beyond ces: three alternative classes of flexible
TRANSCRIPT
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 1 of 37
Beyond CES: Three Alternative Classes of Flexible Homothetic Demand Systems
Kiminori Matsuyama1 Philip Ushchev2
December 19, 2017, Keio University December 20. 2017, University of Tokyo
1 Department of Economics, Northwestern University, Evanston, USA. Email: [email protected]
2 National Research University Higher School of Economics, Russian Federation. Email: [email protected]
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 2 of 37
Homothetic preferences: A general case 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
Thedirectutilityfunction푢(퐱)is푙푖푛푒푎푟ℎ표푚표푔푒푛푒표푢푠 Theindirectutility푉(퐩,ℎ)canberepresentedas
푉(퐩,ℎ) ≡ 푚푎푥퐱∈ℝ
{푢(퐱)|퐩퐱 ≤ ℎ} =ℎ
푃(퐩)
o ℎ is consumer’s income o 푃(퐩) is an ideal price index
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 3 of 37
Homothetic demands and elasticities; A general case The demand system associated with 푃(퐩):
푥 =ℎ푝ℰ (푃)
The inverse demand system associated with 푢(퐱):
푝 =ℎ푥ℰ (푢)
ℰ (푃) and ℰ (푢) are the elasticities defined by:
ℰ (푃) ≡휕푃휕푝
푝푃
, ℰ (푢) ≡휕푢휕푥
푥푢
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 4 of 37
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
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 5 of 37
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
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 6 of 37
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
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 8 of 37
Homothetic demand systems
with a single aggregator (HSA)
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 9 of 37
HSA demand systems
Consider a mapping 풔(풛) = 푠 (푧 ), … , 푠 (푧 ) from ℝ to ℝ , A homothetic demand system with a single aggregator (HSA) is
given by:
푥 =ℎ푝푠
푝퐴(퐩)
, 푖 = 1, … ,푛
where 퐴(퐩) is a common price aggregator defined as a solution to
푠푝퐴
= 1
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 10 of 37
Example 1: Cobb-Douglas
Set 푠 (푧 ) = 훼 , where 훼 , … ,훼 are positive constants such that
훼 = 1
In this case, we obtain the Cobb-Douglas demand system 푃(퐩) = 푐 ∏ 푝 , but 퐴(퐩) is indeterminate
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 11 of 37
Example 2: CES
We obtain the CES demand system if we set 푠 (푧 ) = 훽 푧
Here 휎 > 0 is the constant elasticity of substitution
The price aggregator 퐴(퐩) is proportional to the ideal price index:
퐴(퐩) = 훽 푝 = 푐푃(퐩).
NB: this need not be true in general!
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 12 of 37
Example 2: CES The functions 푠 (푧 ) = 훽 푧 are:
increasing when 0 < 휎 < 1 (the goods are gross complements) decreasing when 휎 > 1 (the goods are gross substitutes) constant when 휎 = 1 (the Cobb-Douglas case)
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 13 of 37
Example 2: CES and its restrictive nature Definition: Good 푖 is essential (or indispensable) if 푥 = 0 implies 푢(퐱) = 0
(or equivalently, if 푝 → ∞ implies 푃(퐩) → ∞). Good 푖 is inessential (or dispensable), otherwise. Under CES Each good is inessential if 휎 > 1. A good is essential only if 휎 ≤ 1 CES cannot capture situations when only some goods are
essential: if one good is essential, all goods must be essential The very distinction of a good being essential or inessential is
redundant: gross complements (respectively, substitutes) are always essential (inessential) goods
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 14 of 37
Integrability Question What are the restrictions to be imposed on the functions 푠 (∙) for
a “candidate” HSA demand system to be compatible with rational consumer behavior?
The answer is given by the following Proposition
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 15 of 37
A characterization of HSA
Proposition 1. Consider a mapping 풔(풛) = 푠 (푧 ), … , 푠 (푧 ) from ℝ to ℝ , which is normalized by ∑ 푠 (1) = 1 and satisfies the conditions:
푧 푠 (푧 ) < 푠 (푧 ), 푠 (푧 )푠 푧 ≥ 0 Then: (i) there exists a unique monotone, convex, continuous and homothetic
preference ≿ over ℝ , such that the candidate HSA demand system associated with 풔(풛) is generated by ≿
(ii) the preference ≿ is described by the following ideal price index
ln푃(퐩) = ln퐴(퐩) +푠 (휉)휉
d휉
/ (퐩)
(iii) when 푛 ≥ 3, 퐴(퐩) = 푐푃(퐩) iff ≿ is a CES preference
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 16 of 37
Budget-share mapping as a primitive The budget-share mapping 퐬(퐳) is the primitive of the HSA
system 퐴(퐩) itself cannot serve as a primitive (see Example 5 below) 퐴(퐩) need not be proportional to 푃(퐩) o 퐴(퐩) captures the cross-price effects in the demand system o 푃(퐩) captures the welfare consequences of price changes
The condition 푛 ≥ 3 is important, as under 푛 = 2 all homothetic
preferences are HSA
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 17 of 37
Self-duality of the HSA demand systems
Consider a mapping풔∗(풚) = 푠∗(푦 ), … , 푠∗(푦 ) from ℝ to ℝ , The inverse HSA demand system is given by
푝 =ℎ푥푠∗
푥퐴∗(풙)
, 푖 = 1, … ,푛
where 퐴∗(풙) is a common quantity aggregator defined as a solution to
푠∗푥퐴∗
= 1
The two classes of HSA demand systems are self-dual to each other with a one-to-one correspondence between 퐬(퐳) and 퐬∗(퐲), defined by 푠∗ = 푠 (푠∗/푦 )
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 18 of 37
Example 3a: Separable translog The translog ideal price index is given by
ln푃(퐩) = 훿 ln 푝 −12
훾 ln푝 ln 푝 − ln 푐,
Here 훿 > 0, while (훾 ) is symmetric and positive semidefinite The following normalizations hold for all 푖 = 1, … , 푛:
훿 = 1, 훾 = 0
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 19 of 37
Example 3a: Separable translog In general, the translog demand system is not HSA However, assume additionally the following separability:
훾 =훾훽 (1 − 훽 ), 푖 = 푗−훾훽 훽 , 푖 ≠ 푗 훽 = 1
By setting 푠 (푧 ) = 훿 − 훾훽 ln 푧 , we get:
푥 =ℎ푝푠
푝퐴(풑)
=ℎ푝
훿 − 훾훽 ln푝퐴(풑)
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 20 of 37
Example 3a: Separable translog The price aggregator 퐴(퐩) is the weighted geometric mean of
prices:
ln퐴(퐩) = 훽 ln 푝
The price index 푃(퐩) differs from the price aggregator 퐴(퐩):
푃(퐩) = 푐 ⋅ exp 훿 ln푝 −훾2
훽 (ln푝 ) − 훽 ln푝 ≠ 퐴(퐩)
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 21 of 37
Example 5: A Hybrid of Cobb-Douglas and CES Consider a convex combination of Cobb-Douglas budget shares
and CES budget shares:
푠 (푧) = 휀훼 + (1 − 휀)훽 푧 Here 0 < 휀 < 1, while 훼 and 훽 are such that
훼 ≥ 0, 훽 > 0, 훼 = 훽 = 1
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 22 of 37
Example 5: A Hybrid of Cobb-Douglas and CES The price aggregator 퐴(퐩) is independent of 휀:
퐴(퐩) = 훽 푝
The ideal price index is given by
푃(퐩) = 푐 푝 훽 푝
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 23 of 37
Example 5: A Hybrid of Cobb-Douglas and CES When 휎 > 1, all goods are still gross substitutes, and yet, if 훼 > 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 휎 > 1 . With a small 휀, the demand system can be
approximated by CES. Were the demand system CES (휀 = 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
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 24 of 37
Implicitly additive homothetic
preferences
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 25 of 37
HDIA preferences A preference ≿ over ℝ is said to be homothetic with direct
implicit additivity (HDIA) if 푢(퐱) is implicitly defined as a solution to
휙푥푢
= 1
Here the sufficiently differentiable functions 휙 :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, 휙 (∙) are normalized as follows: ∑ 휙 (1) = 1
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 26 of 37
HDIA preferences Proposition 2. Assume ≿ is a HDIA preference. Then: (i) the Marshallian demands are given by
푥 =ℎ
푃(퐩)(휙 )
푝퐵(퐩)
,
where 푃(푝) is the ideal price index, while 퐵(푝) is another price aggregator:
휙 (휙 )푝퐵
= 1, 푃(푝) = 푝 (휙 )푝퐵(퐩) ;
(ii) when 푛 ≥ 3, we have 퐵(퐩) = 푐푃(퐩) iff ≿ is a CES preference.
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 27 of 37
HIIA preferences A preference ≿ over ℝ is said to be homothetic with indirect
implicit additivity (HIIA) if 푃(퐩) is implicitly defined as a solution to
휃푝푃
= 1
Here the sufficiently differentiable functions 휃 :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, 휃 (∙) are normalized as follows: ∑ 휃 (1) = 1
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 28 of 37
HIIA preferences Proposition 3. Assume a preference ≿ is HIIA. Then:
(i) the Marshallian demands are given by
푥 =ℎ
퐶(풑)휃
푝푃(풑)
,
where 푃(풑) is the ideal price index, while 퐶(풑) is another price aggregator:
퐶(풑) ≡ 푝 휃푝푃(풑)
;
(ii) when 푛 ≥ 3, we have 퐶(풑) = 푐푃(풑) iff ≿ is a CES preference.
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 29 of 37
Comparing HSA, HDIA, and HIIA
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 30 of 37
Three alternative ways of departure from CES Proposition 4. Assume that 푛 ≥ 3. Then: (i) HDIA ∩ HSA = CES; (ii) HIIA ∩ HSA = CES; (iii) HDIA ∩ HIIA = CES.
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 32 of 37
Thank you for your attention!
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 33 of 37
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
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 34 of 37
Example 3b: Modified translog Separable translog is incompatible with gross complementarity To overcome this, consider the following modification:
푠 (푧 ) = max{훿 + 훾훽 ln 푧 , 훾훽 } Here 훿 and 훽 are all positive and such that
훽 = 훿 = 1, 0 < 훾 < min,…,
훿훽
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 35 of 37
Example 3b: Modified translog The price aggregator A(퐩) has the same form as under the
separable translog:
ln퐴(퐩) = 훽 ln 푝
The price index 푃(퐩) is given by:
푃(퐩) = 푐 ⋅ exp 훿 ln푝 +훾2
훽 (ln푝 ) − 훽 ln푝 ≠ 퐴(퐩)
©Kiminori Matsuyama and Philip Ushchev, Three Alternative Classes
Page 36 of 37
Example 4: Linear expenditure shares Another natural extension of Cobb-Douglas is a demand system
with linear expenditure shares:
푠 (푧 ) = max{(1 − 훿)훼 + 훿훽 푧 , 0} Here δ < 1, 훼 > 0, 훽 > 0, and ∑ 훼 = ∑ 훽 = 1 The goods are o gross complements when 0 < 훿 < 1 o gross substitutes when 훿 < 0