ISSN 0104-8910
OLIGOPOLlSTIC COMPETITION UNDER KNIGHTIAN UNCERTAINTY
Hugo Pedro Bof/
Sérgio Ribeiro da Costa Werlang
OLIGOPOLISTIC COMPETITION
UNDER KNIGHTIAN
UNCERTAINTY
June 1996
Hugo Pedro Boff*
Sergio Ribeiro da Costa Werlang**
ABSTRACT
This artic/e applies a theorem of Nash equilibrium under uncertainty (Dow & Werlang, 1994) to the c/assic Coumot model of oligopolistic competition. It shows, in particular, how one can map ali Coumot equilibrum (which inc/udes the monopoly and the null solutions) with only a function of uncertainty aversion coefficients of producers. The e.ffect of variations in these parameters over the equilibrium quantities are studied, also assuming exogenous increases in the number of matching firms in the game. The Cournot solutions under uncertainty are compared with the monopolistc one. It shows principally that there is an uncertainty aversion levei in the industry such that every aversion coejJicient beyond it induces firms to produce an aggregate output smaller than the monopoly output. At the end of the artic/e equilibrium solutions are specialized for Linear Demand and for Coumot duopoly. Equilibrium analysis in the symmetric case allows to identify the uncertainty aversion coefficient for the whole industry as a proportionaJ lack of information cost which would be conveyed by market price in the perfect competition case (Lerner Index).
• Economic Depanment Federal Univenity of PII'BDA (B1'IZi1). The author ICImowledgcs Phd scholarship grantcd by CAPESlPICD (National Educllion Ministry) .
•• EPGF1Getu1io Vargas Foundation -8ank ofBahia (BraziI).
1- INTRODUCTION
In the rational modeI of choice under Knightian I uncertainty as proposed
by SchmeidIer (1989) and Gilboa (1987), agents reckon reciprocalIy their production
alternative sets by subadditive probability functions, and maximize their expected utility,
using Choquet's integraI2. When the subadditive distributions (say, P) are convex (i.e.
exhibit uncertainty aversion) this modeI is equivalent to the maximin model (SchmeidIer,
1989 and SchmeidIer & Gilboa, 1989) where the relevant distribution is built up as the
infimum probability among every additive distribution beIonging to the core of P, for each
choice in the alternative set.
For every pairwise choice A, B in the alternative set, convexity of P
In
the prospect of the axiomatic theory of SchmeidIer and Gilboa, the differencec(P,A) = 1-P(A) - P(A 1:) is interpreted as an ''uncertainty aversion" coefficient
associated to the event A and more recently Dow & Werlang (1994) extended to this
framework the notion of Nash Equilibrium for a one-shot game with 2 agents, and
dem.onstrated a theorem which ensures the existence ofNash equilibrium under uncertainty
(NEU) for each pair (c) ,c2 ) ofuncertainty aversion ofplayers.
In this article (sec. 11), we state
first
the defmition and then the existencetheorem for NEU as it is presented in the Dow & Werlang paper. In the sequeI (sec. 111),
we apply the theorem to the cIassical Cournot modeI of non-cooperative oligopoIistic
competition game with N producers, where quantities (controI variables) are chosen over
compact and convex sets. We will show, in a general case, how equilibrium output
responds to exogenous variations in uncertainty aversion parameters. Also, such reactions are examined when the number N of competitors in the game exogenously increase.
Furthermore, at the end of the section we set up an important result for this paper: it is
I F.H.Knigbt (1921): Agcnts didn 't know lhe probability distribution functions oflhe control variables they face.
2 For aSOUDd cxposition on subadditivc probabilitics anel lhe applications ofthis modcl in tinancc, $CC SimonscD 4: WcrIang (1991) and
shown that there is an uncertainty aversion leveI for the industry beyond which agents are induced to adopt prudent behaviour in such a way that industry output wiIl be smaller than monopoly output. Applications are made for constant returns to scale, constant price-elasticity and linear market demand (sec. IV). For the later case, a mapping for NEU's is drawn as a function ofuncertainty aversion coefficients when N
=
2 (Cournot duopoly). In the last section (sec. V) the principal results are summarized and some economic insights immediately inferred from them are emphazised ..11 - NASH EQUILlBRIUM UNDER UNCERTAINTY (NEU)
Let r:(AI,A2 , ｕｉＧｾＩ＠ a 2 person game where AI and A2 are their alternative sets and uI' ｾＬ＠ their utility functions. A NEU for the game r is detined by a pair of mixed subadditive strategies (Pl'P2 ) for which there exist
supports
Supp(PI), Supp(P2)3 such that for each ai E ｓｵｰｰＨｾＩＬ＠ aj maximizes the expected utility ofplayer i, given that Pj represents the conjecture of player i about the choices of player j over A j:aj E Arg max EPjUj (a,.) (i,j = 1,2)
•• Sapp[p.]
Notice that this definition includes the standard Nash equilibrium (SI,S2)' where (SI ,S2) are some proper additive probability functions. The following theorem allows one to map NEU' s through the uncertainty aversion coefficients (cl , c2) of players, under the hypothesis that they are constants and (PI, P2) are uniform squeezes of additive Nash strategies (SI,S2) for a transformed game r·:(AI,A2,u;,U;), i.e., Pj = (l-cj)Sj, where
U;
is defined in such a way that Ep Üj=
EsU;,
for i, j=
1,2.
J J
ｾｾｾ＠ ＭｾＭＭｾｾＭＭｾｾｾｾｾＭＭＭｾ＠
-Theorem [Dow & Werlang, 1994]
Let r:(A1,A2,UI,u2) be a fmite 2 person game. For alI ＨｃｉＧｾＩ＠ e[O,lr there exists a NEU (PI,P2 ), such that cl is the uncertainty aversion coefficient associated to P2 and ｾ＠ the
uncertainty aversion coefficient associated to PI.4
In the application of the theorem we have to keep in mind two aspects which are implicit in the definition of NEU:
I. The expression Pj = (1-c j )Sj requires consistency between pIayer j guessing about the choices of pIayer i(Pj) and the optimal standard strategies Sj of player i in the game
r·
(without uncertainty). This requirement turns out to be particularIy strong in a N-person game (N)2) when the guesses of each collusion ofN-1 partners must match the optimal strategy ofthe pIayer Ieft out ofthat collusion (as in the application beIow); 2. The rationality which is implicit in the NEU definition does not impIy logical omniscience, meaning that agents may not righty deduct alI implications of some action previousIy known by them (Dow & Werlang, 1994, p. 320).m -
OUGOPOLISTIC COMPETITION3.1 - The Model
Consider first a competitive oligopoly à la Coumot producing a unique homogenous good and composed ofN firms(N ｾ＠ 2) having production techonologies, each one described by stabIe cost function +j(qj) (Le., with constant input prices; i=I,2, ... N) and facing an inverse market demand R=R(Q) if OS Q S a( a
>
O) and R=O if Q>
a, whereN
Q = Lqj is the industry output (qj ｾ＠ O). In order to make analysis easy, suppose (without j-I
4 Tbc demonstrlltion of lhe tbeorem relies on lhe resu1t when Pisa unifonn squeezc of 111 Idditive disIribution
lost of generality) the functions ｾｪＬ＠ R are C2 (two times continually differentiable) and assume alI marginal costs and marginal revenues are non-decreasing and non-increasing i.e., ｾＢｾ＠ O and (Rqj)" S O respectively (i= 1 ,2, ... N)5. Under these assumptions, the profit functions nj(qj;q_J = qjmax{R(Q),O} ＭｾｩＨｱＩ＠ will be concave over qi e[O,a] given
Note
P
i,P_
j for marginal and joint subadditive probability functions (respectiveIy) give the guesses of the N-l partners about producer i production pIans ＨｾＩ＠ and, reciprocally, the producer i conjectures about the production plans of bis N-l challengers (P-i).Consider an economy without uncertainty
r*:([O,a], ... [O,a],n;,n;, ... ＬｮｾＩＮ＠ which is going to be explained below.
In
accordance with previous remarks given in seco 11, the application ofthe Dow & Werlang theorem for a game with N ｾ＠ 3 pIayers wiIl require some adaptations which can be covered by the foIlowing two assumptions:HJ: Strong consistency between the optimal joint strategies S_i of N - 1 producers in the game
r*
and the fum i conjectures (P-i) on these production strategies,in the following sense: P-i = (1-C)S_i;
where cj is fum i' s constant uncertainty aversion coefficient associated to its guesses P-i; H2 : Reciprocally, strong consistency between producer i optimaI marginal strategy
(Si) in the game
r*
and the joint conjecture made by the N-I producers (Pi) about this strategy, in the following sense: Pi = (l-c_i)Si;where C_i is the constant uncertainty aversion coefficient of the N-l firms, which is associated to their joint conjecture
p;.
65 These properties ensures lhe existence ofCoumot-Nash equilibrium in lhe game r· for each N. However, they are bascd 00 some
resnined ISSUlDptioDS 00 prefcrences anel 1Cchnologies (scc Soonenschcin .t. Rober1s, 1977).
6 NOIice tbat when there is DO possibUity of collusioo IDlODg finos which are fonning guesses aod ta1dng lCtÍODS indepcndeody (15 in lhe Cournot game) then under H1 anel H2' we obtain: (1-e.Xl-c-t)=(1-c.)(1-c_.), V'i,ltJn parâc:u1ar, e. =c. or c-t =c-t< ore. = c-t(V'i,lt) ímplies ci
=
c_i=
c (V'i) , i.e. commoo knowlege IDlODg finos (we used hcrc lhe CODVCDtional definitioo of indepcndency amoog random vlriab1cs with additive distribution. For ao Iltanpt to define jndependence between events in lhe DOD-additive case, scc Dimitri, 1995).Since HI and H2 match personal guesses P_i (expectations) with actual
strategies S_i of collusions, they can be seen as rational expectation hypotheses.7
Hypothesis H I allows one to show that Nash strategies in the game
r*
areequivalent to the guesses under uncertainty in the original game
r,
since we have, theequality: Es)TI; (qi,q-i)] = Ep-iTIi(qi,q-i), i=I,2, ... ,N. Hypothesis H2 allows one to
ca1culate the equilibrium conjectures Pi after getting actual Nash strategies Si in the
transfonned game
r*
(given c_JSo, under Ht ,H2, we obtain, by Choquet's integral:
(1)
In order to give the minimum value on the right side of the equation, we
assume the worst that may happen to
rum
i is that its N-l competitors should reduce tozero its residual demand R -I - ｱｾｩ＠ 1, in such a way that the producer must endure
loss +i(qi). So, min TIi (qi;q-i) = -+i (qi)· Hence, noting
q_IE[O,a]N-1
(2)
The equation (1) shows that NEU' s in the game
r
are proportional to maximinNash strategies in the game
r*8.
Hereafter we are on1y going to look for pure Nash strategies in the game
r*
insofar as we deal with TI; instead of ES_I TI; .
The function TI; is a continuous transformation of TIi and preserves its properties
(c2
class, bounding and concavity). Since the altemative sets[O,a]
are compact and7 Ac:curaIe clefinitions ofrational cxpecWion hypolheses used in geueraI cquilibriwn thcory CID be foUDcl in GrossmIn (1981b) IDCI
bIacr (1989).
8 Notic:c that lhe Me" coeffieienlS CID be viewed ei1hcr as indicators for lhe behavioral attitude of lhe agcnIS &ciog uncer1ainty, or as signs of lheir lack of information. In lhe fim case, it is supposed there is • conunon set of exogenous vlriables (macroeconomic:,
convex sets of R, the point fIX theorem (Kakutani 1941) applies in order to ensure the existence of a Cournot-Nash equilibrium for the game
r*9 .
The equilibrium quantities (qj,q_j) for the economy
r*
must verify thefirst order condition:
ｮｾ＠
ôqjl (qj ,q_j ,cj )
=
O (i=
1,2, ... ,N)From the properties assumed above on 'j and qjR, the second order condition for maximizing
n;
is verified. The use of (2) in the frrst order equation leads to the expression:ｉｉｾ＠
ôq:
=
(l-cj)[qjR'(Q)+ R(Q)]-,\ (qj)=
Owhich shows that the frrst order equation system only admit non-trivial solutions for cj E [0,1).
3.2 - Uncertainty Avenion Effects at Eguilibrium
Since the maximization of TI; (instead of TIi ) implies producer i is going to choose among defensive production strategies, it seems natural to expect Cournot-Nash equilibrium under uncertainty exhibits an aggregate output Qc smaller than the standard product Qo (Qc < Qo). Indeed (as it wiIl be shown in the next section 3.3), alI producers are trying to charge the price payed by consumers the competitive uncertainty costs they are facing.
The diagram below gives an illustration of the price-quantity equilibrium production for a producer of an industry under uncertainty, when the N firms are
9 Tbe unicity ofthc equilibrium is hcrc assured ifeither
n;
are S1ricdy qUlSi-concave and lhe mapping ofbest reply qj=
qj(q_l) , •an.
sucb 1bat _ I (ql (q_1 ),q_l) = o are proper con1rW:tions over [O,a); or if qi e (O, a) and lhe Jacobim JIIIIrix lSSOCiIted with lhe ôqi
•
an.
fintorder condition ｾ＠ = O is anegative quasi-definitc JIIIIrix for ql e(O,a), i = 1,2, ... N. [See Friedmm, 1990, chap.3). ÔQi
identicals ＨｾＺ＠ ］ｾＩＬ＠ and there is simmetry ofinformation among producers (cj =c), i = 1,2,ooo,No Here,
Q
= Nq and it is assumed R(Nq) >ｾＨｱＩ＠
(no free-entry)oq
+'
R--°th l-c
W1 qc
=-R'
Rc...---'t--'"
Ro
ＮＮＭＭＭＭｾ｟Ｋ｟｟＠
....
...!...
l-c
+-I
I
I
I
+-I'
1
J q
I
I
I J
R(Nq)
ｾＭＭＭＭＭＭＭＭＭＭｾＭＭｾＭＭＭＭＭＭＭＭｾｱ＠
ｒＭｾＧ＠
qo = Ｍｾ［＠ Rc = R(Nqc); Ro = R(Nqo)o
N o that °f th H o otice 1 e esslan [ ôq Ô
2n;
Ô ]IJ-. N (ev uate m eqw al d o °lob o 1 num pomts O) o IS a re ar gulI 'qj
matrix, then by implicit function theorem the equilibrium functions qj(c.,ooo,cN)(i= 1,2,o,N) exist and are of class Cl over [O,I)N.lt will be supposed this condition holds hereo
Now we will exam more closely the effects of uncertainty aversion variations on the equilibrium productions qj (c. , o o o, cN), q_j ( c. , o o o, cN ) o Notice frrst that qj(c.,ooo,cN) must be interpreted as conjecturaI (or virtual) equilibriums only, since the actual calculations presuppose mutual omniscience among producers, which is not likely in the real worldo However, this device is theoretically useful for the analysis of the uncertainty variation effects on the producers' optimal decisions, as it will be made clear belowo10
-Recalling the ith equation of the above system:
art
aqjl (qj(cp ... ,cN),q_j(cP".,cN),cJ =
o
(3) The differentiation on both sides of this equation with respect to Cj and c jgives (i "* j) :
a
2 · TI· 8q. ( o ..,2 TI· • \'( I aq.J
o-TI· ..,2._ _ I _ I + 1 ---=!... + I _ O
aq?
ac
j 8qjaq-i)ac
i aqié3ci - (4)(5)
We state now the following proposition for firms ij participating in the market, i.e., such that TI; (q(c» ｾ＠ O and TI/q(c» ｾ＠
o.
Proposition 1: Take the Cournot competition model under Knightian uncertainty as
presented in section 3.1. The uncertainty aversion effects on the Cournot-Nash equilibrium quantities (qj>O) of afirm i matching the game
r
*
and on the equilibrium industry output (Q), for Cj"* 1"* cp are given by the following equations:aj = R" [QR'(1- cj ) - (Q-q)cp
;]+
R'ｾＱＭ
cj)(N+
l)R'-Ncp;] (6)q;R'+R -(1-c;)[q;R"+R'p;
-(1- c; )R'-cp;" (7)
aqi
=
(1-Ci )[qiR"+R']ajOCj (1-ci)R'-cp; (8)
For ali i,j (i "*j) such that qj, qj >0.
and aj = " ｾＭｬＭ 1 . 4Ij(qJ
a ,
8qji( .. j) Ci C j
For ci=Cj=l, we have qi=qj=O and then alI derivatives vanish. Proof.: See Appendix 1.
Here and Iater on we omit the arguments Q and qi in the notation of R and
41;
functions and their derivatives (respectiveIy).From equations (6)-(8), we notice that uncertainty aversion effects on production (and price) at equilibrium depend directly oniy on the retums to scale of factors (4Ii' disregarding whether there is fixed costs or "sunk costs" in the
frrm.
IiHowever, without some additional hypotheses nothing can be said about the signs of the above derivatives.
Particular Cases:
a) Constant Returns to Scale
(41;
= O ; j = l,2, ... ,N)In this case, cr) given by (6) becomes:
crj = (1-c)(N
+
ＱＩｦＬｶｾＱ＠
R"+R'] (6a)Proposition 2: Under constant returns to scale lechnologies.
if
lhe "modified"decreasing marginal revenue condition ..JL.R"+R'<O .V+l
derivatives (6)-(8) Df Proposition 1 have lhe following signs:
crj < O
Sign(8Q;) =
ｓｩｧｮｾ＠
(1-Ti )R"+R'] 8c;Sign(8q;) = -sign[q;R"+R'] (i
*
j)8cj
where
r.
=%
is lhe markel share Df firm i.holds. lhe
Proof.: As we assume positive marginal revenue (qjR'+R >0), the above sign relations are obtained in a straightforward manner taking into account of (6a) and setting
cp;"
=
cp;
=
O in the equations (7)-(8).QED
Notes:
(i) When market demand is non-convex 12 (i.e.,R"<O) we have
(i
*
j) and Oq;/ ôC; < O(ii) In the convex case
(R"
ｾ＠0),
the non-increasing marginal revenue hypothesis(Rq;)"S,O, i=l;2., ... ,N can also be written as (tmaxq;}R"+R'S,O. Hence,
non-increasing marginal revenue assumption implies condition (Q/N+I)R"+R' <O if and decreasing marginal revenue is implyied by that condition if
(iii) If condition (Q/N+I)R"+R'<O for Proposition 2 applies, the sign of ôq; / ôC; is always negative for firms that are not too smalls
ｾ［＠ ｾ＠
,v",.1 ;(iv) If market demand is linear (R"=O), Proposition 2 applies and ôq;/ ôC; < O ; Oq;/ ôCj > O;
1
(v) When R(Q) = Q-li (O < Ô s, 1): constant price- elasticity (=
-8)'
Proposition 2 also applies and a proper substitution of R, R' and R" in (6a) and (7)-(8) withcp;"
=
cp;
=
O gives:aj
=
Nｾ＠ ｬｃｾ｣ｊｪ＠
-k)
wherefj
=q/Q
ôqj
］ＨｾｘＭｉＭｘｲｩＭｾｴＢｔＫＨｬＫￔＩｲｩＩ＠
ÔC· N+I l-c. Ô}N
I J
ôqj (ô+
1)Q (
IX
I)
ôc
j =-(N+I)(I-cj) rj-ã rj - l +ô
So, since rj s,..!. we have, successively: cri S, O, ôqj S, O and
Ô ôcj
12 Recall that the non-convexity ofthe aggregate demand assumption is not related to the non-convexity on preferences among consumers hypothesis', since the assumption depends only on the sign of the third derivative ofthe utility function (see Mas-Collel et a1ii, 1995, sec.l0.c).
· (8q;J . (
1)
Slgn -
aC
=Slgn r.-j J 1+3
Comments:
1. From note (v), externalities caused by an increase in the uncertainty aversion of producer j (Cj) over the production quantities of his competitors (qi) wiIl depend on bis own market power (rj = qj) in the convex demand case. Since.! S _1_ < 1 , if
Q
2 1+3producer j controIs Iess than 50% of the market (qj <
ｾＩＬ＠
the externalities wiIl be negative and the production quantities of bis competitors should foIlow the reduction of bis own production, at equilibrium. Notice that producer j's partners can only improve their absoIute market positions if firm j is the dominant firm (rj > .! );2
2. As could be expected, under the normal conditions of Proposition 2 (i.e., constant returns technoIogies and the 'modified' non-increasing marginal revenue assumption), a rise in the uncertainty aversion of producers always reduces the equilibrium output ofthe whole industry.
b) Egual Firms and Symmetrv of Information
Suppose then, there exists symmetry of infonnation among producers in such a way that:
cj = c (j = I, ... ,N). Therefore, the finns' production at equilibrium, are identical:
qj=q; Q=Nq (j=I,2, ... ,N). Note ｋ］ｾＢＨｱＩｾｏＮ＠ So
aj = IK (0'-
ｾＩＮ＠
The substitution ofthese values in (6) leads to:-c uC
( R'-l-c N
ｾｘｑ＠
R'+R)0'=
I
(
) I
R'[(l-c)(N + I)R'-NK]+R"Q[R'(I-c)-(N
ＭｬＩＨｾＧｎＩ｝Ｍｉｾ｣ｬ＠
(l-c)ｾ＠ ｾ＠
R"+2R' -KJ
CoroUary: When firms are identical and there is symmetry of information among producers. under the conditions of Proposition 2. an increase in the uncertainty aversion
in the industry (c) reducesfirms' equilibrium quantities (q;=q). i=1.2 .... N
Proof.: Let
Xi
=Zi
+±
:qi
stands for the total effect on finn i's quantity of ani )(.;) Cj
increasing in the uncertainty aversion coefficients of alI producers in the industry. Setting qj=QlN=q for identical finns, Cj=C and ｾｩＧ＠ =
4>;
= O (for the symmetry of infonnationand the constant return assumptions, respectively), in (6a),(7) and (8) equations, we fmd:
I [ qR'+R ]
X = (I )(N 1 Q = O' which is negative (from Proposition 2).
-c +) -R"+R' N+\
QED
Comment: Under finns and infonnation symmetry assumptions, there wiIl also be
symmetry of the effects of uncertainty aversion variations on the equilibrium quantities
between individual fmns and the whoIe industry (X = O' ).
3.3 - Lar&e Number ofFirms
Consider now a sequence of games
r
N (and ｲｾＩ＠ for integers N ｾ＠ 2. Weare looking for the effects of uncertainty parameter variations on the equilibrium
quantities q; as the number of firms matching the one-shot game
r
increases. For this reason we wiIl assume that the games ｲｾ＠ are independent and that for each N (given exogenousIy), there is no free entryl3, in such a way that, at equiIibrium the inequalities R(Q)ｾ＠ ｾ｡ｸｻＴ＾ｩＨｱｩＩｽ＠
always hoIds for alI N (assuming equality never hoIds jointly for2sISN qi
alI firms).
First, we consider the case of different firms (4)i ;é 4>j) with information
asymmetry among producers (Cj;é c), i = 1,2,... Assume first, the uncertainty
parameters ci don't change as the number N of players matcbing
r*
becomes larger. In this case, since the industry output is bounded (Q ｾ＠ a), if the sums aj =ＮｉＨＭｉｾ＠
)Ｔ＾ｾＨｱｊ＠
ｾｪ＠
are also bounded (''\fj
=
1,2, ... )14, then a quick inspection'("J) cj ti\,; j
of (6) shows that O'j ｾ＠ O as N ｾｃｘＩ＠ (V). Therefore, from (7) and (8) we wiIl have: aqj ｾ＠ qjR'+R <O and 8qj ｾ＠ O.
àcj (l-cJR'-4>7
8c
j Comment:At equiIibrium, uncertainty aversion increases for one of the producers, affect neither the optimal quantities of bis competitors, nor the aggregate production of the industry. Increase in the uncertainty parameter only makes worse the market position of the producer bimself. So, an increasing number of firms pIaying the game
r*
tends to reduce virtual depressive effects of competitive uncertainty on the aggregate production (henceforth, on the rise of the equilibrium market price). At the same time, the uncertainty aversion extemalities of one producer on the optimal production quantities of bis competitors are counterbalanced by a better distribution of the additional uncertainty in the industry among a greater number of firms.However, one shouId expect that uncertainty aversion coefficients Ci decreases as the number N of matcbing firms increases. Then, it is easy to show that
13 We thank Luiz Guilherme Schymura de Oliveira (EPGEJFGV -Brazil) for this remark. Indeed. we can suppose there exists "sunk costs" in the induslry, or high f1xed costs (+; (O) ｾ＠ K > O, \f;, N) .
14 The assumption is immediatcly verifled when technologies exhibit constant returns to scale (+i = O).
derivatives ôqj, given In (7) also vanish as N increases for any sequence {C;N)} 8cj
converging to zero as N ｾ＠ +00. For instance, let ｣ｾ＠ = ｣ｾＬ＠ 0< c j < 1, and qj Ｈ｣ｾ＠ ,c:j) be the corresponding Cournot equilibrium for each N, i = 1,2, ... ,N. When N ｾ＠ +00, as q;
are C', we have qj
(C;
,c:j) ｾ＠ q;(O,O) and, of course, derivatives8qj •• ôqj.. N-I . N I
ｾ＠ (Cj ,C_j) = - . (Cj ,c_j)NC j also vanlsh (Ncj - wiIl tend to zero as N ｾ＠ 00).
()\;j 8cj
Lemer Index:
Notice now that, from the frrst order condition for
n;
maximization wer· 1
...J...
=
-I--p..j - cj] where À,j=
(Re - ｾ￭Ｉ＠/
Re is the price-cost Lemer index, E isE -Cj
have:
the demand price-elasticity and ri = qi is the market share. If frrm i's production is
Q
negligible with respect to the industry output and if price-elasticity of demand is high
r·
enough, therefore ...J... ==
°
imply cj == À,j . In the symmetric case, the expressionE
c
=
À,e=
(Re - ｾＧＩ＠ / Re allows an interesting interpretation: the uncertainty aversion "c"equals the cost of the lack of information in the industry (as a proportion of the unit
selling price) which would be conveyed by the equilibrium market price in the perfect
competitive case. Indeed, if
Ro
stands for the competitive price, with (Ro = ｾＧＩ＠ we have: c = Re -Ro.
Recall that in the symmetric caseＨｾｪ＠
=ｾ［＠
cj = c), if the demand is· notRc
replicated, as the number of firms entering the game increases, the optimal market shares
r
=
q/Q become smaller. Hence the Cournot-Nash product converges to the equilibriumoutput of a perfect competitive industry composed by price-taker firms of negligble
sizel5. Thus, Re = Ro and there wiIl not be uncertainty aversion in the industry (c
=
O), as the residual demand facing firms (Q -Lqj)
exhibits perfect elasticity, in such aｾｩ＠
way that producers wiIllose any control over price.
15 The Ipplication of the folk theorem to the competitive equilibrium case is clone more easily without any demand replication.
Novshek (1980) shows the convergence of Coumot-Nash equilibrium to the competitive equilibrium in a partia! ana!ysis with
enclogenous N and demand replieation. In a general equilibrium approach when increases both the number of firms and lhe number of consumers Hart (1985) shows thal, for a large elass of cases. if firms are small relative to industry size. the imperfect competition
equilibrium output converges approximatcly. to the perfeet competition output
3.4 - Comparison with Monopolv
The uncertainty aversion degree for the N firms in a Coumot market may
be high enough to induce producers to take conservative production decisions which Iead to an industry output Iess than or equaI to the monopoly result. The next proposition establish one ofthe principal results of our paper.
Proposition 3: Take the Cournot oligopolistic competion model under Knigthian
uncertainty as presented in section 3.1. Under the information symmetry assumption
among producers (c;=c, i=},2, .. ,N), ifthefollowing two conditions holds: (i)cjI;'(O) > O for
some firm i; (ii) Â(O) ｾ＠ Â", / N, then there exists an uncertainty aversion levei for the
•
industry c E [0,1) setting the equality between the oligopoly output (QJ and the
•
monopoly output (Qm). The value of c is given in the equation:
• NJ..( c·) - Â.",
c
=N-Â.", (11)
where
À,..
=
ｬＭ｣ｪｬｾＨｑ＠ .. )j R.. is the monopoly Lerner Index (R.=
R(Q ..»
andÂ(c) = 1-s(c)j R", is the monopoly price margin over marginal costs' average
in lhe
O/igopo/y(S(C)
ｾ＠ ｾ＠ ｾＫＺＨｱＬＨｃＩＩＩＮ＠
Proof.:
Writing the ith equation of system (3) at point Qc = Q. we have, for Ci E [0,1):
A
r·
(l-c)R(Qm)[l- ｾ｝＠ = cjI;.(q) (9)
E
A N
where ; = Qq; ,
{q;}
is the sequence verifying (9), i=I,2, .. ,N withIq;
=Q.
and E isc I
r
OQ.(R) R1
the module of the demand price-eIasticity E =
i
aR
.
QJ
evaluated at the monopolyprice Rm = R(Qm)' Assume E> 1. Similar to equation (9) for the monopoly case we have:
(10)
where. ｾＺｮＨｑｭＩ＠ is the monopoly marginal cost.
Assuming informational symmetry among producers (ci = c) and summing up both sides
1 N
(1-c)R(Qm)[N -=]=
ｉｾｩＨｱｩＩＧ＠
E i=l the previous equation, leads to:1
(1-E) s(c)
c
=1- ( - )(1-
ｾｅＩ＠
sm1 N
where s(c) = N
ｾｾ［ＧＨｱ［Ｈｃﾻ＠
of (9) over
=
1,2, ... ,N, arnves at: Solving for R(Qm) in (lO) and replacing its value inand Now, let
Â,..
=
1-sm / Rm with Rm=
R(Q.) and define À(c)=
1-s(c) / Rm' From (lO) we also have À. =1/
E. Direct substitution of the elasticity and costs by À values in the later equation gives the result in (11). Now define g( c) = [M..( c) -Â".] /
[N -Â".] .
SinceｾＯ＠
and q;(c) are C1 , g also is C1 over [0,1), with g(c)S 1. ForÀ(O) =
À". /
N, c* = O is a solution. For À(O) >À". /
N, we have g(O»O. Now, forI;
> O, since ｾ［Ｇ＠ are nondecreasing, ｾ＠/(q;
(1-I;
»
ｾ＠ ｾ＠ ;'(0) > O for some i according to the condition (i). Hence, À(1-I; )
< 1 implies g(I-I; )
< 1. Since g is continuous over [0,1-1;], there exists in this case (at least) one fixed point c·>O satisfying equation(lI).
QED
Notes:
(i) Recall that g'(c) is not necessarily monotone over [O,l), which leave open the possibility of multiplicity of solutions c·;
R N-À".
(ii) Condition (ii) above is equivalent to the inequality
R:
SN
_
J"
where
I"
= l/E o is the inverse of the price-elasticity calculated at oligopolyN
equilibrium point with c=O: Ro=R(Qo),
Lqj
(O) = Qo . ICorollary: Under the conditions of Proposition 2, there is only one criticai uncertainty aversion levei c(N) setting the equa/ity Qm=Qc- The value c(N) is given by:
N4
-
À",c(N) = N - À", (11a)
where "'"
=
1-(SN / R",) and SN=
(l /ｎＩｌｾＺ＠
whereｾＯ＠
don't depend on qi.j
Moreover, c e [0,1] and c ｾ＠ c( N) imply Qc::;; Q",.
Proof.: The constant return to scale hypothesis means that term s, in Proposition 3, no longer depends on c, which ensures the unicity of c(N) from the rhs of (lI). For
c e [0,1) , if the 'modified' decreasing marginal revenue assumption holds
Ｈｎｾｉ＠
R"+
R' < O), we have by Proposition 2, cr < O. Thus, the second part of the resultproceeds straigthforward from the fisrt one. For the case c=l, we have, Qc = O::;; Q •.
QED.
Notes:
(i) For the inverse market demand R(Q) = Q-6 (O < ô::;; I), (see note (v) in section 3.2a), under constant returns to sca1e we obtain
J".
=ô. Thus, from (lIa):(I-ô) SN
c(N)=I-N (N-Ô)s",;
(ii) The value of c(N) decreases as the oligopoly' s average marginal cost
(s N ) increases, for a fixed N. It means that the oligopoly price margin over the average marginal cost in the industry, say
ｾ＠
= 1-SN /Rc
= 1-(R./ Rc
XI-À-,v), also(lIa): c(N)
=
1-[Rc{l-ｾＩｉ＠
Rm(l-Àm / N)] with Rc=
R(Q(c». Expression (lIa) also shows c(N) increases as Àm increases.(iii)If S N doesn't change significantly as the number of firms N matching
r
increases, sinceo
ｾ＠ À... ｾ＠ 1 it is easy to verify from ( 11 a) that c(N) wiIl also increase.Comments: 1. Proposition 3 means that there is an uncertainty aversion leveI in the industry which induces firms that match non-cooperatively for a market share to behave as a monopoly. This result is to be compared with that of the conventional analysis (without uncertainty) showing an increasing industry output as the number of fmns exogenously increases;
2. The expression of c(N) given in (lIa) doesn't depend on whether there are fixed costs. This seems likely a strong result since the standard regulation market policies are supported by non-cooperative Coumot-Nash stragegies for the firms of an industry with cost barriers to entry. As it is well-known, the market power of each fmn
1
-in this framework (for -instance, -in the symmetric case, equal to N = E À, where E is the price elasticity and
I
the Lemer index, both evaluated at point R=
mean cost) is a decreasing and continuous function of the number of operating firms. If we assume a rising uncertainty aversion in the industry, the mean production per fmn must decrease while the market shares ri= qj(c) may not undergo significant changes. Thus, anti-trustQ(c)
regulation laws to prevent merger or other forms of acquisitions of fmns which joint market share overshoots some specified device (as in the Clayton Act in the USA) tums to be powerless and inoperative in achieving the purpose of protecting consumers against monopoly power.
IV - LINEAR DEMAND
In this section an application of the previous equilibrium relations is done for the case of a linear market demando As we are looking only for qualitative results we choose the simple demand form: R(Q) = max{a- Q,O} and assume identical and constant retums to scale technologies for ali of the N fIrms
in
the industry: CPj(qj) = S constant, with s < a .For the standard game r·:([O,a], ... [O,a],n;, ... ＬｮｾＩ＠ the use of ｮｾ＠ given
in
(2) and the frrst order conditions (3) leads to the equilibrium solutions::total output.
:production frrm i.
:profIt fIrm i.
where:
N 1
ｾ］ｌＭ
j_1 l-cj
The production and profIt for monopoly (qM and n M) can be directly obtained putting N
=
1 and cj=
O (i=
1,2, ... ,N) in the above equation, which gives:1 1
qM
=2(a-s)
;
n
M=4(a- s
f
Notice that the condition
(1\
ｾ＠ O) for the ith frrm taking part in the market is ci Sc
i=
1 - sN ,which wiIl require a not toa high uncertainty aversion. A frrma Ｋｓｾ｟ｩ＠
without uncertainty (c j = O), wiIl always go into the market; indeed, +00 ｾ＠ ｾ｟ｪ＠ ｾ＠ N -I and a > s implies 1
ｾ＠
c
jｾ＠
1- (ｳｾ＠
= cN > O for N < +00. Hence, a sufficienta-s
+sNcondition for ith frrm wishing to produce is ci :S cN •
Consider now the necessary (and suffIcient) conditions ci <
c
i above. Let anFUNDAÇÃO GETÚLIO ｖａｾａｓ＠
increase ordering of the N firms according to the levei of their uncertainty aversion degrees, in such a way that: c(l) ｾ＠ c(2) ｾ＠ .... ｾ＠ c(N).
c(i)
ｾ＠
c(j) = 1- sN can be written, successively,｡ＫｳｾＭＨｪＩ＠
ｾＭＨｩＩ＠ ｾ＠
Nｾ＠
J
ｎｾＭ
:) or,(N + 1)5 c(j) S 1-
=
Co｡Ｋｾｳ＠
Thus, the conditions
. {. (N
+
I)S}
So, If we define n = max I E {2,3, ... , N} C(i) ｾ＠ 1- a + ｾｳ＠ ' the number n
wiIl be the maximum number of firms operating in the market, and Ctn) ｾ＠ Co wiIl be the
highest uncertainty levei which can be supported by market demand and tecnological conditions. Hence, N-n firms wiIl be prevented to operate in the market.
4.1 - Symmetric Eguilibrium and Monopoly Comparisons
It 15 easy to verify that Qc ｾ＠ q M if (and only if)
N+I
-ｾＯｎ＠ ｾ＠
a/s-
2N (ais-I) ］ｾＮ＠ Thus, the oligopoly product of the N firms wiIl not be greater than to the monopoly output, ifj3
ｾ＠MN
<ais.
A particularly interesting case occurs when there is informational symmetry among producers ci
=
c(i = 1,2, ... ,N), where C is then interpreted as the uncertainty levei for the whole industry. In this case, we have successively, Q］ＨｾＩＨ｡＠
__
S
),
4=_1
(a __
s);
fI=(-1
)2(a __
S
)2.
Thus, ifc N+l l-c N+I l-c N+I l-c
_1_ <
ais,
Le, if c < 1-si
a, then ali fmns may operates in the market. Otherwise, the l-coligopoly productions wiIl not be greater than the monopoly output if (_1_)
ｾ＠
j3,
which l-c. I·
25
16unp les c
ｾ＠
1-ｾ＠
I) {
I ) = c(N) .1--
+
1+-N N
16 Of coursc, lhe valuc c(N) givcn herc could bc dcduced dircctly from equation (11 a). In order to do 50, notice that -E
= -
a+s and a-s4.2 - Cournot Duopoly
Hereafter we particularize the Cournot game for two firms in order to
bring out the regions where NEU points are possible as the only function of the
uncertainty aversion parameters (cl,C2 ) ofthe producers.
Putting N=2 in the equilibrium values for the production and profit given
in the previous section we get the foIlowing Cournot values
(Qc,q;
efI;) :
Qc
=2[a
｟ｾＨ｟Ｑ＠
+_1
J]
3 2 1-c] 1-c2
q;
=
ｪｾ＠
-sB;]
i=
1,2fI;
=.!.[a-Qc-s]=.!.[a-
(_3 __
1 _ _ 1J][a-
(3_-
1 __ 1J]
9 9",1-c; 1-c] 1-c2 "' 1-c; 1-c2
Wl'th B.
,
= _2 ___ 1_. " J' l' -12-
,l-c. l-c. , J
The maps for the Nash Equilibrium under uncertainty (NEU) in the game
r:([0,a], ... [0,a],nl ,. .. ,n2 ) wiIl be depicted by the condition
q;
ｾ＠ O. This impliesi,j = 1,2
Notice that f( cj ) above defined is an increasing, continuous and convex
a-s
function in the range [0,1], i.e., f',f">O. Moreover, f(O)=-- and fadmits a fixed a+s
point c = 1-
si
a
= ê .In the plan (c j , C i ) we represent below the regions where NEU points are
allowed. A: joint Cournot production (qj ,q;) > (0,0). The limit cases are: Aj: firm j
monopoly: (qr ,O); Ai: firm i monopoly: (O,qr); Ao: null production: (0,0).
Notice that for regions Aj, Aj we must exclude the points on the dotted
lines. So, the Dowand Werlang theorem (Sec. 11) ensures an A-type NEU for every
(Cj,Ci) e[0,1)2 corresponding to the region A where this point is located.
I
ｾＫＭＭＭ
a -s
a+s
A
A.
J
a -5
a+s
'"
cI
One may also be interested in the uncertainty aversion threshold beyond
which fmn i' s production would not be greater than its corresponding ratio in the
1 1
monopoly output (q M
=
-(a - s» equally shared among the two fmns: ｱｾ＠=
_qM.2
2
s(l-c-)
Indeed, it is easy to see that
ti;
ｾ＠ ｱｾ＠
if ciｾ＠
1- (1- c j)(a +J3S) + l' (i,j = 1,2). Notice that these inequalities generate 4 analogous regions as those depicted in the former diagram for NEU: the frrst (A') where(ti) ,ti; )
ｾ＠ (qr ,q;M); the second (A;) whereti;
ｾ＠
q;M andti}
< qr; the thirdＨａｾＩＬ＠
whereti;
< q;M andti}
ｾ＠
q:; the fourthＨａｾＩ＠
where both Cournot productions are less than monopoly production (ti)
,ti;
)
< (q}M ,q;M).Notice that for the symmetrical case cl = c2 = c, the Cournot production, wiIl be smaller a-s
than the monopoly output if c ｾ＠ - - . a+3s
4.3 - Further Steps
Of course, the present analysis can be applied to alI decision problem
embedded in a Knightian uncertainty environment which can be modelled as a one-shot
game. For instance, in the present oligopolic game, cases where uncertainty affects only
the market demand curve (as in the kinked demand model) or production is labelled by quality standardsl7 may motivate research for smilar treatments.
In order to extend the application of the Schmeidler-Gilboa decision model in game theory an important challenge is to set a theorem for sequential games analogous to the Dow & Werlang theorem. Such an issue, for instance, could allow one to examine the conditions on the uncertainty parameters assuring the Nash property for the Stacklberg equilibriuml8 .
v -
CONCLUSIONSThe present work allows us to emphasize the following points:
a In an oligopolic industry when fIm1s competing à la Cournot (in quantities) are facing competitive Knightian uncertainty, the Nash equilibrium of the market can be mapped as a function of the unique uncertainty aversion parameters of producers. In the duopoly case, when the market demand is linear, the monopoly and the null solutions appear as particular solutions, when the uncertainty aversion coefficients among producers are divergent (high or low) or high and close together, respectively;
b. In Proposition 1 we obtain the effects on individual increases in the uncertainty aversion parameters (cJ on the equilibrium production of fIm1S (qJ and the industry output (Q). Under constant returns to scale technologies and a slight modified decreasing marginal revenue assumption
ｾｾｉ＠
R"+
R' <O),
a rising in Ci always reduces industry output (Proposition 2).8q; / 8c; < O and 8q; / 8c) > O when market
Moreover, we obtain demand is concave, R" ｾ＠ O
anti
8q;/
8cj < O whenever fIm1 i market share (ri) is such that17 For instancc, such models can bc found in Latfont& Morcaux (1991), Crampcs & Hollander (I99S), TlfOle (1989).
18 In lhe standard game (without unccrtainty) the sub-game pcrfection propcrty is obtaincd whcn one considcrs instaIling costs for
lhe entcring firms. For a Brazilian ovcrview of the litenllurc, $CC Schymura de Oliveira (1991).
ri ｾ＠
(N
+
1)-1
in the convex demand case (R">O). For constant price-elasticity of the market demand, the sign of oq i / oc j depends both on the market share of fmn j (rj)and the price-elasticity of the market demand;
c. Under the assumptions of the Proposition 2, if fmns are identical and there is symmetry of information among producers, the effects of uncertainty aversion variations of an individual producer on the industry output, or the effects of uncertainty aversion variations in the industry on an individual fmn output are both identical (and negative);
d. When the perfect competitive equilibrium comes into focus, in the symmetric case, the uncertainty aversion parameter for the whole industry can be viewed as a lack of information cost which producers wiIl add to the competi tive market price charged to consumers;
e. Proposition 3 stresses the main result of our paper: under the assumption of
information symmetry among producers and two light technical conditions, it shows there is an industry uncertainty leveI c· which turns the oligopoly output (Qc) equal to the monopoly output (QnJ. Moreover, under the assumptions of the Proposition 2, the criticai leveI, c(N) is unique, and c e[O,I] anti c ｾ＠ c(N) imply Qc ｾ＠ Q •. The critical uncertainty aversion leveI c(N) increases as both the Lemer Index of the oligopoly
(ic)
and/or that of the monopoly (Â..) increase. The criticai leveI normally increases when the number offrrms matching the game (N) increases (exogenously);f. The result of Proposition 3 may be compared with the result obtained by Grossman (198Ia), which shows that a industry monopolist operating with cost barriers to entry, may be encouraged to choose as the optimal supply strategy, the pure competitive
strategy when he is facing threats from entering firms in the market. Thus, the
accounting for the competitive uncertainty aversion of producers in our approach and
the threats to entry from competing firms in the Grossman approach lead us to
conclude that neither the number of operating firms nor the individual market shares
can no longer be seen as sufficient indicators for the degree of competitiveness in the
industry;
g. Such a theoretical result also has a significant empirical appeal, since it helps to
explain the defensive behaviour of oligopolistic fmns which is usually observed
during periods of high and persistent monetary or institutional instability (as in Brazil,
1962-1965 and 1985-1993). In spite ofthe fact that our static model doesn't enable us
to quote in a stylized form the changes in the industry's competition standarels (which
are driven by the efforts of firms towarels the introduction of a more rational on
production process and more differentation on the gooels produced) it is an observed
fact that, increases in the competitive uncertainty in oligopolic industry are almost
always followed by reductions in the aggregate product and hence, increases in
pricesl9 ;
h. The application of the model for a simple linear market demand explicitly stresses
how the range of the uncertainty aversion leveis of producers plays the implicit role of
an entry barrier for proponent firms to operate in the oligopolic industry (section 4).
From a normative macroeconomic point-of-view our principal result suggests also that
with an high uncertainty aversion environment in the industry, every anti-trust law
enacted to prevent merger of firms which are supported by the usual market shares in
that industry, can turn out to be inoperative to protect consumers from the unpleasant effects of monopoly power.
19 Rqarding priccs. analogous conclusions are obtaincd with a high intlation environmcnt from a mark-up princing model (Frenkel. 1979) wherc the risles bome by producers Iies with the possibilily of market losses and future variable costs underatimations.
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Appendix 1
Taking the second derivatives ofthe profit function
rr;(qpq_J
given in (2)with respect to q's and c's and substituting these derivatives in (4) and (5) leads to:
ｾＱＭｃ［Ｉ｛ｱｩｒＢＨｑＩＫＲｒＧＨｑＩ｝ＭＴｉＢｩ＠
(q;)}Z: + )+(1-c;)[q;R"(Q)
+
R'(Q)]l'N_tＺｾＧ＠
= q/R'(Q)+
R(Q)}
(4a)
tl-Cj)[q jR"(Q)+2R'(Q)]-4I'\ (qj)
ｽＺｾ＠
+J
+(l-cj)[qjR"(Q)+R'(Q)]l'N_t
ｾｾｪ＠
=0J
(5a)
Noting O'j =
ｬｾ｟ｴ＠
ôq-j + ôqj = ôQ for the effect on the total output ofOcj Ocj ôcj
variations in the uncertainty aversion coefficient of producer i and, after arranging terms.
equations (4a) and (5a) can be written as:
(1- cd[qjR"(Q) + R'(Q)pj + [(1- cj)R'(Q)
-41;
(qj)] :.i = [qjR'(Q) + R(Q)] (4b)•
(l-cj)[qjR"(Q)+ R'(Q)pj +[(1-cj)R'(Q)-4I; (qi)]
Ｚｾ＠
= O (5b)J
Summing up (5b) in i(i:l: j). we obtain:
O'j[(Q-qj)R"(Q)+(N -I)R'(Q)]+[O'j -
ｾｾ｝ｒＧＨｑＩ＠
=Ｎｾ｢［ＨｱｩＩ＠
Ｚｾ＠
J ＮＨｾｊＩ＠ c. J
'" 1 . âq.
aj = L.J -=-4Ij (qJ !l.,.' • the above equation can be written as:
j{ .. j)I cj U\,;j
oro noting
âq. ôq.
O'j[(Q-qj)R"(Q)+NR'(Q)]-_J R'(Q)=aj . Now solving for _J
Oc' J ÔC' J
in (4b) with i = j and replacing the result in the above equation leds to:
｡ｊＨＱＭ｣ｪＩｒＧＨｑＩＭＴｉｾＨｱｪＩ｝Ｋ＠ R'(Q)[qjR'(Q)+ R(Q)]
ｾ］＠ • •
The value of cr j given in (6) enables one to calculate the derivatives
ôqj and ôqj from the former equations (4b), (5b):
oc
joc
jôqj = qjR'+R-(l-cj)[qjR"+R'pj
ôcj (1-cj )R'-q,; (7)
and
ôqj (l-cj)[qjR"+R']crj
=
-(l-cj)R'-q,; (8)
QED
ERSAIOS ECORÔMICOS DA EPGE
200. A VISÃO TEÓRICA SOBRE MODELOS PREVIDENCIÁRIOS: O CASO BRASILEIRO -Luiz Guilherme Schymura de Oliveira - Outubro de 1992 - 23 pág. (esgotado)
201. HIPERINFLAÇÃO: CÂMBIO, MOEDA E ÂNCORAS NOMINAIS - Fernando de Holanda Barbosa - Novembro de 1992 - 10 pág. (esgotado)
202. PREVIDÊNCIA SOCIAL: CIDADANIA E PROVISÃO - Clovis de Faro - Novembro de 1992 - 31 pág. (esgotado)
203. OS BANCOS ESTADUAIS E O DESCONTROLE FISCAL: ALGUNS ASPECTOS -Sérgio Ribeiro da Costa Werlang e Armínio Fraga Neto - Novembro de 1992 - 24 pág. (esgotado)
204. TEORIAS ECONÔMICAS: A MEIA-VERDADE TEMPORÁRIA - Antonio Maria da Silveira - Dezembro de 1992 - 36 pág. (esgotado)
205. THE RICAROIAN VICE AND THE INDETERMINATION OF SENIOR - Antonio Maria da Silveira - Dezembro de 1992 - 35 pág. (esgotado)
206. HIPERINFLAÇÃO E A FORMA FUNCIONAL DA EQUAÇÃO DE DEMANDA DE MOEDA - Fernando de Holanda Barbosa - Janeiro de 1993 - 27 pág. (esgotado)
207. REFORMA FINANCEIRA - ASPECTOS GERAIS E ANÁLISE DO PROJETO DA LEI COMPLEMENTAR - Rubens Penha Cysne - fevereiro de 1993 - 37 pág. (esgotado)
208. ABUSO ECONÔMICO E O CASO DA LEI 8.002 - Luiz Guilherme Schymura de Oliveira e Sérgio Ribeiro da Costa Werlang - fevereiro de 1993 - 18 pág. (esgotado)
209. ELEMENTOS DE UMA ESTRATÉGIA PARA O DESENVOLVIMENTO DA AGRICULTURA BRASILEIRA Antonio Salazar Pessoa Brandão e Eliseu Alves -Fevereiro de 1993 - 370pág. (esgotado)
210. PREVIDÊNCIA SOCIAL PÚBLICA: A EXPERIÊNCIA BRASILEIRA - Hélio Portocarrero de Castro, Luiz Guilherme Schymura de Oliveira, Renato FrageIli Cardoso e Uriel de Magalhães - Março de 1993 - 35 pág - (esgotado).
211. OS SISTEMAS PREVIDENCIÁRIOS E UMA PROPOSTA PARA A REFORMULACAO DO MODELO BRASILEIRO - Helio Portocarrero de Castro, Luiz Guilherme Schymura de Oliveira, Renato FrageIli Cardoso e Uriel de Magalhães Março de 1993 43 pág. -(esgotado)
212. THE INDETERMINATION OF SENIOR (OR THE INDETERMINATION OF WAGNER) AND SCHMOLLER AS A SOCIAL ECONOMIST - Antonio Maria da Silveira - Março de 1993 - 29 pág. (esgotado)
213. NASH EQUILffiRIUM UNDER KNIGHTIAN UNCERTAINTY: BREAKING DOWN BACKW ARO INDUCTION (Extensively Revised Version) - James Dow e Sérgio Ribeiro da Costa Werlang - Abril de 1993 36 pág. (esgotado)
214. ON THE DIFFERENTIABILITY OF THE CONSUMER DEMAND FUNCTION - Paulo Klinger Monteiro, Mário Rui Páscoa e Sérgio Ribeiro da Costa Werlang Maio de 1993
215. DETERMINAÇÃO DE PREÇOS DE ATIVOS, ARBITRAGEM, MERCADO A TERMO E MERCADO FUTURO Sérgio Ribeiro da Costa WerIang e Flávio Auler Agosto de 1993 -69 pág. (esgotado).
216. SISTEMA MONETÁRIO VERSÃO REVISADA - Mario Henrique Simonsen e Rubens Penha Cysne - Agosto de 1993 - 69 pág. (esgotado).
217. CAIXAS DE CONVERSÃO - Fernando Antônio Hadba - Agosto de 1993 - 28 pág.
218. A ECONOMIA BRASILEIRA NO PERÍODO MILITAR - Rubens Penha Cysne - Agosto de 1993 - 50 pág. (esgotado).
219. IMPÔSTO INFLACIONÁRIO E TRANSFERÊNCIAS INFLACIONÁRIAS - Rubens Penha Cysne - Agosto de 1993 - 14 pág. ( esgotado).
220. PREVISÕES DE Ml COM DADOS MENSAIS Rubens Penha Cysne e João Victor Issler -Setembro de 1993 - 20 pág. (esgotado)
221. TOPOLOGIA E CÁLCULO NO Rn - Rubens Penha Cysne e Humberto Moreira -Setembro de 1993 - 106 pág. (esgotado)
222. EMPRÉSTIMOS DE MÉDIO E LONGO PRAZOS E INFLAÇÃO: A QUESTÃO DA INDEXAÇÃO - Clovis de Faro - Outubro de 1993 - 23 pág.
223. ESTUDOS SOBRE A INDETERMINAÇÃO DE SENIOR, voI. 1 - Nelson H. Barbosa, Fábio N.P. Freitas, Carlos F.L.R. Lopes, Marcos B. Monteiro, Antonio Maria da Silveira (Coordenador) e Matias Vernengo - Outubro de 1993 - 249 pág (esgotado)
224. A SUBSTITUIÇÃO DE MOEDA NO BRASIL: A MOEDA INDEXADA - Fernando de Holanda Barbosa e Pedro Luiz VaUs Pereira - Novembro de 1993 - 23 pág.
225. FINANCIAL INTEGRATION AND PUBLIC FINANCIAL INSTITUTIONS - Walter Novaes e Sérgio Ribeiro da Costa WerIang - Novembro de 1993 - 29 pág
226. LAWS OF LARGE NUMBERS FOR NON-ADDITIVE PROBABILITIES - James Dow e Sérgio Ribeiro da Costa Werlang - Dezembro de 1993 - 26 pág.
227. A ECONOMIA BRASILEIRA NO PERÍODO MILITAR - VERSÃO REVISADA - Rubens Penha Cysne - Janeiro de 1994 - 45 pág. (esgotado)
228. THE IMP ACT OF PUBLIC CAPITAL AND PUBLIC INVESTMENT ON ECONOMIC GROWTH: AN EMPIRICAL INVESTIGATION - Pedro Cavalcanti Ferreira - Fevereiro de
1994 - 37 pág. (esgotado)
229. FROM THE BRAZILIAN PAY AS VOU GO PENSION SYSTEM TO CAPITALIZATION: BAILING OUT THE GOVERNMENT - José Luiz de Carvalho e Clóvis de Faro - Fevereiro de 1994 - 24 pág.
230. ESTUDOS SOBRE A INDETERMINAÇÃO DE SENIOR - voI. II - Brena Paula Magno Fernandez, Maria Tereza Garcia Duarte, Sergio Grumbach, Antonio Maria da Silveira (Coordenador) - Fevereiro de 1994 - 51 pág.( esgotado)
231. ESTABILIZAÇÃO DE PREÇOS AGRÍCOLAS NO BRASIL: AVALIAÇÃO E PERSPECTIVAS - Clovis de Faro e José Luiz Carvalho - Março de 1994 - 33 pág. (esgotado)
232. ESTIMATING SECTORAL CYCLES USING COINTEGRATION AND COMMON FEATURES - Robert F. Engle e João Victor Issler - Março de 1994 - 55 pág. (esgotado)
233. COMMON CYCLES IN MACROECONOMIC AGGREGATES - João Victor Issler e F arshid Vahid - Abril de 1994 - 60 pág.
234. BANDAS DE CÂMBIO: TEORIA, EVIDÊNCIA EMPÍRICA E SUA POSSÍVEL APLICAÇÃO NO BRASIL - Aloisio Pessoa de Araújo e Cypriano Lopes Feijó Filho - Abril de 1994 - 98 pág. ( esgotado)
235. O HEDGE DA DÍVIDA EXTERNA BRASILEIRA - Aloisio Pessoa de Araújo, Túlio Luz Barbosa. Amélia de Fátima F. Semblano e Maria Haydée Morales - Abril de 1994 - 109 pág. (esgotado)
236. TESTING THE EXTERNALITIES HYPOTHESIS OF ENDOGENOUS GROWTH USING COINTEGRATION - Pedro Cavalcanti Ferreira e João Victor Issler - Abril de 1994 - 37 pág. ( esgotado)
237. THE BRAZILIAN SOCIAL SECURITY PROGRAM: DIAGNOSIS AND PROPOSAL FOR REFORM - Renato Fragelli; Uriel de Magalhães; Helio Portocarrero e Luiz Guilherme Schymura - Maio de 1994 - 32 pág.
238. REGIMES COMPLEMENTARES DE PREVIDÊNCIA - Hélio de Oliveira Portocarrero de Castro, Luiz Guilherme Schymura de Oliveira, Renato Fragelli Cardoso, Sérgio Ribeiro da Costa Werlang e Uriel de Magalhães - Maio de 1994 - 106 pág.
239. PUBLIC EXPENDITURES, TAXATION AND WELFARE MEASUREMENT - Pedro Cavalcanti Ferreira - Maio de 1994 - 36 pág.
240. A NOTE ON POLICY, THE COMPOSITION OF PUBLIC EXPENDITURES AND ECONOMIC GROWTH - Pedro Cavalcanti Ferreira - Maio de 1994 - 40 pág. (esgotado) 241. INFLAÇÃO E O PLANO FHC - Rubens Penha Cysne - Maio de 1994 - 26 pág. (esgotado) 242. INFLATIONARY BIAS AND STATE OWNED FINANCIAL INSTITUTIONS - Walter
Novaes Filho e Sérgio Ribeiro da Costa Werlang - Junho de 1994 -35 pág.
243. INTRODUÇÃO À INTEGRAÇÃO ESTOCÁSTICA - Paulo Klinger Monteiro - Junho de 1994 - 38 pág. (esgotado)
244. PURE ECONOMIC THEORIES: THE TEMPORARY HALF-TRUTH - Antonio M. Silveira - Junho de 1994 - 23 pág. (esgotado)
245. WELFARE COSTS OF INFLATION - THE CASE FOR INTEREST-BEARING MONEY AND EMPIRICAL ESTIMATES FOR BRAZIL - Mario Henrique Simonsen e Rubens Penha Cysne - Julho de 1994 - 25 pág. (esgotado)
246. INFRAESTRUTURA PÚBLICA. PRODUTIVIDADE E CRESCIMENTO - Pedro Cavalcanti Ferreira - Setembro de 1994 - 25 pág.
247. MACROECONOMIC POLICY AND CREDffiILITY: A COMPARATIVE STUDY OF THE F ACTORS AFFECTING BRAZILIAN AND IT ALIAN INFLATION AFTER 1970 -Giuseppe T ullio e Mareio Ronci - Outubro de 1994 - 61 pág. ( esgotado)
249. CUSTOS DE BEM ESTAR DA INFLAÇÃO - O CASO COM MOEDA INDEXADA E ESTIMATIVAS EMPÍRICAS PARA O BRASIL - Mario Henrique Simonsen e Rubens Penha Cysne - Novembro de 1994 - 28 pág. (esgotado)
250. THE ECONOMIST MACIDA VELLI Brena P. M. Femandez e Antonio M. Silveira -Novembro de 1994 - 15 pág.
251. INFRAESTRUTURA NO BRASIL: ALGUNS F ATOS ESTILIZADOS - Pedro Cavalcanti Ferreira - Dezembro de 1994 - 33 pág. (esgotado)
252. ENTREPRENEURIAL RISK AND LABOUR'S SHARE IN OUTPUT - Renato Fragelli Cardoso - Janeiro de 1995 - 22 pág.
253. TRADE OR INVESTMENT ? LOCATION DECISIONS UNDER REGIONAL INTEGRATION - Marco Antonio F.de H. Cavalcanti e Renato G. Flôres Jr. - Janeiro de 1995 - 35 pág.
254. O SISTEMA FINANCEIRO OFICIAL E A QUEDA DAS TRANFERÊNCIAS INFLACIONÁRIAS - Rubens Penha Cysne - Janeiro de 1995 - 32 pág. (esgotado)
255. CONVERGÊNCIA ENTRE A RENDA PERCAPITA DOS ESTADOS BRASILEIROS -Roberto G. Ellery Jr. e Pedro Cavalcanti G. Ferreira - Janeiro 1995 - 42 pág.
256. A COMMENT ON liRA TIONAL LEARNING LEAD TO NASH EQUILffiRIUM" BY PROFESSORS EHUD KALAI EHUD EHUR - Alvaro Sandroni e Sergio Ribeiro da Costa Werlang - Fevereiro de 1995 - 10 pág.
257. COMMON CYCLES IN MACROECONOMIC AGGREGATES (revised version) - João Victor Issler e Farshid Vahid - Fevereiro de 1995 - 57 pág.
258. GROWTH, INCREASING RETURNS, AND PUBLIC INFRASTRUCTURE: TIMES SERIES EVIDENCE (revised version) Pedro Cavalcanti Ferreira e João Victor Issler -Março de 1995 - 39 pág.(esgotado)
259. POLÍTICA CAMBIAL E O SALDO EM CONTA CORRENTE DO BALANÇO DE PAGAMENTOS -Anais do Seminário realizado na Fundação Getulio Vargas no dia 08 de dezembro de 1994 - Rubens Penha Cysne (editor) - Março de 1995 - 47 pág. (esgotado) 260. ASPECTOS MACROECONÔMICOS DA ENTRADA DE CAPITAIS -Anais do Semináril,
realizado na Fundação Getulio Vargas no dia 08 de dezembro de 1994 - Rubens Penha Cysne (editor) - Março de 1995 - 48 pág. (esgotado)
26l. DIFICULDADES DO SISTEMA BANCÁRIO COM AS RESTRIÇÕES ATUAIS f COMPULSÓRIOS ELEVADOS - Anais do Seminário realizado na Fundação Gelulil Vargas no dia 09 de dezembro de 1994 - Rubens Penha Cysne (editor) - Março de 1995 47 pág. (esgotado)
262. POLÍTICA MONETÁRIA: A TRANSIÇÃO DO MODELO ATUAL PARA O MODEL( CLÁSSICO - Anais do Seminário realizado na Fundação Getulio Vargas no dia 09 d dezembro de 1994 - Rubens Penha Cysne (editor) - Março de 1995 - 54 pág. (esgotado) 263. CITY SIZES AND INDUSTRY CONCENTRATION - Afonso Arinos de Mello Franc
Neto - Maio de 1995 - 38 pág. (esgotado)
