Existence and regularity of equilibria in a general equilibrium model with private provision of a public good

Journal of Mathematical Economics 41 (2005) 617–636

Existence and regularity of equilibria in a general equilibrium model with private provision of a public good ¨ Antonio Villanaccia,∗ , E. Unal Zenginobuzb a

Department of Mathematics for Decision Theory (Di.Ma.D), University of Florence, via Lombroso 6/17, 50134 Firenze, Italy b Department of Economics, Bo˘ gazi¸ci University, 34342 Bebek, Istanbul, Turkey

Received 14 May 2001; received in revised form 25 December 2004; accepted 31 December 2004 Available online 8 March 2005

Abstract We prove existence and generic regularity of equilibria in a general equilibrium model of a completely decentralized pure public good economy. Competitive firms using private goods as inputs produce the public good, which is privately provided by households. Previous studies on private provision of public goods typically use one private good, one public good models in which the public good is produced through a constant returns to scale technology. As two distinguishing features of our model, we allow for the presence of several private goods and nonlinear production technology. In that framework, we use an homotopy argument to prove existence of equilibria and we show that economies are generically regular. © 2005 Elsevier B.V. All rights reserved. Keywords: General equilibrium model; Private provision of public good; Existence; Generic regularity

1. Introduction In this paper, we prove the existence and regularity of equilibria in a general equilibrium model of a completely decentralized pure public good economy. The model studied extends ∗

Corresponding author. Tel.: +39 055 4796 817; fax: +39 055 4796 800. ¨ E-mail addresses: [email protected] (Antonio Villanacci); [email protected] (E. Unal Zenginobuz).

0304-4068/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2004.12.005

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¨ Zenginobuz / Journal of Mathematical Economics 41 (2005) 617–636 A. Villanacci, E.U.

the standard pure exchange model with private goods by allowing households to make voluntary purchases of (or “privately provide”) a public good that is produced by competitive firms using private goods as inputs. The interest in a general equilibrium model with private provision of public goods lies in the fact that it serves as a benchmark extension of an analysis of completely decentralized private good economies to public good economies. Moreover, there are some relevant situations in which public goods are in fact privately provided, e.g., private donations to charity at a national and international level, campaign funds for political parties or special interests groups, and certain economic activities inside a family. Previous studies on private provision of public goods typically use one private good, one public good models in which the public good is produced through a constant returns to scale technology. With only one private good, assuming constant returns to scale amounts to assuming linear production function, and as a result there is no loss of generality in normalizing prices of both the private and the public good to one. These assumptions also allow taking profits of firms equal to zero, with the implication that the presence of firms basically plays no role in the model. Therefore, as clearly explained by Bergstrom et al. (1986), the model reduces to a simple game where households are the players and the unique strategy variable is the choice of voluntary contribution. Then, proving existence is straight forward (see Bergstrom et al., 1986, Theorem 2, p. 33).1 As two distinguishing features of our model we have the presence of several private goods and we allow for nonlinear production technology for the public good. Therefore, the abovedescribed price normalization will typically not be possible. Moreover, competitive firms will typically have non-zero profits, which are to be distributed among households. We are, therefore, analyzing a full-fledged general equilibrium model. For given prices firms maximize profits. For given prices, initial endowments of private goods, ownership distribution of firms, as well as other households’ choices of private provision of public goods, each household chooses a vector of private good consumption levels and a voluntary contribution level for the public good so as to maximize utility. Finally, markets for private and public goods clear. We show both existence and generic regularity of equilibria in this model. We provide a proof of existence using a very simple homotopy argument, whose main characteristics consist in linking the true model with a fictitious one where the public good is treated as a private good. This strategy of proof turns out to be useful to show existence of equilibria in some related models as well.2 1 The literature on voluntary contributions stems from Warr (1983), who studied the question of neutrality of taxes in a partial equilibrium public good model. Bergstrom et al. (1986) studied the same issue in a simple general equilibrium model with one private and one public good. The large literature that follows Bergstrom et al. (1986) mainly uses their framework and focuses on questions other than existence of equilibria in a full-fledged general equilibrium framework. A recent contribution by Cornes et al. (1999) on existence and uniqueness of equilibria in public good models also uses the framework in Bergstrom et al. (1986), i.e. a one private good, one public good economy with no relative price effects. The survey of papers on the existence of equilibria in private public good provision models by Cornes et al. (1999) confirms our observation that explicit work on existence of equilibrium in these models have indeed been limited to one public good, one private good case with no relative price effects. 2 In a separate paper we prove, using the same approach, the existence of equilibria in a similar model where the public good is produced by a public firm subject to a budget constraint (Villanacci and Zenginobuz, 2004).

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Regularity is an indispensable tool for carrying out comparative statics analyses. We show that for any vector of utility and transformation functions and ownership shares, in an open and full measure subset of the endowments, there is a finite number of equilibria and a local smooth dependence of the equilibrium variables on the exogenous variables. To show regularity we need to strengthen the assumptions on utility functions and production technology with which we prove existence. Previous work on existence of competitive equilibrium with externalities in consumption and production relate to the model we study here. Public goods can be seen as a special case of consumption externalities. McKenzie (1955) studies a model of externalities with production, but he considers a linear production technology. Arrow and Hahn (1971) allow for decreasing returns to scale in production, but some of their assumptions do not apply to our case. Shafer and Sonnenschein (1975) show the existence of competitive equilibria in a pure exchange framework, hence their results do not apply to our model which incorporates production.3 Section 2 presents the setup of the model. In Section 3 the existence of equilibrium is proved. Section 4 contains the proof of generic regularity of the equilibria.4 2. Setup of the model We consider a general equilibrium exchange model with private provision of a public good. There are C, C ≥ 1, private commodities, labelled by c = 1, 2, . . . , C. There are H households, H > 1, labelled by h = 1, 2, . . . , H. Let H = {1, . . . , H} denote the set of households. Let xhc denote consumption of private commodity c by household h; ech embodies similar notation for the endowment in private goods. The following standard notation is also used: CH H • xh ≡ (xhc )C c=1 , x ≡ (xh )h=1 ∈ R++ .

CH H • eh ≡ (ech )C c=1 , e ≡ (eh )h=1 ∈ R++ . g • pc is the price of private good c, with p ≡ (pc )C c=1 ; p is the price of the public good. g Let pˆ ≡ (p, p ). • gh ∈ R+ is the amount of public good that consumer h provides. Let g ≡ (gh )H h=1 , G ≡
H g , and G ≡ G − g . \h h h=1 h

The preferences over the private goods and the public good of household h are represented by a utility function uh : RC ++ × R++ → R. Note that households’ preferences are defined over the total amount of the public good, i.e. we have uh : (xh , G) → uh (xh , G). 3 The definition of equilibrium by Shafer and Sonnenschein (1975) is problematic due to the way they treat the (artificial) distinction between an household’s actual consumption and her intended consumption (see pp. 84 and 85). Arrow and Hahn (1971) take account of this issue. On the other hand, they postulate the existence of a consumption vector whose utility is independent of other agents’ choices, an assumption which fails to hold in our model (see p. 132 for details on both problems). 4 While the paper was under revision for publication, we realized that Florenzano (2003) proves existence in a model with externalities which is general enough to include the case we analyze. However, the paper offers a different technique of proof which does not allow to address the generic regularity analysis we present.

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Assumption 1. uh (xh , G) is a C2 , differentiably strictly increasing (i.e., for every (xh , G) ∈ 5 RC+1 ++ , Duh (xh , G)  0), differentiably strictly quasiconcave function (i.e., ∀(xh , G) ∈ C+1 C+1 R++ , ∀v ∈ R \{0}, if Duh (xh , G)v = 0, then vD2 uh (xh , G)v < 0,6 and for each u ∈ R the closure (in the standard topology of RC+1 ) of the set {(xh , G) ∈ RC+1 ++ : uh (xh , G) ≥ u} C+1 is contained in R++ . Let U be the set of utility functions uh satisfying Assumption 1. There are F firms, indexed by subscript f , that use a production technology represented g g by a transformation function tf : RC+1 → R, where tf : (yf , yf ) → tf (yf , yf ). g

g

Assumption 2. tf (yf , yf ) is a C2 , differentiably strictly decreasing (i.e., Dtf (yf , yf )  0), and differentiably strictly quasiconcave function,7 with tf (0) = 0. g

For each f , define yˆ f ≡ (yf , yf ), yˆ ≡ (ˆyf )Ff =1 and Yf ≡ {ˆyf ∈ RC+1 : tf (ˆyf ) ≥ 0} and
Y ≡ Ff =1 Yf . Let also t ≡ (tf )Ff =1 and pˆ ≡ (p, pg ). The following assumption is sufficient to get boundedness of the set of feasible allocations. Assumption 2’ (Bounded reversibility). Y ∩ (−Y ) is bounded. Let T be the set of vectors t of transformation functions satisfying Assumptions 2 and 2’. Remark 1. We avoided to assume irreversibility (i.e., Y ∩ (−Y ) = {0}) or no free lunch because either of the two together with free disposal and possibility of inaction would imply that the gradients of the transformation functions of all firms computed at zero are collinear, a quite restrictive assumption. Using the convention that input components of the vector yˆ f are negative and output components are positive, the profit maximization problem for firm f is: For given pˆ ∈ RC+1 ++ , ˆy Maxyˆ ∈RC+1 pˆ s.t. tf (ˆy) ≥ 0(αf )

(1)

where we follow the convention of writing the Kuhn-Tucker multipliers next to the associated constraints. Remark 2. From Assumption 2, it follows that if problem (1) has a solution, it is unique and it is characterized by Kuhn-Tucker (in fact, Lagrange) conditions. For vectors y, z, y ≥ z (resp. y  z) means every element of y is not smaller (resp. strictly larger) than the corresponding element of z; y > z means that y ≥ z but y = z. 6 Observe that since u is differentiably strictly quasiconcave then the following condition holds: if h          ∀(xh , G ), (xh , G ) ∈ RC+1 ++ , ∀λ ∈ (0, 1), uh ((1 − λ)(xh , G ) + λ(xh , G )) > min{uh ((xh , G )), uh ((xh , G ))}. We use this condition in the proof of Theorem 9. g 7 Observe that since t C+1 , f is strictly quasiconcave then it is also the case that ∀(yf , yf ) ∈ R  2 g g  D tf (yf , yf ) Dtf (yf , yf ) det = 0. We use this condition in the proof of Theorem 9 and Lemma 12. g Dtf (yf , yf ) 0 5

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Definition 3. sfh is the share of firm f owned by household h. sf ≡ (sfh )H ∈ RH and s ≡ h=1
H F FH H (sf )f =1 ∈ R . The set of all shares of each firm f is S ≡ {sf ∈ [0, 1] : h=1 sfh = 1}. s ≡ (sf )Ff =1 ∈ S F . The set of shares sh ≡ (sfh )Ff =1 of household h is [0, 1]F . sfh ∈ [0, 1] denotes the proportion of profits of firm f owned by household h. The definition of S simply requires each firm to be completely owned by some households. Household’s maximization problem is then the following: For given pˆ ∈ RC+1 ++ , sh ∈ (C+1)F , [0, 1]F , eh ∈ RC , G ∈ R , y ˆ ∈ R + \h ++ Max(xh ,gh )∈RC

uh (xh , gh + G\h )

s.t.

−p(xh − eh ) − pg gh + pˆ

++ ×R

gh ≥ 0 (µh ) gh + G\h > 0

F 

sfh yˆ f ≥ 0

f =1

(λh )

(2)

Remark 4. From Assumption 1, it follows that problem (2) has a unique solution characterized by Kuhn-Tucker conditions. Market clearing conditions are: H H F    − xh + eh + yf = 0 h=1

−

H 

gh +

h=1

h=1 F  f =1

f =1 g

yf = 0


Remark 5. With Assumption 1 on uh , at an equilibrium allocation we must have h gh > 0,
F g and hence f =1 yf > 0. Moreover, since gh ≥ 0 for all h, there exists h such that gh > 0. Since in the existence proof we consider an arbitrary (s, u, t) ∈ S F × UH × T which remains fixed throughout the analysis, we define an economy as an endowment vector e ∈ RCH ++ . Summing up consumers’ budget constraints, which from Assumption 1 hold with equality, we get       H H F H F      g eh + gh − −p  xh − sfh yf  − pg  sfh yf  = 0 h=1

h=1

f =1

h=1

f =1

i.e., the Walras law. Therefore, the market clearing condition for good C, for example, is redundant. Moreover, we can normalize the price of private good C without affecting the budget constraints of any household. With little abuse of notation, we denote the normalized private and public good prices with p ≡ (p\ , 1) and pg , respectively. Using Remarks 2 and 4, we can give the following definition:

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Definition 6. A vector (x, g, p\ , pg , yˆ ) is an equilibrium for an economy e ∈ RCH ++ if: 1. firms maximize, i.e., for each f, yˆ f solves problem (1) at pˆ ∈ RC+1 ++ ; g 2. households maximize, i.e., for each h , (xh , gh ) solves problem (2) at p\ ∈ RC−1 ++ , p ∈ C F (C+1)F R++ , eh ∈ R++ , G\h ∈ R+ , sh ∈ [0, 1] , yˆ ∈ R ; and 3. markets clear , i.e., (x, g, yˆ ) solves −

H 

\

xh +

h=1

−

H 

gh +

h=1 \

H  h=1 F  f =1

\

eh + g yf

F  f =1

\

yf = 0 (3)

=0

\

\ C−1 . where xh , eh ∈ RC−1 ++ and y ∈ R

3. Existence Given our assumptions, we can now characterize equilibria in terms of the system of Kuhn-Tucker conditions to problems (1) and (2), and market clearing conditions (3). Define

H  (C+1)F F CH H ≡R × R++ × R++ × g ∈ R : gh > 0 h=1

×R

H

× RH ++

× RC−1 ++

× R++

ξ ≡ (ˆy, α, x, g, µ, λ, p\ , pg ) ∈  and dim  , F :  × RCH ++ → R

F : (ξ, π) → left-hand side of (4) below

... (h.1) Dxh uh (xh , gh + G\h ) − λh p = 0 (h.2) Dgh uh (xh , gh + G\h ) − λh pg + µh = 0 (h.3) min{gh , µh } = 0 (h.4) −p(xh − eh ) − pg gh + pˆ

F 

sfh yˆ f = 0

f =1

(f.1) .pˆ. + . αf Dtf (ˆyf ) = 0 (f.2) tf (ˆyf ) = 0 ... H H F    \ \ \ (M.1) − xh + eh + yf = 0 h=1

(M.2) −

H  h=1

gh +

h=1 F 

f =1

f =1

g

yf = 0

(4)

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Observe that (ˆy, x, g, p\ , pg ) is an equilibrium associated with an economy e if and only if there exists (α, µ, λ) such that F (ˆy, α, x, g, µ, λ, p\ , pg , e) = 0. With innocuous abuse of terminology, we will call ξ ≡ (ˆy, α, x, g, µ, λ, p\ , pg ) an equilibrium. To show existence we are going to use the following homotopy theorem. Theorem 7. Let M and N be C2 boundaryless manifolds of the same dimension, and let f, g : M → N be such that f is C0 and g is C0 and C1 in an open neighbourhood of g−1 (y), y is a regular value for g, #g−1 (y) is odd, and there exists a continuous homotopy H from f to g such that H −1 (y) is compact. Then f −1 (y) = ∅. Theorem 7 is a consequence of a degree theorem presented, for example, in Geanakoplos and Shafer (1990). A self-contained account of the proof of the theorem can, for example, be found in Villanacci et al. (2002). To apply Theorem 7, we consider a model in which the public good is treated as a private good and we construct a well-behaved vector of endowments (and the associated unique equilibrium). We then introduce the needed homotopy H, which “links” the true model to the above-mentioned fictitious one. From Assumption 2’, it follows that Y ∩ RC+1 is bounded (a “bounded free lunch” + 8 property) and that Y is closed. Then there exists a point yˆ ∗ on the boundary of Y in RC+1 + . 9 to Y at y ∗ . Then ∀ˆ Let a price pˆ ∗ ∈ RC+1 in the normal cone ˆ y ∈ Y , ++ pˆ ∗ yˆ ≤ pˆ ∗ yˆ ∗ Since Y (ˆy∗ ) ≡ {(ˆyf )Ff =1 ∈ ×Ff =1 Yf : ∃ (ˆyf∗ )Ff =1 ∈ Y (ˆy∗ )


F

ˆf f =1 y

= yˆ ∗ }, we then have

such that ∀f, ∀yf ∈ Yf ,

pˆ ∗ yˆ f ≤ pˆ ∗ yˆ f∗

(5)

Suppose that (5) does not hold, i.e., ∀(˜yf )Ff =1 ∈ Y (ˆy∗ ), ∃f˜ and y˜ f˜ ∈ Yf such that pˆ ∗ y˜ f˜ > pˆ ∗ yˆ f∗ . But then choose y˜ f ≡

y˜ f˜ if f = f˜ yˆ ∗ if f = f˜ f

and define y˜ ≡ (y˜ f )f . Then pˆ ∗ y˜ > pˆ ∗ yˆ ∗ a contradiction. Therefore, from (5), ∀f , yˆ f∗ is a profit maximizer at pˆ ∗ and from Remark 2 we have that ∀f , there exists α∗f ∈ R++ such that pˆ ∗ + αf Dtf (ˆy∗ ) = 0 tf (ˆy∗ ) = 0 8 9

Closedness of Y can be obtained using Lemma A.1 and Corollary 2, p. 69, in Bergstrom et al. (1986). The normal cone to a convex set C at a is the set {∀x∗ ∈ Rn : ∀x ∈ C, (x − a)x∗ ≤ 0}.

(6)

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Then choose (xh∗ , gh∗ ) in order to solve max(xh ,gh ) uh (xh , gh ) s.t. −p∗ xh − pg∗ gh + 1 ≥ 0 whose associated first-order conditions are Duh (xh∗ , gh∗ ) − λ∗h pˆ ∗ = 0 −p∗ xh − pg∗ gh + 1 = 0

(7)

with λ∗h > 0. Define r∗ ≡

H  h=1

xh∗ −

F  f =1

yf∗

and γ ∗ ≡

H  h=1

gh∗ −

F  f =1

g∗

yf

(8)

For each e ∈ RCH ++ , the needed homotopy is then defined as follows. H :  × [0, 1] → Rdim  , H : (ξ, τ) → (left-hand side of system (9) below) ... (h.1) Dxh uh (xh , (1 − τ)G + τgh ) − λh p = 0

(h.2) Dgh uh (xh , (1 − τ)G + τgh ) − λh pg + µh = 0 (h.3) min{gh , µh } = 0 (h.4) −pxh +

− pg g

F  f =1

∗ h + p[(1 − τ)eh + τxh g

+

F  f =1

sfh (yf − τyf∗ )] + pg [τgh∗

∗g

sfh (yf − τyf )] = 0

... (f.1) pˆ + αf Dtf (ˆy) = 0

(9)

(f.2) tf (ˆy) = 0 ... (M.1) −

H  h=1

(M.2) −

H  h=1

\

xh + gh +

F  f =1 F  f =1

\

yf + (1 − τ)

H  h=1

\

eh + τr \∗ = 0

g

yf + τγ ∗ = 0

Remark 8. ξ is a solution to H(ξ, 0) = 0 if and only if ξ is an equilibrium at e, i.e., F (ξ, e) = 0.

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Define also G :  → Rdim  ,

G : ξ → H(ξ, 1)

Theorem 9. For every economy e ∈ RCH ++ , an equilibrium exists. Proof. The proof consists in checking that the assumptions of Theorem 7 are verified, which is done below in four steps. Step 1: There exists ξ ∗ such that G(ξ ∗ ) = 0 Define ξ ∗ = (ˆy∗ , α∗ , xh∗ , g∗ , µ∗ = 0, λ∗h , pˆ ∗ ) G(ξ ∗ ) = 0 is clearly true because of the definition of ξ ∗ and (6), (7) and (8). Step 2: {ξ ∗ } = G−1 (0). Suppose there exists ξ  ≡ (y , yg , α , x , g , µ , λ , p\ , pg ) ∈ G−1 (0), with ξ  = ξ ∗ . Observe that (y , yg , x , g ) satisfies equations (M.1) and (M.2) in system (9) for τ = 1. Moreover, summing up equations (h.4) with respect to h, and using Definition (8) and equation (M.2) , we get −

H  h=1

xh + r ∗ +

F  f =1

yf = 0

(10)

Consider now (x , g , y , yg ) ≡ 21 (x , g , y , yg ) + 21 (x∗ , g∗ , y∗ , yg∗ ), then (x , g , y , yg ) satisfies (10) and (M.2) and tf (y , yg ) ≥ 0, since tf is a quasiconcave function. Now, observe from equations (h.1) − (h.4) that (xh , gh ) solves Max(xh ,gh )∈RC

uh (xh , gh )

s.t.

p x

++ ×R

h

+ pg g

h

gh ≥ 0

≤

pg gh∗

+ p xh∗

+

F  f =1

∗   sfh pˆ (ˆyf − yˆ f∗ )

Since f (y∗ , yg∗ ) = 0 and yˆ  = arg max pˆ  yˆ f

s.t. f (ˆy) ≥ 0

we have that pˆ  yˆ f − pˆ  yˆ f∗ ≥ 0 

Therefore, uh (xh , gh ) ≥ uh (xh∗ , gh∗ ), with gh∗ > 0. Finally, since the utility function is strictly quasiconcave, we have uh (xh , gh ) > min{uh (xh , gh ), uh (xh∗ , gh∗ )} ≥ uh (xh∗ , gh∗ )

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Note that uh (xh , gh ) > uh (xh∗ , gh∗ ) implies that p∗ xh + pg∗ gh > p∗ xh∗ + pg∗ gh∗ . Furthermore, since yf∗ is a solution of the firm problem at price pˆ ∗ and tf (yf ) ≥ 0, by definition of yf and by quasiconcavity of tf , pˆ ∗ yˆ f ≤ pˆ ∗ yˆ f∗ . Summarizing and using the definition of ξ  , we have p∗ xh + pg∗ gh > p∗ xh∗ + pg∗ gh∗   − xh + yf + r ∗ = 0 h

−

 h

(11) (12)

f

gh +

 f

g

yf + γ ∗ = 0

(13)

g

g∗

− p∗ yf − pg∗ yf ≥ −p∗ yf∗ − pg∗ yf ≥ 0

(14)

From (11) and (14),         g xh − r ∗ − yf  + pg∗  gh − γ ∗ − yf  > 0 p∗  h

f

h

(15)

f

But (15), contradicts (12) and (13). Thus (x , y , y , yg ) = (x∗ , g∗ , y∗ , yg∗ ), and the uniqueness of multipliers and prices follows easily. Step 3: Dξ G(ξ ∗ ) has full rank. Dξ G(ξ ∗ ) is                     

..

 . D2 uh (xh∗ , gh∗ )

pˆ ∗ 1

pˆ ∗ ..

. α∗f D2 tf (ˆy∗ ) Dtf (ˆy∗ )

Dtf (ˆy∗ ) ..

−I˜ T 

IC−1

I˜ T 

  where I˜ ≡  0 . 01 We want to show that ker DHξ G(ξ ∗ ) = {0}. Defining ) ≡ (()uh , )µh , )λh )h=1 , ()tf , )αf )Ff =1 , )pˆ \ ) ∈ (RC+1 × R × R)H × (RC+1 × R)F × RC

.

  λ∗h I˜            I˜       

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we prove that if Dξ G(ξ ∗ ) · ) = 0, then ) = 0. The proof proceeds through the following steps: (a) There exists h such that )uh = 0; (b) ∀h, Duh (xh∗ , gh∗ ) · )uh = 0;

(c) ∀f , Dtf (ˆy∗ ) · )tf = 0; (d) h )uh · D2 uh (xh∗ , gh∗ ) · )uh + f )tf · α∗f D2 tf (ˆy∗ ) · )tf = 0. But then (a)–(c), and the fact that uh and tf are differentiably strictly quasiconcave contradict (d). −1 Step 4: For every e ∈ RCH ++ , H (0) is compact. Take an arbitrary sequence (ξ v , τ v )v∈N in H −1 (0) . The main difficulties are in showing that, up to subsequences, {((xn , gn ), yˆ n ) : n ∈ N} are contained in a bounded set, and that ¯ ∈ RCH+1 (xn , Gn ) → (¯x, G) ++ . First of all observe that, since {τ v : v ∈ N} ⊆ [0, 1], up to a subsequence, (τ n )n∈N converges. Then, so does (rn )n∈N , where  n

n

r¯ ≡ (1 − τ )

H 

 n ∗

n ∗

eh + τ r , τ γ

h=1

say to r. Defining c

r˜ c ≡ max({¯r cn : n ∈ N} ∪ {r })

for c = 1, . . . , C, g,

and

n

g ˆ r˜ ≡ (˜rc )C c=1 , r˜ ≡ (˜r , r˜ )

we have {((xn , gn ), yˆ n ) : n ∈ N} ⊆ Ar˜ˆ     H (C+1)F ≡ (x, g, yˆ ) ∈ RCH :− xh + yf + r˜ ≥ 0, ++ × R+ × R  h

−

 h

gh +

 f

g

f

yf + r˜ g ≥ 0 and ∀f, tf (ˆyf ) ≥ 0

  

which is proved to be bounded in Lemma A.1. Therefore, yˆ and gh converges to elements of R(C+1)F and R, respectively, and x converges to an element of RCH . ¯ ∈ RCH+1 to prove that, for sufficiently large n, In order to show that (¯x, G) ++ , it is enough
n C+1 n ¯ for each h, there exists eh ∈ R++ such that uh (xh , h gh ) ≥ uh (e¯ h ). The above result can be proved as follows. Observe that from equations (h.1) and (h.2) in system (9), xhn solves the problem Max(xh ,gh ) uh (xh , gh + (1 − τ)G\h ) s.t. pn xh + pgn gh ≤ pn [(1 − τ n )eh + τ n xh∗ ] + τ n pgn gh∗   F    + pˆ n sfh (ˆyfn − τ n yf∗ )   f =1

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¨ Zenginobuz / Journal of Mathematical Economics 41 (2005) 617–636 A. Villanacci, E.U.

Moreover, ∀n, (1 − τ n )eh + τ n xh∗ belongs to a compact set contained in RC ++ and the term in the curly brackets is nonnegative, because yˆ fn is a solution to firm f problem at prices pˆ n , ∗g 0, yf ∈ Yf , Yf is convex, and therefore pˆ n yˆ fn ≥ τ n pˆ n yˆ f∗ . 4. Regularity In this section, we prove that there is a large set of the endowments (the so-called regular economies) for which associated equilibria are finite in number, and that equilibria change smoothly with respect to endowments - see Theorem 11 below. To do this, we need first of all to add the following assumptions. 2 Assumption 3. ∀h, uh is differentiably strictly concave, i.e., ∀(xh , G) ∈ RC+1 ++ , D uh (xh ) is negative definite.

Assumption 4. For all h and (xh , G) ∈ RC+1 ++ , 

Dxh xh uh (xh , G) Dxh uh (xh , G) det DGxh uh (xh , G) DG uh (xh , G)

 = 0

The above assumption has an easy and appealing economic interpretation. Let wh ∈ R++ denote the wealth of household h, and gh : RC+2 ˆ wh , G\h ) → ++ × R+ → R+ , (p, ˆ wh , G\h ) denote the private contribution function of household h, i.e. part of the gh (p, solution function to problem max(xh ,gh ) uh (xh , gh + G\h ) s.t. pxh + pg gh ≤ wh and gh ≥ 0

(16)

Lemma 10. Assumption 4 is equivalent to for each (p, ˆ wh , G\h ) ∈ RC+2 ++ × R+

such that gh (p, ˆ wh , G\h ) > 0,

Dwh gh (p, ˆ wh , G\h ) = 0 Proof. See Appendix A. Call U˜ the subset of U whose elements satisfy Assumptions 3 and 4. Define pr : F −1 (0) → RCH ++ ,

pr : (ξ, e) → e

We can state the main theorem of this section. H Theorem 11. For each (s, u, t) ∈ S F × U˜ × T, there exists an open and full measure subset R of RCH ++ such that

1. there exists r ∈ N such that Fe−1 (0) = {ξ i }ri=1 ; 2. there exist an open neighborhood Y of e in RCH ++ , and for each i an open neighborhood Ui of (ξ i , e) in F −1 (0), such that Uj ∩ Uk = ∅ if j = k, pr −1 (Y ) = ∪ri=1 Ui and pr|Ui : Ui → Y is a diffeomorphism.

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629

A key ingredient in the proof of the above theorem is to show that zero is a regular value for f. An obvious immediate problem is that the min function used in defining the equilibrium function f is not even differentiable when both the constraint and the multiplier are equal to zero, which can be called a “border line” case. We therefore show that border line cases occur outside an open and full measure subset D∗ of the economy space. H

Lemma 12. For each (s, u, t) ∈ S F × U˜ × T, there exists an open and full measure subset ∗ D∗ of RCH ++ such that ∀e ∈ D and ∀ξ such that F (ξ, e) = 0, it is the case that ∀h ∈ H, either gh > 0 or µh > 0 Proof. Define the set C ≡ {(ξ, e) ∈ F −1 (0) : ∃ h such that gh = µh = 0} ∗ and observe that D∗ = RCH ++ \ pr(C). Since C is a closed set, openness of D follows from properness of pr, which can be proven following the same arguments contained in Step 4 of Theorem 9. The proof of full measure proceeds in three steps.

1. Let P be the family of all partitions of H into three subsets H1 , H2 and H3 , with H3 = ∅. In the equilibrium system (4), in place of min{gh , µh } = 0 substitute µh = 0 for h ∈ H1 , gh = 0 for h ∈ H2 , and gh = 0 and µh = 0 for h ∈ H3 (see the first column of Table 1 for demonstration of how this is done). 2. Define dim +#H3 FH1 ,H2 ,H3 :  × RCH ++ → R

which associates the left-hand side of the equilibrium system (4) modified as explained above to each (ξ, e) in the domain. 3. Define the set BH1 ,H2 ,H3 ≡ {e ∈ RCH ++ : ∃ξ such that FH1 ,H2 ,H3 (ξ, e) = 0} and observe that10 D∗ ⊇ RCH ++ \

∪

{H1 ,H2 ,H3 }∈P

BH1 ,H2 ,H3

(17)

That BH1 ,H2 ,H3 is of measure zero follows from the parametric transversality theorem (see, e.g. Hirsch, 1976, Theorem 2.7, p. 79) and from the fact that zero is a regular value for FH1 ,H2 ,H3 and #H3 > 0, which is shown below. In Table 1, the components of FH1 ,H2 ,H3 are listed in the first column, the variables with respect to which derivatives are taken are listed in the first row, and in the remaining bottom We cannot use the equality sign in (17), because BH1 ,H2 ,H3 contains economies e such that FH1 ,H2 ,H3 (ξ, e) = 0 and for which some components gh and µh of ξ may be negative. 10

630

pˆ + αf Dtf tf (ˆyf ) Dxh1 uh1 + −λh1 p DG uh1 + −λh1

pg

−pzh1 + −pg gh1 µh1 Dxh2 u + −λh2 p

yˆ f

αf

α f D 2 tf Dtf

Dtf

xh 1

h

Dxh11 xh1

+ µh 1
+ pˆ f sfh1 yˆ sfh1 pˆ

h1 DGx h1

−p

DG uh3 + −λh3

+ −pg g

−pzh3 µh3 gh3 −z\ + y\ −G + yg

h3

λh1

h

−p

Dxh1

G 1 h1 DGG −pg h

Dxh2

2 h2

D G u h 2 − λ h 2 p g + µh 2
−pzh2 + −pg gh2 + pˆ f sfh2 yˆ sfh2 pˆ gh2 Dxh3 uh3 + −λh3 p pg

gh 1

−pg

µh1

DGG h

3 h3

+ µh 3
+ pˆ f sfh3 yˆ sfh3 pˆ

λh 2

µh 2

xh 3

h

1

h

h

Dxh22 xh2

Dxh2

2 DGx h

DGG

h

G

2

2 h2

−pg 1 h Dxh3 G 3 h3

DGG

G

DGG

gh 3

λh 3

h

G 1 h1 DGG

−p Dxh3

gh 2

Dxh1

1 G

xh 2

Dxh1

1

G

Dxh1

1

G

h

h

−p

Dxh2 h

−pg 1

2

µh3

\

eh 1

eC h1

p\

1

G

1 h

Dxh33 h3 h3 DGx h3

−p

h

Dxh3

G 3 h3 DGG −pg

−p −pg 1 1

1 01

−I0

−I0 1

−I0 1

eC h3

2 DGG

1 I0

eC h2

I0 1

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Table 1

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631

right corner the corresponding partial Jacobian of D(ξ,e) FH1 ,H2 ,H3 (ξ, e) is displayed. Call M that partial Jacobian. Observe that in Table 1, h1 ∈ H1 , h2 ∈ H2 , and h3 ∈ H3 , and zh ≡
xh − eh , z ≡ h zh , z\ ≡ (zc )c=C ; for h1 , h2 , h3 , Dxhh xh ≡ Dxh xh uh (xh , G), Dxhh G ≡ h h ≡D Dxh G uh (xh , G), DGx ≡ DGxh uh (xh , G), DGG GG uh (xh , G). Note also that, to simh plify the exposition of the proof, only one household from each set is presented in the Table 1. We want to show that ker M T = {0}, i.e., cM = 0 ⇒ c = 0. Define 1 2 3 4 1 2 3 4 c ≡ ((cf 1 , cf 2 )Ff =1 , . . . , (ch1 , ch1 , ch1 , ch1 ), . . . , (ch2 , ch2 , ch2 , ch2 ), . . . , 1 2 3 4 5 , ch3 , ch3 , ch3 , ch3 ), cm1 , cm2 ) (ch3

with 1 , c2 , c3 , c4 )H ∈ (RC × R × R × R)H , c5 ∈ (cf 1 , cf 2 )Ff =1 ∈ (RC+1 × R)F , (ch1 h1 h1 h1 h=1 h3 C−1 , cm2 ∈ R. R, cm1 ∈ R Then cM = 0 can be written as follows  3 s 3 s 3 s (f.1) cf 1 αf D2 tf + cf 2 Dtf + ch1 ˆ + ch2 ˆ + ch3 ˆ  fh1 p fh2 p fh3 p     + cm1 (I0) + cm2 (01)= 0     f 1 Dt = 0  (f.2) c  f    h1  (h .1) c1 Dh1 3 2  1  h1 xh1 xh1 + ch1 DGxh1 − ch1 p − cm1 (I0)= 0    h3 h1 h2 h2  3 g 1 Dh1 2 1 2 1  (h1 .2) ch1  xh1 G + ch1 DGG − ch1 p + ch2 Dxh2 G + ch2 DGG + ch3 Dxh3 G     2 Dh3 + c  + ch3 m2 = 0  GG     1 p − c 2 pg = 0  (h1 .3) −ch1  h1    2 + c4 = 0  (h .4) c  1 h1 h1    h2 3 1 Dh2 2   (h .1) c 2 x  h2 xh2 + ch2 DGxh − ch2 p − cm1 (I0) = 0 h2  2    h1 h2 h2 3 g 1 Dh1 2 1 2 4  (h2 .2) ch1  xh1 G + ch1 DGG + ch2 Dxh2 G + ch2 DGG − ch2 p + ch2    h3 1 Dh3 2 + ch3 xh3 G + ch3 DGG + cm2 = 0 (18)   (h2 .3) −c1 p − c2 pg = 0   h2 h2    2 =0  (h .4) c 2  h2    h3 3 1 Dh3 2  (h3 .1) ch3  xh3 h3 + ch3 DGxh − ch3 p − cm1 (I0) = 0   3   h3 h1 h2 h2 1 Dh1 2 1 2 1  (h3 .2) ch1  xh1 G + ch1 Dxh1 G + ch2 Dxh2 G + ch2 DGG + ch3 Dxh3 G     2 Dh3 − c3 pg + c5 + c  + ch3  m2 = 0 GG h3 h3    1 2 g  (h3 .3) −c p − c p = 0   h3 h3   2 + c4 = 0   (h .4) c 3  h3 h3    3 p\ + c (I0) = 0  (M.1) c  m1 h1    3 =0  (M.2) c  h1    3  (M.3) c   h2 = 0   3 =0 (M.4) ch3

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3 = c3 = c3 = 0. From (M.1), c From (M.2), (M.3) and (M.4), ch1 m1 = 0. From h2 h3 1 = 0, c2 = 0 and c1 = 0, c2 = Assumption 4, (h1 .1), (h1 .3), (h3 .1), (h3 .3), ch1 h1 h3 h3 4 = 0 and c4 = 0. From (h .4), c2 = 0. Sub0. From (h1 .4) and (h3 .4), ch1 2 h3 h2 4 = 0 and c5 = 0. From tracting (h1 .2) from (h2 .2) and from (h3 .2), we get ch2 h3 2 Assumption 4, D uh is negative definite and therefore each principal submatrix is neg1 = 0. From (h .2), ative definite and therefore of full rank. Then, from (h2 .1), ch2 1 cm2 = 0. Finally, from Assumption 2, (cf 1 , cf 2 ) = 0.

We can now prove the main theorem. Proof of Theorem 11. Full measure follows from Sard’s Theorem when it is ap−1 plied to the projection defined on the differentiable equilibrium manifold F|×D ∗ (0). ∗ In fact, if (ξ, e) ∈  × D and F (ξ, e) = 0, f is differentiable at (ξ, e) since either µh > 0 or gh > 0. Consequently, 0 is a regular value of F|×D∗ . This easily follows since the computation is the same as the one in Lemma A.1, except that there is no h in H3 ; hence, the associated system of equations is simpler than system (18), given that we have only to deal with the equations corresponding to households h1 and h2 .

Acknowledgement We would like to thank the editor and three anonymous referees for their helpful comments. In particular, we are very grateful to one of the referees for very stimulating and thoughtful comments and suggestions that helped improve the paper greatly. We would also like to express our gratitude to late Murat Sertel for a helpful discussion on the paper.

Appendix A

Lemma A.1. Ar˜ˆ is bounded. Proof. Since


H

h=1 xh

 0 and


H

h=1 gh

> 0, if (x, g, yˆ ) ∈ Ar˜ˆ , then

  F     ˆ ≡ yˆ  ≡ (ˆy )F ∈ R(C+1)F : ∀f, tf (ˆy ) ≥ 0 and ˆ yˆ ∈ Y y ˆ + r ˜  0 f f =1 f f   r˜ˆ

f =1

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633

Moreover, for each h ∈ H, if (x, g, yˆ ) ∈ Ar˜ˆ , then H 

0  xh 

F 

xh ≤

h=1



yf + r˜ and 0 ≤ gh ≤

f =1

gh ≤

h

 f

g

yf + r˜ g

ˆ r˜ is bounded, which is Therefore, to show the desired result it is enough to prove that Y ˆr˜ ˆ simply by Y ˜ and we omit “” in any vector describing done below, where we denote Y the production. ˜ is not bounded. Then, we can find a sequence (yn )n∈N ≡ ((y n )F )n∈N in Suppose Y f f =1 ˜ such that y n  → +∞. We can partition {1, . . . , F } into two subsets F ∞ = ∅ and F 0 Y such that, up to a subsequence, if f ∈ F ∞ , then yfn  → +∞, and if f ∈ F 0 , then there exists kf ∈ R+ such that yfn  → kf . Recall that from Assumption 2’, ˆ

∃a > 0

such that ∀y ∈ (Y ∩ (−Y )), y < a

(A.1)

Take a ≡ 2Fa > 0. Then for sufficiently large n, for each f ∈ F∞ , yfn  > 0, and we can  n ay define the sequence y nf . Then f

∀f ∈

ayfn ∞ F , n yf 

n∈N

→ y¯ f

and ¯yf  = a

(A.2)

Since 0, yfn ∈ Yf , and Yf is convex, for sufficiently large n and each f ∈ F ∞ , yan  ∈ (0, 1) f

and ayn  f ∈ Yf  n yf 

(A.3)

Since, from Assumption 2, Yf is closed, from (A.2) and (A.3) y¯ f ∈ Yf

(A.4)

˜ From the definition of Y, 




yfn 

f ∈F ∞

f  ∈F ∞

ayfn

yfn  yfn 

+

 f  ∈F 0




ayfn f ∈F ∞

yfn 

+


ar

f ∈F ∞

yfn 

0

(A.5)

Up to a subsequence, we also get λfn

≡


Moreover,  f  ∈F ∞

yfn 

f ∈F ∞

λnf  →

yfn 

 f  ∈F ∞

→ λ¯ f  ∈ [0, 1]

λ¯ f  = 1

(A.6)

(A.7)

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634

Therefore, ∃f ∗ ∈ F ∞ , such that λ¯ f ∗ ≥ F1 . (Suppose otherwise, then ∀f ∈ F ∞ , λ¯ f <

and 1 = f  ∈F ∞ λ¯ f  < f  ∈F ∞ F1 ≤ 1.) Then, from (A.2), λ¯ f ∗ y¯ f ∗  = |λf ∗ | · yf ∗  ≥

1 2Fa > a F

1 F,

(A.8)

Taking limits of both sides of (A.5), we get 

λ¯ f y¯ f ≥ 0

f ∈F ∞

Since ∀f, 0, yf ∈ Yf , and Yf is convex, we have λ¯ f ∗ y¯ f ∗ ∈ Y

(A.9)

Moreover, 

λ¯ f ∗ y¯ f ∗ +

λ¯ f y¯ f ≥ 0

f ∈F ∞ \{f ∗ }

and therefore 

−λ¯ f ∗ y¯ f ∗ ≤

λ¯ f y¯ f ∈ Y

f ∈F ∞ \{f ∗ }

Then, from free disposal − λ¯ f ∗ y¯ f ∗ ∈ Y

(A.10)

But, from (A.9), (A.10) and (A.1) λ¯ f ∗ y¯ f ∗  < a contradicting (A.8). Proof of Lemma 10. The result follows from an application of the implicit function theorem to the first-order conditions of the household maximization problem. Computing the partial Jacobian of the left-hand sides of the first-order conditions of problem (16) with respect to (xh , gh , λh , wh ), we get 

Dxh xh uh

  Dxh G uh −p

Dxh G uh DGG uh −pg

−pT −pg 0

0



 0 1

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635

Since D2 uh is negative definite, we get 

D 2 uh ˆ wh , G\h )] = − [Dwh (xh , gh , λh )(p, −pˆ

−pˆ 0

−1 (C+2)×2

  0 1

(C+2)×1

Then,11 

D 2 uh −pˆ

−1

−pˆ 0

−1 −1  −1 −1   (I + pδ ˆ −1 ˆ D 2 uh ) D 2 uh pδ ˆ h D 2 uh h p −1  2 −1 −1 δh pˆ D uh δh

 =

(C+2)×2

−1  pˆ ∈ R++ , and where δh ≡ −pˆ D2 uh [Dwh (xh , gh )(p, ˆ wh , G\h )] =

−δ−1 h

D uh 2

  !−1 p pg

Define D uh 2

!−1

 ≡

Dxx

DGx

DxG

DGG

−1

 ≡

B11 B21

B12



B22

−1 D )−1 exists (see Horn and Johnson, 1985, 0.8.5, p. 21). Observe that (DGG − DxG Dxx Gx Then, using first-order conditions and the fact that gh > 0 and therefore µh = 0, we get

−δh · Dwh gh = B21 p + B22 pg −1 = (DGG − DxG Dxx DGx )−1

1 −1 (−DGx Dxx D x uh + D G uh ) λh

Finally, again from 0.8.5, in Horn and Johnson (1985), we get  det

Dxx

D x uh

DxG

D G uh

 −1 = det Dxx · det[DG uh − DGx Dxx D x uh ]

the desired result.  11

Given An×n ≡

A11

A12

A21

A22

 with A11 a square n1 -dimensional matrix, A22 a square n2 -dimensional matrix,

n1 + n2 = n, both A11 and C ≡ A22 − A21 A−1 11 A12 invertible, then it can be easily checked that  −1

A

≡

−1 −1 −1 −1 A−1 11 (I + A12 C A21 A11 ) −A11 A12 C

−C−1 A21 A−1 11

C−1



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