Economics Letters 55 (1997) 327–332

A note on the Boyd-McKenzie theorema , b*Ning Sun , Sho-Ichiro Kusumoto

aDoctoral Program in Policy and Planning Sciences, University of Tsukuba, Ibaraki 305, JapanbInstitute of Policy and Planning Sciences, University of Tsukuba, Ibaraki 305, Japan

Received 23 January 1997; accepted 19 March 1997

Abstract

This note constructs two examples to show that McKenzie’s method of introducing the so-called ‘artificial entrepreneurialfactors’ is not necessarily valid for the infinite horizon model of Boyd and McKenzie. 1997 Elsevier Science S.A.

Keywords: Artificial entrepreneurial factors; Constant returns to scale; Diminishing returns to scale; Existence theorem;Infinite horizon

JEL classification: D51

1. Introduction and summary

In economic models, it is often assumed that the production technology exhibits constant returns toscale (c.r.s.). On the other hand, diminishing returns to scale (d.r.s.) technologies would naturallyoccur when there are some fixed factors, such as managerial ability and entrepreneurship, which arenot marketed. However, McKenzie (1959) has shown the equivalence of the d.r.s. and c.r.s. models inthe case of finitely many goods, as far as the existence of a competitive equilibrium is concerned. Inthis sense, it can be justified to assume c.r.s. technology. The purpose of this note is to argue that ananalogous justification may not be valid if one considers models with infinitely many goods.

For this, we reexamine an article of Boyd and McKenzie (1993), wherein they established anexistence theorem of competitive equilibrium in an infinite horizon model with general consumptionsets and a single c.r.s. technology. They asserted: diminishing returns can be accommodated in theirc.r.s. model by introducing artificial entrepreneurial factors as in the case of finitely many goods.However, we shall show, by giving two examples, that this statement is not valid. Moreover, we shallpoint out that, for the induced c.r.s. model to satisfy their assumptions, there must be an additionalrestriction on the consumption sets, that is, the tail of any vector in the trading set be replaced by zerowhile remaining in the trading set. This Boyd and McKenzie discarded.

*Corresponding author. Fax: (81-298) 55-3849; e-mail: sun@shako.sk.tsukuba.ac.jp

0165-1765/97/$17.00 1997 Elsevier Science S.A. All rights reserved.PII S0165-1765( 97 )00104-3

328 N. Sun, S. Kusumoto / Economics Letters 55 (1997) 327 –332

2. The problem

Boyd and McKenzie (1993) considered an infinite horizon economy on the product topological`n nspace s 5P 5 (t) with a finite number of traders and a single c.r.s. technology. An importantt51

feature of their model is the generality of consumption sets. For the details on their model, readersmay refer to their article. We shall extend their model to a d.r.s. production economy

h h j hj% 5 (C ,P ,Y ,u , h 5 1, . . . , H, j 5 1, . . . , J),

h nin which there are H traders and J producers, each trader h is characterized by a trading set C ,sh h hand a preference correspondence P :C →C , each producer j is characterized by a convex production

j n hjset Y ,s , and each u denotes the hth trader’s liability share of firm j.Boyd and McKenzie made seven assumptions for their c.r.s. model, notably:

1. The production set Y is a closed convex cone with vertex at the origin that contains no straightlines.

h h h H h¯ ¯ ¯ ¯ ¯ ¯7. For all h, there is x [C 2Y with x #0. Moreover, x5o x <0 and x 5x for all s and t.h51 t sh h h h hFor any x [C , let z [R (x )2Y and d.0. Then there is a t such that for each t.t , there0 0

h h h h h h n¯is an a.0 with (z 1de , z , . . . , z , ax , . . . )[R (x )2Y, where e 5(1, . . . ,1)[5 .1 1 2 t t11 1

J h J hj jLet Y5o be the aggregate production set of economy %, and let Z 5o u Y be the sharej51 j51

set of the hth trader. Then, the following two assumptions are natural extensions of their Assumptions1 and 7 when d.r.s. technologies are possible.

j j19. Each Y is a closed convex set, and 0[Y . The aggregate production set Y is also closed,nY>s 5h0j, and Y contains no straight lines.1

h h h h H h¯ ¯ ¯ ¯ ¯ ¯79. For all h, there is x [C 2Z with x #0. Moreover, x5o x <0 and x 5x , for all s andh51 t sh h h h h ht. For any x [C , let z [R (x )2Z and d.0. Then there is a t such that for each t.t ,0 0

h h h h h h h¯there is an a.0 with (z 1de , z , . . . , z , ax , . . . )[R (x )2Z .1 1 2 t t11

jIt is clear that if each Y is a cone, Assumptions 19 and 79 are equivalent to Assumptions 1 and 7,respectively. We shall assume Assumptions 19 and 79 for our d.r.s. model.

Since Boyd and McKenzie asserted that the existence of competitive equilibrium for a d.r.s.economy can be proved by introducing artificial entrepreneurial factors, the profit of a firm, which isequal to the equilibrium price of the entrepreneurial factor, should be well defined and finite. Thus wedefine:

h hCOMPETITIVE EQUILIBRIUM: A competitive equilibrium for the d.r.s. production economy %5(C , P ,j hj 1 J 1 H nY , u , h51, . . . , H, j51, . . . , J) is a (H1J11)-tuple (p, y , . . . , y , x , . . . , x ) with p[s which

obeys:

h J hj j h h J hj y1. For each h, px #o u py and z[P (x ) implies pz. o u pj ,j51 j51j j j j j2. For each j, y [Y , py is well defined and finite, and limsup pz(t)#py for any z[Y , wheret→`

z(t)5(z , . . . , z , 0, . . . ),1 t

N. Sun, S. Kusumoto / Economics Letters 55 (1997) 327 –332 329

H h J j3. o x 5o y ;h51 j51

McKenzie (1959) showed the concordance of a d.r.s. production model with a c.r.s. model in thecase of finitely many goods in the following senses: (i) every d.r.s. production economy can beinduced to a c.r.s. production economy by introducing artificial entrepreneurial factors; (ii) under localnon-satiation assumption, a d.r.s. economy has a competitive equilibrium if and only if the induced

1c.r.s. economy does ; and (iii) if a d.r.s. economy satisfies the Arrow-Debreu type assumptions thenthe induced c.r.s. economy satisfies the McKenzie assumptions. Thus, the Arrow-Debreu theorem canbe proved by the McKenzie theorem.

Boyd and McKenzie asserted that diminishing returns can be accommodated in their infinitehorizon c.r.s. model by introducing artificial entrepreneurial factors. One may understand thisstatement in the following sense: if a d.r.s. production economy satisfies Assumptions 19 and 79, withtheir other Assumptions 2, 3, 4, 5 and 6, then one can use the McKenzie method and theBoyd-McKenzie theorem to show that there is a competitive equilibrium defined above.

Of course, by introducing artificial entrepreneurial factors, every d.r.s. production model as definedabove induces a Boyd-McKenzie model

h h˜ ˜ ˜ ˜% 5 (C ,P ,Y,h 5 1, . . . ,H ).

where

h J n h1 hJ hC 5 h(w,x) [ 5 3 s uw $ (2u , . . . , 2u ) and x [ C j,

h h h h h h h h h˜ ˜ ˜P (w ,x ) 5 h(w,x) [ C ux [ P (x )j for every (w ,x ) [ C ,

Jj J j j j jY 5 closureh(2t,y)ut 5 (t ) [ 5 , 'y [ Y for all j with y 5O t y j.1

j51

It can also be shown that under local non-satiation assumption, a d.r.s. economy has a competitiveequilibrium if and only if the induced c.r.s. economy does.

However, the first example in Section 3, that satisfies all Assumptions 19, 2, 3, 4, 5, 6 and 79, doesnot have a competitive equilibrium. Therefore, if one wants to establish an existence theorem in thed.r.s. model, their is a need for strengthening and/or adding some assumptions, as far as one keeps thedefinition of the competitive equilibrium. The second example further shows that even if a d.r.s.production economy satisfies some extra assumptions, under which the existence of the competitiveequilibrium can be proved by some other existence theorem (see Sun and Kusumoto, 1996), theexistence of equilibrium can not be proved by applying the Boyd-McKenzie theorem to the inducedc.r.s. economy. In other words, although the induced c.r.s. economy has a competitive equilibrium, theBoyd-McKenzie theorem fails to detect one.

1 Without local non-satiation assumption, there may be an equilibrium in the induced c.r.s. economy in which someentrepreneurial factors are not fully employed by the firms since a satiated individual consumes some of them.

330 N. Sun, S. Kusumoto / Economics Letters 55 (1997) 327 –332

3. Two counterexamples

Example 1: We shall consider a d.r.s. production economy with one trader and one producer. Thereare two commodities at each period, the first commodity denotes labor and the second denotes

2consumption good. Thus the commodity space is s . The trader’s trading set is given by C5hx[2 ts ux $21 and x $2 21, for t51, 2, . . . j, where x represents quantity of the ith commodity in1,t 2,t i,t

` 2t tperiod t. The trader’s preferences are represented by utility function U(x)5o 2 arctan (x 22 1t51 2,t

1). The producer’s production set is defined by

t2 t 1 / 2Y 5 hy [ s uy # 0 and y # 2 (2y ) , for t 5 1,2, . . . j.1,t 2,t 1,t

Finally, the trader’s liability share u of the firm, of course, is equal to 1.

(i) Assumptions 19, 2, 3, 4, 5, 6 and 79 satisfied: It is readily checked that this d.r.s. productioneconomy satisfies Assumptions 19, 2, 3, 4, 5, and 6. We shall show that this example also satisfies

tt 1 / 2Assumption 79. Let f (l)52 l denote the production function at period t, where l denotes the inputt t(1 / 2 )219of labor. Note that for each t the derivative f (l)5l #2 for all l[[0.5, 1]. Hence the productiont1 2¯plan y5((20.9, 2 20.2), (20.9, 2 20.2), . . . )[Y. Thus, we have

1 2¯ ¯x 5 ((20.1, 2 0.8), (20.1, 2 0.8), . . . ) 5 ((21, 2 2 1), (21, 2 2 1), . . . ) 2 y [ C 2 Y

5 C 2 Z.

¯ ¯ ¯Clearly, x <0 and x 5x for all s and t. Therefore, the first part of Assumption 79 is satisfied. On thet s

other hand, for any x[C, let z[R(x)2Z5R(x)2Y and d .0. Then there exist x9[C and y[Y suchthat U(x9)$U(x) and z5x92y. Note that: x91de [C, U(x91de ).U(x), and U is continuous in C in1 1

the product topology. So there is a t such that x9[t]1de [C and U(x9[t]1de ).U(x) for all t .t ,0 1 1 0t 11 t 129 9where x9[t]5(x , . . . , x , (21, 2 21), (21, 2 21), . . . )[C. Next we see that y[t]5( y , . . . ,1 t 1

t 11 t 12y , (20.9, 2 20.2), (20.9, 2 20.2), . . . [Y. Then, let a 51, we have (z 1de , z , . . . , z ,t 1 1 2 t

¯ax , . . . )5x9[t]1de 2y[t][R(x)2Z. That is, the second part of Assumption 79 is also satisfied.t 11 1

1 2 1 2* *(ii) No competitive equilibrium: Let x 5((21, 2 ), (21, 2 ), . . . ), and y 5(21, 2 ), (21, 2 ), . . . ).* *It is evident that ( y , x ) is the unique Pareto optimal allocation of the economy. On the other hand,

* *by calculating the marginal utilities at x5x and the marginal productions at y5y , we see that21 21 22 22*p 5(1,1), (2 , 2 ), (2 , 2 ), . . . ) is the only possible candidate of its supporting prices.

* * * * * * * *However, in the state of ( p , y , x ), the profit of the firm p y is infinite. Thus, ( p , y , x ) is nota competitive equilibrium, and, therefore, this example does not have any competitive equilibrium.

(iii) Artificial entrepreneurial factors introduced: If we introduce an artificial entrepreneurial factor in˜ ˜ ˜ ˜this example, then the induced c.r.s. production economy %5(C, P, Y ) does not satisfy Assumption 7,

and does not have any competitive equilibrium.˜ ˜˜First, notice that for any small d .0, C>(12d )Y5[. This implies that there is no x [C2Y with

˜x <0. That is, % does not satisfy the first part of Assumption 7. We see that in the finite dimensionalcase, if x4y then ax4y for some a [(0,1). It follows from this property that, in the case of finitely

H h˜ ˜many goods, the inferiority assumption, i.e., interior (o C >Y )±[, is automatically satisfied inh51

the induced c.r.s. economy. However, this property does not hold in this example.

N. Sun, S. Kusumoto / Economics Letters 55 (1997) 327 –332 331

2˜ ˜Next, for any x[C, we have z5(21,x)5(21,x)2(0,(0,0),(0,0), . . . )[C2Y ,R3s . However,˜˜ ˜for any t, z(t)5(21,x(t))[⁄ C2Y. Therefore, % does not satisfy the second part of their Assumption 7.

The readers may wonder if our Assumption 79 is too weak. One can show in general, by theexistence theorem of Sun and Kusumoto, that if a d.r.s. production economy satisfies Assumption 70,instead of 79, and one additional Assumption 8 given below, then there is a competitive equilibrium.

h h h h¯ ¯70. There exists a real number k (0,k,1) such that, for all h there is x [C 2kZ with x #0.H h h h h h h h¯ ¯ ¯ ¯Moreover, x5o x <0 and x 5x , for all r and t. For any x [C , let x [R (x )2Z andh51 t r

h h hd.0. Then there is a t such that for each t.t , there is an a.0 with (z 1de , z , . . . , z ,0 0 1 1 2 th h h h¯ax , . . . )[R (x )2Z .t11

j j j j j j j8. For each j, if y [Y then y (t)[Y for all t, where y (t)5(y , . . . , y , 0, . . . ).1 t

The first part of Assumption 70 is slightly stronger than that of Assumption 79. Assumption 8 iscalled the Exclusion Assumption by Bewley (1972).However, the next example will show that, evenif a d.r.s. production economy satisfies all Assumptions 19, 2, 3, 4, 5, 6, 70 and 8, the induced c.r.s.economy need not satisfy the second part of Assumption 7. Therefore, the existence of competitiveequilibrium cannot be proved by introducing artificial entrepreneurial factors and using the Boyd-McKenzie theorem.

2Example 2: Consider a d.r.s. production economy on the space s with one trader and one producer.2The trader’s trading set is C5hx[s ux $21 and x $0.5, for all tj. The trader’s preferences are1,t 2,t

` 2trepresented by utility functions U(x)5o 2 arctan(x 20.5). The producer’s production set ist51 2,t2 1 / 2given by Y5hy[s uy #0 and y #(2y ) , for all tj.1,t 2,t 1,t

It is easy to verify that this example satisfies all of the above assumptions, hence the existencetheorem of Sun and Kusumoto can be applied. Specifically, let x5(21,1), (21,1), . . . ), y5(21,1),

21 0 22 21 23 22(21,1), . . . ), p5(2 ,2 ), (2 ,2 ), (2 ,2 ), . . . ). Then ( p,y,x) is a competitive equilibrium. The˜ ˜ ˜corresponding competitive equilibrium in the induced c.r.s. economy is given by (p,y,x ), where

˜ ˜ ˜p5(1, p), y5(21,y), x5(21,x).˜However, in the induced economy %, for each x[C, the trader can consider such a trading plan

˜˜ ˜ ˜ ˜z5(21,x)[C2Y. But, for any t, z(t)5(21,x(t))[⁄ C-Y. That is, the induced c.r.s. economy % does˜not satisfy the second part of Assumption 7. Therefore, in the induced economy % the existence of

˜ ˜ ˜competitive equilibrium (p,y,x ) cannot be proved by the Boyd-McKenzie theorem.To see why the second part of Assumption 7 fails in the induced model, we note that for every

h h h1x [C , in the induced c.r.s. economy trader h can always consider such a trading plan (-u , . . . ,h1 h˜ ˜-u , x)[C -Y. If the induced model satisfies the second part of Assumption 7, then there must be a t0

h1 h1 h J n˜ ˜ ˜such that (-u , . . . , -u , x(t))[C -Y for all t .t . Note that (R 3s )>Y5[. Thus we must have0 1hx(t)[C . Therefore, if the induced c.r.s. model satisfies Assumption 7, the original d.r.s. economy

must satisfy the following restriction on the consumption sets,

h h hfor every x [ C , there is a t such that x(t) [ C for all t . t ,0 0

which is similar to an assumption used by Stigum (1973). Namely, if one wish to strengthen ourAssumptions 19, 2, 3, 4, 5, 6 and 79 so that the Boyd-McKenzie theorem can be used, this additionalrestriction on the consumption sets must be included. However, such a restriction would be against theoriginal motivation of Boyd and McKenzie (Boyd and McKenzie, 1993, pp. 1).

332 N. Sun, S. Kusumoto / Economics Letters 55 (1997) 327 –332

Acknowledgements

We are very grateful to Atsushi Kajii for his helpful comments and suggestions.

References

Bewley, T., 1972. Existence of equilibria in economies with infinitely many commodities. Journal of Economic Theory 4,514–540.

Boyd, J.H., McKenzie, L.W. III, 1993. The existence of competitive equilibrium over an infinite horizon with production andgeneral consumption sets. International Economic Review 34, 1–20.

McKenzie, L.W., 1959. On the existence of general equilibrium for a competitive market. Econometrica 27, 54–71.Stigum, B.P., 1973. Competitive equilibria with infinitely many commodities (II). Journal Economic Theory 6, 415–445.Sun, N., Kusumoto, S.I., 1996. Infinite horizon equilibria with convex production (Institute of Policy and Planning Sciences,

University of Tsukuba) Discussion Paper No. 689; read by N. Sun at the Annual Meeting of the Japanese Economic andEconometric Association held at Osaka, 1996.