Non-CES Aggregators: A Guided Tour

The constant-elasticity-of-substitution (CES) aggregator and its demand system are ubiquitous in business cycles theory, macroeconomic growth and development, international trade and other general equilibrium fields; this is because the CES aggregator has many knife-edge properties that help to keep the analysis tractable in the presence of many goods and factors. However, this also makes it hard to tell which properties of CES are responsible for certain results. Furthermore, it is necessary to relax some of those properties for certain applications. In this article, I review several classes of non-CES aggregators, each of which removes some properties of CES and keeps the rest to introduce some flexibility while retaining the tractability of CES as much as possible. These classes are named after the properties of CES they keep. I explain how these classes are related to each other and discuss their relative strengths and weaknesses to indicate which classes are suited for which applications.


INTRODUCTION
The scope of this article is summarized in Figure 1.The figure shows that the constant-elasticityof-substitution (CES) aggregator is an intersection of many different classes of aggregators, and that we could depart from CES in many different directions.
We all know CES and love using it.CES is ubiquitous in business cycles theory, macroeconomic growth and development, international trade and other general equilibrium fields.Most researchers in these fields use CES almost anywhere they need some kinds of aggregators (preferences, production functions, matching functions, externalities, etc.), since CES has many knife-edge properties that help to keep the analysis tractable even in the presence of many goods and factors.But precisely because it has many properties, it is hard to tell which properties are responsible for particular results.Moreover, it is desirable or even necessary to relax some of them for certain applications.Yet, we may want to relax just a few at a time, while keeping the rest.This is important not only to retain the tractability of CES as much as possible, but also to understand the implications of departing from CES in different directions.
A large number of studies have already attempted to depart from CES.However, I find many of them problematic for several reasons.First, many people tend to use a particular alternative to CES repeatedly for all purposes.For example, Stone-Geary is a favorite of many scholars whenever they need non-CES, even though Stone-Geary is just one of many possibilities and it has its own limitations.Translog is another example.Quite often, I can think of better options, depending on the goal of the analysis.Second, the relation between different classes of non-CES aggregators is poorly understood.For example, some studies use a demand system that belongs to the direct explicit additivity class, as defined below, and claim that it is general enough to encompass all homothetic demand systems.In some other studies, the authors use CES and yet claim that the results are more general because they carry over to any demand system that satisfies Landscape of the non-CES world.Abbreviations: CES, constant elasticity of substitution; DEA, direct explicit additivity; DIA, direct implicit additivity; HSA, homothetic with a single aggregator; HDIA, homothetic direct implicit additivity; HIIA, homothetic indirect implicit additivity; IEA, indirect explicit additivity; IIA, indirect implicit additivity.
a particular set of assumptions, despite the fact that CES is the only demand system that satisfies that set of assumptions.
The fact that such claims are frequently made and repeated by others indicates the need for a guided tour that collects in one place many results on different classes of non-CES, which are scattered in the literature of the past 60 years.However, my aim is not just to write a guided tour by explaining how they are related.I also aim to highlight some key features of different classes-both their strengths and weaknesses-and to indicate which classes are well-suited for what purposes, providing a sort of user's guide.
Before proceeding, three caveats should be mentioned.First, most demand systems reviewed here have found many applications, but my main goal is to explain the relation between different demand systems and their relative merits.For this reason, several applications are cited but their findings are not discussed in detail, unless they shed lights on the relative merits of different demand systems.Furthermore, no applications to monopolistic competition are even cited for the reasons explained in the concluding section.Second, although the materials covered here are technical in nature, I try to keep the discussion as nontechnical as possible.I offer some intuition behind the main results, but I provide no formal proofs and skip most derivations.Also, some regularity assumptions, such as continuity and differentiability, are not explicitly stated.This review should thus never be considered as a substitute for consulting the references cited.Finally, my goal is to clarify.Hence, I do not hesitate to drop some original terminologies in favor of alternatives, whenever I judge that they are so uninformative and/or misleading that they have become constant sources of confusion. 1

STANDARD CES
Let us start with CES of the following form and its monotone transformation: 2 1.
For the moment, let us interpret U (x) as the direct utility function.Thus, x i ≥ 0 is consumption of good i ∈ I = {1, 2, . . ., n} , with x = (x 1 , . . ., x n ) ∈ R n + being the consumption vector; β i > 0 is the share-shift parameter of i ∈ I; and σ ∈ (0, 1) ∪ (1, ∞) is the (constant) elasticity of substitution. 3Let p = (p 1 , . . ., p n ) ∈ R n + denote the price vector.Then, maximizing U (x) subject to the budget constraint, px = ∑ i∈I p i x i ≤ E yields the CES demand where P(p) is the cost-of-living index given by P(p) ≡ min From these, the budget share of i ∈ I and the indirect utility function are obtained as Some notable properties of the standard CES are the following.
■ Income elasticity of demand for each good is 1.No good is neither a necessity nor a luxury.This is due to the homotheticity of CES.
■ Marginal rate of substitution between any two goods, and hence their relative inverse demand, p i /p j = ∂U (x)/∂x i ∂U (x)/∂x j = [(x i /x j )/(β i /β j )] −1/σ , is independent of the quantity of a third good.This is due to the direct explicit additivity of CES.Furthermore, x i and x j enter only as the ratio x i /x j .
■ Relative demand between any two goods, x i /x j = (β i /β j )(p i /p j ) −σ , is independent of the price of a third good.This is due to the indirectly explicit additivity of CES, as defined below.Furthermore, p i and p j enter only as the ratio p i /p j .
■ The elasticities of substitution between all pairs of goods are identical across all pairs, and the price elasticity of demand for each good, holding P(p) fixed, is constant and identical.
■ If the goods are gross substitutes, they cannot be essential under CES.■ Demand for any good remains strictly positive when its relative price becomes arbitrarily high (no choke price).
■ Demand for any good goes up unbounded when its relative price becomes arbitrarily low (no satiation).
■ With σ ̸ = 1, one could set β i = 1 by choosing the unit of measurement of each good appropriately; the standard CES can be assumed to be symmetric without loss of generality.
These features of CES make it highly tractable, which explains its popularity.CES possesses a high degree of symmetry.The impact on the relative demand and the relative price between the two goods can be studied independently of what happens to other goods.This feature makes CES tractable even when it is defined over an arbitrarily large number of goods.No choke price/no satiation means that we do not need to worry about a corner solution.Knowing the local properties of demand, say whether the goods are gross complements or gross substitutes, is enough to know its global properties (say, whether the goods are essentials or not).Moreover, being characterized effectively by one parameter, σ , simplifies the task of estimating and calibrating.
However, precisely because CES has so many properties, it is hard to tell which ones are responsible for certain results.Moreover, these features make CES restrictive and inflexible. 5We DEA: direct explicit additivity certainly do not need all these features every time we need some types of aggregators somewhere in our models.Yet, we do not need to drop all of them.Instead, we may want to drop just a few at a time, not only to retain the tractability of CES as much as possible, but also because which features should be dropped and which features should be kept depend on the goal of the analysis.For some purposes, we may need goods to differ in their income elasticities, not in their price elasticities; for other purposes, we may need goods to differ in their price elasticities, not in their income elasticities.For some applications, we may need a mixture of gross complements and gross substitutes, or a mixture of essentials and inessentials; for some other applications, we may want to avoid the local properties of demand systems dictating their global properties, and so on.
The questions are then, How do we depart from CES and make it more flexible in some dimensions while maintaining the restrictive features in the others to keep its tractability as much as possible?And how do we achieve this systematically?With these goals in mind, I organize the remainder of this review by different classes of non-CES, each of which is defined and named by a particular property of the standard CES it maintains.

DIRECT EXPLICIT ADDITIVITY AND INDIRECT EXPLICIT ADDITIVITY 6
Let us start with the following three properties.(In what follows, M[•] denotes a monotone transformation.) ■ Direct explicit additivity (DEA): Preferences are called DEA if the direct utility function, U (x), is explicitly additive, that is, where ūi (•), i ∈ I, satisfy some additional conditions to ensure that U (x) is strictly increasing and quasi-concave.
any multi-sector models in which the intersectoral demand is given by the representative consumer with CES preferences and each sector produces its output using a CES production function or a variety of goods aggregated by CES-such as the multi-sector extension of Eaton & Kortum's (2002) model by Costinot et al.  (2012) and Caliendo & Parro (2015)-effectively use nested CES.And it works well for some purposes.For example, the relative demand for skilled versus unskilled labor depends on the price of capital, if capital and skilled labor are in the same nest and unskilled labor is not (Krusell et al. 2000).Nevertheless, nested CES inherits much of the restrictive features of CES since CESs are its building blocks.Moreover, any flexibility of nested CES is entirely due to how goods are partitioned into different nests and not to the flexible functional forms.For example, the elasticities of substitution between all pairs of goods within the same nest are identical, relative demand between two goods in the same nest is independent of the prices of a third good, some combinations of essential and inessential goods are ruled out, essential goods cannot be gross substitutes, and so on.Further, we can use any of the homothetic non-CES aggregators discussed below as a building block in a nested structure.Such nested homothetic non-CES can do everything nested CES can do and more. 6Following Hanoch (1975), I distinguish four types of additivity: direct explicit additivity (DEA), indirect explicit additivity (IEA), direct implicit additivity (DIA), and indirect implicit additivity (IIA).Being the first type of additivity introduced in the literature, DEA is often called simply "additive" without any qualifier.This common practice unfortunately created the false impression that IEA, DIA, and IIA were special cases of DEA.Quite to the contrary, DEA is a special case of DIA and is disjoint with IEA and IIA with the sole exception of CES, as shown in Figure 1.Likewise, IEA is often called simply "indirectly additive," which created the false impression that IIA was a special case of IEA.Again, quite to the contrary, IEA is a special case of IIA.These common practices have become frequent sources of confusion.To avoid such confusion, I refer to these two classes of preferences only by DEA and IEA.

IEA: indirect explicit additivity
■ Indirect explicit additivity (IEA): Preferences are called IEA if the indirect utility function, U (p/E ), is explicitly additive, that is, where vi (•), i ∈ I, satisfy some additional conditions to ensure that U (p/E ) is strictly decreasing and quasi-convex, or, equivalently, that P(p/E ) is strictly increasing and quasi-concave.
■ Homotheticity: Preferences are called homothetic if the direct utility function U (x) can be represented as a monotone transformation of a linear homogenous function of x as follows: , where X (x) satisfies X (λx) = λX (x) for any λ > 0.
Clearly, from Equation 1 and Equation 3, CES satisfies all three properties and hence belongs to the three classes labeled as DEA, IEA, and Homothetic in Figure 1.Furthermore, CES is the only intersection of the DEA and Homothetic classes (Bergson's Law).Samuelson (1965) showed that CES is also the only intersection of the DEA and IEA classes.Berndt & Christensen (1973,  theorem 6) showed that CES is also the only intersection of the IEA and Homothetic classes.These three classes are thus pairwise disjoint with the sole exception of CES, as shown in Figure 1, and hence they offer three alternative ways of departing from CES.
In the remainder of this section, we discuss DEA and IEA in detail.We will turn to the Homothetic class in Section 5.

Direct Explicit Additivity
From Equation 6, it is easy to show that DEA satisfies the following properties.
■ Marginal rate of substitution between any two goods, and hence their relative inverse demand, is independent of the quantity of a third good: ) .
From this expression, the inverse demand curve for good i ∈ I can be derived as ■ The relative inverse demand is not a function of x i /x j , and hence a proportional increase in x i and x j changes p i /p j , unless ūi (•), i ∈ I, are all power functions with a common exponent, i.e., with the sole exception of CES.
This in turn implies the following point.
■ DEA is homothetic if and only if it is CES (Bergson's Law), as indicated in Figure 1.
Many non-CES commonly used in the literature belong to the DEA class.

Example 1: quasi-linear.
Let where u i (x i ), i ̸ = k are all strictly concave.The income elasticity of k is one, and those of i ̸ = k are zero.
3.1.2.Example 2: distance to the bliss points.Let for 0 < x i < b i where δ > 0. (This one does not satisfy strict monotonicity.)For δ = 1, this is the negative of the quadratic loss function.
3.1.3.Example 3: Stone-Geary. 7Let , where xi > 0 may be interpreted as the subsistence level of consumption of good i and − xi > 0 as the nontransferable endowment of good i.With the budget constraint, ∑ n i=1 p i x i ≤ E, the demand takes the form of ∑ n i=1 i (p) = 0 for E large enough to ensure that m i > 0 for all i ∈ I. Until recently, Stone-Geary was by far the most commonly used of nonhomothetic preferences in the growth, trade, and development fields (see, e.g., Caselli & Ventura 2000; Kongsamut et al.  2001; Markusen 1986, 2013; Matsuyama 1992, 2009).In fact, it was so common that some people use "Stone-Geary" as synonymous with "nonhomothetic." Some key properties of Stone-Geary are the following.
■ The budget share of i (its average propensity to consume) is decreasing in E (i.e., a necessity) for i (p) > 0 and increasing in E (i.e., a luxury) for i (p) < 0.
■ The marginal propensity to consume, ∂ (p i x i )/∂E = B i (p), is independent of E, which allows for aggregation across households with different expenditure.
■ It is asymptotically homothetic so that nonhomotheticity is quantitatively important only for poor households/countries.This feature not only is inconsistent with the evidence of stable slopes of Engel's curves (Comin et al. 2021) but also makes Stone-Geary difficult to fit the long-run data (Buera & Kaboski 2009).
■ The price elasticity of demand for a necessity (a luxury) is increasing (decreasing) in E.
■ The key parameters, xi , are defined in quantity of good i, hence they are not unit-free.One could thus choose the unit of each good so that xi = 1, 0, or −1 without loss of generality.In other words, Stone-Geary cannot meaningfully distinguish more than three goods in terms of their income elasticities.
■ If two or more goods have a subsistence level of consumption-say, x1 > 0 and x2 > 0the domain of this utility function cannot be extended unambiguously to 0 ≤ x 1 < x1 and 0 ≤ x 2 < x2 .
which can be viewed as a limit of Stone-Geary as σ → 0 and xi Examples 2, 3, and 4 are often called the Pollak family (Pollak 1971) family or linear expenditure systems (LES). 8They all imply that the marginal propensity to consume each good is constanthence they have nice aggregation properties across households with different total expendituresand that they are all asymptotically homothetic. 93.1.5.Example 5: constant ratios of elasticities of substitution.Let or equivalently, Houthakker (1960) called this "direct addilog."Let η i be the income elasticity of i and σ i j be the Allen/Uzawa 10 elasticity of substitution between i and j.Then, for any i ̸ = j ̸ = k ∈ I, we have where σ ≡ ∑ n l=1 m l σ l is the budget-share weighted average of {σ l }.Notice that σ i j is not constant, because σ is not.Yet the ratio σ ik /σ jk is constant, σ i /σ j .For this reason, Mukerji (1963) called it constant ratios of elasticities of substitution (CRES).Note that η i /η j is also constant, σ i /σ j .For this reason, Caron et al. (2014) called it constant relative income elasticity (CRIE).Unlike the Pollak family, nonhomotheticity does not disappear as the expenditure goes up.However, the (constant) ratio of income elasticities between any two goods is always equal to the (constant) ratio of their price elasticities.This makes it unclear whether any results obtained by departing from CES within CRES = CRIE should be interpreted as due to the income elasticity differences, as Fieler (2011) and Caron et al. (2014, 2020) did, or as due to the price elasticity differences. 11ndeed, this is a general feature of DEA, as shown by Houthakker (1960), Goldman & Uzawa  (1964), and Hanoch (1975, equation 2.11), among others. 8They are not to be confused with linear demand systems (LDS), derived from linear-quadratic ∑ n i, j=1 γ i j x i x j .LDS are not DEA, unless γ i j = γ ji = 0 for all i ̸ = j. 9LES is not the only demand system that allows for aggregation across households with different total expenditures.One example is price independent generalized linearity (PIGL) proposed by Muellbauer (1975, 1976)  and recently applied by Boppart (2014).Another is the hierarchical demand system discussed below.These demand systems are not asymptotically homothetic. 10Hicks originally defined the elasticity of substitution for n = 2.For n > 2, there are related but alternative definitions.Allen/Uzawa is one (see, e.g., Uzawa 1962).Morishima (1967) is another.Readers are referred to Blackorby & Russell (1981, 1989) on this issue. 11Although Fieler (2011) and Caron et al. (2014, 2020) performed some robustness checks using alternative classes of nonhomothetic preferences, income and price elasticities are still tightly linked under those alternatives, with the exception of isoelastic nonhomothetic CES (Example 7) used by Caron et al. (2020).
3.1.6.Pigou's Law.Under DEA, for any i ̸ = j ̸ = k ∈ I, we have Clearly, Bergson's Law is a special case.Pigou's Law also explains why, with quasi-linear preferences (Example 1), the income elasticity of k is 1 and those of i ̸ = k are 0, and why, with Stone-Geary (Example 3), the relative price elasticity of luxury goods must be decreasing in the total expenditure, because their income elasticities are also decreasing in the total expenditure, due to its asymptotic homotheticity.It is also the reason behind the (well-known but counterintuitive) result that the optimal commodity taxation, which should tax the goods with lower price elasticity more heavily, should tax the goods with lower income elasticity more heavily (see, e.g., Auerbach  1985, Chari & Kehoe 1999).
Pigou's Law not only is rejected empirically (Deaton 1974) but also suggests a limitation of using DEA as an attempt to introduce more flexibility to CES.Under DEA, the effects of the income elasticity differences across goods cannot be disentangled from those of the price elasticity differences.

Indirect Explicit Additivity
From Equation 7, it is easy to show that IEA satisfies the following properties.
■ Relative demand for any two goods is independent of the price of any other goods, because From this expression, the demand curve for good i ∈ I can be derived as ■ Relative demand is neither independent of E nor a function of p i /p j , hence a change in E and a proportional increase in p i and p j shift x i /x j , unless vi (•), i ∈ I, are all power functions with a common exponent, i.e., with the sole exception of CES.
This, in turn, implies the following point.
■ IEA is homothetic if and only if it is CES, as indicated in Figure 1.

Example 6: constant differences of elasticities of substitution.
Let or equivalently, Houthakker (1960) called this indirect addilog.Hanoch (1975) called it constant difference of elasticity of substitution (CDES).Jensen et al. (2011) discuss its properties and the history of its use in detail.Analogously to CRES = CRIE, one may also call it constant difference of income  Hanoch (1975, equation 3.11).Let η i denote the income elasticity of i and σ i j denote the Allen/Uzawa elasticity of substitution between i and j.Then, under IEA, we have

Indirect Pigou's Law: Houthakker (1960),
Again, as for DEA, the effects of the income elasticity differences and those of the price elasticity differences cannot be disentangled under IEA.
Both (direct and indirect) Pigou's Laws show a limitation of explicit additivity and of the DEA and IEA classes of demand systems.Of course, there are many preferences that belong neither to DEA nor IEA.For example, the linear-quadratic direct utility function, or constant differences of elasticities of substitution (CDES), augmented by the Stone-Geary subsistence consumption shifters, 2017), is not explicitly additive due to the presence of the interactive terms.While these interactive terms add more flexibility, these functional forms still impose tight links between the income and price elasticities.So is the Almost Ideal Demand System (AIDS) proposed by Deaton & Muellbauer (1980a) and applied recently by, e.g., Fajgelbaum & Khandelwal (2016), in which both income and price elasticities are controlled by the same parameters.

DIRECT IMPLICIT ADDITIVITY, INDIRECT IMPLICIT ADDITIVITY, AND IMPLICIT CES
The restrictive nature of explicit additivity motivated Hanoch (1975) to introduce the weaker notion of implicit additivity, which makes it possible to control for the income and price elasticity differences across goods separately.

Direct Implicit Additivity and Indirect Implicit Additivity
Let us now introduce two weaker properties, direct implicit additivity (DIA) and indirect implicit additivity (IIA), and the two classes of demand systems they define.13■ Direct implicit additivity: Preferences are called DIA if the direct utility function, U (x), is implicitly additive, that is, where ũi (•, •), i ∈ I, satisfy some additional conditions for strict monotonicity and quasiconcavity of U (x). DEA is a subclass of DIA, with ũi (x i , U ) = ūi (x i )g(U ).
■ Indirect implicit additivity: Preferences are called IIA if the indirect utility function,U (p/E ), is implicitly additive, that is, where ṽi (•, •), i ∈ I, satisfy some additional conditions for strict monotonicity and quasiconvexity of U (p/E ).IEA is a subclass of IIA, where ṽi ( Implicit additivity has clear advantages relative to explicit additivity. 14It allows us to control for the price elasticity difference and the income elasticity difference across goods separately.For DIA, the price elasticity depends on the curvature of ũi (x i , U ) with respect to x i , and the income elasticity depends on the curvature of ũi (x i , U ) with respect to U ; in particular, the two elasticities can be controlled separately for ũi (x i , U ) = ūi (x i )g i (U ).Similarly, for IIA, the price elasticity depends on the curvature of ṽi (p i /E, U ) with respect to p i , and the income elasticity depends on the curvature of ṽi (p i /E, U ) with respect to U ; in particular, the two elasticities can be controlled separately for ṽi (

Nonhomothetic CES
Due to such flexibility of implicit additivity, the standard CES is not the sole member of the intersection of DIA and IIA.Indeed, Hanoch (1975) showed that implicit CES defined below satisfies both DIA and IIA.Furthermore, implicit CES is the only demand system that satisfies both DIA and IIA, as illustrated in Figure 1. 15 4.2.1.Implicit CES.More formally, preferences are called implicit CES if the direct utility function, U (x), is defined implicitly as where σ (U ) > 0; ̸ = 1, and β i (U ) > 0, i ∈ I, are functions of U and must satisfy some additional conditions to ensure that U (x) is strictly monotonic and quasi-concave (see Fally 2022, section A4).Its indirect utility function, U (p/E ), is written implicitly as ≡ 1; and U = U (p/E ) and P = P(p, U ) satisfy the identity PU = E.
14 Some people seem to view that any implicitly defined direct or indirect utility functions as in Equation 8and Equation 9 are illegitimate.My response is that many commonly used functions are defined implicitly; for example, log is defined as an inverse of an exponential function, and arctangent is defined as an inverse of a tangent function. 15I am not aware of any existing proof of this.However, it follows from the proof of proposition 4(iii) of Matsuyama & Ushchev (2017).Though this proposition states that the homothetic restrictions of DIA and IIA (HDIA and HIIA, defined below) imply homothetic CES, homotheticity does not play any role in the proof.
This class of preferences is nonhomothetic whenever ∂ ln β i (U )/∂ ln U depend on i and/or σ (U ) depend on U .Nevertheless, they are CES in that the Hicksian demand generated is indistinguishable from those generated by the standard CES, because the Hicksian demand is calculated for a fixed level of the utility. 16mong this class, the following parametric family found many applications in the structural transformation literature (see, e.g., Bohr et al. 2021, Comin et al. 2021, Cravino & Sotelo 2019,  Fujiwara & Matsuyama 2022, Lewis et al. 2022, Matsuyama 2019, Sposi et al. 2021).

Example
where σ > 0 ensures global quasi-concavity, while global monotonicity requires where indirect utility, U = U (p/E ), is implicitly given by where (ε i − σ )/(1 − σ ) > 0, the condition for global monotonicity, ensures that U = U (p/E ) is strictly increasing in E, and the cost-of-living index, P = P(p, E ), is implicitly given by From Equation 11, we obtain the familiar double-log CES demand systems, ln with an additional term representing the income effect, with the constant slope ε i − ε j that is, unlike Stone-Geary, consistent with the empirical evidence of the stable slopes of the Engel's curve (Comin et al. 2021). 18One could also show that

PIGL:
price independent generalized linearity which means that good i ∈ I is a necessity if and only if ε i < ε and is a luxury if and only if ε i > ε, where ε ≡ ∑ n k=1 m k ε k is the budget-share weighted average of ε i , i ∈ I. Thus, unlike DEA or IEA, the income elasticities of demand for different goods, η i , i ∈ I, can be controlled by the parameters ε i , i ∈ I, separately from the constant elasticity of substitution parameter, σ , which governs the price elasticity.
To explore further, let us index the goods such that ε That is, the goods are ordered such that higher-indexed goods have higher income elasticities.Then, from Equation 12and Equation 13, we obtain the following.
■ A larger U = E/P shifts the budget shares, m i , i ∈ I, toward more income-elastic, higherindexed goods in a monotone likelihood way.
■ The income elasticity of i ∈ I, η i , declines monotonically in U = E/P and hence in E as follows.
°We have °For 2 ≤ i ≤ n − 1 (with n ≥ 3), we have η i > 1for a small E and η i < 1 for a large E, since for the poor but a necessity (η i < 1) for the rich.Even though the ratio of the budget shares of two goods is monotonic in U = E/P and hence in E, the budget share of good i is hump-shaped.This means that isoelastic nonhomothetic CES can capture the situations like a private jet being a luxury for most people but a necessity for the billionaire, or air conditioners or smartphones being necessities for most but luxuries for the poor. 19This feature makes Example 7 well suited for explaining the rise and fall of industry, and more generally structural transformation, where sectoral shares exhibit hump-shaped paths over the course of development (see Bohr et al. 2021, Comin et al. 2021, Fujiwara & Matsuyama  2022, Matsuyama 2019).In contrast, Stone-Geary and other LES, CRES = CRIE, and AIDS cannot capture such situations, because whether a good is a necessity or a luxury is independent of the household expenditure. 20A downside of this feature is that nonhomothetic CES does not aggregate easily across households with different expenditures, unlike LES or price independent generalized linearity (PIGL). 21 19 Bnks et al. (1997) showed evidence that the budget shares of alcohol and clothing are hump-shaped in the total expenditure.This motivated them to propose quadratic AIDS, an extension of AIDS in which the budget shares are quadratic in log total expenditure, which violates global monotonicity.In contrast, Example 7 generates hump shapes without violating global monotonicity. 20This explains why Kongsamut et al. (2001), who used Stone-Geary, were unable to generate the humpshaped path of the manufacturing share in spite of having three sectors. 21Nonhomothetic demand systems in which some goods are luxuries for the poor and necessities for the rich, with nice aggregation properties, exist in the form of hierarchical demand systems (see, e.g., Buera &  Kaboski 2012a,b; Foellmi & Zweimueller 2006; Matsuyama 2000, 2002).In these demand systems, goods are ranked according to priority, and as the income goes up, the household expands the range of goods by going down on the shopping list.For example, let U (x) = ∑ ∞ j=1 β j min{x j , x j }, where x j is the saturation level of good j.If β j /p j is monotone decreasing, households buy goods j ∈ {1, . . ., J} up to the saturation levels and some of good J + 1, where J is determined by j=1 p j x j .Thus, as E rises, J goes up.
This means that each good is a luxury for poor households and a necessity for rich households.Alternatively, )u(x 2 ) + . . . .Then, if x j = 0, we have ∂U (x)/∂x k = 0 for any j < k.Then, demand is hierarchical for any prices, and each good is a luxury for the poor and a necessity for the rich.The hierarchical systems have easy aggregation properties but also their own limitations (i.e., most goods are either consumed at their saturation levels or not at all).

HOMOTHETIC AND LINEAR HOMOGENEOUS FUNCTIONS: A QUICK REFRESHER
We now turn to homothetic non-CES.Departing from the standard CES without giving up homotheticity is important for several reasons.First, when we model a competitive industry, we often need to assume that its production technologies satisfy constant returns to scale (CRS). 22This means that we need to have a linear homogeneous (hence homothetic) function.Second, think of any level of aggregation that defines a composite good.For example, "food" is not a physical object.Instead, it is a category of goods, say, bread, fish, fruits, meat, vegetable, etc.Most of these goods are in turn a composite of finer categories of goods.For example, "fruits" consists of apples, bananas, oranges, etc., and "vegetable" consists of carrots, cucumbers, onions, potatoes, tomatoes, etc.In order to give a cardinal (i.e., quantity) interpretation to any composite of goods, so that the statement like "a 10% increase in food consumption" makes sense, an aggregator that maps a quantity vector of component goods into a quantity of the composite must be linear homogeneous (hence homothetic).Third, we often write down a general equilibrium model in which an overall demand system of the economy is given by multi-layers of the demand systems with nested structures.Then, assuming demand systems to be nonhomothetic anywhere except in the highest tier would create a technical problem, because that would prevent us from solving an overall demand system by breaking it down to smaller problems and solving them sequentially using a multi-stage budgeting procedure. 23Fourth, we often abstract from nonhomotheticity for tractability.For example, the homotheticity assumption may be necessary for ensuring the existence of a steady state in dynamic general equilibrium.Moreover, homothetic functions not only are used for utility and production functions but also are used often for matching functions and externality terms to keep the model scale-free.For all these reasons, it is useful to have linear homogeneous aggregators, for which we may not want our choice to be restricted to the standard CES.
Let us recall next the definitions of homothetic and linear homogeneous functions and their basic properties, which can be found in any graduate-level microeconomics textbooks (see, e.g., Jehle & Reny 2010, Mas-Colell et al. 1995).

Homothetic and Linear Homogeneous Functions: A General Case
An aggregator, X (x) : where M[•] is a monotone transformation with linear homogeneous X (x).Conversely, any homothetic H (x) can be expressed as In what follows, for concreteness, let us interpret x ∈ R n + as a quantity vector of the factors of production and X (x) as a CRS production function.Then, with a factor price vector, p ∈ R n + , we define the unit cost function as 22 Recall that the CRS technology of a competitive industry is consistent with the firm-level technologies subject to increasing returns due to some fixed costs and decreasing returns due to some managerial constraints.
As any introductory textbook shows, the U-shaped average cost curve of a firm leads to the constant average cost of an industry as industry size changes with the number of firms in the industry. 23Of course, for some applications (e.g., Fajgelbaum et al. 2011, Flam & Helpman 1987), it is essential to have sector-level nonhomothetic demand, but this needs to be combined with some specific assumptions on intersectoral demand to keep the model tractable.which is linear homogeneous, monotone, and quasi-concave in p ∈ R n + .Furthermore, if X (x) is monotone and quasi-concave, it can be recovered from P(p) as Due to this duality, either X (x) or P(p) can be used as a primitive of the CRS technology.

Homothetic Demands and Budget Shares: A General Case
Let us denote the factor demand by the competitive producers by x(p) ≡ argmin For a strictly quasi-concave X (x), Shepherd's lemma tells us that from which the budget share of factor i can be written as a function of p ∈ R n + : From Euler's theorem on linear homogeneous functions, these shares are added up to 1, ∑ n i=1 m i = 1, and each of them is homogeneous of degree zero in p.
The inverse factor demand can be given by p(x) ≡ argmin For a strictly quasi-concave P(p), we have from which the budget share of factor i can be written as a function of x ∈ R n + : Again, from Euler's theorem on linear homogeneous functions, these shares are added up to 1, ∑ n i=1 m i = 1, and each of them is homogeneous of degree zero in x.

Three Classes of Linear Homogeneous Functions: An Overview
As shown in Equation 14and Equation 15, the budget shares can be written as functions of homogeneity of degree zero in p ∈ R n + or in x ∈ R n + .This also means that they could generally depend up to (n − 1)-relative prices or (n − 1)-relative quantities. 24For the tractability, some restrictions may be imposed so that the budget shares depend on a few relative prices or quantities.
To this end, Matsuyama & Ushchev (2017) consider three properties of demand systems called homothetic with a single aggregator (HSA), homothetic direct implicit additivity (HDIA), and homothetic indirect implicit additivity (HIIA), each of which is used to define a class of homothetic functions, because of the following advantages.
■ For n > 2, HSA, HDIA, and HIIA are pairwise disjoint with the sole exception of CES, as shown in Figure 1.Thus, they offer three alternative ways of departing from CES without giving up the homotheticity.
24 Indeed, it is easy to construct homothetic demand systems with n-factors that depend on (n − 1)-relative prices or quantities.For example, the nested CES of n-factors, X n (x n ), x n ∈ R n + , given recursively by ■ They contain some existing families of homothetic functions.■ Each is tractable because the budget share of each factor is a function of one relative price (for HSA) or of two relative prices (for HDIA and HIIA) for any number of factors, which drastically reduces the dimensionality of the problem.
■ The price elasticity of each factor is a function of one relative price in each class.This allows for a natural extension of the definition of gross substitutes and gross complements.
■ Each is defined nonparametrically and hence is flexible.This provides a template to construct many different types of homothetic functions that relax some features of CES.Some examples are listed below.
°Different factors have different but constant price elasticities.°Factors can be gross substitutes and yet essential. 25Any combination of essential and of inessential factors is possible.°For HDIA and HIIA, any combination of gross substitutes and gross complements is possible, and a factor can be a gross substitute or a gross complement, depending on the relative prices.
We now formally define each of the three classes and explain their properties in some detail.

HOMOTHETIC WITH A SINGLE AGGREGATOR
The CRS production function X (x) and its unit cost function P(p) are called HSA if the budget share of each factor as a function of p ∈ R n + can be written as where s i : R + → R + is a function of a single variable, and A(p) is linear homogeneous in p, defined implicitly and uniquely 26 by the adding-up constraint, which ensures, by construction, that the budget shares of all factors are added up to 1. Equation 16and Equation 17 state that the budget share of a factor is a function of its relative price, z i ≡ p i /A(p), defined as its own price, p i , divided by the common price aggregator, A(p).Notice that A(p) is independent of i; it is the average factor price against which the relative price of every factor is measured.In other words, one could keep track of all the cross-price effects in the demand system by looking at a single aggregator, A(p), which is the key feature of HSA. 27The unit cost function, P(p), behind this HSA demand system can be obtained by integrating Equation 16, which yields ln 25 For any X (x) and P(p), factor i is essential (or indispensable) if x i = 0 implies X (x) = 0 (or equivalently, if p i → ∞ implies P(p) → ∞) and inessential (or dispensable) otherwise.The notion of essentials should not be confused with that of necessities, which are defined as the goods whose income elasticities are less than 1. 26 The unique solution requires that s i be either nonincreasing in all i with 27 The HSA class is the homothetic restriction of what Pollak (1972) refers to as generalized additively separable demand systems.However, we prefer to call it HSA, instead of homothetic with generalized additivity, because it does not contain any demand systems with additivity (whether direct or indirect or explicit or implicit) with the exception of CES, as seen in Figure 1.
where c 1 is an integral constant. 28By applying Antonelli's integrability theorem [Antonelli 1971  (1886), Hurwicz & Uzawa 1971; see also Mas-Colell et al. 1995, chap.3; Jehle & Reny 2010, chap.2], Matsuyama & Ushchev (2017, proposition 1-i) show that the demand system is well defined by Equation 16and Equation 17 and that the unit cost function, P(p), satisfies the linear homogeneity, monotonicity, and strict quasi-concavity if z i s ′ i (z i ) < s i (z i ) and s ′ i (z i )s ′ j (z j ) ≥ 0. By defining the price elasticity function, these conditions can be further rewritten as This guarantees the integrability of the HSA demand system, that is, the existence of the underlying CRS technology, X (x) or P(p), that generates this HSA demand system.
It is also important to note that, for n > 2, P(p) ̸ = cA(p) for any constant c > 0 with the sole exception of CES. 29 This can be verified by differentiating Equation 17to obtain for all i ∈ I.This should not come as a surprise.After all, A(p) is the average factor price, capturing the cross-price effects in the demand system, while P(p) is the unit cost of production, capturing the productivity (or welfare) effects of factor price changes.There is no reason to think a priori that they should move together.Because the budget share of i ∈ I is a function of a single relative price, z i , s i (z i ), the notion of gross substitutes and gross complements under CES can be extended naturally.That is, we call factor i ∈ I a gross substitute (complement) when s i (z i ) is strictly decreasing (strictly increasing) in z i .In other words, factor i ∈ I is a gross substitute when ζ i (z i ) ≡ 1 − z i s ′ i (z i )/s i (z i ) > 1, and factor i ∈ I is a gross complement when 0 < ζ i (z i ) < 1.Notice that one of the integrability conditions, [1 − ζ i (z i )][1 − ζ j (z j )] ≥ 0, implies that HSA does not allow for a mixture of gross substitutes and gross complements.However, a factor with ζ i (z i ) = 1 can coexist either with gross substitutes or with gross complements. 30efore proceeding to some examples, let us point out that there exists an alternative and yet equivalent definition of HSA.That is, the CRS production function X (x) and its unit cost function P(p) are called HSA if the budget share of factor i as a function of x ∈ R n + can be written as ) , 18.
28 Note that this constant cannot be pinned down.First, A(p), the "average factor price," does not depend on the unit of measurement of the final good.In contrast, P(p) is the cost of producing one unit of the final good when the factors prices are given by p; hence, it depends not only on the units of measurement of factors but also on those of the final good.Second, a change in TFP, though it affects P(p), leaves the relative factor demand, hence A(p), unaffected. 29The condition n > 2 is necessary.If n = 2, the budget share of both factors is always a function of one relative price.Hence, all homothetic functions are HSA.In other words, HSA are restrictive only for n > 2. 30 One could also show that both Allen/Uzawa and Morishina elasticities of substitution between i and j are greater than one if ζ i (z i ), ζ j (z j ) > 1 and smaller than one if ζ i (z i ), ζ j (z j ) < 1.

TFP: total factor productivity
where s * i : R + → R + is a function of a single variable, its relative quantity, y i ≡ x i /A * (x), and A * (x) is the common quantity aggregator defined implicitly and uniquely 31 by The CRS production function X (x) behind this HSA demand system can be obtained by integrating Equation 18, which yields Matsuyama & Ushchev (2017) show that the two definitions define the same class of homothetic functions, with the one-to-one correspondence between s i (z i ) and s * i (y i ), defined by ) .
Note that differentiating either of these equalities yields p), which cannot be a constant with the sole exception of CES for n > 2.
We now turn to several examples of HSA.16and Equation 17 we have

Example 8: CES as a Special Case of HSA
and from Equation 18and Equation 19 we have where Z > 0 is an integral constant, which can be interpreted as total factor productivity (TFP).Note that both A(p) and A * (x) are independent of TFP, which is true for any HSA demand system.Indeed, TFP shocks to the CRS production function do not affect its relative factor demand.Note also that A(p)/P(p) = X (x)/A * (x) = Z is constant.This is true only for CES, as already pointed out.
For σ > 1, s i (z i ) is globally strictly decreasing for i ∈ I, which means that every factor is always a gross substitute, and yet s i (z i ) > 0 for any z i < ∞, meaning that it has no choke price.Moreover, 31 The unique solution requires that s * i be either nonincreasing in all i with the generic condition for factor i being inessential, which can be expressed as automatically holds so that every factor is inessential.For σ < 1, s i (z i ) is globally strictly increasing for i ∈ I, so that ∫ ∞ c 1 (s i (ξ )/ξ )dξ = ∞, which means that every factor i is always a gross complement and essential.
Under generic HSA, it is easy to verify that, when s i (z i ) is globally strictly increasing for all i ∈ I (the case of gross complements), all factors must be essential.On the other hand, when s i (z i ) is globally strictly decreasing for all i ∈ I (the case of gross substitutes), there are four possibilities: Under CES with σ > 1, only the third case is allowed.We now turn to examples for the first case, in which some gross substitutes factors can be essential.

Example 9: Hybrids of Cobb-Douglas and CES Under HSA
Consider the HSA demand system, Equation 16and Equation 17, given by Then, we have This is a convex combination of Cobb-Douglas and CES since it is Cobb-Douglas for ε = 1 and CES for ε = 0. Similarly, consider the HSA inverse demand system, Equation 18 and Equation 19, given by Then, we have This is another convex combination of Cobb-Douglas and CES. 32In both cases, all factors are gross substitutes for σ > 1, and yet factor i is essential if α i > 0 and inessential if α i = 0. Thus, 32 These two convex combinations are not equivalent, because Note also that neither of them is a nested CES, because α i β i ̸ = 0 for some i. some gross substitutes are essential.Furthermore, any combination of essential and inessential factors can coexist.
To see implications, consider a model of international trade where each country produces the single nontradeable consumption good using tradeable factors under the HSA technologies described above.With a small ε, the demand system can be approximated by CES with trade elasticity σ .If it is CES (ε = 0), autarky would lead to a small welfare loss with a moderately large σ > 1.Yet, for an arbitrarily small but positive ε > 0, the welfare loss of autarky, measured in the cost-of-living index, is infinity if a country has no domestic supply of an essential factor. 33ore broadly, when gross substitutes are essential with their price elasticities converging to 1 as they become increasingly scarcer, the welfare impacts of large shocks-say, sanctions or pandemic-induced lockdowns-would be large.This offers a caution against assessing the impacts of large changes by using the empirical evidence obtained by local changes as "disciplines" under the straitjacket of CES.
The next example features gross substitutes with the choke prices.

Example 10: "Separable" Translog
Two often-used non-CES are translog unit cost functions and translog production functions (Christensen et al. 1973, 1975).They are isolated from CES and not an extension of CES.Nevertheless, it is worth discussing them here, because they have two subfamilies that belong to HSA.First, consider the translog unit cost function where δ i > 0; (γ i j ) is symmetric and non-negative semidefinite, which can be normalized as ∑ n j=1 δ j = 1; and ∑ n j=1 γ i j = 0.In general, the translog unit cost function is not HSA.However, under the following "separability" condition, satisfied by symmetric translog } .
If γ = 0, this is Cobb-Douglas.If γ > 0, all factors are gross substitutes with the choke prices zi A(p), where zi = exp(δ i /γ β i ), and inessential.For p i < zi A(p) for all i, we have Next, consider the translog production function where δ i > 0; (γ i j ) is symmetric and non-negative semidefinite, which can be normalized as ∑ n j=1 δ j = 1; and ∑ n j=1 γ i j = 0.In general, the translog production function is not HSA.However, under the following "separability" condition, satisfied by symmetric translog it is HSA with , 0 } .
If γ = 0, this is Cobb-Douglas.If γ > 0, all factors are gross complements with the saturation points, ȳi A * (x), where ȳi = exp(δ i /γ β i ), and essential.For x i < ȳi A * (x), for all i, we have These calculations reveal the restrictive nature of the translog aggregators, which seems unnoticed by many in spite of their popularity as an alternative to CES.In the case of the translog unit cost function, it allows only for gross substitutes and inessential factors with the choke prices.In the case of the translog production function, it allows only for gross complements and essential factors with the saturation point. 34

Example 11: HSA Demand Systems with Constant but Different Price Elasticities)
where either σ i ≤ 1 for all i, or σ i ≥ 1 for all i.Then, A(p) and A * (x) are given implicitly by 34 Both translog unit cost and production functions, as well as their nonhomothetic counterparts, AIDS, are often touted as flexible.But they are flexible only in the sense that they offer local approximations to any aggregators up to their second derivatives.Such approximations may be good enough for studying the impacts of small shocks to a competitive economy, where all firms are price takers.However, they should be used with great caution when studying the impacts of large shocks or even those of small shocks if some firms have the price-setting powers, since the results would then depend on the global properties and/or the third or higher derivatives of the aggregators.
and we have ln Both Allen/Uzawa and Morishima elasticities of substitution between each pair are variable unless σ i = σ for all i ∈ I.However, holding A(p) or A * (x) fixed, the own price elasticity of each factor is constant but different, because Furthermore, for a large n, the impact of a change in p i on A(p) and the impact of a change in x i on A * (x) are negligible.Hence, the own price elasticity when all other prices are fixed, or when all other quantities are fixed, is approximately constant and converging to σ i , as n → ∞. 35 Thus, this example, along with Example 13 and Example 15, discussed below, can isolate the role of price elasticity differences across factors without giving up homotheticity, unlike Example 5 under DEA, which are subject to Pigou's Law, and Example 6 under IEA, which are subject to indirect Pigou's Law.

Definition
The CRS production function X (x) and its unit cost function P(p) are called HDIA if X (x) can be written as where ϕ i : R + → R, i ∈ I, are strictly increasing and strictly concave and satisfy This ensures the unique existence of monotonic and quasi-concave X (x).Clearly, HDIA is the homothetic restriction of DIA, as shown in Figure 1. 36olving P(p) ≡ min x∈R n + {px|X (x) ≥ 1} subject to Equation 20 yields the HDIA demand system ) , 21. 35 The own price elasticity of each factor is constant without these qualifications for the case of a continuum of factors, with the summations in Equation 16and Equation 17 or those in Equation 18and Equation 19 being replaced by the integrals. 36According to the original definition of DIA by Hanoch (1975), its homothetic restriction HDIA should be written as ∑ n i=1 ϕ i (x i /X (x)) = 1.However, the right-hand side of Equation 20 can be any constant, and we set it equal to zero, which has two advantages.First, one could restrict all ϕ i to be strictly increasing and concave without loss of generality.Second, multiplying all ϕ i by a positive constant would not change the function defined.For example, the Kimball aggregator, a special case of HDIA, is typically defined as ∑ n i=1 ϕ(x i /X (x)) = 1, where ϕ is strictly increasing and concave.This definition imposes not only symmetry but also gross substitutability.Furthermore, the function defined changes if ϕ is multiplied by a positive constant.
where B(p) is the Lagrange multiplier associated with the minimization problem; it is a linear homogeneous function in p ∈ R n + , implicitly defined by )) ≡ 0, and P(p) is the unit cost function, related to B(p) as follows: ) .
The HDIA inverse demand system can be obtained by differentiating Equation 20 as follows: where C * (x) is a linear homogeneous function in x ∈ R n + defined by ) .
From these, we also obtain ) and Notice that Equation 21and Equation 22 suggest that the budget share of i under HDIA depends on the two different relative prices, p i /B(p) and p i /P(p), or on the two relative quantities, x i /X (x) and x i /C * (x), unless P(p)/B(p) = C * (x)/X (x) = c for a constant c > 0. In other words, HDIA belongs to HSA if and only if the budget share of i can be written as a function of p i /P(p) or x i /X (x) only.This means that HDIA and HSA do not overlap with the sole exception of CES for n > 2, as shown in Figure 1.

Price Elasticity Function Under HDIA
Even though the budget share of i under HDIA depends on two different relative quantities, one of them, x i /C * (x), enters proportionately.Thus, the price elasticity depends solely on ≡ i /X (x), as follows: Cobb-Douglas is a special case, ζ D i ( i ) = 1, where CES is a special case, ζ D i ( i ) = σ , where Note that, under CES with gross substitutes, ζ D i ( i ) = σ > 1, ϕ i ( i ) is unbounded from above and bounded from below, while under CES with gross complements, ζ D i ( i ) = σ < 1, ϕ i ( i ) is unbounded from below and bounded from above.Thus, even though the price elasticity function, ζ D i ( i ), is defined locally, the assumption that it is globally constant imposes a strong restriction on its global property.Cobb-Douglas, ζ D i ( i ) = 1, is the borderline case, where ϕ i ( i ) is unbounded both from below and from above.
In what follows, we call factor-i a gross substitute if Recall that, for HDIA to be well defined by Equation 20, ϕ i ; i ∈ I, only need to be strictly increasing and strictly concave and satisfy . Hence, unlike HSA, HDIA does not impose any restriction on the price elasticity functions, ζ D i ( i ); i ∈ I, except that all need to be positive.In particular, it is possible to have , and hence gross substitutes and gross complements can coexist.Indeed, ζ D i ( i ) − 1 may switch signs, and hence factor i could switch from being gross substitute to being a gross complement as i changes.

Essential Versus Inessential Factors Under HDIA
Recall that factor i is essential if x i = 0 implies X (x) = 0 and inessential otherwise.Under HDIA, this means that factor i is essential if and only if ϕ i (0) + ∑ n k̸ =i ϕ k ( k ) < 0 for all k > This condition is always satisfied under CES with gross complements or under Cobb-Douglas, because ϕ i ( i ) is unbounded from below.On the other hand, this condition is never satisfied under CES with gross substitutes, because ϕ i ( i ) is bounded from below and ϕ k ( k ) is unbounded from above.This is the reason why factors are inessential if and only if they are gross substitutes under CES.
However, gross substitutes can be essential under HDIA.To see this, let ϕ i ( i ) = β i g( i ), where is strictly increasing and strictly concave, and we have −∞ < g(0) < 0 < g(∞) < ∞.Then, factors i = 1, . . ., j are essential and factors i = j + 1, . . ., n are inessential for This example suggests that HDIA can have j essential factors and n − j inessential factors, where j = 0, 1, . . ., n.Furthermore, the price elasticity function, ζ D i ( i ) = −g ′ ( i )/ i g ′′ ( i ) can be arbitrary, and hence the factors could be gross substitutes or gross complements, except asymptotically, as i → 0 or as i → ∞.
It is also easy to construct an example using a convex combination of Cobb-Douglas and CES, as follows.
where 0 < ε < 1, α i ≥ 0 and β i > 0, and The constant ratios of elasticities of substitution homothetic (CRESH) class proposed by Hanoch  (1971) and recently applied by Berlingieri et al. ( 2022) is a special case of this example, where σ i > 1 for at least some i.The properties are similar to Example 11 under HSA, except that there is no need to impose the restriction that either σ i ≤ 1 for all i or σ i ≥ 1 for all i.

Definition
The CRS production function, X (x), and its unit cost function, P(p), are called HIIA if P(p) can be written as where θ i : R + → R, i ∈ I, are strictly increasing and concave and satisfy This ensures the unique existence of monotonic and quasi-concave P(p).Clearly, HIIA is the homothetic restriction of IIA, as shown in Figure 1. 37he HIIA inverse demand system can be obtained by differentiating Equation 23, where C(p) is a linear homogenous function in p ∈ R n + defined by Solving X (x) ≡ min p∈R n + {px|P(p) ≥ 1} subject to Equation 23 yields the HIIA inverse demand system where B * (x) is the Lagrange multiplier associated with the above minimization problem; it is a linear homogeneous function in x ∈ R n + , implicitly defined by )) = 0, 37 According to the original definition of IIA by Hanoch (1975), its homothetic restriction HIIA should be defined as ∑ n i=1 θ i (p i /P(p)) = 1.However, the right-hand side of Equation 23 can be any constant and we set it equal to 0, which has two advantages.First, one could restrict all θ i to be strictly increasing and strictly concave without loss of generality.Second, multiplying all θ i by a positive constant would not change the function defined.and X (x), the production function, is related to B * (x) as follows: ) .
From these, we also obtain Notice that Equation and Equation 25suggest that the budget share of i under HIIA depends on the two different relative prices, p i /C(p) and p i /P(p) or on the two relative quantities, x i /B * (x) and x i /X (x), unless C(p)/P(p) = X (x)/B * (x) = c for a constant c > 0. In other words, HIIA belongs to HSA if and only if the budget share of i can be written as a function of p i /P(p) or x i /X (x) only.This means that HIIA and HSA do not overlap with the sole exception of CES for n > 2, as shown in

Price Elasticity Function Under HIIA
Even though the budget share of i under HIIA depends on two different relative prices, one of them, p i /C(p), enters proportionately.Thus, the price elasticity depends solely on z i ≡ p i /P(p), as follows: CES is a special case, Note that, under CES with gross substitutes, ) is unbounded from below and bounded from above, while under CES with gross complements, ζ I i (z i ) = σ < 1, θ i (z i ) is unbounded from above and bounded from below.Thus, even though the price elasticity function, ζ I i (z i ), is defined locally, the fact that it is constant imposes a strong restriction on its global property.Cobb-Douglas, ζ I i (z i ) = 1, is the borderline case, where θ i (z i ) is unbounded both from below and from above.
In what follows, we call factor i a gross substitute if ζ I i (z i ) > 1 and a gross complement if ζ I i (z i ) < 1. Recall that, for HIIA to be well defined by Equation 23, θ i ; i ∈ I, only need to be strictly increasing, and strictly concave, and satisfy ∑ n i=1 θ i (0) < 0 < ∑ n i=1 θ i (∞).Hence, unlike HSA but similar to HDIA, HIIA does not impose any restriction on the price elasticity CDESH: constant differences of elasticities of substitution homothetic functions, ζ I i (z i ); i ∈ I, except that they all need to be positive.In particular, it is possible to have ζ I i (z i ) > 1 > ζ I j (z j ), and hence gross substitutes and gross complements can co-exist.Indeed, ζ I i (z i ) − 1 may switch signs, and hence factor i could switch from being a gross substitute to being a gross complement as z i changes.

Essential Versus Inessential Factors Under HIIA
Recall that factor i is essential if p i → ∞ implies P(p) → ∞, and inessential otherwise.Under HIIA, this means that factor i is essential if and only if θ i (∞) + ∑ n k̸ =i θ k (z k ) > 0 for all z k > 0. This condition is always satisfied under CES with gross complements or under Cobb-Douglas, because θ i (z i ) is unbounded from above.On the other hand, this condition is never satisfied under CES with gross substitutes, because θ i (z i ) is bounded from above and θ k (z k ) is unbounded from below.This is why factors are inessential if and only if they are gross substitutes under CES.
However, gross substitutes can be essential under HIIA.To see this, let θ i (z i ) = β i g(z i ), where β i > 0 is decreasing in i, ∑ n i=1 β i = 1, g(z i ) is strictly increasing and strictly concave, and we have −∞ < g(0) < 0 < g(∞) < ∞.Then, factors i = 1, . . ., j are essential and factors i = j + 1, . . ., n are inessential for This example suggests that HIIA can have j essential factors and n − j inessential factors, where j = 0, 1, . . ., n.Furthermore, the price elasticity function, ζ I i (z i ) = −z i g ′′ (z i )/g ′ (z i ) > 0 can be arbitrary, and hence the factors could be gross substitutes or gross complements, except asymptotically, as z i → 0 or as z i → ∞.
It is also easy to construct an example using a convex combination of Cobb-Douglas and CES, as follows.
where 0 < ε < 1, α i ≥ 0, β i > 0, and This corresponds to what Hanoch (1975) called homothetic CDE, but we prefer to call it constant difference of elasticities of substitution homothetic (CDESH) to make it parallel to his terminology of CRESH.The properties of CDESH are similar to Example 11 under HSA and Example 13 under HDIA, except that, unlike in Example 11 but like Example 13, there is no need to impose the restriction that either σ i ≤ 1 for all i or σ i ≥ 1 for all i.

CONCLUDING REMARKS
Instead of recapitulating what has been covered, let me mention briefly one important topic I was unable to cover in this article due to the space limitation.Following Dixit & Stiglitz (1977, section I) and Melitz (2003), most monopolistic competition models assume the CES demand system, which implies that all firms face demand curves with constant and common price elasticity and hence charge the exogenous and common markup rate.One of the most active areas of research today is to allow for endogenous and/or heterogeneous markup rates by replacing CES with non-CES, most of which belong to the classes of non-CES reviewed in this article.
To apply non-CES to monopolistic competition models, one must confront a whole set of additional issues.To ensure that no firm has the power to affect the aggregate price indices through its monopoly power over its own variety, we need to redefine the demand systems over a continuum of product varieties.To ensure that marginal revenue for each firm is positive, we need to assume that all products must be gross substitutes.Furthermore, the marginal revenue for each firm needs to be monotonically decreasing in its output (or increasing in its price) along its demand curve to ensure that the profit function is well behaved.To allow for entry and exit and for endogenous product variety, all products must be inessential.It may also be necessary to impose additional restrictions on the demand systems to ensure the existence and uniqueness of freeentry equilibrium.These are just some of the additional considerations that affect the pros and cons of using different classes of non-CES. 38Addressing all these issues adequately and reviewing this rapidly growing literature on monopolistic competition under non-CES calls for an entirely separate treatment, which I hope to do in the near future.

DISCLOSURE STATEMENT
The author is not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS
This article is based on some graduate lectures I gave at Northwestern University and University of Tokyo, as well as the invited lecture at the 2022 Summer Workshop hosted by Shanghai University of Finance and Economics.I would like to thank the students and participants for their questions, as well as D. Baqaee, G. Barlevy, A. Burstein, J. Kaboski, S. Kikuchi, E.G.J. Luttmer, M. Mestieri, P. Ushchev, and J. Vogel for their comments on earlier drafts and slides.The usual disclaimer applies.

LITERATURE CITED
Antonelli GB. 1971 (1886).Sulla teoria matematica della economia politica [On the mathematical theory of political economy].In Preferences, Utility, and Demand: A Minnesota Symposium, ed.JS Chipman, pp.333-46.New York: Harcourt Brace Jovanovich Figure1 ), due to the following relation between income and price elasticities imposed by IEA, which I call, for want of a better name, indirect Pigou's Law.
12: a Hybrid of Cobb-Douglas and CES under HDIA.Let ϕ

Figure 1 .
Comparing Equation 21 and Equation 24 or Equation 22 and Equation 25 also suggests that HDIA and HIIA can overlap if and only if both P(p)/B(p) = C * (x)/X (x) and C(p)/P(p) = X (x)/B * (x) are positive constants, which implies HDIA and HIIA do not overlap with the sole exception of CES for n > 2, as shown in Figure 1.
14: a hybrid of Cobb-Douglas and CES under HIIA.Let The implications are similar to Example 9 under HSA and Example 12 under HDIA.8.3.2.Example 15: HDIA demand system with constant but different elasticities.Let

.3. Example 13: HDIA demand system with constant but different elasticities. Let ϕ
The implications are similar to Example 9 under HSA and Example 14 under HIIA.