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8414 mic Research a er1991 ~~

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No. 9141

CAPITAL ACCUMUTATION AND EN1RY DEI'ERRENCE:A CLARIFYING NOTE

by Chaím Fershtman ' ~~. 3Zend Aart de Zeeuw `~ ,X j~j

August 1991

~SSN o924-~815

CAPITAL ACCUMULATION AND ENTRY DETERRENCE: A CLARIFYIN(i NOTB~

Chaim FershtmanTel-Aviv University, Israel

Aart de ZeeuwTilburg Univeraity and Free University Amsterdam, The Netherlands

Abstract

This note clarifies aome of the entry deterrence aspects of capitalaccumulation. Since accumulating capital takes time the focus of this noteis on the importance of time in the analysis of entry deterrence. While thepost-entry game is modelled as a capital accumulation differential game Forwhich we solve for the feedback equilibrium, we also add a time dimension tothe pre-entry game assuming that the entry decision ia subject to entry pre-paration that also takes time. This preparation period affects the analysisof entry deterrence and the possibility and the attractiveness of entrydeterrence.

Department of EconomicsTilburg UniversityP.o. sox 901535000 LE TilburgThe Netherlands

July 1991

- The first version of this paper was written while the first author visitedthe CentER of Economic Research at Tilburg University.We would like to thank the CentER for their hospitality.

z

1. Introduction

In his seminal work on the supply side of new markets Spence (1977,1979) analyses the strategic interaction of early entrants in their gamewith potential entrants. The result is in spirit similar to the von Stackel-berg oligopoly model but the strategic asymmetries are now induced by thehistory 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 theterminology of Schelling (1960) these investments are an advence commitmentor credible threat. Commitmenta to future investments are ruled out, becausethese commitments are not credible and because otherwise the fundamentalasymmetry between established firms and potential entrents disappears. Dixit(1980) argues that this also means that an excess capacity strategy cannotbe sustained. After entry a Cournot~Nash game will be played between all thefirms in the market. Dixit stresses in his paper that it is important todistinguish a pre-entry stage and a post-entry stage. Although the rules ofthe post-entry game are exogenous (such as a Cournot~Nash oligopoly), theestablished firm can influence the outcome of that game to its advantage bybuilding favourable initíal conditions in the pre-entry stage. Fudenberg andTirole (1983) analyse the post-entry game in continuous time with an infi-nite horizon. They decive the set of subgame perfect equilibria for invest-ment strategies which are only a function of the state (that is the currentcapital stocks). By first allowing the firms to coordinate their strategiesand by then invoking some ad hoc argument they can reduce the set of perfectequilibria to one. This is necessary to be able to discuss entry deterrence.Reynolds (1987) derives the subgame-perfect (or feedback Nash) equilibriumfor the game in which investments are reversible but capacity is subject toadjustment coats. This equilibrium ia unique. Reynolds compares it with theopen-loop Nash equilibrium in which the investment strategies are only afunction of time.

In this note the post-entry game is modelled as the capital accumula-tion geme in Reynolds' paper. However, since investment commitmenta in ad-vance of actual investments are not credible, the feedback Nash or subgame-perfect Markov equilibrium ia in our view the only reasonable solution

3

concept. In Dixit's terminology the rules of the post-entry game lead to thefeedback Nash equilibrium. The values of this game are a function of theinitiel capacity levels. Suppose there is one incumbent and one potentisl

entrant. At the start of the post-entry game the capacity level of theentrant ia zero and the capacity level of the incumbent dependa on its

investments before entry. The novelty in this note ia to model explicitly atime-lag between the potential entrant's preliminary decision to enter andthe actual entry to the market. The incumbent can use thia period to invest

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

position for the post-entry game. When the investment costs are convex, thelength of the time-lag pla,ys 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 incumbentfirm the possibility to realize monopoly profits in the pre-entry stage,

even if it is not a natural monopoly but an artificial monopoly in theterminology of Eaton end Lipsey (1981). It is therefore better to apeak inthis case of a natural monopoly end 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-entry

capital accumulation differential game is derived. Section 4 analyses thepre-entry decision problem of the incumbent firm. In section 5 a stretegicnatural monopoly is introduced es a consequence of small but non-zero pre-paration costs and section 6 concludes the paper.

2. The framework

Consider an industry in which there is one incumbent firm and one po-tential entrant. It is assumed that entry cannot be decided upon and carriedout instantaneously. The lag between the entry decision and the actual entryis denoted by the preparation time te ) 0. Moreover, the entry decision isnot a commitment. A firm can reconsider this decision at the last momentbefore the preparatíon time has elapsed. Considering entry is not necessa-rily costless and we denote this cost as fe x 0. Clearly, if fe - 0, the po-tential entrant will ennounce its intention right awey end leave the actual

4

entry decision to time te or later. We atart our analysis by considering thecase of zero preparation costs. In section 5 we consider the case in whichthe potential entrant has to bear some preparation costs.

Both firms in our model accumulate some form of capital according tothe standard capital accumulation dynamics

Ki(t) s Ii(t) - bKi(t), Ki(0) s KiO, i~ 1,2, (1)

where Ki denotea the capital level, Ii the inveatment level and b is acommon depreciation factor. The entrant can atart to accumulate capital onlyafter it enters. There is no capital accumulationperiod. Investment is costly end we let Ci(Ii) be theia a convex and increasing function.

We assume that the instantaneous profits at timea function of the state variablea [K1(t),K2(t)]. Thiafirms do not compete through pricea or quantities but

during the preparationcost of ínvestment. Ci

t cen be expreased asis not to say that thethat a reduced form

can be used. It is assumed that the profit function R1(K1(t),K2(t)) is anincreasing and concave function of Ki end a decreasing function of K~, i-1,2, j f i. The objective of each firm is to maximize its discounted streamof profits net of investment costs

J {Ri(Ki(t).Kf(t)) - Ci(Ii(t))) e-rt dt,0

(2)

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

Entry is associated with some fixed (sunk) cost of entry, denoted byFe. That is fe is paid at the beginning of the preparation period while Feis paid et the time of the entry itself. Clearly, the potential entrent willenter the market, if, at the moment of entry, the discounted atream of pro-fits net of inveatment costs at least covers the entry fee Fe. The post-entry game is described by (1)-(2), starting at time te. This can be calleda capital accumulation differential game (see, e.g., Fershtman and Muller(1984) and Reynolds (198~)). Suppose that the incumbent firm is denoted es

5

firm 1 and the entrant firm as firm 2. On the assumption that prior to entrythe entrant dces not accumulate capital, the initial condition of the capi-tal accumulation game is [K1(te),O]. If the value functions V1 of the post-entry game exíst, the potential entrent will choose to enter at time te onlyif

VZ(K1(t).0) - Fe ) 0. (3)

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

Kd(Fe) :- inf {K1~V2(K1,0) - Fe 5 0} (4)

entry is blocked. Therefore we define Kd(Fa) as the capital deterrence le-vel.

An incumbent firm may ignore the possibility of entry. This firm maximizes(2) subject to (1) with KZ always equal to 0. Ne denote the optimal capitalaccumulation path of this firm as Km(.).

Definition 1: M incumbent fírm will be called a natural monopoly, if

Km(t) 2 Kd(Fe) for every t 2 te. (5)

Note that the position of a firm as a natural monopoly depends on both theentry cost FQ and the length of the preparation period te.

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 hes to reachthe deterrence level. The shorter the preparation time te is the more costlyit will be to reach Kd(Fe). Moreover, the incumbent's decision problem isnot just whether to deter entry or to accommodate entry. Even if this firmdecides to accommodate, it is of importance which capital level it reachesby time te as this level affects its profits in the post-entry game.

6

3. The post-entry game: capital accumulation game

As was already stated in section 2, after entry the two firms are enga-ged in a capital accumulation differential geme. Because in the general casesuch a game is not analytically tractable, a linear-quadratic structure isadopted here. The cost functions are given by

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

and the profit functions are given by

ITi(Ki.K~) ' Ki(a - Ki - K~). i.j ' 1.2, i~ j.

It follows that the firms try to maximize

(6)

(7)

J{Ki(t)(a - Ki(t) - Kf(t)) -~ cIi(t)} e'rt dt i,~ - 1.2. i~~. (8)0

subject to (1), where the initiel time is taken to be 0 in order to simplifynotation, although the actuel initial time is te.

The capital accumulation differential game (8)-(1) ia identical to the onein Reynolds (1987). Furthermore, the game is in structure very similar tothe dynamic duopoly with sticky prices (Fershtman end Kamien, 198~) and to amodel of competitive arms accumulation (van der Ploeg and de Zeeuw, 1990).Because it seems reasonable to assume that the firms can condition theirinvestments on the current capital levels and that the firms can not committhemselves to future investments, the feedback Nash (Starr and Ho, 1969) orsubgame-perfect Markov equilibrium has to be derived. The outcome can befound in Reynolds (1987), but the appendix of this paper presents thisoutcome and the derivation in a much more transparent way.

The capital accumulation game can therefore be summarized by a value func-tion Vi(K1,K2), which is continuous in its arguments with ~Vi~~l{i ~ 0 and

~Vi~dlCj ( 0, i~ j, and which gives the value for player i of the capitalaccumulation game that starts with the initial condition (K1,K2).

4. The incumbent decision problem

At date t- 0 the incumbent has to decide whether to follow a capitalaccumulation path that prevents entry or a path that accommodatea entry. Theincumbent has to compare its profits given by Lhe outcome of problem 1, whenentry is accommodated, and its profits given by the outcome of problem 2,when entry is deterred.

Problem 1: Accommodating Entry

temaximize f (K1(t)(a - K1(t)) -} cIi(t)} e rt dt t V1(K1(te),O) e-rte,(9)I1(.) 0

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

The trade-off for the incumbent in problem 1 is to realize monopoly profitsin the pre-entry stage, on the one hand, and to reach a favourable initialposition in the post-entry stage, on the other hand.

Problem 2: Deterring Entry

maximize f{K1(t)(a - K1(t)) -} cIi(t)} e-rt dt, (10)I1(') JO

subject to (1), K1(0) - K10 and K1(t) 2 Kd(FQ) 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 ia sometimescalled an artificisl monopoly.

8

Let Va(K10,te,Kd(Fe)) be the value of the control problem for the

incumbent when entry is accommodated (problem 1) and Vd(K10,te,Kd(Fe)) be

the 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))

Although we choose not to provide specific solutions to problems 1 and 2, we

argue that both Va(.) and Vd(.) increase with te. There are two reasons for

this. Firstly, a higher te implies that the incumbent can enjoy a longer

monopolistic period. Secondly, the incumbent now has more time either to

achieve a favourable starting position for the post-entry game in case itchooses to accommodate entry, or to achieve the capital deterrence level Kd.

Since the accumulation costs are convex, reaching these levels is less cost-

ly when there is more time. Clearly, the effects of changing te in Va(.) end

Vd(.) are not identical. Therefore, it is possible that for a given pair

(t ,Kd(Fo)) the incumbent will chooae to deter entry WhorcnH for a shorter

preparation period, i.e. te C te, the optimal strategy is to accommodate

entry. This happens when the extra investment costs to reach the capital

deterrence level Kd(Fe) in e shorter period te outweigh the losses that are

suffered in problem 1 due to s ahorter preparation period.

te

A

FeFigure 1

We can conclude that the length of the preparation period is an importantfactor in determining whether the incumbent's optimal strategy is to deter

entry or not. In particular we expect that the set of posaible pairs (te,Fe)will be divided as in Figure 1, such that for (te,Fe) E A entry is deterred

whereas for (te,Fe) ([ A entry is accommodated.

9

Intuitively, for a high Fe the level of deterrence capital Kd is lowand thus the incumbent only needs a short preparation period to accumulatethis level. Reducing Fe leads to a higher Kd, which implies that the incum-bent will only accumulate this level if the preparation period is suffi-ciently long. The area A covers both the situationa that correspond to anatural monopoly and to an artificial monopoly.

Likewise, whether the incumbent firm is a natural monopoly or not alsodepends on the length of the preparation period te. It can very well be thatKm(t) x Kd(Fe), t 2 te, which implies that the incumbent firm is a naturalmonopoly for the preparation period te, whereas Km(te) t Kd(Fe) holds for ashorter preparation period te C te, which implies that in that case theincumbent firm is not a natural monopoly.

5. The strategic natural monopoly

The standard definition of natural monopoly describes the case where afirm, by acting as a monopolist that maximizes profits, while ignoring thepossibility of entry, in fact deters entry. That is the profit maximizingcapital accumulation path Km(.) lies on or above the capital deterrencelevel Kd(Fe) from time te onwards, so that entry is automatically deterred.

Consider now the case in which the preparation costs fe - E ) 0, andsuppose that the incumbent firm is not a natural monopolist. The question iswhether 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 capitaldeterrence level Kd(Fe) by Lime te, so that entry ia deterred. Given such areaction the optimal strategy of the potentisl entrant is not to start thepreparations for entry at all end 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 ia not to saythat the incumbent firm ignores the possibility of entry. Since there ís apreparation period te, the incumbent realizea that if necessary once entry

10

will 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 entry 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 coststhe incumbent firm's capital accumulation path follows the path Km(.), evenif the íncumbent firm is not a natural monopoly. We denote such a market asa strateQic natural monovoly (SNM). The part of the area A in Figure 1, thatcorresponds to an artificisl monopoly in the case of zero preparation costs,now becomes the area of a strategíc natural monopoly with monopoly profítsfor the incumbent firm at all times. Clearly, the position of a firm as aSNM depends again on both the entry cost Fe and the length of the prepara-tion 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 assumptionof convex investment costs the length of the pre-entry period proves to bean important factor in the analysis of entry deterrence. Furthermore, it isshown that in the case of non-zero preparation costs the artificial monopolycan in fact realize monopoly profits in the pre-entry stage because of itscredible threat to accumulate extra capitsl in order to deter entry end toburden the potential entrent with these preparation costs when it starta toprepare for entry.

These observations may lead to a more detailed analysis of the entrydeterrence 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.

11

References

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

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

Fershtman, C. and M.I. Kamien ( 198~), "Dynamic duopolistic competition withsticky prices", Econometrica 55. 5. 1151-1164.

Fershtman, C. and E. Muller (1984), "Capital accumulation games of infiniteduration", Journal of Economic Theorv 33. 322-339.

Fudenberg, D. and J. Tirole (1983), "Capital as a commitment: strategicinvestment to deter mobility", Journal of Economic Theorv 31, 227-250.

van der Plceg, F. and A.J. de Zeeuw (1990), "Perfect equilibrium in a modelof competitive arms accumulation", International Economíc Review 31, 1,131-146.

Reynolds, S.S. (1987), "Capacity investment, preemption and commitment in aninfinite horizon model", International Economic Review 28, 1, 69-88.

Schelling, T.C. (1960), The Strategy of Conflict, Harvard University Preas,Cambridge, Massachusetts.

Spence, A.M. (197~), "Entry, capacity, investment, and oligopoliatic pri-cing", The Bell Journal of Economics 8, 2, 534-544.

Spence, A.M. (19~9), "Investment strategy and growth in a new market", TheBell Journal of Economics 10, 1, 1-19. -

Starr, A.W. and Y.C. Ho (1969), "Further properties of nonzero-sum differen-tial games", Journal of Optimization Theory and Applications 3, 4, 20~-219.

12

Appendix

Consider the differential game (i - 1,2)

maximise r{-} cIi(t) t} x'(t)Qix(t) f qix(t)}e-rt dt (A.1)Ii(.) J~

subject to z(t) - Ax(t) t B1I1(t) t B2I2(t), x(0) - x~, (A.2)

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

A:-[-0 -S,~ Bl'-[0,; B2:-1~J ; Q1:-1-i -ÓJ : Q2:'I-~ -ZJ ; q1:-[p]; q2:-~áJ.

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

rVi(x) - max {-} cIi a} x'Qix 4 qix t Vix(x)(Ax 4 Blul 4 B2u2)}. (A.3)

The equilibrium strategies are given by

Ii(x) - (l~c)BiVix(x). (A.4)

The dynamic programming equations become (i,j - 1,2; i~ j)

rVi(x) - -} (l~c)Vix(x)BiBiVix(x) . } x'Qix . qix t

vixl.x)(Ax ~ (l~c)BiBivix(x) . (l~c)BjB~Vjx(x))}. (A.5?

With the quadratic value functions

V1(x) -} x' pl p3l x..... ; V2(x) ' } x' f p2 p3 l x. .... (A.6)3 2J L 3 1J

13

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

P1 t 2p3 -(2btr)cpl - 2c - o, (A.7)

2P1P3 ' P3P2 - Í 2b~r)cP3 - c - , (A.8)

P3 t 2p1P2 - (2b'r)cp2 - o. (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

Acl S~P1Ic p3~c'- - P3Ic -bfpl~c)' (a.1o)

The eigenvalues of Acl are -b i pl~c : p3~c, ao that the closed-loop aystemis stable if and only if

pl t p3 - bc ~ o and Pl - P3 - bc ~ o. (A.11)

Equation (A.7) describes an ellipse in the (pl,p3)-plane. Flirthermore, foreach value of p2 equation (A.8) describes a hyperbola i n the (p p)-planeand equation (A.9) a parabola. Definin 1~ 3g p:- J(2c .}(2b.r)2c2), the use ofpolar coordinates

P1 - } (2b'r)c , p sin 9; P3 - ~J2 p cos ~. -n ~ ~ s n. (A.12)

and the elimination of p2 leads to the equation

p2 cos3 ~o - 8p2 sin2 p cos p. 4J2 c sin p s 0.

Defining a :- J2 p2~c - 2,~2 .},~2 (2b,r)2c, (A.13) yields eventuslly

(A.13)

tan3 y~ - a tan2 ~ . tan ~ . (1~8)a - o. (A.14)

14

Consider the function f given by

f(Y) .' y3 - aY2 t Y f(1~8)a, a 2 2J2. (A.15)

Some straightforward calculus shows that the function f has one negativeroot y~ and two positive roots y2 and y3 with }J2 C y2 C y3. Consider y2 asa t'unction of a. It i s easy to see that y2(a) -},I2 for a - 2,~2. Implicitdifferentiation yields

yz(a) - Cyz(a) - (1~8)]If~(Y2(a)) C 0. (A.16)

It follows that the smallest positive root y2 of the function f satisfies},~2 C y2 C},~2. Since f(-},~2) C 0, the negative root yl of the function fsatisfies yl ) -},I2.The largest positive root y3 of the function f is given by

Y3(a) - 2,I{(a2~9) - (ll3)} cos (w~3) ~ (a~3).

where

(A.17)

v - arccos [(a3~27) - (11a~48)]ILJ{(a219) - (1~3))]3. 0 ~ y ( (nl2).

ClaimThe largest posítive root y3 of the function f satisfies the stabilityconstraints (A.il), but the negative root yl and the smallest positive rooty2 of the function f do not satisfy the stability constraints (A.11).

ProofWith the polar coordinates (A.12) the stability constraints (A.11) are givenby-n ~pCOand

~tan ~I )(}J2 z.} rc]~z with z ) 0 given by z2 a(},~2 z .} rc)2 - p2.

It follows that the stability constraints are given by

~tan ~~ ) L2a t rJl3J2 ac - r2c2)]~[2,I2 a- r2c]. (A.18)

15

Because

~tan p~ -[2a t rJ(3J2 ac - r2c2)]IL8 i 4bc(ó~r)] ~

2a~[8 t 4bc(b~r)l ~ ~J2.

and because yl )-}J2 and }J2 ( y2 ~ }J2, these roots of the function f donot satisfy the stability constrainta. FLrthermore,

Y3(a) ~ J{(a219) - (1~3)} ~ (a~3). (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 i s larger then this maximal value.It follows that the largest positive root y3 of the function f satisfies thestability constraints. Q.E.D.

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

P1 -~(2bir)c t J{L~J2 ac y3(a)]I[1 4 Y3(a)]}, (A.20)

P3 - J{[}J2 acll[1 ~ Y3(a)]}. (A.21)

where a- 2J2 .}J2 ( 2S.r)2c, and y3(a) i s given by (A.17).The parameter p2 of the value functions can then be calculated from (A.9),which yields

P2 '-} J{L}J2 ac]~LY3(a) t Y3(a)]}. (A.22)

The linear terms of the quadratic value functions

vl(x) - .... f [Py P5]x . .... . v2(x) ' .... 4 [P5 P4]x ; .... (A.6)

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

16

{P1 ~ P3 - (btr)c}p4 . p3p5 . a~ - p, (A.23)

(P2 . p3)p4 . {P1 - (bir)c}p5 - 0.

Finally, the equilibrium strategies become

li(x) -(l~c){P1K1 ` P3K~ i P4}. i.~ - 1.2. i~~- (A.4)

Discussion Paper Series, CentER, Tilburg University, The Netherlands:(For previous papers please

No. Author(s)

9025 K. Kamiya andD. Talman

y026 P. Skott

9027 c. Dang anaD. Talman

9028 J. Bai, A.J. Jakemanand M. McAleer

9029 Th. van de Klundert

903o Th. van de Klundertand R. Gradus

9031 A. Weber

9032 J. Osiewalski andM. Steel

9033 C. R. Wichers

9034 C. de Vrtes

9035 M.R. Baye,D.W. Jansen and Q. Li

9036 J. Driffill

9037 F. van der Ploeg

9038 A. Robson

9039 A. Robson

9040 M.R. Baye, G. Tianand J. Zhou

9o41 M. Burnovsky andI. Zang

consult previous discussion papers.)

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Integrsting Direct Metering and ConditionalDemand Analysis for Estimating End-Use Loads

Evaluating Pseudo-R2's for Binary ProbitModels

More on the Grouped HeteroskedasticityModel

9059 F. van der Ploeg Channels of International Policy Transmission

No. Author(s)

9060 H. Bester

9061 F. van der Plceg

9062 E. Bennett andE. van Damme

9063 S. Chib, J. Osiewalskiand M. Steel

9064 M. Verbeek andTh. Nijman

9065 F. van der Ploegand A. de Zeeuw

9066 F.C. Drost andTh. E. Nijman

9067 Y. Dai and D. Talman

9068 Th. Nijman andR. Beetsma

9069 F. van der Ploeg

9070 E. van Damme

9071 J. Eichberger,H. Haller and F. Milne

9072 G. Alogoskoufis andF. ven der Ploeg

9073 K.C. Fung

9101 A. van Soest

9102 A. Barten andM. McAleer

9103 A. Weber

9104 G. Alogoskoufis andF. van der Ploeg

9105 R.M.W.J. Beetsma

Title

The Role of Collateral in a Model of DebtRenegotiationMacrceconomic Policy Coordination during theVarious Phases of Economic and MonetaryIntegration in Europe

Demand Commitment Bargaining: - The Case ofApex Games

Regression Models under Competing CovarianceMatrices: A Bayesian Perspective

Can Cohort Data Be Treated as Genuine PanelData?

International Aspects of Pollution Control

Temporal Aggregation of GARCH Processes

Linear Stationary Point Problems on UnboundedPolyhedra

Empirical Tests of a Simple Pricing Model forSugar Futures

Short-Sighted Politicians and Erosion ofGovernment Assets

Fair Division under Asymmetric Information

Naive Bayesian Learning in 2 x 2 MatrixGames

Endogenous Growth and Overlapping Cenerations

Strategic Industrial Policy for Cournot andBertrand Oligopoly: Management-LaborCooperation as a Possible Solution to theMarket Structure Dilemma

Minimum Wages, Earnings and Employment

Comparing the Empirical Performance ofAlternative Demand Systems

EMS Credibility

Debts, Deficits and Growth in InterdependentEconomies

Bands and Statistical Properties of EMSExchange Rates

No. Author(s)

91G6 C.N. Teulings

910~ E. van Demme

9108 E. van Damme

9109 G. Alogoskoufis andF. van der Plceg

9110 L. Samuelson

9111 F. van der Ploeg andTh. van de Klundert

y112 Th. Nijman, F. Palmand C. Wolff

9113 H. Bester

9114 R.P. Gilles, G. Owenand R. van den Brink

9115 F. van der Plceg

9116 N. Rankin

9117 E. Bomhofe

9118 E. BomhoFf

9119 J. Osiewalski andM. Steel

9120 S. Bhattacharya,J. Glazer andD. Sappington

y121 J.W. Friedman andL. Samuelson

9122 S. Chib, J. Osiewalskiand M. Steel

9123 Th. ven de Klundertand L. Meijdam

y124 S. Bhattacharya

9125 J. Thomas

Title

The Diverging Effects of the Business Cycleon the Expected Duration of Job Search

Refinements of Nash Equilibrium

Equilibrium Selection in 2 x 2 Games

Money and Growth Revisited

Dominated Strategies and Commom KnowledgePolitical Trade-off between Growth andGovernment Consumption

Premia in Forward Foreign Exchange asUnobserved Components

Bargaining vs. Price Competition in a Marketwith Quality Uncertainty

Games with Permission Structures: TheConjunctive Approach

Unanticipated Inflation and GovernmentFinance: The Case for an Independent CommonCentral Bank

Exchange Rate Risk and Imperfect CapitalMobility in an Optimising Model

Currency Convertibility: When and How? AContribution to the Bulgarian Debate!

Stability of Velocity in the G-~ Countries: AKalman Filter Approach

Bayesian Marginal Equivalence of EllipticalRegression Models

Licensing and the Sharing of Knowledge inResearch Joint Ventures

An Extension of the "Folk Theorem" withContinuous Reaction Functions

A Bayesian Note on Competing CorrelationStructures in the Dynamic Linear RegressionModel

Endogenous Growth and Income Distribution

Banking Theory: The Main Ideas

Non-Computable Rational ExpectationsEquilibria

No. Author(s)

9126 J. Thomasand T. Worrall

912~ T. Gao, A.J.J. Talmanand Z. Wang

9128 S. Altug andR.A. Miller

9129 H. Keuzenkamp endA.P. Barten

Title

Foreign Direct Investment and the Risk ofExpropriation

Modification of the Kojima-Nishino-ArimaAlgorithm and its Computational Complexity

Human Capital, Aggregate Shocks and PanelData Estimation

Aejection without Falsification - On theHistory of Testing the Homogeneity Conditionin the Theory of Consumer Demand

913o G. Maileth, L. Samuelson Extensive Form Reasoning in Normal Form Gamesand J. Swinkels

9131 K. Binmore andL. Samuelson

Evolutionary Stability in Repeated GamesPlayed by Finite Automata

9132 L. Samuelson andJ. Zhang

9133 J. Greenberg andS. Weber

9134 F. de Jong andF. van der Ploeg

9135 E. Bomhoff

9136 H. Bester andE. Petrakis

913~ L. Mirman, L. Samuelsonand E. Schlee

9138 c. Dang

9139 A. de Zeeuw

914o B. Lockwood

9141 C. Fershtman andA. de Zeeuw

Evolutionary Stability in Asymmetric Games

Stable Coalition Structures with Uni-dimensional Set of AlternativesSeigniorage, Taxes, Government Debt andthe EMS

Between Price Reform and Privatization -Eastern Europe ín Transition

The Incentives for Cost Reduction in aDifferentiated Industry

Strategic Information Manipulation inDuopolies

The DZ-Triangulation for ContinuousDeformation Algorithms to Compute Solutionsof Nonlinear Equations

Comment on "Nash and Stackelberg Solutions ina Differential Game Model of Capitalism"

Border Controls and Tax Competition in aCustoms Union

Capital Accumulation and Entry Deterrence: AClarifying Note

P(~ F3(~X 9(11.r,~ ~,M(1 I F TII RI IRr TNF NFTI~FRLAND;Bibliotheek K. U. Brabantbi~ i~~~~~~~ s W ~~~ ~i~

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