The Lagrange multipliers and existence of competitive equilibrium in an intertemporal model with endogenous leisure

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The Lagrange multipliers and existence of competitive

equilibrium in an intertemporal model with endogenous

leisure

Manh-Hung Nguyen, San Nguyen Van

To cite this version:

Manh-Hung Nguyen, San Nguyen Van. The Lagrange multipliers and existence of competitive equi-librium in an intertemporal model with endogenous leisure. 2005. �halshs-00194723�

Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647 Paris Cedex 13

UMR CNRS 8095

The Lagrange multipliers and existence of competitive equilibrium in an intertemporal

model with endogenous leisure

Manh Hung NGUYEN, CERMSEM

San NGUYEN VAN, CERMSEM

The Lagrange Multipliers and Existence of

Competitive Equilibrium in an Intertemporal

Model with Endogenous Leisure

¤

Nguyen Manh Hung

_{, San Nguyen Van}

CERMSEM,Université de Paris 1,

MSE,106-112 Bd de l’Hopital, 75647 Paris, France.

January 25, 2005

Abstract

This paper proves the existence of competitive equilibrium in a single-sector dynamic economy with elastic labor supply. The method of proof relies on some recent results (see Le Van and Saglam [2004]) concerning the existence of Langrange multipliers in in…nite dimen-sional spaces and their representation as a summable sequence. JEL Classi…cation: C61, C62, D51, E13, O41

Keywords: Optimal Growth Model, Lagrange Multipliers, Competi-tive Equilibrium, Elastic labor supply.

¤_{We are grateful to Cuong Le Van for suggesting the problem to us and proposing} the method of using Lagrange multipliers. We are also grateful to Yiannis Vailakis for comments and suggestions.

y_{E-mail: manh-hung.nguyen@malix.univ-paris1.fr} z_{E-mail: san.nguyen-van@malix.univ-paris1.fr.}

1 Introduction

Since the seminal work of Ramsey [1928], optimal growth models have played a central role in modern macroeconomics. Classical growth theory relies on the assumption that labor is supplied in …xed amounts, although the original paper of Ramsey did include the disutility of labor as an argument in consumers’ utility functions. Subsequent research in applied macroeconomics (theories of business cycles ‡uctuations) have reassessed the role of labor-leisure choice in the process of growth. Nowadays, intertemporal models with elastic labor continue to be the standard setting used to model many issues in applied macroeconomics.

Lagrange multipliers techniques have facilitated considerably the analysis of constraint optimization problems. The applications of those techniques in the analysis of intertemporal models inherits most of the tractability found in a …nite setting. However, the passage to an in…nite dimensional setting raises additional questions. These questions concerned both with the extension of the Lagrangean in an in…nite dimensional setting as well as the representation of the Lagrange multipliers as a summable sequence.

Our purpose is to prove existence of competitive equilibrium for the ba-sic neoclasba-sical model with elastic labor using some recent results (see Le Van and Saglam [2004]) concerning the existence of Lagrange multipliers in in…nite dimensional spaces and their representation as a summable sequence. The issue of endogenous labor supply in intertemporal models have been analyzed before. See, for example,Greenwood and Hu¤man [1995], Cole-man(1997), and Datta et al. [2002]. These models analyze economies with distortions and exploit the existence and uniqueness of stationary equilib-rium paths. Recently, Le Van and Vailakis [2004] have proved the existence of a competitive equilibrium in a version of a Ramsey model in which leisure enters the utility function by exploiting the link between Pareto optima and competitive equilibria. Their method of proof is in the line of Le Van and Vailakis [2003] for the optimal growth model with inelastic labor supply. To develop methods for studying the existence of competitive equilibrium in a one sector growth model with leisure, this paper takes somewhat di¤erent approach than attempting to impose Inada conditions on the utility function. Following the early work of Peleg and Yaari [1970], Bewley [1972] studied the existence of equilibrium in an economy in which 1 _{is the commodity}

space and the method of using the limit of equilibria of …nite dimensional economies. The most important development since Bewley’s work was

pro-vided by Mas-Collel [1986], by using Negishi’s approach when the commodity space is a topological vector lattice. Many others works can be found in Flo-renzano [1983], Aliprantis et al. [1990], Mas-Collel and Zame [1991], Dana and Le Van [1991], Becker and Boyd [1997]... These methods constitute a general and elegant approach to the question of optimality and existence but they require a high level of abstraction. Our approach here extends the re-sults of Le Van and Saglam [2004] to a model with endogenous leisure. The proof of existence of equilibrium we give is more simple than in Le Van and Vailakis [2004] and require less stringent assumptions ( no Inada conditions for the utility function and the production function, no constant return to scale for the production function).

The organization of the paper is as follows. Section 2 provides the suf-…cient conditions on the objective function and the constraint functions so that Lagrangean multipliers can be presented by an 1

+sequence of multipliers

in optimal growth model with the leisure in the utility function. In section 3, we prove the existence of competitive equilibrium by using these multipliers as sequences of price and wage systems.

2 The Lagrange Multipliers in Optimal Growth

Model

Consider an economy in which a representative consumer has preferences de…ned over processes of consumption and leisure described by the utility

function _X1

=0

_(  )

In each period, the consumer faces two resource constraints given by + +1 ·  ( ) + (1 ¡ )

+  = 1 8

where  is the production function,  2 (0 1) is the depreciation rate of capital stock and  is labor. These constraints restrict allocations of

com-modities and time for the leisure.

maxX1 =0 _(  ) s.t. + +1 · ( 1 ¡ ) + (1 ¡ ) 8 ¸ 0  ¸ 0 ¸ 0 ¸ 0 1 ¡ ¸ 0 8 ¸ 0 0 ¸ 0 is given

We make the following assumptions:

A1: ( ) 2 R+ if ( ) 2 R2+, ( ) = ¡1 if ( ) 2 R2+ and (0 0) = 0

A2:The function  is strictly increasing, concave and di¤erentiable in R2 ++

A3: ( ) 2 R+ if ( ) 2 R2+  ( ) = ¡1 if ( ) 2 R2+ and

(0 ) =  ( 0) = 0

A4: The function  strictly increasing, concave and di¤erentiable in R2 ++

Further, (0 1)   and (+1 1) = 0

We say that a sequence (  )=011 is feasible from 0 if it satis…es

the constraints

8 ¸ 0 + +1· ( 1 ¡ ) + (1 ¡ )

 ¸ 0 ¸ 0 ¸ 0 1 ¡ ¸ 0

0  0 is given

It is easy to check that, for any initial condition 0  0 a sequence k =

(0 1 2   ) is feasible i¤ 0 · +1·  ( 1) + (1 ¡ ) for all . The

class of feasible capital paths is denoted by ¦(0)A pair of

consumption-leisure sequences (c l) =((0 0) (1 1) ) is feasible from 0  0 if there

exists a sequence k 2 ¦(0) that satis…es 0 · ++1·  ( 1¡)+(1¡)

and 0 ·  · 1 for all 

De…ne ( ) = ( ) + (1 ¡ ) Assumption A4 implies that

0(+1 1) = (+1 1) + (1 ¡ ) = 1 ¡   1

0

(0 1) = (0 1) + (1 ¡ )  1

From above, it follows that there exists   0 such that: (i) ( 1) =  , (ii)    implies ( 1)  , (iii)    implies ( 1)   Therefore for any k 2 ¦(0) we have 0 · · max(0 ) Thus, k 2 1+ which in turn implies

c 2 1

Denote x = (c k l) and de…ne F(x) = ¡P1 =0 _(  ) ©1 (x) = + +1¡  ( 1 ¡ ) ¡ (1 ¡ ) 8 ©2 (x) = ¡ ©3(x) = ¡ ©4(x) = ¡ ©5(x) = ¡ 1 8 ©= (©1 ©2 ©3+1 ©4 ©5) 8

The social planner’s problem (P) can be written as: min F(x) s.t©(x) · 0 x 2 1 + £ +1£ +1 where: F : 1 + £ +1£ +1! R [ f+1g © = (©)=01 : 1+ £ 1+ £ 1+ ! R [ f+1g Let:  = (F) = fx 2 1 + £ +1£ +1jF(x)  +1g ¡ = (©) = fx 2 1 + £ +1£ +1j©(x)  +1 8g

The following Theorem is due to Le Van and Saglam [2004]. Theorem 1 Let x 2 1_{ y 2 }1_{  2 .} De…ne  (x y) = ½    ·      

Suppose that two following assumptions are satis…ed:

T1: If x 2  y 2 1 _{satisfy 8 ¸ 0, x}_{(x y) 2  then (x}_{(x y)) !}

(x) when  ! 1 T2: If x 2 ¡ y 2 ¡ and x_{(x y) 2 ¡ 8 ¸ } 0. Then, a) ©(x(x y)) ! ©(x) as  ! 1 b) 9 s.t. 8 ¸ 0 , k©(x(x y))k · 

Let x¤ _{be a solution to (P) and x}0 _{2  satisfy the Slater condition:}

sup

 ©(x

0_{)  0}

Suppose x_(x¤_{ x}0_{) 2  \ ¡ Then, there exists ¤ 2 }1

+ such that

8x 2 1_{ F(x) + ¤©(x) ¸ F(x}¤_{) + ¤©(x}¤_{) 8x 2 ( \ ¡)}

and ¤©(x¤_{) = 0}

Proof. See Theorems 1 and 2 in Le Van and Saglam [2004]. We make use of Theorem 1 and obtain :

Proposition 2 If x¤ _{= (c}¤_{ k}¤_{ l}¤_{) is a solution to the following problem:}

min ¡X1 =0 _(  ) (Q) s.t. + +1¡ ( 1 ¡ ) ¡ (1 ¡ )· 0 ¡ · 0 ¡· 0 ¡· 0 ¡ 1 · 0

Then there exists ¸ = (¸1 ¸2_{ ¸}3_{ ¸}4_{ ¸}5_{) 2 }1

+£ +1 £ +1 £ +1 £ 1+ such that: 8x = (c k l) 2 1 + £ +1£ +1 1 X =0 _(¤  ¤) ¡ 1 X =0 1 (¤ + +1¤ ¡ (¤ 1 ¡ ¤) ¡ (1 ¡ )¤) +X1 =0 2 ¤ + 1 X =0 3 ¤+ 1 X =0 4 ¤ + 1 X =0 5 (1 ¡ ¤) ¸ X1 =0 _(  ) ¡ 1 X =0 1 (+  ¡ ( 1 ¡ ) ¡ (1 ¡ )) +X1 =0 2  1 +X =0 3  1 +X =0 4 + 1 X =0 5 (1 ¡ ) (1) and 1 (¤ + +1¤ ¡ (¤ 1 ¡ ¤) ¡ (1 ¡ )¤) = 0 (2)

2 ¤ = 0 (3) 3 ¤ = 0 (4) 4 ¤ = 0 (5) 5 (1 ¡ ¤) = 0 (6) 0 2 _ 1(¤ ¤) ¡ f1g + f2g (7) 0 2 _ 2(¤ ¤) ¡ 12(¤ ¤) + f4g ¡ f5g (8) 0 2 1 1(¤ ¤) + f(1 ¡ )1g + f3g ¡ f1¡1g (9)

where (¤ ¤)  (¤ ¤) respectively denote the projection on the 

com-ponent of the subdi¤erential of function  at (¤

 ¤) and the function  at

(¤  ¤)

Proof. We …rst check that Slater condition holds. Indeed, since 0

(0 1)  1

then for all 0  0 there exists some 0  b  0 such that: 0  b  (b 1)

and 0  b  (0 1)Thus, there exists two small positive numbers  1 such

that:

0  b +   (b 1 ¡ 1) and 0  b +   (0 1 ¡ 1)

Denote x0 _{= (c}0_{ k}0_{ l}0_{) such that c}0 _{= (   ) k}0 _{= (0 b b ), l}0 ₌

(1 1 1 ) We have ©1 0(x0) = 0+ 1¡ (0 1 ¡ 0) ¡ (1 ¡ )0 =  + b ¡ (0 1 ¡ 1)  0 ©1 1(x0) = 1+ 2¡ (1 1 ¡ 1) ¡ (1 ¡ )1 =  + b ¡ (b 1 ¡ 1)  0 ©1 (x0) =  + b ¡ (b 1 ¡ 1)  0 8 ¸ 2 ©2 (x0) = ¡  0 8 ¸ 0

©3 0(x0) = ¡0  0; ©3 (x0) = ¡b  0 8 ¸ 1 ©4 (x0) = ¡1  0 8 ¸ 0 ©5 (x0) = 1¡ 1  0 8 ¸ 0

Therefore the Slater condition is satis…ed. Now, it is obvious that, 8 x_(x¤_{ x}0_{) belongs to }1

+ £ +1£ +1

As in Le Van-Saglam 2004, Assumption T2 is satis…ed. We now check Assumption T1

For any ex 2  eex 2 1

+ £ 1+ £ +1 such that for any  x(ex eex) 2  we

have F(x_{(ex eex)) = ¡}X =0 _(e  e) ¡ 1 X = +1 _(ee  ee) As eex 2 1 + £ 1+ £ +1 sup

 jeej  +1  there exists   0 8 jeej ·  Since

 2 (0 1) we have 1 X = +1 _{( 1) = ( 1)} X1 =+1 _{! 0 as  ! 1}

Hence, F(x_{(ex eex)) ! F(ex) when  ! 1Taking account of the Theorem}

1, we get (1) - (6)

Finally, we obtain (7) - (9) from the Kuhn-Tucker …rst-order conditions.

3 Competitive Equilibrium

De…nition 3 A competitive equilibrium for this model consists of an allo-cation fc¤_{ l}¤_{ k}¤_{ L}¤_{g 2 }1

+ £ 1+ £ 1+ £ +1 price sequence p¤ 2 1+ for the

consumption good, a wage sequence w¤ _{2 }1

+ for labor and a price   0 for

the initial capital stock 0 such that:

i) (c¤_{ l}¤_{) is a solution to the problem}

max X1 =0 _(  )  p¤_{c · w}¤_{L + ¼}¤_{+ } 0

where ¼¤ _{is the maximum pro…t of the …rm.}

ii) (k¤_{ L}¤_{) is a solution to the …rm’s problem}

¼¤ _{= max}X1 =0 ¤ [ ( ) + (1 ¡ )¡ +1] ¡ 1 X =0 ¤ ¡ 0  0 · +1·  ( ) + (1 ¡ )  ¸ 0 8 iii)Markets clear 8 ¤  + ¤+1 =  (¤ ¤) + (1 ¡ )¤ ¤  + ¤ = 1 and 0¤ = 0

Theorem 4 Let (c¤_{ k}¤_{ l}¤_{) solve Problem (Q). Take}

¤

 = 1 for any  and   0

There exists 2

(¤ ¤) 2 2(¤ ¤) such that fc¤ k¤ L¤ p¤ w¤ g is a

competitive equilibrium with ¤

 = 12(¤ ¤).

Proof. Consider ¸ = (¸1 ¸2_{ ¸}3_{ ¸}4_{ ¸}5_{) of Proposition2. Conditions (7),(8),(9)}

in Proposition2 show that (¤

 ¤) and (¤ ¤) are nonempty. Moreover,8

there exists 1 (¤ ¤) 2 1(¤ ¤), 2(¤ ¤) 2 2(¤ ¤), 1(¤ ¤) 2 1(¤ ¤)and 2 (¤ ¤) 2 2(¤ ¤) such that _1 (¤ ¤) ¡ 1 + 2 = 0 (10) _2 (¤ ¤) ¡ 12(¤ ¤) + 4 ¡ 5 = 0 (11) 1 1(¤ ¤) + (1 ¡ )1 + 3 ¡ 1¡1 = 0 (12) De…ne ¤

 = 12(¤ ¤)  +1We now prove that w¤ 2 1+

We have +1 X1 =0 _(¤  ¤) ¡ 1 X =0 _{(0 0) ¸}X1 =0 _1 (¤ ¤)¤ + 1 X =0 _2 (¤ ¤)¤ which implies ₁ X =0 _2 (¤ ¤)¤  +1 (13)

and +1 X1 =0 1  (¤ ¤)¡ 1 X =0 1 (0 0) ¸ 1 X =0 1 1(¤ ¤)¤+ 1 X =0 1 2(¤ ¤)¤ which implies 1 X =0 1 2(¤ ¤)¤  +1 (14) Given  we multiply (11) by ¤

 and sum from 0 to . We then obtain

8 X =0 _2 (¤ ¤)¤ =  X =0 1 2(¤ ¤)¤ +  X =0 5 ¤¡  X =0 4 ¤ (15) Observe that 0 ·X1 =0 5 ¤ · 1 X =0 5   +1 (16) 0 ·X1 =0 4 ¤ · 1 X =0 4   +1 (17) Thus, since ¤  = 1 ¡ ¤ from (15), we get  X =0 _2 (¤ ¤) =  X =0 _2 (¤ ¤)¤+  X =0 1 2(¤ ¤)¤ +X =0 5 ¤ ¡  X =0 4 ¤

Using (13),(14),(16),(17) and letting  ! 1 we obtain 0 · X1 =0 _2 (¤ ¤) = 1 X =0 _2 (¤ ¤)¤+ 1 X =0 1 2(¤ ¤)¤ +X1 =0 5 ¤ ¡ 1 X =0 4 ¤  +1

Consequently, from (11), P1 =0 1 2(¤ ¤)  +1 i.e. w¤ 2 1+ So, we have fc¤_{ l}¤_{ k}¤_{ L}¤_{g 2 }1 + £1+£1+£+1 with p¤ 2 1+and w¤ 2 +1

We show that (k¤_{ L}¤_{) is solution to the …rm’s problem.}

Since ¤  = 1 ¤ = 12(¤ ¤), we have ¼¤ ₌X1 =0 1 [ (¤ ¤) + (1 ¡ )¤¡ +1¤ ] ¡ 1 X =0 1 2(¤ ¤) ¤ ¡ 0 Let : ¢ =  X =0 1 [(¤ ¤) + (1 ¡ )¤¡ ¤+1] ¡  X =0 1 2(¤ ¤) ¤ ¡ 0 ¡ Ã_X =0 1 [ ( ) + (1 ¡ )¡ +1] ¡  X =0 1 2(¤ ¤) ¡ 0 !

By the concavity of  , we get ¢ ¸  X =1 1 1(¤ ¤)(¤¡ ) + (1 ¡ )  X =1 1 (¤¡ ) ¡  X =0 1 (+1¤ ¡ +1) = [111(¤1 ¤1) + (1 ¡ )11¡ 10](1¤¡ 1) +  +[1 1(¤ ¤) + (1 ¡ )1 ¡  ¡11 ](¤ ¡ ) ¡ 1( +1¤ ¡  +1) By (4) and (12), we have: 8 = 1 2   [1 1(¤ ¤) + (1 ¡ )1 ¡ 1¡1](¤¡ ) = ¡3(¤¡ ) = 3 ¸ 0 Thus, ¢ ¸ ¡1( +1¤ ¡  +1) = ¡1¤+1+ 1+1 ¸ ¡1¤+1 Since ¸1 _{2 }1 + sup   ¤  +1  +1 we have

lim

 !+1¢ ¸ lim !+1¡  1

 +1¤ = 0

We have proved that the sequences (k¤_{ L}¤_{) maximize the pro…t of the}

…rm.

We now show that ¤ _{solves the consumer’s problem.}

Let fc Lg satisfy 1 X =0 1 · 1 X =0 ¤ + ¤+ 0 (18)

By the concavity of , we have: ¢ =X1 =0 _(¤  ¤) ¡ 1 X =0 _(  ) ¸X1 =0 _1 (¤ ¤)(¤ ¡ ) + 1 X =0 _2 (¤ ¤) (¤ ¡ )

Combining (3 ),(6),(10),(11) yields that ¢ ¸X1 =0 (1  ¡ 2)(¤ ¡ ) + 1 X =0 (1 2(¤ 1 ¡ ¤) + 5 ¡ 4)(¤ ¡ ) = X1 =0 1 (¤ ¡ ) + 1 X =0 2 ¡ 1 X =0 2 ¤+ 1 X =0 (¤  + 5)((¤ ¡ ) ¡X1 =0 4 ¤ + 1 X =0 4  ¸ X1 =0 1 (¤ ¡ ) + 1 X =0 (¤  + 5)((¤¡ ) = 1 X =0 1 (¤ ¡ ) + 1 X =0 ¤ (¤¡ ) + 1 X =0 5 (1 ¡ ) ¸ X1 =0 1 (¤ ¡ ) + 1 X =0 ¤ (¡ ¤)

Since ¤ ₌X1 =0 1 [(¤ ¤) + (1 ¡ )¤¡ +1¤ ] ¡ 1 X =0 ¤ ¤ ¡ 0 and ¤ +1¡ (1 ¡ )¤ =  (¤ ¤) ¡ ¤

it follows from ( 18) that

1 X =0 1  · 1 X =0 ¤ + 1 X =0 1 [(¤ ¤) ¡ (¤ ¤) + ¤] ¡ 1 X =0 ¤ ¤ · X1 =0 ¤ (¡ ¤) + 1 X =0 1 ¤

Consequently, ¢ ¸ 0 that means ¤ _{solves the consumer’s problem.}

Finally, the market clears at every period, since 8 ¤

++1¤ ¡(1¡)¤ =

(¤

 ¤) and 1 ¡ ¤ = ¤

References

[1] Aliprantis C. D. and D. J. Brown and O. Burkinshaw, [1990], Existence and Optimality of Competitive Equilibria, Berlin Heidelberg NewYork, Springer.

[2] Becker R. A. and J. H. Boyd III, [1997], Capital Theory, Equilibrium Analysis and Recursive Utility, Malden Massachusetts Oxford, Black-well.

[3] Bewley T. F., [1972], Existence of Equilibria in Economies with In…nitely Many Commodities, Journal of Economic Theory, 4, pp. 514-540. [4] Dana R-A. and C. Le Van, [1991], Equilibria of a Stationary Economy

with Recursive Preferences, Journal of Optimization Theory and Appli-cations, 71, pp. 289-313.

[5] Datta M. and L. J. Mirman and K. L. Re¤ett, [2002], Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Cap-ital and Labor, Journal of Economic Theory, 103, pp. 377-410.

[6] Dechert W.D., [1982], Lagrange Multipliers in In…nite Horizon Discrete Time Optimal Control Models, Journal of Mathematical Economics, 9, pp. 285-302.

[7] Coleman W.J., [1997], Equilibria in distorted in…nite-horizon economies with capital and labor, Journal of economic Theory, 103, 446-461. [8] Florenzano M., [1983], On the Existence of Equilibria in Economies with

an In…nite Dimensional Commodity Space, Journal of Mathematical Economics, 12, pp. 207-219.

[9] Greenwood G. and G. W. Hu¤man, [1995], On the Existence of Non-Optimal Equilibria in Dynamic Stochastic Economies, Journal of Eco-nomic Theory, 65, pp. 611-623.

[10] Le Van, C. and Y. Vailakis, [2003], Existence of a Competitive Equi-librium in a One Sector Growth Model with Heterogeneous Agents and Irreversible Investment, Economic Theory, 22, pp. 743-771.

[11] Le Van C. and H. Saglam, [2004], Optimal Growth Models and the Lagrange Multiplier, Journal of Mathematical Economics, 40, pp. 393-410.

[12] Le Van, C. and Y. Vailakis, [2004], Simple Proof of Existence of Equi-librium in a Single-Sector Growth Model with Elastic Labor, Discussion Paper.

[13] Mas-Colell A., [1986], The Price Equilibrium Existence Problem in Topological Vector Lattices, Econometrica, 54, pp. 1039-1053.

[14] Mas-Colell A. and W. Zame, [1991], Equilibrium Theory in In…nite Di-mensional Spaces, in W. Hildenbrand and H. Sonnenschein (eds.), Hand-book of Mathematical Economics, Vol. IV, Amsterdam, North Holland. [15] Peleg B. and M. E. Yaari, [1970], Markets with Countably Many

[16] Ramsey F., [1928], A Mathematical Theory of Saving, Journal of Eco-nomic Theory, 38, pp. 543-559.