Beyond CES: Three Alternative Classes of Flexible Homothetic Demand Systems

Transcription

1 Beyond CES: Three Alternative Classes of Flexible Homothetic Demand Systems Kiminori Matsuyama 1 Philip Ushchev 2 October Department of Economics, Northwestern University, Evanston, USA. k-matsuyama@northwestern.edu 2 National Research University Higher School of Economics, Russian Federation. ph.ushchev@gmail.com Page 1 of 36

2 Homothetic preferences Common across many fields of applied general equilibrium, preferences are homothetic and technologies are CRS A preference over R is called homothetic if any two indifference sets can be mapped one into the other by a uniform rescaling The direct utility function u(x) is linear homogeneous The indirect utility V(p, h) can be represented as V(p, h) max{u(x) px h} = h x R o h is consumer s income o P(p) is an ideal price index P(p) Page 2 of 36

3 Why are homothetic preferences and CRS technologies important? Under identical homothetic preferences, aggregate consumption behavior is derived from utility maximization of a representative consumer, even though incomes may vary across households Perfect competition is valid only when the industry has CRS technologies Simple behavior of budget shares: o holding the prices constant, the budget share of each good (or factor) is independent of the household expenditure (or the scale of operation by industries) o this allows us to focus on the role of relative prices in the allocation of resources Ensure the existence of a balanced growth path in multi-sector growth models Page 3 of 36

4 CES and its restrictive features In practice, most models assume that preferences/technologies also satisfy constant-elasticity-of-substitution (CES) property, which implies that the price elasticity of demand for each good/factor is constant and identical across goods/factors relative demand for any two goods/factors is independent of the prices of any other goods/factors the marginal rate of substitution between any two goods is independent of the consumption of any other goods in the case of gross substitutes (complements) all goods are inessential (essential) in a monopolistically competitive setting, each firm sells its product at a markup independent of the market environment Page 4 of 36

5 Our paper In this paper, we characterize three alternative classes of flexible homothetic demand systems In each of the three classes, the demand system only depends on one or two price aggregators for any number of goods Each of these classes contains CES as a special case Yet, they offer three alternative ways of departing from CES, because non-ces demand systems in these three classes do not overlap Each of these three classes is flexible in the sense that they are defined non-parametrically Page 5 of 36

6 Homothetic demand systems with a single aggregator (HSA) Page 6 of 36

7 HSA demand systems A homothetic demand system with a single aggregator (HSA) is given by: x = h s p p, i = 1,, n A(p) The functions s : R R are normalized as follows: s (1) = 1 A(p) is a common price aggregator defined as a solution to s p A = 1 Page 7 of 36

8 Example 1: Cobb-Douglas Set s (z ) = α, where α,, α are positive constants such that α = 1 In this case, we obtain the Cobb-Douglas demand system P(p) = c p, but A(p) is indeterminate Page 8 of 36

9 Example 2: CES We obtain the CES demand system if we set s (z ) = β z Here σ > 0 is the constant elasticity of substitution The price aggregator A(p) is proportional to the ideal price index: A(p) = β p = cp(p). NB: this need not be true in general! Page 9 of 36

10 Example 2: CES The functions s (z ) = β z are: increasing when 0 < σ < 1 (the goods are gross complements) decreasing when σ > 1 (the goods are gross substitutes) constant when σ = 1 (the Cobb-Douglas case) Page 10 of 36

11 Example 2: CES and its restrictive nature Under σ > 1, the budget share s (z ) = β z is not only decreasing but also goes to zero when z. Each good is inessential. A good can be essential (or indispensable) only if σ 1 Thus, the CES cannot capture situations when only some goods are essential: if one good is essential, all goods must be essential Under the CES, the very concept of a good being essential or inessential is redundant: gross complements (respectively, substitutes) are always essential (inessential) goods Page 11 of 36

12 Question What are the restrictions to be imposed on the functions s ( ) for a candidate HSA demand system to be compatible with rational consumer behavior? The answer is given by the following Proposition Page 12 of 36

13 A characterization of HSA Proposition 1. Consider a mapping s(z) = s (z ),, s (z ) from R to R, which is normalized by s (1) = 1 and satisfies the conditions: z s (z ) < s (z ), s (z )s z 0 Then: (i) there exists a unique monotone, convex, continuous and homothetic preference over R, such that the candidate HSA demand system associated with s(z) is generated by (ii) the preference is described by the following ideal price index ln P(p) = ln A(p) + s (ξ) dξ ξ /(p) (iii) when n 3, A(p) = cp(p) iff is a CES preference Page 13 of 36

14 Budget-share mapping as a primitive The budget-share mapping s(z) is the primitive of the HSA system A(p) itself cannot serve as a primitive (see Example 5 below) A(p) need not be proportional to P(p) o A(p) captures the cross-effects in the demand system o P(p) captures the welfare consequences of price changes The condition n 3 is important, as under n = 2 all homothetic preferences are HSA Page 14 of 36

15 Self-duality of the HSA demand systems The inverse HSA demand system is given by x p = h s x A, i = 1,, n (x) The functions s : R R are defined by s = s (s /y ) A (x) is a common quantity aggregator defined as a solution to s x A = 1 The two classes of HSA demand systems are self-dual to each other with a one-to-one correspondence between s(z) and s (y) Page 15 of 36

16 Example 3a: Separable translog The translog ideal price index is given by ln P(p) = δ ln p 1 2 γ ln p ln p ln c, Here δ > 0, while (γ ) is symmetric and positive semidefinite The following normalizations hold for all i = 1,, n: δ = 1, γ = 0 Page 16 of 36

17 Example 3a: Separable translog In general, the translog demand system is not HAS However, assume additionally the following separability: γ = γβ (1 β ), i = j γβ β, i j By setting s (z ) = δ γβ ln z, we get: β = 1 x = h p s p A(p) = h p δ γβ ln p A(p) Page 17 of 36

18 Example 3a: Separable translog The price aggregator A(p) is the weighted geometric mean of prices: ln A(p) = β ln p The price index P(p) differs from the price aggregator A(p): P(p) = c exp δ lnp γ 2 β (lnp ) β lnp A(p) Page 18 of 36

19 Example 5: A Hybrid of Cobb-Douglas and CES Consider a convex combination of Cobb-Douglas budget shares and CES budget shares: s (z) = εα + (1 ε)β z Here 0 < ε < 1, while α and β are such that α 0, β > 0, α = β = 1 Page 19 of 36

20 Example 5: A Hybrid of Cobb-Douglas and CES The price aggregator A(p) is independent of ε: A(p) = β p The ideal price index is given by P(p) = c p β p Page 20 of 36

21 Example 5: A Hybrid of Cobb-Douglas and CES When σ > 1, all goods are still gross substitutes, and yet, if α > 0, good i is essential Implication: consider international trade between two countries, and suppose that some of the essential goods can be produced only in one country Trade elasticity is σ > 1. With a small ε, the demand system can be approximated by CES. Were the demand system CES (ε = 0), autarky would lead to a relatively small welfare loss But the welfare loss of autarky (measured by the price index change) is infinity for the country which cannot produce such essential goods Page 21 of 36

22 Implicitly additive homothetic preferences Page 22 of 36

23 HDIA preferences A preference over R is said to be homothetic with direct implicit additivity (HDIA) if u(x) is implicitly defined as a solution to φ x u = 1 Here the sufficiently differentiable functions φ : R R are o either strictly increasing and strictly concave (goods are gross substitutes) o or strictly decreasing and strictly convex (goods are gross complements) Moreover, φ ( ) are normalized as follows: φ (1) = 1 Page 23 of 36

24 HDIA preferences Proposition 2. Assume is a HDIA preference. Then: (i) the Marshallian demands are given by x = h P(p) (φ ) p B(p), where P(p) is the ideal price index, while B(p) is another price aggregator: φ (φ ) p B = 1, P(p) = p (φ ) p B(p) ; (ii) when n 3, we have B(p) = cp(p) iff is a CES preference. Page 24 of 36

25 HIIA preferences A preference over R is said to be homothetic with indirect implicit additivity (HIIA) if P(p) is implicitly defined as a solution to θ p P = 1 Here the sufficiently differentiable functions θ : R R are o either strictly decreasing and strictly convex (goods are gross substitutes) o or strictly increasing and strictly concave (goods are gross complements) Moreover, θ ( ) are normalized as follows: θ (1) = 1 Page 25 of 36

26 HIIA preferences Proposition 3. Assume a preference is HIIA. Then: (i) the Marshallian demands are given by x = h C(p) θ p P(p), where P(p) is the ideal price index, while C(p) is another price aggregator: C(p) p θ p P(p) ; (ii) when n 3, we have C(p) = cp(p) iff is a CES preference. Page 26 of 36

27 Comparing HSA, HDIA, and HIIA Page 27 of 36

28 Three alternative ways of departure from CES Proposition 4. Assume that n 3. Then: (i) HDIA HSA = CES; (ii) HIIA HSA = CES; (iii) HDIA HIIA = CES. Page 28 of 36

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30 Thank you for your attention! Page 30 of 36

31 Homothetic demands and elasticities; A General Case The demand system associated with P(p): x = h p E (P) The inverse demand system associated with u(x): p = h x E (u) Here E (P) and E (u) are the elasticities defined by: E (P) P p p P, E (u) u x x u Page 31 of 36

32 HSA are GAS HSA demand systems are the homothetic restriction of what Pollak (1972) refers to as generalized additively separable (GAS) demand systems We prefer to call HSA instead of homothetic generalized additively separable, because it does not nest the demand systems generated by additively separable preferences. We provide sufficient conditions for the candidate HSA demand system to actually be a demand system generated by some continuous and convex homothetic preference Page 32 of 36

33 Example 3b: Modified translog Separable translog is incompatible with gross complementarity To overcome this, consider the following modification: s (z ) = max{δ + γβ ln z, γβ } Here δ and β are all positive and such that β = δ = 1, 0 < γ < min,, δ β Page 33 of 36

34 Example 3b: Modified translog The price aggregator A(p) has the same form as under the separable translog: ln A(p) = β ln p The price index P(p) is given by: P(p) = c exp δ lnp + γ 2 β (lnp ) β lnp A(p) Page 34 of 36

35 Example 4: Linear expenditure shares Another natural extension of Cobb-Douglas is a demand system with linear expenditure shares: s (z ) = max{(1 δ)α + δβ z, 0} Here δ < 1, α > 0, β > 0, and α = β = 1 The goods are o gross complements when 0 < δ < 1 o gross substitutes when δ < 0 Page 35 of 36

36 Example 4: Linear expenditure shares The price aggregator A(p) is the weighted arithmetic mean of prices: A(p) = β p The ideal price index is given by P(p) = c[a(p)] p A(p) Page 36 of 36