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Capital accumulation and entry deterrence

Fershtman, C.; de Zeeuw, Aart

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for88~ 3 ~mic Research1993 II III I III I I III V I I II Inl l III II A ll l IIII I I III!II I

Capital Accumulation and EntryDeterrence: A Clarifying Note

byChaim Fershtman

andAart de Zeeuw

Reprinted from G. Feichtinger (ed.), DynamicEconomic Models and Optimal Control,

Amsterdam: Elsevier Science Publishers B.V.(North-Holland), 1992

,~,~~~ ' Reprint Series~,~OJP ~ no. 128

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1993

G~ j ~~~~7~ifor

Economic Research

Capital Accumulation and EntryDeterrence: A Clarifying Note

byChaim Fershtman

andAart de Zeeuw

Reprinted from G. Feichtinger (ed.), DynamicEconomic Models and Optimal Control,

Amsterdam: Elsevier Science Publishers B.V.(North-Holland), 1992

Reprint Seriesno. 128

Dynamic Economic Models and Optimal ControlG. Feichtinger ( editor)OO 1992 Elsevier Science Publishers B.V. All ríghts reserved.

CAPITAL ACCUMULATION AND ENTRY DETERRENCE: A CLARIFYING NOTE

Chaim Fershtmana and Aart de Zeeuwb

aDepartment of Economics, Tel Aviv University, Ramat Aviv 69978, Israel

bDepartment of Economics, Tilburg University, P.O. Box 90153. 5000 LE

Tilburg, The Netherlands, and Department of Economics, Free University,

P.O. Box 7161, 1007 MC Amsterdam, The Netherlands

281

AbstractThis note clacifies some of the entry deterrence aspects of capital accu-

mulation. Since accumulating capital takes time the focus of this note is

on the importance of time in the analysis of entry deterrence. While the

post-entry game is modelled as a capital accumulation differential game for

which we solve for the feedback equilibrium, we also add a time dimension

to the pre-entry game assuming that the entry decision is subject to entry

preparation that also takes time. This preparation period affects the ana-

lysis of entry detercence and the possibility and the attractiveness of

entry deterrence.

1. INTRODUCTION

In his seminal work on the supply side of new markets Spence [1-2] ana-

lyses the strategic interaction of early entrants in [heir game with po-

tential entrants. The result is in spirít similar to the von Stackelberg

~The first version of this paper was written while the first author visitedthe CentER of Economic Research at Tilburg University. We like to thank theCentER for ttieir hospitality. Furthermore, we are grateful for the usefulcomments of Eric van Damme and two anonymous referees.

282 C. Fershtman and A.J. de Zeeuw

oligopoly model but the strategic asymmetries are now induced by the his-

tory of the market. The crucial point is that the incumbents can makeirrevocable investments in product-specific capital which deter entry orwhich lead to a favourable position on the market after new entries. In the

terminology of Schelling [3] these investments are an advance commitment orcredible threat. Commitments to future investments are ruled out, becausethese commitments are not credible and because otherwise the fundamentalasymmetry between established firms and potential entrants disappears.Dixit [4] argves that this also means that an excess capacity strategy

cannot be sustained. After entry a Cournot~Nash game will be played betweenall the firms in the market. Dixit [4] stresses in his paper that it isimportant to distinguish a pre-entry stage and a post-entry stage. Although

the rules of the post-entry game are exogenous (such as a Cournot~Nasholígopoly), the established firm can influence the outcome of that game toits advantage by building fevourable initíal conditions in the pre-entrystage. Fudenberg and Tirole [5] analyse the post-entry game in continuoustime with an infinite horizon. They derive the set of subgame perfectequilibria for investment strategies, which are only a function of thestate (that is the current capital stocks). By first allowing the Firms tocoordinate their strategies and by then invoking some ad hoc argument theycan reduce the set of perfect equilibria to one. This is necessary to beable to discuss entry deterrence. Thepot [6] also discusses in essentiallythe same setting the multiplicity of these feedback equilibria. Reynolds[7J derives the subgame-perfect (or feedback Nash) equilibrium for the

línear-quadratic game, in which now investments are reversible but capacityis subject to adjustment costs. It is not proven that this equilibrium isunique. In particular equilibria may exist in which the strategies are notlinear, but differential game theory has not yet found a way to identifythese equilibria. Reynolds [7] compares the feedback Nash equilibrium withthe open-loop Nash equilibrium in which the investment strategies are onlya function of time. It is important to note here that Fudenberg and Tirole[5] consider irreversible investments as opposed to Reynolds (~].

In this note the post-entry game is modelled as the capital accumulationgame in Reynolds' [~] paper. However, since investment commitments in ad-vance of actual investments are not credible, the feedback Nash or subgame-perfect Markov equilibriun is in our view the only reasonable solution

Caaral Accumulanon and Ernry Derenence 283

concept. In Dixit's [4] terminology the rules of the post-entry game lead

to the feedback Nash equilibrium. The values of thís game are a function of

the initial capacity levels. Suppose there is one incumbent and one po-

tential entrant. At the start oF the post-entry game the capacity level of

the entrant is zero and the capacity level of the incumbent depends on its

investments before entry. The novelty in this note is to model explicitly a

time-lag between the potential entrant's preliminary decision to.enter and

the actual entry to the market. The incumbent can use this period to invest

in extra capacity in order to deter entry or to achieve a good starting

position for the post-entry game. When the investment costs are convex, the

length of the time-lag plays a crucial role in the decision problem of theincumbent. A second novelty in this note is the introduction of preparation

costs for the potential entrant. It is shown that this gives the incumbent

firm the possibility to realize monopoly profits in the pre-entry stage,even if it is not a natural monopoly but an artificial monopoly in the

terminology of Eaton and Lipsey [8]. It is therefore better to speak in

this case of a natural monopoly and a strategic natural monopoly.

The note is organized as follows. Section 2 sets out the framework of the

analysis. In section 3 and the appendix the equilibrium of the post-entrycapital accumulation differential game is derived. Section 4 analyses the

pre-entry decision problem of the incumbent firm. In section 5 a strategic

natural monopoly is introduced as a consequence of small but non-zero pre-

paration costs and section 6 concludes the paper.

2. TNE FRAMEWORK

Consider an industry in which there is one incumbent firm and one poten-

tial entrant. It is assumed that entry cannot be decided upon and carried

out instantaneously. For example, a company in telecommunications has to

apply for a licence before it can enter the market in a specific country,

and it usually takes a lot of time before the company is licensed to enter

(if ever). Furthermore, the firm has to perform some market research and

has to establish contacts with customers. All these activities are public

information. The lag between the entry decision and the actual entry is

284 C. Fershlman and A.J. de Zeeuw

denoted by the preparation time te ) 0. Moreover, the entry decision is not

a commitment. A firm can reconsider this decision at the last moment before

the preparation time has elapsed. Considering entry is usually not costless

and we denote this cost as Fe 2 O. Applying for a licence is relatively

cheap (fe - 0) but marketing certainly forms a cost (fe ) 0). These costs

are distinguished from the other entry costs, which will be introduced in

a next paragraph. Clearly, if fe - 0, the potential entrant will announce

his intention right away and leave the actual entry decision to time te or

later. We start our analysis by considering the case of zero preparation

costs. In section 5 we consider the case in which the potential entrant has

to bear some preparation costs. We assume for simplicity that firms are

symmetric regarding depreciation rate, discount factor and cost function.

Both firms in our model accumulate some form of capital according to the

standard capital accumulation dynamics

Ki(t) - Ii(t) - bKi(t). Ki(o) - Ki0' 1- 1'2~ (1)

where Ki denotes the capital level, Ii the investment level and b is acommon depreciation factor. The entrant can start to accumulate capitalonly after it enters. There is no capital accumulation during the pre-paration period. Investment is costly and we let Ci(Ii) be the cost ofinvestment. C. is a convex and increasing function.i

We assume that the instantaneous profits at time t can be expressed as afunction of the state variables [K1(t).K2(t)]. This is not to say that the

firms do not compete through prices or quantities but that a reduced formcan be used. It is assumed that the profit function Ri(K1(t),KZ(t)) is anincreasíng and concave function of Ki and a decreasing function of K~, i-1,2, j~ i. The objective of each firm is to maximize its discounted streamof profits net of investment costs

( {rti(Ki(t),Kj(t)) - Ci(Ii(t))} e-rt dtJO

(2)

where r is the common discount rate, subject to the capital accumulationdynamics (1).

Caprtal Accumulation and Entry Deterrence 285

Entry is associated with some fixed ( sunk) cost of entry, denoted by Fe.That is fe is paid at the beginning of the preparation period while Fe ispaid at the time of the entry itself. For example, Fe stands for the costsof establishing the production facilities. Clearly, the potential entrantwill enter the market, if, at the moment of entry, the discounted stream ofprofits net of investment costs at least covers the entry fee Fe. The post-entry game is described by (1)-(2), starting at time te. Thís can be calleda capital accumulation differential game (see, e.g., Fershtman and Muller[9] and Reynolds [~]). Suppose that the incumbent Firm is denoted as firm 1and the entrant firm as firm 2. On the assumption that prior to entry theentrant does not accumulate capital, the initial condition of the capitalaccumulation game is [K1(te),0]. If the value functions Vi of the post-

entry game exist, the potential entrant will choose to enter at time teonly if

~2(KL(te).0) - Fe ) 0. (3)

This implies that if the incumbent firm achieves at time te at least the

capital level

Kd(Fe) :- inf {K1IV2(K1,0) - Fe S 0} (4)

entry is blocked. Therefore we define Kd(Fe) as the capital deterrencelevel.

An incumbent firm may ignore the possibility of entry. This firm maximizes

(2) subject to (1) with K2 always equal to 0. We denote the optimal capital

accumulation path of this firm as Km(.).

Definition 1: An incumbent firm will be called a natural monopoly, iF

Km(t) 2 Kd(FP) for every t~ te. (5)

Note that the position of a firm as a natucal monopoly depends on both the

entry cost Fe and the length of the preparation period te.

286 C. Fershtman and A.J. de Zeeuw

Given the deterrence level Kd(Fe), the incumbent Firm is facing a standardentry deterrence problem. The firm can accumulate capital up to Kd(Fe) andblock entry, or it can accommodate entry. Since the investment costs areconvex, the decision of the incumbent depends on the time it has to reachthe deterrence level. The shorter the preparation time te is the more cost-ly it will be to reach Kd(Fe). Moreover, the incumbent's decision problemis not just whether to deter entry or to accommodate entry. Even if thisfirm decides to accommodate, it is of importance whích capital level itreaches by time te as this level affects profits in the post-entry game.

j. THE POST-ENTRY CAME: CAPITAL ACCUMULATION GAME

As was already stated in section 2, after entry the two firms are enga-ged in a capital accumulatíon differential game. Because in the generalcase such a game is not analytically tractable, a linear-quadratic struc-ture is adopted here. The cost functions are given by

C(Ii) -} cIi, c) 0, i- 1,2 (6)

and the profit functions are given by

Ri(Ki.Kj) - Ki(a - Ki - Kj). i.j - 1.2. i~ j.

It follows that the firms try to maximize

(7)

J{Ki(t)(a - Ki(t) - Kj(t)) -} cZi(t)} e-rt dt, i,j - 1.2. i~ j. (8)0

subject to (1), where the initial time is taken to be 0 in order to sim-plify notation, although the actual initial time is Le.

The capital accumulation differential game (8)-(1) is identical to the onein Aeynolds (~]. Furthermore, the game is in structure very similar to thedynamic duopoly with sticky prices (Fershtman and Kamien [10]) and to a

Captal Accumulaóon and Entry Oefenence 287

model of competitive arms accumulation (van der Ploeg and de Zeeuw [11]).

Because it seems reasonable to assume that the firms can condition their

investments on the current capital levels and that the firms cen not commit

themselves to future investments, the feedback Nash (Starr and Ho [12]) or

subgame-perfect Markov equilibrium has to be derived. The outcome can be

found in Reynolds [7], but the eppendix of this paper presents this outcome

and the derivation in a much more transparent way.

The capital accumulation game can therefore be summarized by a value Func-

tion Vi(K1,KZ), which is differentiable ín its arguments with ~Ví~~Ki ) 0

and ~Vi~~Kj C 0, i~ j, and which gives the value for player i of the

capital accumulation game that starts with the initial condition (K1,K2).

4. INCUMBENT'S DECISION PROBL.EM

At date t- 0 the incumbent has to decide whether to follow a capital

accumulation path that prevents entry or a path that accommodates entry.

The incumbent has to compare íts profits given by the outcome of problem 1,

when entry is accommodated, and its profits given by the outcome of problem

2, when entry is deterred.Problem 1 is given by:

Problem 1: Accommodating Entry

et e

maximize ({K1(t)(a - K1(t)) - 1 cIi(t)} e-rt dt . V1(K1(te).0) e-rt .(9)

I1(.) )~

subject to (1), K1(0) - K10 and K1(te) ( Kd(Fe).

The trade-off for the incumbent in problem 1 is to realize monopoly profits

in the pre-entry stage, on the one hand, and to reach a favourable initial

position in the post-entry stage, on the other hand.

Problem 2 is given by:

288 C. Fershfman ar~d A.J. de Zeeuw

Problem 2: Deterring Entry

maximize ({K1(t)(a - K1(t)) -} cIi(t)} e-rt dt, (10)11(') JO

subject to (1), K1(0) - K10 and K1(t) 2 Kd(Fe) for every t 2 te.

When the last constraint is not binding, problem 2 corresponds to a naturalmonopoly as defined in definition 1. Otherwise, the incumbent is sometimescalled an artificial monopoly.

Let Va(K10,te,Kd(Fe)) be the value of the control problem for theincumbent when entry is accommodated (problem 1) and Vd(K10,te,Kd(Fe)) bethe value when the incumbent chooses to deter entry (problem 2). Clearly,the incumbent will deter entry iff Vd(K10'te'Kd(Fe)) ~ Va(K10'te

Kd(Fe))

Since we can not provide at this stage explicit solutions to problems 1 and2, we can not give rigorous proofs of the statements that follow. We argue,however, that both Va(.) and Vd(.) increase with te. There are two reasonsfor this. Firstly, a higher te implies that the incumbent can enjoy a lon-ger monopolistic period. Secondly, the incumbent now has more time eitherto achieve a Favourable starting position for the post-entry game in caseit chooses to accommodate entry, or to achieve the capital deterrence levelKd. Since the accumulation costs are convex, reaching these levels is lesscostly when there is more time. Clearly, the effects of changing te inVa(.) and Vd(.) are not identical. Therefore, it is possible that for a

e d egiven pair (t ,K (F )) the incumbent will choose to deter entry whereas fora shorter preparation period, i.e. te ~ te, the optimal strategy is toaccommodate entry. This happens when the extra investment costs to reachthe capital deterrence level Kd(Fe) in a shorter period te outweigh thelosses that are suffered in problem 1 due to a shorter preparation period.We can conclude that the length of the preparation period is an importantfactor in determining whether the incumbent firm's optimal strategy is todeter entry or not. In particular we expect that the set of possible paírs(te,Fe) will be divided as in Figure 1, such that for (te,Fe) E A entry isdeterred whereas for (te,Fe) ~ A entry is accommodated.

Capilal Accumulanon and Entry Dete~ence 289

te

Fe

Figure 1.

Intuitively, for a high Fe the level of deterrence capital Kd is low and

thus the incumbent only needs a short preparation period to accumulate this

level. Reducing Fe leads to a higher Kd, which implies that the incumbent

will only accumulate this level if the preparation period is sufficiently

long. The area A covers both the situations that correspond to a natural

monopoly and to an artificial monopoly.

Likewise, whether the íncumbent firm is a natural monopoly or not also

depends on the length of the preparation period te. It can very well bethat Km(t) Z Kd(Fe), t 2 te, which implies that the incumbent firm is a

natural monopoly for the pceparation period te, whereas Km(te) ~ Kd(Fe)

holds for a shorter preparation period te C te, which implies that in that

case the incumbent firm is not a natural monopoly.

5. THE STRATEGIC NATURAL MONOPOLY

The standard definition of natural monopoly describes the case where a

firm, by acting as a monopolist that maximízes profits, while ignoring the

possibility of entry, in fact deters entry. That is the profit maximizing

capital accumulation path Km(.) lies on or above the capital deterrence

level Kd(Fe) from time te onwards, so that entry is automatically deterred.

Consider now the case in which the preparation costs fe ) 0, and suppose

that the incumbent firm is not a natural monopolist. The question is now

290 C. Fershfman and A.J. de Zeeuw

whether the potential entrant will announce entry in such a case and thuspay fe ) 0. The answer depends on what the potential entrant expects theincumbent's reaction to be to an entry announcement. If the announcement isdone at time t- 0 and if Vd(K10,te,Kd(Fe)) ) Va(K10,te,Kd(Fe)), then theincumbent will react to such an announcement by accumulating the capital

deterrence level Kd(Fe) by time te, so that entry is deterred. Given such areaction the optimal strategy of the potential entrant is not to start thepreparations for entry at all and thus to avoid the costs fe ) 0. The onlysubgame perfect equilibrium in this case is that the incumbent firm acts asa monopolist, accumulating capital according to Km(.). That is not to saythat the incumbent firm ignores the possibility of entry. Since there is apreparation period te, the incumbent realizes that if necessary once entrywill be announced it can start to accumulate capital to the level Kd(Fe) inorder to deter entry. Moreover, since preparation to enter is not costlessand entcy will be deterred, the potential entrant will not start to preparefor entry, and the incumbent can make monopoly profits all the way.This analysis implies that in the case of non-zero preparation costs the

incumbent firm's capital accumulation path follows the path Km(.), even ifthe incumbent firm is not a natural monopoly. We denote such a market as astrategic natural monopoly (SNM). The part of the area A in Figure 1, thatcorresponds to an artificial monopoly in the case of zero preparationcosts, now becomes the area of a strategic natural monopoly wíth monopolyprofits for the incumbent firm at sll times. Clearly, the position of afirm as a SNM depends again on both the entry cost Fe and the length of thepreparation period te.

6. CONCLUSION

The role of capital in entry deterrence is well documented in the lite-rature. The standard setting for such an analysis has been a two-stage gamewhere in the first stage capital is built, usually with linear costs.

In this note we claim that the pre-entry stage should also be modelledwith a specific attention to the role of time. Under the standard assump-tion of convex investment costs the length of the pre-entry period provesto be an important factor in the analysis of entry deterrence. Furthermore,

Capdal Accumulalion and Enfry Delenence 291

it is shown that in the case of non-zero preparation costs the artificialmonopoly can in fact realize monopoly profits in the pre-entry stage,because of its credible threat to accumulate extra capital in order todeter entry and to burden the potential entrant with these preparationcosts when it starts to prepare for entry.These observations may lead to a more detailed enalysis of the entry

deterrence problem in which the preparation period can also be one of thefirm's strategic variables. This analysis is, however, beyond the scope ofthis note and will be subject of further research.

7. REFERENCES

1 A.M. Spence, "Entry, capacity, investment, and oligopolistic pricing",The Bell Journal of Economics, 8, 2(1977) 534-544.

2 A.M. Spence, "Investment strategy and growth in a new market", The BellJournal of Economics, 10, 1(1979) 1-19.

3 T.C. Schelling, The Strategy of Conflict, Harvard University Press,Cambridge, Massachusetts, 1960.

4 A. Dixit, "The role of investment in entry-deterrence", The EconomicJournal, 90 (1980) 95-106.

5 D. Fudenberg and J. Tirole, "Capital as a commitment: strategic invest-

ment to deter mobility", Journal of Economic Theory, 31 (1983) 227-Z50.

6 J. Thepot, "The strategic investment problem revisited: a differential

game approach", Working Paper, 8708, Faculté des Sciences Economiques,

Université Louis Pasteur, Strasbourg (1987).

7 S.S. Reynolds, "Capacity investment, preemption and commitment in aninf'inite horizon model", International Economic Review, 28, 1(1987) 69-

88.

292 C. Fershtman and A.J. de Zeeuw

8 B.C. Eaton and R.C. Lipsey, "Capital, commitment, and entry equili-brium", The Bell Journal of Economics, 12 (1981) 593-604.

9 C. Fershtman and E. Muller, "Capital accumulation games oF infinite

duration", Journal of Economic Theory, 33 (1984) 322-339.

10 C. Fershtman and M.I. Kamien, "Dynamic duopolistic competition withsticky prices", Econometrica, 55. 5(1987) 1151-1164.

11 F. van der Ploeg and A.J. de Zeeuw, "Perfect equilibrium in a model ofcompetitive arms accumulation", International Economic Review, 31, 1(1990) 131-146.

12 A.W. Starr and Y.C. Ho, "Further properties of nonzero-sum differentialgames", Journal of Optimization Theory and Applications, 3, 4(1969)207-219.

APPENDIX

Consider the differential game (i - 1,2)

maximise ({-} cIi(t) a} x'(t)Qix(t) 4 qix(t)}e-rt dt (A.1)li(.) Jo

subject to z(t) - Ax(t) t B1I1(t) 4 62I2(t), x(o) - x0, (A.2)

where the state x consists of the capital stocks [K1,K2]', and where

A:-[-0 -b] : B1: - I Ó J : B2:' I ~J : Ql : - L-i -ÓJ : Q2: - I -0 -ZJ : 91: - I OJ : q2: - I aJ .

The value functions are denoted as V. The feedback Nash equilibrium for thedifferential game results from the dynamic programming equations (i - 1,2)

rVi(x) - max {-} CIi ~} x'Qix } qix ~ Vix(x)(Ax } Blul ' B2u2)}. (A.3)

Capital Accumulation and Enlry Deterrence 293

The equilibrium strategies are given by

Ii(x) - (llc)Bi~ix(x). (A.4)

Ttie dynamic programming equations become ( i,j - 1,2; i f j)

rv.(x) - -} (l~c)v: ( x)B.B:V. ( x) . } x'Q.x ~ q:x .i ix i i ix i i

~ix(x)(Ax ~ (l~c)BidiVix(x) . (l~c)ajawjx(x))}. (A.5)

With the quadratic value functions

vl(x) - 3 x'I pl p3 J x ..... : v2(x) - : x'I p2 p3 J x ..... (A.6)L 3 2 L 3 1

equation of the quadratic terms of (A.5) yields the system of equations

pi ~ 2p3 - (2bfr)cpl - 2c - 0, (A.7)

2P1P3 t P3PZ -(2b}r)cp3 - c- 0. (A.8)

p3 ~ 2p1p2 - (2btr)cp2 - 0. (A.9)

When the equilibrium strategies of (A.4) are substituted in the system(A.2) the closed-loop system results with the state-transition matrix

b~P Ic p3lcAc1:- - 1p3~c -btPl~c . (A.10)

The eigenvalues of Acl are -b t Pl~c t p3~c, so that the closed-loop systemis stable if and only if

pl t p3 - bc t o and pl - p3 - bc t o. (A.11)

294 C. Fershtman and A.J. de Zeeuw

Equation (A.7) describes an ellipse in the ( pl,p3)-plane. Furthermore, foceach value of p2 equation (A.8) describes a hyperbola in the ( pl,p3)-planeand equation ( A.9) a parabola. Defining p :- J(2c ~}(2b.r)2c2), the use ofpolar coordinates

pl -}(2b~r)c f p sin p; P3 -},~2 p cos ~, -rt C y 5 rt. (A.12)

and the elimination of p2 leads to the equation

p2 cos3 ~- 8p2 sin2 ~o cos yv t 4J2 c sin ~- o. (A.13)

Defining a:- ,~2 p2~c - 2.~2 a}f2 (2bir)2c, (A.13) yields eventually

tan3 p - a tan2 y~ tan ~ t(1~8)a - 0. (A.14)

Consider the function F given by

f(Y) :' Y3 - aY2 t Y' (1~8)a. a 2 2,~2. (A.15)

Some straightforward calculus shows that the function f has one negativeroot yl and two positive roots y2 and y3 with },~2 C y2 C y3. Consider y2 asa function of a. It is easy to see that y2(a) - iJ2 for a - 2J2. Implicit

differentiation yields

y2(a) - LYZ(a) - (1~8)~If~(YZ(a)) ( 0. (A.16)

It follows that the smallest positive root y2 of the function F satisfíes},~2 C y2 C}~2. Since f(-}J2) C 0, the negative root yl of the function fsatisfies yl ) -}J2.The largest positive root y3 of the function f is given by

Y3(a) - ZJ{(az~9) - (ll3)} cos (w~3) t (a~3),

where

w- arccos [(a3127) -(l1~148)]~[J{(a219) -(ll3)?J3. 0 C w C(nl2)-

Capifal Accumulalion and Enfry Deferience 295

ClaimThe largest positive root y3 of the function f satisfies the stability con-straints (A.11), but the negative root yl and the smallest positive root y2of the function f do not satisfy the stability constraints (A.11).

ProofWith the polar coordinates (A.12) the stability constraints (A.11) aregiven by -n C p C 0 and

~tan ~~ )[}J2 z 4} rc]~z with z~ 0 given by z2 t (}J2 z t} rc)2 - p2.

It follows that the stability constraints are given by

~tan p~ )[2a . rJ(3J2 ac - r2c2)lIL2J2 a- rzc]. (A.18)

Because

Itan ~I -[2oc . rJ(3J2 ac - r2c2)]~[8 t 4bc(bfr)] ~

Za~[8 ` 4bc(b~r)] ~ }J2,

and because yl )-}Jz and :J2 C y2 C}J2, these roots of the function f donot satisfy the stability constraints. Furthermore,

y3(a) ~ J{(a2~9) - (ll3)} ~ (al3). (A.19)

The right-hand side of (A.19) is minimal for b- 0 and the right-hand sideoF (A.18) is maximal for ó- 0. It is tedious but straightforward to showthat this minimal value is larger than this maximal value.It follows that the largest positive root y3 of the function f satisfiesthe stability constraints. Q.E.D.

The conclusion of this analysis is that the parameters pl and p3 of thevalue functions are given by

296 C. Fershtman and A.J. de Zeeuw

P1 - i(2b'r)c t J{[iJ2 occ y3(o~)J~[1 ' Y3(oc)J}. (A.20)

p3 - ,I{[{,I2 o~cJl[1 ' Y3(a)]}. (A.21)

where a - 2,I2 '~J2 (2b~r)2c, and y3(a) is given by (A.1~).The parameter pz of the value functions can then be calculated from (A.9),which yields

p2 - -i ,l{[}J2 ac7l[Y3(oc) ' Y3(a)J}.

The linear terms of the quadratic value functions

V1(x) - - - ' [P4 PSJx ' - ; V2(x) - . - ' CP5 P4Jx ~ ....

(A.2z)

(A.6)

can be found by equation of the linear terms of (A.5), which yields the

simple system of equations

{P1 ' P3 -(b'r)c}p4 ~ p3p5 t ac - 0.

(P2 . p3)P4 . {P1 - (b'r)c}p5 - 0.

Finally, the equilibríum strategies become

li(x) -(lIc){P1Ki ' P3K~ ' P4}. 1..] - 1.2. i~ J-

(A.23)

(A.24)

(A.4)

Reprint Scrics, CentER, Tilburg Univervily, The Nethcrlands:

No. I G. Marini and F. van der Ploeg, Monetary and fiscal policy in an optimising moclelwilh capital accumulation and finite lives, The Economic Journal, vol. 98, no. 392,19R8, pP. 772 - 7R6.

No. 2 P. van der Ploeg, Intemational policy coordination ín interdependent monetaryeconomies, Journa! of Internationa! F-cnnanics, vol. 25, 1988, pp. I- 23.

No. 3 A.P. Barten, Tlie history of Dutch macroeconomic modelling ( 1936-19R6), in W.

Driehuis, M.M.G. Fase and H. den Harlog (eds.), Challenges jor Macroeconomic

Modelling, Contributions to Economic Analysis 178, Amsterdam: North-Ilolland,

1988, PP. 39 - 88.

No. 4 F. van der Plceg, Disposable income, unemployment, in(lation and state spendingin a dynamic pulitical-economic model, Public Choice, vol. 60, 19R9, pp. 21 I- 239.

No. 5 Th. ten Raa and F. van der Plceg, A statistical approach to the problem of negativesin input-output analysis, Economic Modelling, vol. 6, no. I, 1989, pp. 2- 19.

No. 6 E. van Damme, Renegotiation-proof equilibria in repeated prisoners' dilemma,Jotrrnal ojEconomic THeory, vol. 47, no. I, 1989, pp. 206 - 217.

No. 7 C. Mulder and F. van der Plceg, Trade unions, investment and employment in asmall open economy: a Dutch perspective, in J. Muysken and C. de Neubourg(eds.), Unemployrnent in Europe, London: The Macmillan Press Ltd, 1989, pp. 200- 229.

No. 8 Th. van de Klundert and F. van der Plceg, Wage rigidity and capital mobility in anoplimizing tnodel of a small open economy, De Economist, vol. 137, nr. 1, 1989,

PP. 47 - 75.

No. 9 G. Dhaene and A.P. Barten, When it all began: the 1936 Tinbergen model revisited,Economic Modelling, vol. 6, no. 2, 1989, pp. 203 - 219.

No. 10 F. van der Plceg and A.J. de Zeeuw, Conflict over anns accumulalion in market andcommand economics, in F. van der Plceg and A.J. de Zeeuw ( eds.), Dynamic PalicyGarnes in Economics, Conlributions to Economic Analysis I81, Amster- dam:Elsevier Science Publishers B.V. (North-Holland), 1989, pp. 91 - I 19.

No. II 1. Driffill, Macroeconomic policy games with incomplete informetion: someextensions, in F. van der Plceg and A.1. de Zeeuw ( eds.), Dynamic Policy Ganresirr Economics, Contributions to Economic Analysis I81, Amsterdam: ElsevierScience Publishers B.V. (North-Holland), 1989, pp. 289 - 322.

No. 12 F. van der Ploeg. Towards monetary integration in Europe, in P. De Grauwe et al.,De Europese Moneraire InleRratie.~ vier visies, Wetrnschappelijke Raad voor hetRegeringsbeleid V 66, 's-Gravenhage: SDU uitgeverij, 1989, pp. 81 - 106.

No. 13 R.J.M. Alessie and A. Kapteyn, Consumplion, savings and demography, in A.Wenig, K.F. Zimmermann ( eds.), Uemogrophic Change andEconomic Developmenr,BerlirJF~eidelberg: Springer-Verlag, 1989, pp. 272 - 305.

No. 14 A. Hoque, J.R. Magnus and B. Pesaran, The exact mulli-period mean-squareforecast error for the first-order sutorcgressive model, Journa! ojEconomerrics, vol.39, no. 3, 1488, pp. 327 - 346.

No. I S R. Alessie, A. Kapteyn and B. Melenberg, The effects of liquidity constraints onconsumption: estimation from household panel data, European F.conomic Review,vol. 33, no. 213, 1989, pp. 547 - 555.

No. 16 A. Holly and J.R. Magnus, A note on instrumental variables and maximum likeli-hood estimation procedurcs, Annales d'Économie et de S~atistique, no. 10,April-June, 1988, PP. 121 - 138.

No. 17 P. ten Hacken, A. Kapteyn and I. Woittiez, Unemployment benefits and the labormarket, a microlmacro approach, in B.A. Gustafsson and N. Anders Klevmarken(eds.), Tlre Politicol Economy oj Socia! Security, Contributions to EconomicAnalysis 179, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1989,

PP. 143 - 164.

No. IS T. Wansbeek and A. Kapteyn, Estimation of the error-components model withincomplete panels, Jourrra! ojEconon:etrics, vol. 41, no. 3, 1989, pp. 341 - 361.

No. 19 A. Kapteyn, P. Kooreman and R. Willemse, Some methodological issues in theimplementation of subjective poverty definítions, The JournalojHuman Resources,vol. 23, no. 2, 1988, PP. 222 - 242.

No. 20 Th. van de Klundert and F. van der Plceg, Fiscal policy and finite lives ininterdependent economies wilh rcal and nominal wage rigidity, Oxjord F.conomicPapers, vol. 41, no. 3, 1989, pp. 459 - 489.

Na. 21 J.R. Magnus and B. Pesaran, The exect multi-period mean-square forecast error fortlre first-order autorcgressive model with an intercept, Journa! ojEconometrics, vol.42, no. 2, 1989, PP. 157 - 179.

No. 22 F. van der Ploeg, Two essays on political economy: (i) The political econotny ofovervaluation, The Economíc Journal, vol. 99, no. 397, 1989, pp. 8S0 - SSS; (ii)Election outcomes and the slockmarket, European Journal oJ Polilica! Econonry,vol. S, no. I, 1989, pp. 21 - 30.

No. 23 J.R. Magnus and A.D. Woodland, On the maximum likelihood estirnation ofmultivariale regression models containing serially correlated error components,International Economic Review, vol. 29, no. 4, 1988, pp. 707 - 725.

No. 24 A.J-1. Talman and Y. Yamamolo, A simplicial algorithm for stationary point

problems on polytopes, Malhematies ojOperations Research, vol. 14, no. 3, 1989,pp. 383 - 399.

No. 25 G. van Damme, Stable equilibria and forward induction, .lournul oj FcnnunucTlrenry, vol. 48, no. 2, 1989, pp. 476 - 496.

No. 26 A.P. [3arten and L.J. Bettendorf, Price fonnation of fish: An application of aninversc demand system, European Econonric Revicw, vol. 33, no. 8, 1989, pp. IS09

- I S2S.

No. 27 G. Noldekc and E. van Damme, Signalling in a dynamic labour markct, Review oJF.cnnumic Snrdie.c, voL 57 ( I), no. 189, 1990, PP. 1- 23.

No. 28 P. Kop Jansen and Th. len Raa, The choice of model in the construction of

input-output cocfficients matrices, Internationa! Ccanonric Review, vol. 31, no. I,

1990, pp. 213 - 227.

No. 29 F. van der Ploeg and A.J. de Zecuw, Perfect equilibrium in a model of cornpetitive

anns accumulation, Inlernatiana! Econonric Review, vol. 31, no. I, 199Q pp. 131

- 146.

No. 30 1.R. Magnus and A.D. Woodland, Separability and aggregation, Ecnnontica, vol. S7,no. 226, 1990, pp. 239 - 247.

No. 31 F. van der Plceg, Intemational interdependence and policy coordination ineconomies wilh real and nominal wage rigidity, Greek Ecorronric Revietiv, vol. 10,no. I, lune 1988, pp. I- 48.

No. 32 E. van Damme, Signaling and forward induction in a market entry context,Uperations Researclr Proceedings l989, Berlin-Heidelberg: Springer-Verlag, 1990,

PP. 45 - 59.

No. 33 A.P. Barten, Toward a levels version of the Rotterdam and relaled demand systems,Contriburions ro Operations Research and Economics, Cambridge: MIT Press, 1989,pp. 441 - 465.

No. 34 F. van der Plceg, Intemational coordinalion of monetary policies under eltemativeexcliange-rale regimes, in F. van der Ploeg (ed.), Advanced Leclures in QuanlifativeEconornics, London-Orlando: Academic Press Ltd., 1990, pp. 91 - 121.

No. 35 Th. van de Klundert, On socioeconomic causes of 'wait unemployment', EuropearrEcononric Review, vol. 34, no. 5, 1990, pp. 1011 - 1022.

No. 36 R.1.M. Alessie, A. Kapleyn, 1.B. van Lochem and T.J. Wansbcek, Individual effectsin utility consistent models of demand, in J. Hartog, G. Ridder and J. Theeuwes(eds.). Pane! Dula artd Labor Market S(udies, Amsterdam: Elsevier SciencePublishers B.V. (North-Ilolland), 1990, pp. 2S3 - 278.

No.37 F. van der Plocg, Capital accumulation, inflation and long-mn contlict inintemational objectives, Uxjord Economic Papers, vol. 42, no. 3, 1990, pp. 501 -525.

No. 38 Th. Nijman and F. Palm, Parameter identification in ARMA Processes in the

presence of regular but incomplete sampling, Journa! oj Time Series Arralysis, vol.

I I, no. 3, 1990, pp. 239 - 248.

No. 39 Th. van de Klundert, Wage difierentials and employment in a two-sector model witha dual labour market, Metroeconnmica, vol. 40, no. 3, 1989, pp. 235 - 256.

No.40 Th. Nijman and M.F.J. Steel, Exclusion restrictions in instrumental varieblesequations, Econometric Reviews, vol. 9, no. 1, 1990, pp. 37 - SS.

No. 41 A. van Scest, 1. Woittiez and A. Kapteyn, Labor supply, income taxes, and hoursrestrictions in the Netherlands, Journa! ojHuman Resources, vol. 25, no. 3, 1990,pp. 517 - 558.

No. 42 Th.C.M.J. van de Klundert and A.B.T.M. van Schaik, Unemployment persistenccand loss of productive capacity: a Keynesian approach, Journal of Macro-economics, vol. 12, no. 3, 1990, pp. 363 - 380.

No. 43 Tlt. Nijman and M. Verbeek, Estimation of time-dependent parameters in lineazmodels using cross-sections, panels, or both, Journal ojF.conometrics, vol. 46, no.3, 1990, pp. 333 - 346.

No. 44 E. van Damme, R. Selten and E. Winter, Alternating bid bazgaining with a smallestmoney unit, Games und Economic Behavior, vol. 2, no. 2, 1990, pp. 188 - 201.

No. 4S C. Dang, The D,-triangulation of R' for simplicial algorithms for computingsolutions of nonlinear equalions, Malhenturics ojOperafions Research, vol. Ití, no.l, 1991, PP. 148 - 161.

No. 46 Th. Nijman and F. Palm, Predictive accuracy gain from disaggregate sampling inARIMA models, Journal ojBusiness 8r Bconomic Slafislics, vol. 8, no. 4, 1990, pp.405 - 41 S.

No. 47 J.R. Magnus, On certain moments relating to ratios of quadratic forms in normalvariables: further results, Sankhya: The Indian Joarnalujsrarisrics, vol. S2, seriesB, part. I, 1990, PP. I- 13.

No. 48 M.F.J. Steel, A Bayesian analysis of simultaneous equation models by combiningrecursive analytical and numerical approacltes, Journa! ojEcononrerrics, vol. 48, no.112, 1991, PP. 83 - I 17.

No. 49 F. van der Plocg and C. Withagen, Pollution control and lhe ramsey problem,Environmentul and Resource Economics, vol. 1, no. 2, 1991, pp. 21S - 236.

No. 50 F. van der I'locg, Money and capital irt interdependent economics with overlappinggenerations, Economica, vol. 58, no. 230, 1991, pp. 233 - 256.

No. 51 A. Kapteyn and A. de 7eeuw, Changing incentives for econornic research in theNetherlands, Europeun Econonoic Review, vol. 3S, no. 213, 1991, pp. 603 - 61 I.

No. 52 C.G. de Vries, On Ihe relation between GARCH and stable processcs, Journul ojEcwromerrics, vol. 48, no. 3, 1991, pp. 313 - 324.

No. S3 R. Alessie and A. Kaptcyn, Habit fomration, interdependen[ preferences anddemographic effects in the almost ideal demand system, The I:conumic Journal, vol.101, no. 406, 1991, pp. 404 - 419.

No. 54 W. van Groenendaal and A. de Zeeuw, Control, coordination and con(lict on

international commodity markets, Economic Mndefling, vol. 8, no. I, 1991, pp. 90

- 101.

No. 55 F. van der Ploeg and A.1. Markink, Dynarnic policy in linear modcls with rational

expectations of future events: A computer package, Computer Se'ience in Fconomics

and Management, vol. 4, no. 3, 1991, pp. I75 - 199.

No. 56 H.A. Keuunkamp and F. van dcr Ploeg, Savings, investmenl, govemment finance,

and the currcnt account: The Dutch experience, in G. Alogoskoufis, L. Papademos

and R. Portes (eds.), Exrernul Cbn.clraints an Macroecanonric inlicv: 77m European

Experience, Cambridge: Cambridge University Press, 1991, pp. 219 - 263.

No. 57 Th. Nijman, M. Verbeek and A. van Scest, The e[rciency of rotating-panel designs

in an analysis-of-variance model, Journol af Economelrics, vol. 49, no. 3, 1991, pp.

373 - 399.

No. 58 M.F.J. Steel and 1.-F. Richard, Bayesian multivariate exogeneity analysis - an

application to a UK mnney demand equation, Journa!afEcarometrics, vol. 49, no.

I12, 199I, pp. 239 - 274.

No. 59 Th. Nijman and F. Palm, Generalized least squares estimation of lincar models

conlaining rational future expectations, lnternational Econamic Review, vol. 32, no.

2, 1991, pP. 383 - 389.

No. 60 E. van Damme, Equilibrium selection in 2 x 2 games, Revrsta Gspanofa deEconomia, vol. 8, no. I, 1991, pp. 37 - 52.

No. 61 E. Bennett and E. van Damme, Demand commitment bargaining: the case of apex

games, in R. Selten ( ed.), Gome Equilibrium Models 1fI - Strategic Bargaining,Berlin: Springer-Verlag, 1991, pp. 118 - 140.

No. 62 W. Guth and E. van Damme, Gorby games - a game theoretic analysis of

disarmament campaigns and the defense efficiency - hypothesis -, in R. Avenhaus,

H. Karkar and M. Rudnianski (eds.), Dejense Decisian Making - Analytical Support

and Crisis Management, Berlin: Springer-Verlag, 1991, pp. 215 - 240.

No. 63 A. Rcell, Dual-capacity trading and the quality of the market, Jotrrna! afFirrancial

Intermedintiort, vol. 1, no. 2, 1990, PP. 105 - 124.

No. 64 Y. Dai, G. van der Laan, A.1.J. Talman and Y. Yamamoto, A simplicial algorillrmfor Ihe nonlinear stationary point problem on an unbounded polyltedron, Sionr

Journal of Optimization, vol. 1, no. 2, 1991, pp. I51 - 165.

No.65 M. McAleer and C.R. McKenzie, Keynesian and new classical models of

unemployment revisited, The Economic Journal, vol. 101, no. 406, 1991, pp. 359

- 381.

No. 66 A.1.1. Talman, Generel equilibrium programming, Nieuw Archief voor wiskunde,

vol. 8, no. 3, 1990, pp. 387 - 397.

No. 67 1.R. Magnus and B. Pesaran, The bias of forecasts from a first-order autoregression,

Econometric Theary, vol. 7, no. 2, 1991, pp. 222 - 235.

No. 68 F. van der Plceg, Macrceconomic policy coordination issues during the variousphases of economic and monelary integration in Europe, European Economy - TheEcononrics ojEMU, Commission of the European Communilies, special edition no.I, 1991, PP. 136 - 164.

No. 69 H. Keuzenkamp, A precursor to Muth: Tinbergen's 1932 model of rationalexpectations, Tlie Economic Journal, vol. 101, no. 408, 1991, pp. 1245 - 1253.

No. 70 L. Zou, The target-incentive system vs. the price-incentive system under adverseselection and the ratchet effect, Journal ojPublic Economics, vol. 46, no. 1, 1991,

PP. 51 - 89.

No. 71 E. Bomho(T Between price reform and privatiration: Eastem Europe in transition,Finanzmarkl und Portjolio Management, vol. 5, no. 3, 1991, pp. 241 - 251.

No. 72 E. Bomhoff, Stability of velocity in the major industrial countries: a Kalman filterapproaeh, lnternational Monelary Fund Sla,,(f Papers, vol. 38, no. 3, 1991, pp. 626- G42.

No. 73 E. Bomhoff, Currency convertibility: when and how7 A contribulion to theBulgarian debate, Kredit und Kapilal, vol. 24, no. 3, 1991, pp. 412 - 431.

No.74 H. Keuunkamp and F. van der Ploeg, Puceived constraints for Dutchunemployment policy, in C. de Neubourg (ed.~ The Art oJ Full Employmenl -Unemploymen! Policy in Open Economies, Contributions to Economic Analysis 203,Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1991, pp. 7- 37.

No. 75 H. Peters and E. van Damme, Characterizing the Nash and RaifT'a bargainingsolutions by disagreement point axions, Mathematics ojOpera(ions Research, vol.ltí, no. 3, 1991, pp. 447 - 461.

No. 76 P.J. Deschamps, On the eslimated variances of regression coc(Ticients inmisspecified etror components models, Econometric Theory, vol. 7, no. 3, 1991, pp.369 - 384.

No. 77 A. de Zecuw, Note on 'Nash and Slackelberg solutions in a diffcrential game modelof capitalism', JournalojEconomic Dynamics and Conlrol, vol. 16, no. I, 1992, pp.139 - 145.

No. 78 1.R. Magnus, On the fundamental bordered matrix of linear estimation, in F. van derPloeg (ed.), Advnnced Lectures in Quantitarive Economics, London-Orlando:Academic Press Ltd., 1990, pp. 583 - 604.

No. 79 F. van der Ploeg and A. de Zeeuw, A differential gatne of intemational pollutioncontrol, Systems and Conlrol Leners, vol. 17, no. 6, 1991, pp. 409 - 414.

No. 80 l~h. Nijman and M. Verbeek, The optimal choice of controls and pre-experimen- talobservations, Journa! ojEconontetrics, vol. 51, no. I12, 1992, pp. 183 - 189.

No. 81 M. Verbeek and Th. Nijman, Can coliort data be treated as genuine panel data?,Empirical Economics, vol. 17, no. I, 1992, pp. 9- 23.

No. 82 E. van Uamme and W. Guth, Equilibrium seleclion in the Spence signaling game,in R. Selten ( ed.), Gonre Equilihrium Mndcls II - MeUrods, Mnrals, and Mnrkers,Bcrlin: Springcr-Vcrlag, 1991, pp. 263 - 288.

No. 83 R.P. Gilles and P.ILM. Ruys, Characteriration of economic agents in arbitrarycommwricalion structures, Nieuw Archief voor Wi.ckunde, vol. 8, no. 3, 1990, pp.325 - 345.

No. 84 A. de Zeeuw and F. van der Ploeg, Difference games and policy evalaation: aconceptual framework, Osjord Economic Papers, vol. 43, no. 4, 1991, pp. 612 -636.

No. 85 E. van Damme, Fair division under asymmetric inforrnation, in R. Selten (ed.),Ratianal Inreractinn - Fcsayc br llonor of Jolur C. Harsanyi, Bcrlin~Hcidclberg:Springcr-Vcrlag, 1992, pp. 121 - 144.

No. 86 F. de Jong, A. Kemna and T. Kloek, A contribution to cvent study metliodologywith an application to the Dutch stock market, Journa! ofUanking and Finance, vol.16, no. I, 1992, pp. I I- 36,

No. 87 A.P. Barlen, The estimation of mixed demand systems, in R. Bewley and T. VanHoa (eds.), Conlributions !o Consumer Demand and Econometrics, Essays inHonour ojHenri Theil, Basingsloke: The Macmillan Press Ltd., 1992, pp. 31 - 57.

No. 88 T. Wansbeek and A. Kapteyn, Simple estimators for dynamic panel data modelswith errors in variables, in R. Bewley and T. Van Hoa (eds.), Contribution.c loCorrsumer Demand and Eeonometrics, Essays in Honour of Ifenri Theil,Basingstoke: Ttte Macmillan Press Ltd., 1992, pp. 238 - 251.

No. 89 S. Chib, J. Osiewalski and M. Sleel, Posterior inference on the degrees of freedomparameter in multivaríate-1 regression models, Econonrics Lelters, vol. 37, no. 4,1991, PP. 391 - 397.

No. 90 N. Peters and P. Wakker, Independence of irrelevant altematives and revealed grouppreferences, Econometrica, vol. 59, no. 6, 1991, pp. 1787 - I801.

No. 91 G. Alogoskoufis and F. van der Ploeg, On budgetary policies, growth, and extemaldeficits in an interdependent world, Journal oj Ihe Japanese and ~nternationalEconomies, vol. 5, no. 4, 1991, pp. 305 - 324.

No. 92 R.P. Gilles, G. Owen and R. van den Brink, Games with pertnission structures: Theconjunctive approach, Internalional Jaurna! of Game Theory, vol. 20, no. 3, 1992,Pp. 277 - 293.

No. 93 I.A.M. Potters, I.1. Curiel and S.H. Tijs, Traveling salesman games, MathematicalProgramming, vol. 53, no. 2, 1992, pp. 199 - 21 I.

No. 94 A.P. Jurg, M.J.M. Jansen, J.A.M. Potters and S.H. Tijs, A symmetri~tion for finitetwo-person games, Zeitschrift feir Olxrations Research - Melhods and Models oJOperations Research, vol. 36, no. 2, 1992, pp. 1 I I- 123.

No. 95 A. van den Nouweland, P. Borm and S. Tijs, Allocation rules for hypergraphcommunication siluations, Infernaliona! Jowna! oJ Game Theory, vol. 20, no. 3,

1992, PP. 255 - 268.

No. 96 E.J. Bomhoff, Monetary refortn in Eastern Europe, European Economic Review, vol.

36, no. 213, 1992, pp. 454 - 458.

No. 97 F. van der Ploeg and A. de Zeeuw, International aspects of pollution conlrol,Environnrenta! and Resowce Eeonomics, vol. 2, no. 2, 1992, pp. 117 - 139.

No. 98 P.E.M. Borm and S.H. Tijs, Strategic claim games corresponding to an NTU-game,

Gornes and Economie Behavior, vol. 4, no. I, 1992, pp. 58 - 71.

No. 99 A. van Scest and P. Koorcman, Cohercncy of the indirect translog demand system

with binding nonnegativity constraints, Joramal oj Econometrics, vol. 44, no. 3,

1990, pp. 391 - 400.

No. 100 Th. ten Raa and E.N. Wol(Í, Secondary products and the measurematt of

productivity growlh, Regiona! Science ond Uróan Economics, vol. 21, no. 4, 1991,pp. 581 - 615.

No. 101 P. Kooreman and A. Kapteyn, On the empirical implementation of some game

theoretic models of household labor supply, The Jorvnal oJHuman Resources, vol.

25, no. 4, 1990, pp. 584 - 598.

No. 102 H. Bester, Bertrartd equilibrium in a differentiated duopoly, Inlernational Economic

Review, vol. 33, no. 2, 1992, pp. 433 - 448.

No. 103 J.A.M. Potters and S.H. Tijs, The nucleolus of a matrix game and olher nucleoli,Mathemotics of Operations Research, vol. 17, no. I, 1992, pp. 164 - 174.

No. 104 A. Kapteyn, P. Kooreman and A. van Scest, Quantity rationing and concavity in aflexible household labor supply model, Review oJEconomics and Starisrics, vol. 72,

no. I, 1990, pp. 55 - 62.

No. 105 A. Kapteyn and P. Kooreman, Household labor supply: What kind uf data can tellus how many decision makers there areT, European Economic Review, vol. 36, no.213, 1992, pp. 365 - 371.

No. 106 l~h. van de Klundert and S. Smulders, Reconstructing growth theory: A survey, De

Econorni.sr, vol. 140, no. 2, 1992, pp. 177 - 203.

No. 107 N. Rankin, Impcrfect competition, expectations and the multiplc effects of monetarygrowth, The Econornic Joarnal, vol. 102, no. 413, 1992, pp. 743 - 753.

No. 108 J. Greenberg, On the sensitivity of von Neumann and Morgenstem abstract slable

sets: The stable and the individual slable bargaining set, Inlernaliona! Journa! uJ

Garrte Thenry, vol. 21, no. 1, 1992, pp. 41 - 55.

No. 109 S. van Wijnbergen, Trade reform, policy unceriainty, and the currcnt account: A

non-expected-utility approach, American Econ~mic Review, vol. 82, no. 3, 1992, pp.

626 - 633.

No. I10 M. Verbeek and Tli. Nijman, Tesling for selectivity bias in panel data models,Interrra(ional Econumic Review, vol. 33, no. 3, 1992, pp. 681 - 703.

Nn. I I I Th. Nijman and M. Vcrbcck, Nonresponse in pancl data: The impact on estimatesof a life cycle consumption function, Jaurnal nj~pplied Ecnrronretrics, vol. 7, no.3, 1992, pp. 243 - 257.

No. 1 12 I. Bomze and E. van Damme, A dynamical characterization of evolutionarily stableslates, AnnaLc aJOperarion.r Researclr, vol. 37, 1992, pp. 229 - 244.

No. I 13 P.J. Deschamps, Expectations and intertemporal separability in an empirical modelof consumption and investment under uncertainty, Ernpirica! Economic,c, vol. 17, no.3, 1992, pp. 419 - 450.

No. I14 K. Karniya and D. lalman, Simplicial algorithm for computing a core element ina balanced game, Journalnj(he Operarions Research, vol. 34, no. 2, 1991, pp. 222- 228.

No. I I S G.W. Imbens, An efficienl melhod of moments estimator for discrete choice modelswith choice-based sampling, Economelrica, vol. 60, no. 5, 1992, pp. I187 -1214.

No. I 16 P. Borm, On perfectncss concepts for bimatrix games, OR Spekrrum, vol. 14, no.I, 1992, PP. 33 - 42.

No. I 17 A.P. Jurg, f. Garcia lurado and P.E.M. Borm, On modifications of the concepts ofperfect and proper equilibria, OR Spektrum, vol. 14, no. 2, 1992, pp. 85 - 90.

No. I I8 P. Borm, H. Keiding, R.P. McLean, S. Oortwijn and S. Tijs, The compromise valuefor NTU-games, Internariona! Journa! ajGame Theory, vol. 21, no. 2, 1992, pp.175 - 189.

No. I 19 M. Maschler, J.A.M. Potters and S.li. Tijs, The general nucleolus and the reducedgame propetty, lnlerna(ional Jonrrtalof Game Theory, vol. 21, no. I, 1992, pp. 85 -

106.

No. 120 K. Wárneryd, Communication, correlation and symmetry in bargaining, EcarronricsLerrers, vol. 39, no. 3, 1992, pp. 295 - 300.

No. 121 M.R. Baye, D. Kovenock and C.G. de Vries, lt takes two to tango: equilibria in amodel of sales, Games and Economic Behavior, vol. 4, no. 4, 1992, pp. 493 - 510.

No. 122 M. Verbeek, Pseudo panel data, in L. Mátyás and P. Sevestre (eds.), TheEconometrics ojPane! Dala, Dordrecht: Kluwer Academic Publishers, 1992, pp.303 - 315.

No. 123 S. van Wijnbergen, Intertemporal speculation, shortages and the political economyof price reform, The Economic Journal, vol. 102, no. 415, 1992, pp. 1395 - 1406.

No. 124 M. Verbeek and Th. Nijmsn, Incomplete panels and selection bias, in L. Mátyás andP. Sevestre ( eds.), The Econometrics ojPane! Data, Dordrecht: Kluwer AcademicPublishers, 1992, pp. 262 - 302.

No. 125 J.J. Sijben, Monetary policy in a game-theoretic framework, .Iahrbucher firrNationaliikonomie und Starislik, vol. 210, no. 314, 1992, pp. 233 - 253.

No. 126 H.A.A. Verhon and M.J.M. Verhceven, Decision making on pension schemes underrational expectations, Journd oJEconomics, vol. 56, no. I, 1992, pp. 71 - 97.

No. 127 L. Zou, Ownership structurc and e(Ticiency: An incenlive mechanism approach,Journal oJCornparalive Economics, vol. 16, no. 3, 1992, pp. 399 - 431.

No. 128 C. Fershtman and A. de Zceuw, Capital accumulation and entry deten-ence: Aclarifying note, in G. Feichtinger (ed.), !)ymanric Economic Models and OptinralCnntrol, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1992, pp.281 - 296.

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