of income spent on good by θ, that is, θ px (p, I)/I. There exist two upper bounds θ g and θ i such that (i) good is a Giffen good if and only if θ < θ g ; (ii) good is an inferior good if and only if θ < θ i ; (iii) θ g and θ i are independent of (p, I) and satisfy 0 < θ g < θ i. We emphasize that Property III means that Giffen behavior is possible, irrespective of the standard of living, and Property IV means that Giffen behavior is compatible with an arbitrarily low share of income spent on the inferior good,

3 not compatible with high share. So Giffen behavior under this utility function cannot be comprehended through the standard interpretation, and reveals the theoretical possibility such that it is inappropriate to presume Giffen s paradox merely an exceptional phenomenon under extreme circumstances. Let us briefly review the theoretical literature on Giffen s paradox. According to Weber (997), Wold and Jureen (953) are the first to present a specific utility function that generates Giffen behavior. Vandermeulen (97) and Spiegel (994) presented other specific utility functions that generate Giffen behavior. On the other hand, Moffatt (00) shows that for an arbitrarily given ( x, x ) there exists a strictly quasi-concave utility function which generates a backward sloping price offer curve around ( x, x ). However, all of those specific functions lack either some standard properties as a utility function or analytical tractability due to smooth demand functions. And none of preceding researchers fully characterizes the domain of the demand functions to the extent of Property II and Property III above. As we shall show in the subsequent sections, subsets χ, Ω χ (u), χ = g, i, n and the upper bounds θ g and θ i are explicitly derived. Based on Property II, for an arbitrarily given point on the two-dimensional positive commodity space, we can draw the income expansion path and the price offer curve (see Figure 3B in this paper) both of which intersect with each other at the point. Moreover, find that on each income expansion path, Giffen behavior is accompanied by an arbitrarily high level of utility and low expenditure share on the inferior good. Thus, we can examine an asymptotic behavior of the demand for a Giffen good, which is impossible without specifying utility functions as we propose. This paper is organized as follows. Section presents the specific utility function. Section 3 derives the demand functions of good and good from the utility function and proves Property I. Section 4 characterizes the demand functions and proves Property II, Property III, and Property IV. Section 5 concludes. The utility function. The definition and the main assumption Let Ϝ(x, x ) α ln x + β ln x γx x The utility function we consider is defined as { Ϝ(x, x ) for (x, x ) R+ Λ 0, u(x, x ) ln(α/γ) α x β α () α for (x, x ) Λ 0, Vandermeulen (97) also shows that expenditure shares are irrelevant to Giffen behavior, because it depends solely on the properties of the indifference map, that is, the slope of indifference curves along vertical lines. Indeed, none of preceding researchers does, because their specific functions lack global quasi-concavity or generate Giffen behavior only at a finite part of the commodity space. 3

4 where Λ 0 {(x, x ) > (0, 0) : x x α/γ}. Parameters, α, β, and γ, satisfy the following assumption. ASSUMPTION: β > α > β > 0 and γ > 0.. The properties () has all the standard properties as a utility function. First, both u(x, x ) and its partial derivatives with respect to x and x are continuous in (x, x ) over the domain, as stated in the following lemma. Lemma : For any ( x, x ) R ++, lim u(x, x ) = u( x, x ), () (x,x ) ( x, x ) lim u i(x, x ) = u i ( x, x ), (3) (x,x ) ( x, x ) where u i (x, x ) is the partial derivative of u(x, x ) with respect to x i, i =,. Proof. See Appendix. (QED) Second, we prove the following lemma, and then draw some indifference curves of the utility function (see Figure ). Note that the slope of any indifference curve is zero in Λ 0. Lemma : u(x, x ) is non-decreasing and quasi-concave in x and x over the domain R ++. Proof. In fact, the function () is concave in x and x : For any (x, x ) R + Λ 0, u (x, x ) = α x γx > 0, u (x, x ) = β x γx > 0, u (x, x ) x u(x, x ) = α x < 0, u (x, x ) x u(x, x ) = β x < 0, u (x, x ) u(x, x ) = γ < 0, x x 4

5 u u u u = αβ = x x γ [ ][ ] αβ αβ + γ γ x x x x > 0, ( α < αβ, due to ASSUMPTION) and for any (x, x ) Λ 0, u = u = u = 0, u = β α > 0, u = β α x x < 0 It follows that the utility function is quasi-concave over the commodity space R+ and strictly concave on the subset R+ Λ 0. (QED) Third, the utility level on its border, α = γx x, is ( ) ( ) α α α u, x = ln x β α α γx γ Due to ASSUMPTION (β > α), it is monotonically increasing in x along the hyperbola. Since lim u( ) α x 0 γx, x = and lim u( ) α x γx, x =, for any real number u there is (x, x ) in R++ such that u = u(x, x ). In a word, u(x, x ) is a mapping from R++ onto the set of real numbers R. Finally, considering that u(0, x ) = u(x, 0) =, we see that any indifference curve cuts neither horizontal nor vertical axis of coordinates; both goods are indispensable. 3 The derivation of the demand function 3. The utility maximization problem Let us show that for any positive price-income pair (p, I) the utility maximization problem, max u(x, I px ) (4) 0 x I/p has a unique, interior and positive optimal consumption pair (x (p, I), x (p, I)) in R ++ Λ 0. First of all, note that, depending on the values of p and I, we have two types of budget constraints. Let G(x ) α γx (I px ). Type budget constraints such that G(x ) > 0 for any x (0, I/p). An example is BEC in Figure. 5

6 Type budget constraints which are not of type ; that is, each type- budget contstraint satisfies the condition as follows An example is B E C in Figure. x (0, I/p) such that G(x ) 0 Let us derive the optimal consumption pair for each type. 3. Type- budget constraint Taking into account the definition of the utility function, we have Since u(x, I px ) = α ln x + β ln(i px ) γx (I px ) lim u(x, I px ) = lim u(x, I px ) =, x 0 x I/p there is an interior optimal solution to (4) which satisfies the first-order condition 3 d dx u(x, I px ) = α x βp = γ(i px ) I px ] [ p [ α x γ(i px ) β I px γx = 0 (5) To show the uniqueness of the optimal solution, let us prove the following lemma. Lemma 3: For any x (0, I/p) satisfying G( x ) > 0, the second derivative of u(x, I px ) with respect to x is negative, i.e., d u(x, I px ) < 0 (6) x= x dx ] Proof. See Appendix. (QED) 3 d Note that u(.) denotes the total derivative of u(.) with respect to x dx. Letting x = I px, we have the frollowing identity. d dx u(x, x ) = = u(x, x ) + u(x, x ) dx x x dx u(x, I px ) p u(x, I px ) x x 6

7 Lemma 3 implies that the second-order condition holds, and that the optimal solution to the maximization problem (4) is uniquely determined by the firstorder condition for any given positive pair (p, I) which generates a type- budget constraint. Note that the first-order condition (5) can be rewritten to the standard (price) = (the marginal rate of substitution) condition, p = (I px )[α γx (I px )] x [β γx (I px )] (7) Point E in Figure is the unique optimal solution corresponding to a type- budget constraint BEC. The point satisfies (7). 3.3 Type- budget constraint We find from the definition of G(x ) that G(0) = G(I/p) = α, G (0) = γi < 0, and G (I/p) = γi > 0. Thus, there exist x and ˆx such that and 0 < x < ˆx < I/p G(x ) > 0 for x (0, x ) (ˆx, I/p), G(x ) < 0 for x ( x, ˆx ), G(x ) = 0 for x = x, ˆx Based on this sign pattern of G(x ) and Lemma which implies that not only u(x, I px ) but also the first derivative d dx u(x, I px ) are continuous in x (0, I/p), we are now at the position to prove the unique existence of the optimal solution to the maximization problem (4) for any given type- budget constraint. First, it follows from (5) that there exists ε > 0 such that the first derivative, d dx u(x, I px ), is positive for any x (0, ε ). Second, it is also clear from (5) that the first derivative is negative at x and ˆx. In fact, we can find that the first derivative is always negative in the interval ( x, ˆx ). Third, Lemma 3 d ensures us that dx u(x, I px ) is negative for any x > ˆx as well. Since Lemma 3 also holds for the interval (0, x ), we can conclude that the graph of u(x, I px ) is bell-shaped with the single peak between 0 and x, where the first-order condition d u(x, I px ) = 0 dx is established. Point E in Figure is an example of the optimal solution. 7

8 3.4 The first main proposition Based on the foregoing argument, we can assert that, whether the budget constraint is of type or type, the optimal solution is uniquely determined and continuous in p and I. We now derive the first result of this paper. PROPOSITION (Property I): For any given positive income I and prices of good and good, p ( p) and p ( ), the textbook utility maximization problem has a unique, positive, and interior optimal consumption bundle. That is, the demand functions x (p, I) > 0 and x (p, I) I px (p, I) > 0 are defined for any positive pair (p, I). Moreover the demand functions are continuous in p and I. 4 Properties of the demand functions To characterize the demand functions, x (p, I) and x (p, I), we focus on the budget constraint and the equality between the price and the marginal rate of substitution (7). I = px + x, (8) 0 = (β z)px (α z)x, (9) where z γx x. 4 Solving this system of equations for x and x, we obtain x = x = (α z)i p(α + β z), (0) (β z)i α + β z () Combining (0) and (), we have p (α z)(β z) = Ψ(z) () γi z(α + β z) Figure depicts the graph of Ψ(z). As is clear from lim Ψ(z) =, Ψ(α) = 0, (3) z 0+ and Ψ (z)z Ψ(z) [ z(α β) ] = + < for any z (0, α), (4) (α z)(β z)(α + β z) 4 As shown in the preceding section, for any (p, I) the optimal consumption pair (x, x ) exists in R ++ Λ 0, therefore, z (0, α). 8

9 z is uniquely determined for any positive m p/γi. Let us denote the inverse function of Ψ by z(m). It follows from (3) and (4) that lim z(m) = 0, z(0) = α, m and z (m)m z(m) = [ z(m)(α β) ] + (α z(m))(β z(m))(α + β z(m)) (, 0) for any m > 0 (5) Making use of (0), () and the inverse function, we can express the demand functions as I ( α z ( )) p γi x (p, I) = p ( α + β z ( p )), (6) γi I ( β z ( )) p γi x (p, I) = α + β z ( ) p (7) γi In what follows, we shall use these expressions Comparative statics Now that we obtain the demand functions, we shall check how the demand for x (p, I) and x (p, I) may depend on p and I. 4.. Price effects First, the demand for good has a distinctive feature such that the sign of x (p, I)/ depends solely on the value of z(m), as proved in the following lemma. Lemma 4.: For any given price-income pair, (i) x (p, I)/ is negative if z(p/γi ) ( 0, β β(β α)/ ) ; (ii) x (p, I)/ is positive if z(p/γi ) ( β β(β α)/, α ). Proof. The logarithmic differentiation of x (p, I) with respect to p yields p x (p, I) x (p, I) z(m)(α β) = + (α z(m))(α + β z(m)) z (m)m z(m) 5 Needless to say, from the definition of z(.) the equality z(p/γi ) = γx (p, I)x (p, I) holds. 9

10 Making use of (5), we derive p x (p, I) x (p, I) [ { }] { }] (z(m) α) z(m) β [z(m) β(β α) β + β(β α) = z(m)(α β), + (α z(m))(β z(m))(α + β z(m)) (8) where the denominator is positive, due to β > α > z(m). Let us denote the numerator as [ { { }] Ξ(z(m)) (z(m) α) z(m) β }][z(m) β(β α) β + β(β α) It is clear that Ξ(0) = αβ(α + β) < 0. Further, we see that the inequality α > β β(β α) holds, due to ASSUMPTION. 6 Considering that Ξ(0) < 0, we derive ( Ξ(z(m)) < 0 for z(m) 0, β ( Ξ(z(m)) > 0 for z(m) β ), β(β α) ), β(β α), α and (8) implies [ ] x (p, I) sign [ ( ( p ))] = sign Ξ z γi Therefore, Lemma 4. is established. (QED) Second, let us partially differentiate x (p, I) with respect to p. We obtain, from the partial differentiation of (7), x (p, I) = (β α)z (m) γi(α + β z(m)) < 0 6 (β α) p β(β α)/ < 0 holds for» q» q (β α) β(β α) (β α) + β(β α) = (β α) β(β α) «β = (β α) α, which is negative due to ASSUMPTION. 0

11 Finally, since z(.) is the inverse function of Ψ(.), from Lemma 4. we obtain the following result. PROPOSITION. (Property II): (i) If a pair (p, I) satisfies Ψ Ψ ( β β(β α)/ ) < p/γi, then x (p, I)/ is negative at the pair; (ii) If they satisfy Ψ > p/γi > 0, then x (p, I)/ is positive at the pair; good is a Giffen good at the price and income levels; (iii) x (p, I)/ is always negative. 4.. Income effects Next, let us check income effects on the demand for good and good. Similarly to the case of price effects, we first derive the following lemma. Lemma 4.: For any given price-income pair, (i) x (p, I)/ I is positive if z(p/γi ) ( 0, β β(β α) ) ; (ii) x (p, I)/ I is negative if z(p/γi ) ( β β(β α), α ). Proof. See Appendix. (QED) Second, let us partially differentiate x (p, I) with respect to I. We obtain, from the partial differentiation of (7), x (p, I) I = β z(m) α + β z(m) p(β α)z (m) γi (α + β z(m)) > 0 From Lemma 4. we have the following result. PROPOSITION. (Property II): (i) If a pair (p, I) satisfies Ψ Ψ ( β β(β α) ) < p/γi, then x (p, I)/ I is positive at the pair; good is a normal good at the pair; (ii) If they satisfy Ψ > p/γi > 0, then x (p, I)/ I is negative at the pair; good is an inferior good at the pair; (iii) x (p, I)/ I is always positive The diagrammatic explanation of the comparative static analysis Let us diagrammatically explain PROPOSITIONS. and., and then depict the income expansion path and the price offer curve. See Figure 3A, where the horizontal and vertical axes of coordinates measure p and I, respectively; curve OGC and OAH are two quadratic curves Ψ = p/γi and Ψ = p/γi. The propositions above mean that (i) good is a Giffen good at (p, I) in Region I, (ii) good is an inferior good with negatively sloped demand curve at (p, I) in Region II, and good is a normal good at (p, I) in Region III. The signs of x (p, I)/ and x (p, I)/ I depends on whether the value z = γx (p, I)x (p, I), which is uniquely determined by p/γi, is greater or smaller than β β(β α)/ and β β(β α) (Lemma 4. and 4.). Thus,

12 as is shown in Figure 3B, the commodity space (x, x ) > (0, 0) is divided by three hyperbolas; γx x = α, γx x = β β(β α), γx x = β β(β α) Regions I, II, and III in Figure 3B correspond to Region I, II, and III in Figure 3A, respectively. Take any positive point in the commodity space, say (x e, x e ). There is an indifference curve and its tangent line at point e; the latter line is expressed as u (x e, x e ) u (x e, xe )x + x = u (x e, x e ) u (x e, xe )xe + x e (x e, x e ) is the solution to the maximization problem (4) if the price is p E u (x e,xe ) u (x e,xe ) and the income is IE u (x e,xe ) u (x e,xe )xe + x e ; we have x e = x (p E, I E ) and x e = x (p E, I E ). For example, suppose that point e is in Region II in Figure 3B. Then p E /γ(i E ) is in between Ψ and Ψ ; point (p E, I E ) has to be in Region II in Figure 3A, in which case good is an inferior good with negatively sloped demand curve at (p E, I E ), i.e., both x (p E, I E )/ and x (p E, I E )/ I are negative. By this correspondence, points in Figures 3A and 3B have an one-toone relationship Income expansion path and price offer curve 7 Based on the foregoing argument, we can draw the income expansion path and the price offer curve both of which commonly cross a point, say E, in Figure 3A. Point E in Figure 3A corresponds to point e in Figure 3B, and so does the vertical line BAECD in Figure 3A to the income expansion path baecd in Figure 3B and the horizontal line JHEGF in Figure 3A to the price offer curve jhegf in Figure 3B. The income expansion path and the price offer curve bend when they cross the hyperbolas β β(β α) = γx x and β β(β α)/ = γx x, respectively. One can also verify that the Engel curve of good is bell-shaped. Moving up along the vertical line BAECD in Figure 3A, the ratio p/γi monotonically declines, which means that z(p/γi ) (= γx (p, I)x (p, I)) monotonically rises. The path goes up between the hyperbola α = γx x and the vertical axis of coordinates. On the other hand, moving to the right direction along the horizontal line F GEHJ in Figure 3A, the ratio p/γi monotonically rise, which means that z(p/γi ) (= γx (p, I)x (p, I)) monotonically decreases. The price offer curve converges a point on the vertical axis of coordinates, say j in Figure 3B. Note that, because of (7), lim x = Iβ/(α + β). z(p/γi ) 0 7 The definitions of income expansion path and price offer curve are given in standard textbooks of micro eocnomic theory. For example, see Varian (99, pp.6-8).

13 4..5 The independency of Giffen behavior from living standard For an arbitrarily chosen hyperbola ζ = γx x (0 < ζ α), the utility level on it is given by ( ) ( ) α ζ ζ u, x = ln x β α ζ, (9) γx γ which is monotonously increase from to +, as x increase from 0 to +. Therefore, any indifference curve crosses the hyperbola ζ = γx x, which obviously means it has to cross the two border curves β β(β α) = γx x and β β(β α)/ = γx x. Based on this fact, we derive the following result. PROPOSITION 3 (Property III): Define a set of positive pair (p, I) Ω(u) {(p, I) : u = u(x (p, I), x (p, I))} For any utility level u there are three nonempty subsets Ω g (u), Ω i (u), and Ω n (u) such that (i) for any (p, I) Ω g (u) x (p, I) > 0 and x (p, I) I < 0; (0) good is a Giffen good at (p, I) Ω g (u); (ii) for any (p, I) Ω i (u) x (p, I) < 0 and x (p, I) I < 0; () good is an inferior good with negatively sloped demand curve at (p, I) Ω i (u); (iii) for any (p, I) Ω n (u) x (p, I) < 0 and good is a normal good at (p, I) Ω n (u). x (p, I) I > 0; () Proof: See Figure 4A, where a awns is an arbitrarily chosen indifference curve. As we proved just before the statement of PROPOSITION 3, the curve has to cross the two border hyperbolas at w and n. Corresponding to the indifference curve awns in Figure 4A, we can derive the curve AW NS in Figure 4B; points w and n in Figure 4A correspond to points W and N in Figure 4B, respectively. As is clear from what we have showed so far, good is a Giffen good when (x (p, I), x (p, I)) is on aw, an inferior good with negatively sloped demand curve when it is on wn, and a normal good when it is on ns. Therefore, we can define Ω g (u), Ω i (u), and Ω n (u) as follows. Ω g (u) = {(p, I) : (p, I) is on AW in Figure 4B} Ω i (u) = {(p, I) : (p, I) is on W N in Figure 4B} Ω n (u) = {(p, I) : (p, I) is on NS in Figure 4B} (QED) 3

14 REMARK: Formally, the curve AW NS in Figure 4B is expressed as follows. Let x = ς(x ; u) be an indifference curve with the utility level u. Then, the corresponding p and I is expressed as p(x ) I(x ) ( ς(v; u) v ς(v; u) v v=x v=x ) x + ς(x ; u) Any point on the curve AW NS in Figure 4B is expressed as (p(x ), I(x )). For example, point A is (p(x a ), I(x a )) where p(x a ) = 0. Similarly, point W and point N are (p(x w ), I(x w )) and (p(x n ), I(x n )), respectively The expenditure share on a Giffen good Suppose that the consumption pair (x, x ) is optimally chosen for a given (p, I), then from (8) and (9), the share of income spent on good is px I = = (α z)x (β z)x x (α z)x (β z)x x + x α z α + β z α z Let ϑ(z) α+β z, then ϑ(z) ( ) α 0, α + β and ϑ (z) = Based on the above, we obtain the following result. α β < 0 for z (0, α) (3) (α + β z) PROPOSITION 4 (Property IV): Denote a share of income spent on good by θ, that is, θ px (p, I)/I. There exist the following relations between the property of demand for good and θ. (i) Good is a Giffen good if θ (0, θ g ); (ii) Good is an inferior good with negatively sloped demand curve if θ (θ g, θ i ); (iii) Good is a normal good if θ ( ) α θ i, α+β, where θg and θ i are defined as follows. θ g α β + β(β α) α β + β(β α), θ i α β + β(β α) α β + β(β α) 4

15 Proof: Considering the fact that for any (p, I) the expenditure share on good is given by ϑ(z(p/γi )), from (3), Lemma 4., and 4., we can conclude x (p, I) x (p, I) x (p, I) > 0 and < 0 and < 0 and x (p, I) < 0 I ( ( if θ ϑ(α), ϑ β x (p, I) < 0 I ( ( if θ ϑ β β(β α) )), β(β α) ), ϑ ( β β(β α) )), x (p, I) > 0 I ( if θ ϑ ( β β(β α) ) ), ϑ(0), where ϑ(α) = 0, ϑ ( β β(β α)/ ) = θ g, ϑ ( β β(β α) ) = θ i, and ϑ(0) = α/(α + β). (QED) Let us consider the change along an income expansion path. As households income I goes up, which means z(p/γi ) monotonously rises, the expenditure share on good monotonously declines, but at the same time good becomes a Giffen good. However low the expenditure share on good becomes, it remains to be a Giffen good. 5 Concluding remarks In this paper we have proposed a specific but standard utility function under which one good can be an inferior good or even a Giffen good. We show that Giffen behavior generated from the utility function is compatible with an arbitrarily high level of utility and low share of income spent on the inferior good. Thus, this behavior is clearly to be discriminated from the textbook margarine-butter paradigm, so the theoretical possibility emerges such that it is inappropriate to presume Giffen s paradox merely an exceptional phenomenon under extreme circumstances. Finally let us make a couple of technical remarks on the utility function we proposed in this paper. First, we can slightly modify the utility function in such a way that it becomes increasing and strictly quasi-concave in good and good. See Figure 5. The curve ABCD is an arbitrarily chosen indifference curve of the utility function. Choose a small positive number ε that satisfies α [ β β(β α)/ ] > ε. We can draw a curve BF G such that (i) it is tangent to the original indifference curve at B, (ii) it is a convex curve, and (iii) it asymptotically converge to the horizontal part of the original indifference curve ABCD. Define ABF G as a modified indifference curve. Making such 5

16 modification to each indifference curve, we can derive a modified utility function which is increasing and strictly quasi-concave in good and good. Second, one may wonder whether there is a family of utility functions that has the same properties as (). The answer is affirmative. The following utility function does. u(x, x ) α σ (x σ ) + βσ σ [ ( ) ] σ β γ α γ σ σ (x ) γx x for x σ x α/γ, σ σ x σ σ + α βσ σ for x σ x > α/γ, where we assume that all parameters, α, β, γ, and σ, are positive and satisfy ( ) /σ β γ < α γ < β γ. We can check that, as σ, this function converges to (). Appendix Proof of Lemma. Trivial from the definition if α γ x x. Let us focus on any point of the hyperbola α = γ x x. It suffices to prove. lim Ϝ(x β α, x ) = ln(α/γ) α x α { = u( x, x ) } (x,x ) ( x, x ). lim (x,x ) ( x, x ) Ϝ (x, x ) = 0 3. lim (x,x ) ( x, x ) Ϝ (x, x ) = β α x { [ = x ln(α/γ) α x β α α ] } (x,x )=( x, x ) { [ = x ln(α/γ) α x β α α ] } (x,x )=( x, x ) where Ϝ i (x, x ) is the partial derivative of Ϝ(x, x ) with respect to x i, i =,. Let us prove here. The others can be proved in a similar way. Since Ϝ(x, x ) [ β α ln(α/γ) α x α ] = α ln x + β ln x γx x = α ln γx x α [ α ln + β ln x x (γx x α), α ] + β ln x α γ x it is apparent from the continuity of γx x that for any δ > 0 there is some ɛ > 0 such that max { x x, x x } < ɛ Ϝ(x, x ) [ β α ln(α/γ) α x α ] < δ Therefore, is established. (QED) 6

17 Proof of Lemma 3. Considering (5), the differentiation of the first derivative at x yields d u(x, I px ) dx x= x [ βp = (I p x ) γp + α x ] [ p = β (I p x ) γ ] β (I p x ) (I p x ) [ ] βα β x (I p x ) γ Here the term [ βα γ ] is positive if G( x x ) = α γ x (I p x ) > 0 and (I p x) x (0, I/p). For, [ βα βα x (I p x ) γ x = + γ(i p x ) ][ βα x γ(i p x ) ] (I p x ) > > 0, [ βα x + γ(i p x ) ][ α x γ(i p x ) ] (I p x ) where the first inequality comes from α < βα ( ASSUMPTION) and the second inequality is due to G( x ) > 0. Therefore, (6) is established for any x (0, I/p) if G( x ) > 0. (QED) Proof of Lemma 4. The logarithmic differentiation of x (p, I) with respect to I yields Let I x (p, I) x (p, I) I z(m)(β α) = + (α z(m))(α + β z(m)) z (m)m z(m) z(m)(β α)(β z(m)) = z(m)(α β) + (α z(m))(β z(m))(α + β z(m)) [ ] αβ βz(m) + z(m) (α + β z(m)) = [ z(m)(α β) + (α z(m))(β z(m))(α + β z(m)) ] Φ(z(m)) αβ βz(m) + z(m) The sign of x (p, I)/ I is equal to the one of Φ(z(m)). Since Φ(0) = αβ > 0 and Φ(α) = α(β α) < 0, we derive [ ] [ x (p, I) ( ( p ))] sign = sign Φ z I γi > 0 ( p ) if z γi is in the interval ( 0, β β(β α) ) 7

18 and [ ] [ x (p, I) ( ( p ))] sign = sign Φ z I γi < 0 ( p ) if z γi is in the interval ( β β(β α), α ) Therefore, Lemma 4. is established. (QED) References [] Doi, J., K. Iwasa, and K. Shimomura, 006, A note on Giffen goods, Kobe University. [] Mas-Collel, A., M. D. Whinston, and J. R. Green, 995, Microeconomic Theory, Oxford: Oxford University Press. [3] Moffatt, P. G., 00, Is Giffen behavior compatible with the axioms of consumer theory?, Journal of Mathematical Economics 37, [4] Spiegel, U., 994, The case of a Giffen good, Journal of Economic Education 5, [5] Vandermeulen, D. C., 97, Upward sloping demand curve without the Giffen paradox, American Economic Review 7, [6] Varian, H., 99, Microeconomic Analysis, 3rd edition, New York: W.W. Norton & Company. [7] Weber, C. E., 997, The case of a Giffen good: comment, Journal of Economic Education 8, [8] Wold, H. and L. Jureen, 953, Demand Analysis, New York: John Wiley. 8