aa r X i v : . [ ec on . E M ] A ug Injectivity and the Law of Demand ∗ Roy AllenDepartment of EconomicsUniversity of Western [email protected] 19, 2019
Abstract
Establishing that a demand mapping is injective is core first step for avariety of methodologies. When a version of the law of demand holds, globalinjectivity can be checked by seeing whether the demand mapping is constantover any line segments. When we add the assumption of differentiability, weobtain necessary and sufficient conditions for injectivity that generalize classicalGale and Nikaido [1965] conditions for quasi-definite Jacobians. ∗ I thank Nail Kashaev, Salvador Navarro, John Rehbeck, and David Rivers for helpful comments. Introduction
A variety of recently developed methods require, as a first step, that a demand map-ping be injective. Examples include work on endogeneity with market level data(Berry [1994], Berry, Levinsohn, and Pakes [1995], Berry and Haile [2009], Chiappori and Komunjer[2009], Berry and Haile [2014]); simultaneous equations models (Matzkin [2008], Matzkin[2015], Berry and Haile [2018]); multidimensional heterogeneity in a consumer setting(Blundell et al. [2017]); index models (Ahn, Ichimura, Powell, and Ruud [2017]); andnonparametric analysis in trade (Adao et al. [2017]). The applicability of these methods depends on whether the demand mapping is in-jective. This paper studies injectivity using a shape restriction that allows comple-mentarity: the law of demand.
Definition 1. Q : U Ď R K Ñ R K satisfies the law of demand if for each u, ˜ u P U , p Q p u q ´ Q p ˜ u qq ¨ p u ´ ˜ u q ě . Many models imply a version of the law of demand, both in the standard consumerproblem and outside it. In the standard consumer problem, u is the negative of theprice vector. Quasilinear preferences imply the law of demand. Hildenbrand [1983]provides conditions under which the law of demand holds in the aggregate, even if itdoes not hold at the individual level. Outside the standard consumer problem, thediscrete choice additive random utility model (McFadden [1981]) also satisfies the lawof demand. In that model one may interpret u k as the deterministic utility index foralternative k and Q p u q as a vector of choice probabilities.Directly checking whether a demand mapping is injective is nontrivial. This paperprovides necessary and sufficient conditions for injectivity that can simplify this task.The simplest condition states that when demand is continuous and the domain of Chesher and Rosen [2017] and Bonnet et al. [2017] take an alternative approach, working withinverse images that may be multivalued. Fosgerau et al. [2018] provide an injectivity result for a demand system that allows complemen-tarity. They consider quasilinear preferences and so their demand system fits into the setup of thispaper. Hildenbrand [1983] also provides sufficient conditions that ensure a strict law of demand p Q p u q´ Q p ˜ u qq ¨ p u ´ ˜ u q ą u ‰ ˜ u , which clearly implies injectivity. Q can be checked by checking whether Q is constant over line segments. This implies that global and local injectivity areequivalent.The main result of this paper is a nondifferentiable counterpart to the classical in-jectivity results of Gale and Nikaido [1965] for functions with weakly quasi-definiteJacobians. When I specialize the main result by assuming the demand mapping Q is differentiable, I establish a generalization of Gale and Nikaido [1965], Theorem 6w.While Gale and Nikaido [1965] impose invertibility of the Jacobian of Q as a sufficient condition for global injectivity, I provide a necessary and sufficient condition for local(and global) injectivity in terms of certain directional derivatives.Berry, Gandhi, and Haile [2013] have recently shown that demand mappings thatsatisfy a “connected substitutes” property are injective. This connected substitutescondition applies to a number of existing models, including models of market sharesbased on a discrete choice foundation (e.g. Berry and Haile [2014]), but may notapply when there is complementarity. This paper complements their analysis bystudying injectivity using a shape restriction that allows complementarity without areparametrization.The injectivity results of this paper exploit the fact that when Q is continuous andsatisfies the law of demand, the inverse image of any quantity is a convex set. Thisis a classical result in monotone operator theory. To my knowledge, this importantproperty has not been exploited for studying injectivity in the econometrics literature,yet it has several implications that I describe further in the paper. The closestprecedent appears to be in the study of uniqueness of general equilibrium, whereseveral conditions are known to yield convexity of equilibria (e.g. Arrow and Hurwicz[1958], Arrow and Hurwicz [1960], and the discussion in Mas-Colell [1991]). To be clear, this paper only overlaps when the Jacobian is weakly quasi-definite, not just a P matrix as in Theorem 4 in Gale and Nikaido [1965]. Berry et al. [2013] show in several examples that certain models with complements may bereparametrized to fit into their setup. See also Brown and Matzkin [1998] and Beckert and Blundell[2008] for injectivity results that allow complementarity between goods. See e.g. Rockafellar and Wets [2009]. Characterization of Injectivity
This section presents the main results, which provide necessary and sufficient condi-tions for a demand mapping to be injective. We use the following assumption, whichallows us to reduce checking global injectivity to checking local conditions.
Assumption 1. Q : U Ď R K Ñ R K satisfies the law of demand, is continuous, and U is open and convex. Recall that a set U is convex if for u, ˜ u P U and any scalar α P r , s , it follows that αu `p ´ α q ˜ u P U . An important implication of this assumption is that inverse images Q ´ p u q are convex, which I formalize below. As discussed in the Introduction, this isa classical result in monotone operator theory. Lemma 1.
Let Assumption 1 hold. Then for each y P R K , Q ´ p y q “ t u P U | Q p u q “ y u is convex.Proof. If the domain is U “ R K , this is a textbook result, e.g. Rockafellar and Wets[2009], p. 536. When U ‰ R K , this is covered by Kassay, Pintea, and Szenkovits[2009], Theorem 3.5.When K ą
1, continuity cannot be dropped without alternative structure, as thefollowing example illustrates.
Example 1.
Let K “ , U “ R , and A “ t u P R | u ` u ą or u “ u “ u .Let Q p u q “ p t u P A u , t u P A uq , where t u P A u is an indicator function for whether u P A . To show the law of demand is satisfied, note that if u, ˜ u are either both in A orboth in its complement A c , Q does not vary and clearly satisfies the law of demand.Consider then u P A, ˜ u P A c . Then we have p Q p u q ´ Q p ˜ u qq p u ´ ˜ u q “ p u ´ ˜ u q ` p u ´ ˜ u q ě , Nonetheless, Q ´ p , q “ A c is not convex, since both points p´ , q , p , ´ q are in A c , yet their convex combination p , q is not. Using Lemma 1, we obtain the following list of conditions that are equivalent to global4njectivity of Q . Proposition 1.
Let Assumption 1 hold. Then the following are equivalent:i. Q is injective, i.e. for each y P R K there is at most one u P U such that Q p u q “ y .ii. Q is locally injective, i.e. for each u P U there is a neighborhood H Ď R K of u such that the restriction of Q to H is injective.iii. The only line segments in U along which Q is constant are singleton points.Proof. Clearly, (i) ùñ (ii) ùñ (iii). That (iii) ùñ (i) follows from Lemma 1 andthe definition of convexity. Indeed, if Q p u q “ Q p ˜ u q , then the set Q ´ p Q p u qq is convexand must contain u and ˜ u . In particular, Q is constant on any line segment joining u and ˜ u . By assumption, this is only possible if u “ ˜ u .Local injectivity always implies condition (iii), but in general the reverse is not true.An example of a multivariate function that satisfies (iii) but is not locally injective is Q p u , u q “ p u ´ u , u ´ u q . The equivalence of (i) and (iii) is the most powerful partof Proposition 1, since part (iii) is often easy to check. In addition, we can leveragethis equivalence to relax the domain restrictions. We formalize this as follows. Corollary 1.
Let Assumption 1 hold. Let Q H : H Ď U Ñ R K denote the restric-tion of Q to the set H . If H is open or convex, any of the injectivity conditions ofProposition 1 are equivalent when applied to the function Q H .Proof. It is clear that (i) ùñ (ii) ùñ (iii). It remains to show that (iii) ùñ (i).Let Q H p u q “ Q H p ˜ u q for u, ˜ u P H . Let T Ď U be the line segment from u to ˜ u . FromLemma 1, we conclude that Q ´ p Q H p u qq contains T , and hence Q ´ H p Q H p u qq contains T X H . In particular, Q H is constant over T X H .Suppose for the purpose of contradiction that u ‰ ˜ u . If H is either open or convex,the set T X H contains two distinct line segments that are not points, beginning at u and ˜ u , respectively. Since we assumed (iii) holds, we reach a contradiction since Q H is constant over T X H , and thus is constant over these line segments.Note that this proposition assumes Q satisfies the law of demand over the entire set U . This assumption may be satisfied by appealing to economic theory. It allows one5o show injectivity for the restriction Q H for a set H that is either open or convex.The primary reason one may be interested in such restrictions is that one may onlyhave information on Q over a certain region of utility indices (such as H ).The proof of Corollary 1 establishes the following additional result. Corollary 2.
Let Assumption 1 hold. Let Q H : H Ď U Ñ R K denote the restrictionof Q to the set H . If H is open or convex, the following are equivalent for arbitrary u P H :i. For any u, ˜ u P H with u ‰ ˜ u , Q H p u q ‰ Q H p ˜ u q , i.e. the inverse image Q ´ H p Q H p u qq is a singleton.ii. Q H is locally injective at u , i.e. there is a neighborhood N Ď H of u such thatthe restriction of Q H to N is injective.iii. The only line segment in H that contains u and and over which Q H is constantis the point u . The equivalence of (i) and (ii) shows that to check whether an inverse image is asingleton, it is enough to check features of the mapping Q H that are local to a single u . Importantly, this differs from Proposition 1 and Corollary 1, which instead studyhow local injectivity holding for each u implies global injectivity. Thus, Corollary 2further highlights the sense in which injectivity can be reduced to a local condition. Italso conceptually differs from classical papers such as Gale and Nikaido [1965], whichfocus on when a local condition holding everywhere implies global injectivity.Finally, to see that these equivalences do not hold in general, consider H “ U “ R and Q p u q “ u , which is continuous yet violates the law of demand. Then Q H islocally injective at u “
1, but Q ´ H p Q H p qq “ t´ , u . In this section I describe how the local-to-global injectivity result of Proposition 1may be seen as a nondifferentiable version of a classical result due to Gale and Nikaido[1965]. In drawing this relationship, I present a new result complementing their results6or weakly quasi-definite Jacobians, which drops the requirement that the functionhave a convex domain.I now add the assumption that Q is differentiable. Assumption 2.
The function Q : U Ď R K Ñ R K is differentiable, where U is anopen, convex set. To relate to Gale and Nikaido [1965], I introduce some definitions.
Definition 2. A K ˆ K matrix B is positive semi-definite if λ Bλ ě for every λ P R K . If λ Bλ ą for every nonzero λ , then B is positive definite. Definition 3. A K ˆ K matrix B is weakly quasi-definite if p B ` B q{ is positivesemi-definite. If p B ` B q{ is positive definite, then B is quasi-definite. The following result connects the law of demand and quasi-definiteness of Jacobians.
Lemma 2.
Let Assumption 2 hold. The function Q satisfies the law of demand ifand only if its Jacobian is everywhere weakly quasi-definite.Proof. See e.g. Parthasarathy [2006], p. 92.With this lemma and the previous results, we obtain a generalization of Gale and Nikaido[1965], Theorem 6w.
Proposition 2.
Let Assumption 2 hold and suppose Q satisfies the law of demand.Let H Ď U be an open set and let Q H denote the restriction of Q to H . Then Q H isinjective if its Jacobian is everywhere invertible.Proof. From Corollary 1 we see it is enough to establish local injectivity of Q H . Sincethe Jacobian of Q H is everywhere invertible, Q H is locally injective by Proposition 3because its directional derivatives are never zero. Gale and Nikaido [1965] prove this result when H “ U , i.e. over convex domains. The link between the law of demand and results in Gale and Nikaido [1965] has pre-viously been noted (Kassay, Pintea, and Szenkovits [2009], L´aszl´o [2016]). Proposi- Note that if we had assumed Q is continuously differentiable, we could apply the classical inversefunction theorem to establish local injectivity. We have not assumed the derivative is continuous,and hence we use an alternative technique. Theorem 6 in that paper requires instead that U be convex and drops openness, but insteadalso requires that the Jacobian be positive quasi-definite (not just weakly quasi-definite, which isimplied by the law of demand). The previous section uses the invertibility of the Jacobian of Q as a sufficient conditionfor local injectivity of Q . Invertibility of the Jacobian is not necessary for localinjectivity, as illustrated for K “ Q p u q “ u , since the derivative is 0 at 0.Obtaining necessary and sufficient conditions for local injectivity in terms of deriva-tives is nontrivial in general. For Q satisfying the law of demand, by Proposition 1(iii),however, checking global or local injectivity is equivalent to checking whether Q isconstant over any line segment that is not a point. One may write this condition interms of certain directional derivatives. To that end, define the directional derivativeof Q at u in direction v , denoted Q p u, v q , by ˇˇˇˇ lim λ Ó Q p u ` λv q ´ Q p u q λ ´ Q p u, v q ˇˇˇˇ “ Proposition 3.
Let Assumption 2 hold and suppose Q satisfies the law of demand.The following are equivalent for arbitrary u P U :i. For any ˜ u P U with u ‰ ˜ u , Q p u q ‰ Q p ˜ u q .ii. Q is locally injective at u .iii. The only line segment in U that contains u and and over which Q is constant isthe point u .iv. There are no nonzero vectors v P R K such that for all λ P r , s satisfying u ` λv P U , Q p u ` λv, v q is the zero vector.Proof. Since Q is differentiable, it is continuous. The equivalence between (i)-(iii)follows from Corollary 2. Equivalence between (iii) and (iv) follows from the meanvalue theorem. 8ote that if the Jacobian of Q is invertible at u ‰
0, then Q p u, v q cannot be the zerovector for any nonzero v , because directional derivatives satisfy J p u q v “ Q p u, v q ,where J p u q is the Jacobian of Q at u . Thus if Q has an everywhere invertible Jacobian,condition (iii) is satisfied for each u P U . More generally, suppose that failures ofinvertibility of the Jacobian of Q only occur on an isolated set of points. Thenclearly, condition (iii) is satisfied for each u P U .Recall from Proposition 1, global injectivity of Q is equivalent to local injectivity foreach u P U . Thus, global injectivity of Q is also equivalent to conditions (ii) or (iii)of Proposition 3 holding for each u P U .By combining Lemma 2 and Proposition 3, we obtain a generalization of Gale and Nikaido[1965], Theorem 6. This generalization drops the assumption that the Jacobian of Q is everywhere invertible. Corollary 3.
Let Assumption 2 hold and assume the Jacobian of Q is everywhereweakly quasi-definite. The following are equivalent:i. Q is injective.ii. For each u P U , any of the equivalent conditions in Proposition 3 holds. Berry, Gandhi, and Haile [2013] have recently shown that a “connected substitutes”condition implies global injectivity. The present paper shows that a version of the lawof demand, which allows complementarity, also suffices. This approach is not nestedin and does not nest that of Berry, Gandhi, and Haile [2013].The setup of Berry, Gandhi, and Haile [2013] imposes the following properties on thedemand mapping Q : U Ď R K Ñ R K :i. (Strict Own-Good Monotonicity) Let k be arbitrary. For each u, ˜ u P U such that u k ą ˜ u k and u j “ ˜ u j for j ‰ k , it follows that Q k p u q ą Q k p ˜ u q . k be arbitrary. For each u, ˜ u P U such that u k ą ˜ u k and u j “ ˜ u j for j ‰ k , it follows that for all ℓ ‰ k , Q ℓ p u q ď Q ℓ p ˜ u q . Condition (i) states demand increases in its own utility shifter. Condition (ii) statesthat if a utility shifter for good k increases, then all other demands weakly decrease.Condition (i), except with a weak inequality, follows whenever Q satisfies the law ofdemand, but Condition (ii) does not.Berry, Gandhi, and Haile [2013] impose a “connected substitutes” condition, whichdirectly assumes weak substitutability and implies strict own-good monotonicity (seeRemark 1 in Berry, Gandhi, and Haile [2013]). For brevity, I omit a formal statement,and instead describe a key implication of their assumption: for arbitrary u, ˜ u P U , Q p u q ě Q p ˜ u q ùñ u ě ˜ u, where ě denotes the usual partial order in R K , i.e. u ě ˜ u if and only if u k ě ˜ u k for each k . This shape restriction is called inverse isotonicity , and clearly impliesthat Q is injective. This property is essential for establishing injectivity using theapproach of Berry, Gandhi, and Haile [2013].As discussed previously, the methods of this paper are distinct from those of Berry, Gandhi, and Haile[2013]. I provide two examples showing the distinction between inverse isotonicityand the law of demand. First, I show that the law of demand does not imply inverseisotonicity.
Example 2.
Consider a linear demand system Q p u q “ Au , where A “ « ff . The matrix A is symmetric and satisfies row-diagonal dominance (e.g. for each row,the diagonal | | exceeds the sum of the off-diagonal | | ), which are well-known con-ditions that ensures the matrix p A ` A q{ is positive semi-definite. Thus, the law Appendix A describes a characterization of inverse isotonicity in this setting using results inMor´e and Rheinboldt [1973]. f demand is established from Lemma 2. This demand mapping violates weak substi-tutability because the off-diagonals of A are positive. In addition, it violates inverseisotonicity. To see this, consider the two vectors u “ p , q , ˜ u “ p , ´ q . Then Q p u q “ p , q , Q p ˜ u q “ p , q and so Q p ˜ u q ě Q p u q , but we do not have ˜ u ě u . The following example illustrates that inverse isotonicity does not imply the law ofdemand.
Example 3.
Now consider a demand system Q p u q “ « ´ ´ ff « u u ff . The function Q satisfies the connected substitutes property of Berry, Gandhi, and Haile[2013], hence inverse isotonicity. It does not satisfy the law of demand. To see this,consider u “ p , q and ˜ u “ p , q . One obtains Q p u q “ p , q and Q p ˜ u q “ p´ , q .Thus, p Q p ˜ u q ´ Q p u qq p ˜ u ´ u q “ p´ , q p , q “ ´ ă . In this example, the law of demand fails because the substitution effect outweighs theown-good effect.
To shed further light on the distinction between inverse isotonicity and the law ofdemand, it is helpful to note that the law of demand is not an ordinal property. Thisis illustrated in Example 3 by considering the strictly increasing function f p v q “ v { .Consider the transformed mapping˜ Q p u q “ Q pp f p u q , f p u qqq “ « ´ ´ ff « u u ff . The mapping ˜ Q satisfies the law of demand even though the original mapping Q inExample 3 violates the law of demand. Further analysis of the law of demand anda change of variables is covered in Section 7.Finally, it is clear that strict own-good monotonicity, weak substitability, and in-verse isotonicity are all ordinal properties in the following sense: they hold forsome mapping Q p u q if and only if they hold for ˜ Q p u q “ Q p f p u qq where f p u q “ This follows because the symmetrized matrix of coefficients in ˜ Q satisfies diagonal dominance. f p u q , . . . , f K p u K qq , and each f k is strictly increasing. In the standard consumer problem, quasilinear utility is a well-known class of pref-erences that implies the law of demand. I provide an injectivity result that exploitsadditional structure of this model.Suppose an individual maximizes a utility function of the form y ` C p y , . . . , y k q , with budget constraint ř Kk “ p k y k ď M . Suppose p does not vary and is normalized to1. Under local nonsatiation and allowing negative quantities of y (or sufficiently highincome), the maximization problem is equivalent to choosing quantities to maximize ´ K ÿ k “ p k y k ` C p y , . . . , y k q , where y “ M ´ ř Kk “ p k y k has been substituted out. Thus, if the maximizer is uniquewe have Q p u q “ argmax y P R K K ÿ k “ u k y k ` C p y q , where u k “ ´ p k ; more generally, we may take Q p u q to be an element of the argmaxcorrespondence.To see that Q satisfies the law of demand, consider the necessary condition for max-imization K ÿ k “ u k Q k p u q ` C p Q p u qq ě K ÿ k “ u k Q k p ˜ u q ` C p Q p ˜ u qq . An analogous inequality holds with u and ˜ u reversed. Summing up the two analogousinequalities and rearranging establishes that Q satisfies the law of demand. If there are multiple maximizers, this inequality holds for any maximizers. In particular, a lawof demand holds for arbitrary selectors from the argmax correspondence. U need not be convex. Lemma 3 (cf. Rockafellar [1970], Theorems 23.5 and 25.1) . Let Q : U Ď R K Ñ R K satisfy Q p u q P argmax y P R K K ÿ k “ u k y k ` C p y q where U is open. If C : R K Ñ R Y t`8u is concave, upper semi-continuous, andfinite at some point, then the following are equivalent for arbitrary u P U :i. C is differentiable at Q p u q .ii. There is no ˜ u P U such that u ‰ ˜ u and Q p u q “ Q p ˜ u q . This result directly follows from Rockafellar [1970] and so the proof is omitted. Acorollary of this lemma is that if C is everywhere differentiable (and the other con-ditions are met), then Q is globally invertible. A version of this result has beenused in Allen and Rehbeck [2019]; I include this result for completeness, to illustratehow additional structure allows us to further specialize the results, and because thequasilinear structure is widely used. A structure that shares a mathematical relationship with quasilinear utility has beenstudied in Brown and Matzkin [1998]. Suppose now that there is a budget constraintbut the demand is not quasilinear. Formally, the consumer solves the problemmax p y ,y qP R K ` ` K ÿ k “ u k y k ` C p y , y q s.t. K ÿ k “ p k y k ` y ď I, A function f : R K Ñ R Y t`8u is upper semi-continuous if t y | f p y q ě α u is closed for each α . I is income. Brown and Matzkin [1998] study this model with the interpreta-tion that u is a vector of exogenous unobservables. Blundell et al. [2017] study ageneralization that is not covered by our setup.Note that fixing prices and income, this problem differs from the setup of Lemma 3only because of the budget constraint. However, under local nonsatiation the con-straint is satisfied with equality and so the problem reduces tomax y P R K ` K ÿ k “ u k y k ` C ˜ I ´ K ÿ k “ p k y k , y ¸ . Injectivity of this demand system is then covered by Lemma 3. In particular, fixingprices and income, differentiability of the mapping ˜ C p y q “ C ´ I ´ ř Kk “ p k y k , y ¯ andinjectivity are equivalent under certain conditions, as formalized in Lemma 3. Im-portantly, while this argument can provide sharp conditions relating injectivity anddifferentiability, it does not establish smoothness of the inverse. Indeed, differentiabil-ity of ˜ C does not rule out multiple maximizers, and so the demand mapping need noteven be continuous. Brown and Matzkin [1998] provide additional conditions thatensure injectivity and smoothness. Many models outside of the consumer problem that have additively separable unob-servable heterogeneity also share the structure of the quasilinear utility model. Inparticular, they imply a version of the law of demand in utility indices that need notinvolve price. For the additive random utility model, this has been recognized at leastsince the seminal work of McFadden [1981]. Other examples that share this structureare the discrete choice bundles model of Gentzkow [2007], the matching model ofFox, Yang, and Hsu [2018], and a model of decisions under uncertainty considered inAgarwal and Somaini [2018]; see Allen and Rehbeck [2019] for details. Their presentation is slightly different since they consider also a term u y in the utility andthen normalize u “
1. With this normalization, this term can be absorbed into C p y , y q . A related change of variables argument has appeared in Allen and Rehbeck [2019] to study adiscrete choice problem. xample 4 (Additive Random Utility Models (McFadden [1981])) . Let v j “ u j ` ε j denote the latent utility for alternative j . Treat ε “ p ε , . . . , ε K q as a random variableand u as a constant. Normalize the latent utility of the outside good ( j “ ) to and assume there are K inside goods. Suppose the individual chooses an alternativethat maximizes latent utility and let ˜ D p u, ε q P t , u K be a vector of indicators denot-ing denoting which, if any, of the inside goods ( j ą ) is chosen. Then by similararguments as in the quasilinear utility example, necessary conditions for optimalityimply ´ ˜ D p u, ε q ´ ˜ D p ˜ u, ε q ¯ ¨ p u ´ ˜ u q ě . Moreover, letting Q p u q “ E ” ˜ D p u, ε q ı where the expectation is over ε , we have p Q p u q ´ Q p ˜ u qq ¨ p u ´ ˜ u q ě . In this example, Q p u q is the vector of probabilities for choosing each of the K insidegoods. This example illustrates two principles. First, the law of demand is preservedunder expectations. In particular, the law of demand holding at the individual levelimplies it holds at the aggregate level. Second, injectivity results may be used foraggregate data even when injectivity fails at the individual level. Note that for fixed ε , the function ˜ D p¨ , ε q cannot be injective whenever U has more than K ` Q is injective depends on the distribution of ε (Norets and Takahashi[2013]; see also Azevedo, Weyl, and White [2013]). In this paper I treat u as a fixed parameter. If u is treated as an observable random variable,then as long as u is independent of ε (and some technical conditions are met), Q p u q is the conditionalprobability of choosing each alternative, conditional on the shifters u . See Shi et al. [2018] for a related application of this principle for discrete choice panel data. Law of Demand with a Change of Variables
In some models the law of demand does not hold, but holds after a change of variables.We can adapt Proposition 1 to such settings when the change of variables is sufficientlywell-behaved. I formalize that the law of demand holds after a change of variables asfollows.
Assumption 3. ˜ Q p u q “ Q p f p u qq , where Q : U Ď R K Ñ R K satisfies the law ofdemand and is continuous, f : T Ď R K Ñ U Ď R K , and U and T are open andconvex. When f is a homeomorphism , i.e. a continuous function with a continuous inverse,we still obtain that local injectivity implies global injectivity. Proposition 4.
Let Assumption 3 hold with f a homeomorphism. Then the followingare equivalent:i. ˜ Q is injective.ii. ˜ Q is locally injective.Proof. Clearly (i) ùñ (ii), and so we wish to show (ii) ùñ (i). By Lemma 1, the set Q ´ p Q p f p u qqq is convex, hence connected, for each u P U . Since f ´ is continuous,its image of the connected set Q ´ p Q p f p u qqq is connected. Hence, ˜ Q ´ p ˜ Q p u qq “ f ´ p Q ´ p Q p f p u qqqq is connected.By the assumption of local injectivity of ˜ Q , the set ˜ Q ´ p ˜ Q p u qq consists of isolatedpoints. That is, each ˜ u P ˜ Q ´ p ˜ Q p u qq has a neighborhood H such that H X ˜ Q ´ p ˜ Q p u qq “ ˜ u . Since ˜ Q ´ p ˜ Q p u qq is connected, nonempty and consists of isolated points, it canhave exactly one point. Since this is true for arbitrary u , we obtain part (i).An example of a homeomorphism is f p u q “ p f p u q , . . . , f K p u K qq , where each f k isstrictly increasing and continuous and T is rectangular (i.e. the Cartesian productof intervals). Note that we no longer conclude that checking local injectivity of ˜ Q isequivalent to checking whether it is constant on line segments. Recall a set is connected if it cannot be partitioned into two disjoint nonempty sets that areopen in the relative topology. Convex sets are clearly connected.
16e can obtain a sharper result with alternative structure on f . Say that a mapping f : R K Ñ R K is affine if it may be written f p u q “ Au ` b for some K ˆ K matrix A andvector b P R K . Affine mappings need not satisfy the law of demand. Nonetheless,affine mappings have the important property that for a convex set B Ď R K , f ´ p B q is convex. By leveraging Lemma 1, this leads to the following result. Proposition 5.
Let Assumption 3 hold for affine f . Then for each y P R K , ˜ Q ´ p y q “ t u P T | ˜ Q p u q “ y u is convex. In particular, the following are equivalent:i. ˜ Q is injective.ii. ˜ Q is locally injective.iii. The only line segments in U along which Q is constant are points.Proof. The set Q ´ p ˜ Q p u qq is convex. Thus the set ˜ Q ´ p ˜ Q p u qq “ f ´ p Q ´ p ˜ Q p u qqq isconvex because f is affine. The result is then analogous to Proposition 1. This paper leverages a classical result in monotone operator theory to provide sim-ple necessary and sufficient conditions to check when a demand mapping is injective.Specifically, for continuous demand mappings that satisfy the law of demand and thatare defined over an open convex domain, local injectivity and global injectivity areequivalent. In addition, injectivity can be checked by seeing if the demand mappingis constant over any line segments that are not points. I describe the relationshipto a classical result of Gale and Nikaido [1965] for quasi-definite Jacobians, providingnecessary and sufficient conditions for global injectivity in terms of directional deriva-tives. Finally, I show that the law of demand is not nested in and does not nest the“connected substitutes” condition of Berry et al. [2013]. They do precisely when the symmetrized matrix p A ` A q{ A denotes the transpose of A . This can be seen by writing p f p u q ´ f p ˜ u qq p u ´ ˜ u q “ p u ´ ˜ u q A p u ´ ˜ u q and recalling the definition of the law of demand. ppendix A Inverse Isotonicity and Substitution To keep the paper self-contained, I provide a primitive condition that ensures thedemand mapping satisfies inverse isotonicity.
Lemma 4 (Mor´e and Rheinboldt [1973]) . Let Q : U Ď R K Ñ R K , where U is aCartesian product. In addition, assume Q satisfies strict own-good monotonicity andweak substitutability. The following are equivalent:1. Q satisfies inverse isotonicity.2. Q is a P -function, i.e. for u ‰ ˜ u , there is some k such that p Q k p u q ´ Q k p ˜ u qqp u k ´ ˜ u k q ą . We note that while Mor´e and Rheinboldt [1973] prove this result for U a rectangle(i.e. a Cartesian product of intervals), their proofs go through without modifica-tion when U is an arbitrary Cartesian product. P -functions are closely related tofunctions whose Jacobians are P -matrices, whose injectivity properties are studied inGale and Nikaido [1965]. See Mor´e and Rheinboldt [1973] for more details.From this result we conclude that because the connected substitutes assumption ofBerry, Gandhi, and Haile [2013] implies inverse isotonicity, the demand mapping intheir setup is a P -function. This can be deduced from their Lemma 3, which statesthat under their assumptions, if u ‰ ˜ u and I “ t k | u k ą ˜ u k u is nonempty, then ÿ k P I Q k p u q ą ÿ k P I Q k p ˜ u q . This implies that there must be some k P I such that p Q k p ˜ u q ´ Q k p u qqp u k ´ ˜ u k q ą Q must be a P -function. Note that if I is empty we can repeat the argument with u and ˜ u interchanged. eferences Rodrigo Adao, Arnaud Costinot, and Dave Donaldson. Nonparametric counterfac-tual predictions in neoclassical models of international trade.
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