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COMPUTING VALUE CORRESPONDENCES FOR REPEATED GAMES WITH STATE VARIABLES James Conklin Banco de España - Servicio de Estudios Documento de Trabajo nº 9622 COMPUTING VALUE CORRESPONDENCES FOR REPEATED GAMES WITH STATE VARIABLES (*) James Conklin (*) This research was conducted while the author visited the Banco de España. Errors and omis· sions are the author's own. Banco de España - Servicio de Estudios Documento de Trabajo n' 9622 In publishing this series the Banco de España seeks to disseminate studies of interest that will help acquaint readers better with the Spanish economy. The analyses, opinions and findings of these papers represent the views of their authors; they are not necessarily those of the Banco de España. ISSN: 0213-2710 ISBN: 84-7793-499-) Depósito legal: M. 29034-1996 Imprenta del Banco de España Abstract The nature oC repeated interaction has been extensively studied in the repeated garne liter­ ature. Abreu (1988), berger develop a recursive approa.ch to characterizing repeated games by focusing on (1994) Abreu, Pea.rce an,d Stacchetti (1986, 1990), and Cronshaw and Luen­ the present values oí suhgame perfect strategies for each player, V. Judd and Conk1in Cronshaw and Rutherford (1994) and Cronshaw (1996) (1995), have implemented these techniques computationally. Sorne oí the most interesting examples oí strategic interaction, however, arise in environments with state variables in wruch the recursive techniques cited aboye can­ not be employed. In such environments tbe set oí values oí suhgame perfect equilibrium becomes a function of the state variable-the object of inlerest becomes the value correspon­ den ce. This paper presents a general method for computing value correspondences under perfect monitoring and discounting. -3- 1. Introduction The nature oí repeated interaction has been extensively studied in the repeated game liter­ ature. While the theory oí repeated games has produced important qualitative results, one weakness oí this theory is the lack oí quantitative applications. A second weakness is that this theory has yet to be broadly extended to environments with state variables. In an earlier paper we address the first weakness, presenting a method for computing the set oí subgame perfect equilibria in infinitely repeated games witb perfect monitoring. Our approach is re­ cursive, lollowing the lead 01 Abreu (1988) and Abreu, Pearch and Staccbetti (1986, 1990), and Cronshaw and Luenberger (1994). Here, we extend that technique to games with state variables, restricting the state space to have finite elements. The recursive tbeory of supergames introduces the concept of self-generating sets of pay­ offs to characterize the set of equilibria. The numerical method discussed below revolves around efficient ways to approximate and construct these self-generating sets. As in Cron­ shaw and Luenberger (1994) and Judd and Conklin (1995), we allow public randomization. This is advantageous since the resulting elements oC the equilibrium value correspondence are convex, and a convex set can be represented by a simple function denoting the distance Croman interior point to each point on its boundary. We cboose a piecewise linear procedure to approximate these boundary functions. This method is of interest since it generates a "lower bound" to the true solution and is computationally cheap to evaluate and recom­ puteo We iIlustrate the method with an example: computing subgame perfect equilibria Cor government fiscal poliey in a primitive society. This example needs moderate computer power. We give running times -on Pentium­ based personal computers. Furthermore, we used standard, freely available software. Al! the prograros were written in FORTRAN, but could be implemented in any programming language, such as BASIC or Matlab. Section 2 covers notatiofl. Section 3 discusses the methods for approximating convex sets and describes the basic algorithm. Section 4 applies the algorithm to the two examples, and Section 5 concludes. 2. Supergames and Characterization of Equilibrium Payoffs We examine an N-player infinitely repeated game with state variables. A;, i = 1,···, N, will denote the action space of player i in the stage game. X = U�=¡{Xk} will denote the state space of K finite elements. We will call elements of A == Al X A2 X . . . X AN action profiles. A X X -t R. (aJ,·· ·,Gi_1!ai+1,··· ,aN ) and Jet Player i's payoff in the stage game will be II¡ : the standard way, we define a_i = Abusing notation in (1) -5- denote the best response payoff for pIayer i given the actions a_i by his opponents and the state x. Considering that for each value of x we can define a one+period garne, we rnake the following assurnptions: Assumption 1: Ai, i = 1,···,N, is a compact subset of Rm for sorne m. Assumption 2: II¡(·,x), Assumption 3: The stage game has a pure strategy Nash equilibrium for x Assumption 4: The value correspondence is non-empty for a1l values of x E X. i= l, ...,N, is continuous. xk,k _ 1, ... , K. Assumptions 1 and 2 are trivially satisfied by finite-play games. Assumptions 3 and 4 are satisfied by many interesting games, and are essential. Mixed strategy equilibria could be included if actions consisted of publicly observed randomization over pure strategy actions. Prom the stage game, we can construct the corresponding supergame. The action space in the supergame is Aoo == x�lA. We assume that player i aims to maximize his average discounted payoff, which equals 1 - ó ó 00 L 6tIli(at,xt) (2) t=1 where aj is the action profile taken in period t. A history, {a"x,}�=o. By the continuity of ni and compactness of A;, yt, is defined as the sequence lI;(x) _ min n¡(a,x) (3) ni(x) _ ma.>< lI;(a,x) (4) .eA .eA are both finite. Therefore, for any given value of the state, the supergame payoffs are contained in the compact set W(x) = X"II.!!i(X), lI;(x)]. (5) A strategy for player i is a mapping from the history of the game in time t ioto pIayer i's . action space Cor each time t: Ui : A! x XL _ A¡, t 0,1,2, .... A strategy profile is the set strategies oC all pIayers: u = UieN 17;. For each subgame perfect strategy profile 17, we define its value to player i to he :;: Vi(XI,U) = 1-6 00 ó 'lIi(a"x,lu). ó- ¿ t =1 -6- (6) It foHows that for each state there is a value players across aH strategies: U V(x)= set, representing the values that arise for v(x,<1),xEX. all (7) SPE Finally, we define the value corrspondence to be (8) Our objective is to compute the value correspondence V(X). The perspective implicit in the notation aboye does not provide us with a numerically tractable way of viewing the supergame since it focusses on infinite sequences of actions. While such descriptions are useful theotetically and descriptively, they are not useful when it comes to computation since infinite sequences are costly, if at all possible, to represent in a computer. Furthermore, examining the set of possible strategies and checking for subgame perfection is also difficult if one takes an infinite sequence approach. Naturally, we take a dynamic prograrnming approach to the problem, compressing the information in the game's histories into continuation values and extend the approach to garues with state variables. This "dynamic progarnming approach" to supergames with state variables can be ex­ pressed through a map, B. Let W(x) be a ca.ndidate for V(x) for each state x. Then the set of possible payoffs today are those consistent with N.ash play today with continuation values in W(x'), a.nd are defined by the map B(W)(x)= U (a,w(x'))EA xco(W(x')) {(i-ó)lI(a,x)+ów(x')} X'= h(a,x) subject to: i = h(ii;,a_;,x) (1 -ó)lI;(a,x) + ów;(x') � m.ax{ (1 ; ' - ó )lI;(ii;,a_;, x) + óll!;(i), \li E N). (9) (!O) (11) (12) h(a, x) is tbe Iaw oC motion of tbe state variable. Intuitively, a value b(x) is in B(W)(x) iC tbere is sorne action profile, a, and continuation payoff, W(X')EW(x'), such that b(x) is the value oC playing a today and receiving the continuation value w(x') tomorrow, a.nd, for each i, player i will cboose to play a¡ because he believes tbat to do otherwise will yield him the worst possible continuation payoff. The right ha.nd side oC equation 12 represents tbe value oC deCection from the prescribed equilibriurn. For player i playing a¡ implies that the state tomorrow will be x'= h(a,x). However if player i defects a.nd takes action a·, the economy will Dot be in state x' tomorrow but state x· = h(ai,a_i,x). Hence when computing the value oC defection, the punishment value must be drawn Crom the set W(x·). - 7- Proposition: proof: V(X) = B�WO(X) il WO(X) 2 V(X). The numerical algorithm"described in section While this characterization oí V(X) ficulties associated with computing ¡t. 3 provides a constructive prooL is elegant and intuitive, there are important dif­ The maio problem is that V(X) can be aD ugly correspondence, difficult to represent in a computer. To deal with these problems, we alter the supergarne by including public randomization. Intuitively, we assume that a.t the end oí each stage game, a lottery is used to determine how the game continues. Formally, strategies will now map histories of players' a.ctions, the history oí the realizations oí tbe lottery and the current lottery outcome, ioto current actions. Realized stage-game payoffs will differ from expected stage-game payoffs in strategies where non-trivial (non-degenerate) lotteries are employed. Operationally, we imagine the Collowing scenario. tomorrow's continuation value líes in co(W(x')). AH the players know that However, they don't know what that con­ tinuatíon value will be until after a publicly observed random valuable is revealed, whích will signal tomorrow's w(x') E W(x'). ThereCore, co(W). Crom today's perspective, we can ex ante construct a continuation value in While the main motivation Cor adding lotteries is to make the computation easier, it does generalize the notion oC subgame perCect Nash equilibrium in an appropriate and interesting Cashion. Since one aim oC this research is to see what is possible in the absence oC contracting, it is natural to add this public randomizing device. The addition oC lotteries will possibly be inessential. In particular, iC not expand V(X) and V(X) without lotteries is convex, then adding lotteries will our algorithm wiU compute V(X). One couId also further generalize the anaIysis to correlated equilibria. However, that would admit private information and assume sorne coordinating agent, unappealing assumptions in many contexts. 3. Computing the Equilibrium Value Correspondence Choosing.n initia! set B(Wk)(x), WO(X) 2 V(X), the sequence 01 sets {W'(X)}f:o, where W'+I(x) V(x) Cor each x E X. Direct application of this technique, however, = converges to poses two computational obstacles. The first is that Cor many supergames, equilibrium value correspondence V(X) is infinite and thereCore impossible to represent on a computer. We treat this problem by exploiting the convexity of W(X) and B(W)(X): any ray originating from any'interior point of a convex set intersects the boundary of that set only once. convex set can be represented by its boundary, a fact we will exploit. A By virtue oC its "star-shapeness", a convex sets comprising the value correpondence can be represented in a particularly simple fashion; we discuss the details of this in section 2.1. W(X) B(W)(X) can involve exhaustive pairing of actions with all possible value (Wi(X'),!Qo(x')) to derive incentive compatible actions and rewards. This technique is The second computational obstacle we confront is that even if we can represent finitely, generating pairs, -8- too crude for anything but tbe smallest problems. As an illustration, consider constructing B(W)(X) W(X) in direct fasbion: for a given state today, x, and the state tbat will x' h(a,x), applying tbe definition of B(W)(x) for each player, you pair an action ai with two continuation values (Wi(X'),y!.{r(a¡, x» ) E Wi(x') x Wi(.i(ai,a_i,x)). 1f.(w,(x'), !Q;(x(a;,x))) s.tis6es incentive compatibility, (1 - ó)Il,(a,x) + ów,(x') ;:: (1 ó)Il;(a_i,x) + ó",,(x(a;,x)), then we koow the oew poiot b,(a,x, w,(x')) (1 - ó)Il,(a,x) + áw;(x') will lie in B(W)(x). Doing this exhaustively, for every possible combination of action profile and promisc·threat pair, will generate the correspondence B(W)(X). Exhaustive from arise tomorrow, = = pairing is clearly impracticaJ for most games. boundary of 2.2 B(W), ignoring interior points. We avoid it by directly approximating the We describe this method in section 2.1. Section covers more specific details of the algorithm's implementation. 3.1 Piecewise Linear Approximations of Convex Sets Since tbe sets the comprise the elements of tbe value correpondeoce are convex, we need efficient ways to approximate'convex sets in the computer. Also, for tbeoretical reasans apparent below, we wil1 want to use approximations which allow us to have useful and relatively precise inforrnation about the approximation error. For these rcasans we will use piecewise linear approximations. Sinee we will only develop a twa--player implementation of our methods, our discussion will focus on approximating twa--dimensional convex sets, such as those in Figure 1. 3.2 Value Set Iteration for Convex Sets To do thi s, we exploit the sl.r-sh.peness assumption that there exists sorne point of.e.eh 01 the vNa(x,\:) sel B(W)(x,), k = 1,2, . . . K .nd Ihe for every state which is the equilibrium value of at least one subgame perfeet equilibrium whose path passes through tbat state. W(X,l;) and B(W)(x,l;). Any ray originating from 8B(W)(x,\:) only once. Henee, we can deter· mine the boundary at a given direction 9 E [0,211"] by extending a ray from the set center vNa(x,\:). Let a be the length oí the ray and let w(a,9,x,\:) be the poiot at the "tip" of the rayo We want to increase a until w(a,e,x,l;) is no longer in B(W)(x,\:). To determine if w(a, 9, x,\:) E B(W)(x,\:), we check two conditions. First, tbere must be both an action profile and a eontinuation value promise and threat, (a,w(x,\:),Yl(ai, a_j, x¡(a;,a_j" x,\:»), tbat sup· port w(a, 9). Second, the supporting triple (a, w(x,\:),.w.(ai, a_i, XI(ai, a_j,,x,\:) must satisfy incentive compatibility. Maximizing a at each 9 E [0,211"], subject to these two constraints, Refer to Figure 1, where vNa(x,\:) vNa(X,l;) is in intersects the boundary oí the set -9- Figure 1: Star-shapeness - 10- oC B(W) defines a periodic function representing 8B(W)(xlo): ) ( ") In Equation (9), the point tii(a, O,XI:) is written as VNc(xk)+av(O), where v(O) = (cos O,sin O). In order for VNc(Xk) + av(O) to be in B(W)(Xk), this point must be generated by a triple (a, W(xk),1Q(a¡,a_i,x/(ai,a_i" XIo» where vNc(xlo) + av(8) (1 - ó)IT(a, Xk) + ów(x'), for sorne w(x') E W(x'). This irnplies that [(vN·(x.) + ,,¡¡(e) - (1 - S)TI(a,x.)JlS w(x'). Imposing w(x') E W(x'),x' = h(a,xlo) yields condition i. in the equation. If we define a grid of M angles, e � [0, 211"J, this representation reduces the problem of finding the boundary 8B(W)(Xk) in every direction 8 for each state k E {1, ... , K} to M x K constrained maximization problems, a task that can be handled by conventional software and hardware. We compute the total approximation of the boundary function R(8) by first convexifying the vertices defined by the R(9m) and computing the appropriate coefficients for the linear approximation between vertices. Figure 2 illustrates the method applied with an angular grid. Starting at the "center" VN"(Xk), we select an angle 9m E 0. At that angle, we extend a ray to the boundary of B(W)(xlo) by maximizing 0'. At that maximized 0''", tii(a'",Om, Xk) is the interpolant vertex for the piecewise linear approximation of B(W)(xlo). Note that our constraints appear in the set W(x'). For this examplc, the binding con st raint on maximizing a is that we hit aW(x/) - W(x') will afford us no more extravagant promises than w(x'). Direct application of this technique, however poses two computational obstacles. The first is that for many supergames, equilibrium value sets V(x) are infinite and therefore im­ possible to represent on a computer. We treat this problem by invoking lotteries to render W(x) and B(W)(x) convexo Any ray originating from any interior point of a convex set intersects the boundary of that set only once. By virtue of this quality, "star-shapeness", a convex set can be represented by its bounda.ry, hence, in finite fashion. The second compu­ tational obstacle we confront is that even representing W(x) finitely, generating B(W)(x) can involve exha.ustive pairing of actions with value pairs, (wi(x),.lQ¡(i)). This technique is too crude for anything but the smallest problems. As an illustration, consider constructing B(W)(x) from W(x) in direct fashion: applying the definition of B(W), you pair an action profile with a continuation value: (a,w(x') E A x W(X'). If (a,w(x')) satisfies equations 2023, then we know the new point w(a,W(X/), x) (1-ó)Il(a, x) +ÓW(X') will lie in B(W)(X'), Xl = h(a,x). Doing this exhaustively, for every possible action profile and promise pair, will = = I = generate the correspondence B(W)(x). Exhaustive pairing is c1early impractical for most -11- games. We avoid it by using a piecewise linear approximation of the boundary of B(W)(x), ignoring interior points, and directly approximating the boundary of B(W)(x) from the boundary of W(x). 0.1 Speciflcs of Value Set Iteration for the 2-player Case. At this point, we shall discuss the steps in detail for the special case where N = 2, Al> the action-space of player 1, is finite with J elements, and A2 is finite with L elements, and the state space has K elements. Starting with an initial W( X) that we are sure contains the equilibrium value correspondence V(X), we apply the operator B(o) to W until 11 B(W) W II"N.S é, where é is the convergence criterion and 11 II"N. is the root sum of squared differences oC o� at each (Jm between iterations. We summarize the principal steps for the operator B(·) witb the following scheme: o: Initialize 0.1: Select vNa E V(x) for x E 0.2: Set angle grid e = X. {thet." .. . ,OM} � [0,2�] 0.3: Seleet initia] R(O)(x) lengths for eaeh o for 1: x E X. E e, thereby eonstructing a W(x) 2 V(x) For each Xk E X do 2: For each (Jm E e do 2.1: For each action profile (aj,al) E A do o(aj,a,) = max a s.t. (i) 6-1[(VN(X.) + av(Om)) - (1- 6) n(.j,.¡,x.))] E W(x'(aj ,a" Xk)) (1- 6)n (.¡,.¡,x.) + 6w( x') (ii) � (1- 6)W(._;.x.)+ ów(i(aj,ah Xk)) where o(aj,al,Xk)= 2.2: O (iii)· a � O is understood if the constraint set is empty. o·((Jm,Xk) = max(o(at.a¡,xk) ,o(alla2,xk), . .. ,o(aJ,aL)); 2.3: w(a·( Om.x, ) . Om,X' ) = vN·(x.) + a·(Om.x.)v(Om); 3: Determine those elements oC the set UmEMW(O·((Jm,Xk), Om,Xk) which lie in W·, the boundary oí the set 's oonvex hu11; 4: Compute piecewise linear approximation oC the boundary oC compute the convex hull oC the points from (2). -12- B(W)(Xk); specificalIy, 5, W(X) Sel Step B(W)(X). = W has changed little; else go lo 1. in the scheme specifies that the a.Jgorithm "loop" through each angle 1 ... M. We maximize (10). Slop if O' 8"" m = in at each angle, subject to the constraints that appear in Equation These constra.ints are functioos oC action profiles, and sorne profiles will be more constraining than albers. We loop through all action profiles in step profile tbat yields the maximal each iteration on W, a. 1.1, 8m In step a we maximize M x J x L numerical subproblem oí deciding if a paiot is in (aj,a¡) and 1. ti to find tbe a.ction are fixed, meaning that for x K times. Note that in 1.1, there is the W(Xk). These points show that the heart oí the algorithm is a constrained maximization problem. To understand this constrained maximization problem, we study how large before hitting one of the two constraints. l.). With reference to Figure 3, consider a can be constraint w(x'), used to support today's vN (x, ) + av(Óm) = (1- 6)II(a"a"x,) + 6w(x'), musl be in W(x'). break w(x') down into two components, one that varies with 0', and one This constraint didates that tombrrow's value promise, value, w(x"a,Om) Moreover, we can = , that is fixed: w (x' ) _ - vN,(x,) - (1- 6)II(a"a"x,) fj This decomposition appears in Figure approaches tbe boundary of this quantity as ir. Now consider constraint W(x') A3. � " + fj V(Vm) = _ Va( Xk) + � (ió ) fj V Vm (14) We increase a so that along tbe angle (Jm' When w(a,x') = Va (Xk) + lV(Om) w(a,x') strikes the boundary, ¡i.), the incentive compatibility (le) condition. We can rearrange this constraint as the inequality w;(x') � 6-'[(1- 6)(II;(a "a" x,) - 1I;(a_;,x,)) - 6m (x')], Examine Figure player A3. as el a.nd el in Figure a, they Since these constraints are not a function of 1, borizontal for player 2, A3. i = 1,2. and shift as a function of the action profiJe (15) are vertical for a. They appear The le constraints are binding if a given player's value promise, V;,a(Xk)+ ¡Vi, is equal to tbe le constraint. In Figure A3, tbis is shown happening at o' = ó. Therefore, at the angle 8m and the action profiJe (aj,al), Ó is the constrained maximum for step 1.1; tbe le constraint for player 2 and tbe constraint w E W w;(o',x') = for player 1 are not binding. Tbe maximization problem oí step 1.1 boils down to solving a Cew linear equations that identify tbe intersedions oC the ray at angle two le constraints and the boundary oC We do, tbe maximization in step 8m originating at Va, witb tbe 1.1 for alI possible action profiles. impJy different constrained maxima for show bow botb W. a. This is shown in Figure Va and the le constraints shiCt as a function oC Q. I One might exploit game structure to economize on this search - 13 - I Different action profiJes A4. In tbat figure, we eonsider the action profile as one would have to if A were continuou$. � w( ,,�e..l =- 11"'+ ¡( if B{W) =(I-rlTr/a)< (J... I I ,W ----�-�==--� !.!l, --+Figure 2: Extending the ray v + ov at the boundary oí B(W) -14- v.... B{'N) '1f. - .��- 1r""_(J-f)1r/G) Ó ;,-.... Figure 3: Extending the ray at angle () with the a.ction profile fixed - 15 - B(V.¡) j " ...". ... +«,IV" --- "V.....tI% ,.;r � 'Y"'" �t<'JV ¿'v.:(J) ir"" �(r) .. Figure 4: . . '1,r"(3)) Ray extension from varjous action profile s -16- a(l): it shifts Va(t) to the upper Ieft of vNa. As we inerease a at angIe 8m, we find that we encounter pIayer l's re constraint first, it(a(l». Hence, a.ction profile a(l) produces a maximaI a of length al' Moving on to a.ction profile a(2) , Va(2) shifts directly below vNa. The le constraints for this action profile are non.binding, therefore we hit the constraint Va(2) + ¡'v(8m) = w(a) E W first. This determines that the maximal a for a(2) is 0'2' For action profile a(3) we hit pIayer 2's le eonstraint first (the le constraint for pIayer 1 is not depicted) giving us a muimal a which we denote 0'3' Aetion profiles do not always generate a point Va which lies in W. We show this with action profile a(4). As we extend a, the point w(a) enters the set W. Hence, it is possible that although a = O is not in the constraint set, there are larger values of a that do fall in the constraint seto With 8m remaining fixed, after having looped through each action profile we determine the largest value o(aj, ak) and set it equal to o· (step 1.2). We determine the extremal elements of B(W) at each angle Om in cartesian coordinates, w(a"(Om),Om) = v: + o·v(8m) (step 1.3). We then convexify this set of extremal points and then fit an approximation of R(O) (steps 2 and 3). 4. An Example: Credible Fiscal Policy in a Primitive Society. In this example a government must finance a varying but periodic series of (exogenous) expenditures, either through varying the taxes paid by its constituents, or through financing fluctuations in expenditures with the help of a lendee. Government's objective is to maximize the welfare of its constituents. Included in the specification of the government's action space is a budget constraint. If an action violates tbe budget constraint, it is rendered unavailable. Tbougb the government's source of revenue is a tu 00 the productioo of bouseholds, it ca.nnot directly choose allocations of consumption and leisure on behalf of the households. Naturally, these allocations are ehosen by the households themselves, in light of the tax rateo Taxes are distorting and therefore decrease households' welfare beyond losses that would arise tbrougb lump sum taxation. Due to its restriction to a distorting instrument of taxation, the government prefers to smootb taxes, since tbis maximizes household welfare subject to the government financing its expenditures. Tbe lender is risk neutral. He lends· to the govemment out of an endowment he receives each period, w. His payoff, de, is the sum of the endowment he receives each period less tbe net transfer made to the government. One can interpret the lender as a banking house or cartel of banks, assuming also that the coalition is rohust to defection by tbe individual players tbat comprise it. The fiscal problem faced by tbe government reduces to a two..player ga.me between tbe government and its lender. The government is tempted to default on tbe lender once it has debt outstanding¡ the lender protects birnself from such opportunism with the threat of witbdrawing lending services indefinitely in the future sbould tbe government defa.uIt. - 17 - Literature 00 sovereign leoding2 has shown that, perhaps surprisingly, this threat is not sufficient to support lending in all environments. That is, the Folk Theorem does not hold for a11 games of sovereign lending. The key to such "no lending results" is whether the government has access to an alternative channel for smoothing taxes: if the government does not have access to a storage technology, bankers effectively have a sanction technology. · They can deny the government the means by which to smooth taxes in the face of fluctuating expenditures. If the government does have access to a storage technology (e.g., or a state­ contingent form of savings deposity in an environment of uDcertainty), it can self-smooth, and will always be better off not paying lenders, and using the proceeds of default to build a tax-smootbing fundo As an illustration of the algorithm, we replicate these well-known results in the economy described aboye. 4.1 Program 1: the government cannot save The bargaining session works as. íollows: a11 lending matures in one periodo Lending oí tonger maturity must be arranged by successively "rolling over" principal. The government begins by stating how much interest and principal from last period's loaos it will pay back, RIB,_I, where Rt is the gross factor oí return (one plus the interest rate). Simultaneously, the government states how much fresh credit it wants, Bt. Next, the ooalition oí lenders state how much they are willing to lend, E,. New credit extended is Bt min(Bt. Bt), and the net transíer of funds in the period is (B, - RtBt_d.l This specification reduces to an arrangement where the government and its lender bargain over net flows: let T, = (B, R,B,_,) and f, = (B, - R,B,_,). Then T, sign(f,)·min(lf,¡, If,1) il f, and f, are the same sign, T, = O ií T, and T, are of opposite signo The repeated game has a state variable, the leve! of government expenditure. For a given profile oí actions, players' current and expected future payoffs differ according to the value oí the state variable. The discounted present value of a strategy will aloo depend upon the current value of the state. Hence, the discounted value of an equilibrium is actually a set of values, indexed by the states tbat occur aJoog the path arising in that equilibrium. The set of equilibrium values is comprised oí many sets, one associated with each state of the world. Because the game is recursive, given the state of the world, the set of equilibrium values that can be achieved does not depend on time. Formally, the lender's problem is = = 'Bulow and Rogoff (198980, 1989b); Chari and Kehoe (199380, 1993b). sTec:hnically, the timing scheme amounts to a device of exp08ition: the solution algorithm used to compute eq,uilibruim utilizes tbe stage game in its normal form, which is equivalent to a version of tbe game wbere neitber player observes tbe otber's action at tbe time be cbooses. We do not adopt a Rubenstein bargaining set-up for negotiations, common in other of tbe debt literature. -18- V(9o) = m.!'x (16) subject tO: (17) {Te} if if (1 � -8)¿:: 8'd, 1=6 � 1'1 and TI are oC the sa.me sign (18) T¡ and Tt are oí the opposite signo V(9.) (19) In the problem fJ is the discount factor1 (I-b") is a normalization factor that scales the dis­ counted present value of utility.to the range oí single period utility values. Equation 3 is the bargaining rule .on nel transfers between government and hanker explained aboye. Equation 4 imposes subgame perfect behavior on the lender. The economy is inhabiled by a number oí identical households that produce the single consumption good by supplying labor. They have access lo a production technology, f(nd, that allows thero to convert labor. nI, ioto the consumption good. This fundian is increasing in its first derivative and concave. Households optimize over an infinite horizon, {) is the discount factor, and preferences are definoo over consumption and leisure, U(er. lt). The households' basic choice is how much to work versus how much to eat. The formal problem is max {CI'/'} subject to: (J � -8)¿::8'U(c"I,) t:O c, $ It (20) (1 - r,)! (n,) (21 ) 1 - nt. (22) The family is a "tax·taker", that is, it accepts the tax levy Tt as giveo each periodo The family does oot save because it is denied means to do so. Assume that production technology and preferences are 5uch that the revenue maximizing tax rate, T-, always yields' revenues sufficient to cover government expenditure: gt � T·f(nt(T")). Households do oot behave strategically. They go through life equating marginals in their repeated, static problem. The government solves the problem � (! -8) ¿:: 8' U(c"I,) t=O -19- (23) (24) subject to: U,(c" 1,)fU, (e" 1,) = (1 - T, )f'(n,) C , = (1 - T,)f(n,) ir ir (25) (26) Tt and Tt are oC the same sign (27) Tt and Tt are oC tbe opposite signo (1 -ó) L ó' U(c" 1,) ? W(y.) t =. (28) Equation 8 is tbe government's objective Cunction, equal to the discouted present value oC the weighted utility oC its subjects. Ut(Ch lt) > O, U2(Ch lt) > O and U(Ct, lt) is concave. Equation 9 is the government budget constraint. Domestic spending, 9t, must equal the net current flow oC Cunds Crom the lenders plus total revenue taxes. Equations 10 and 11 are the households' incentive compatibility constraints. Tbey determine how government taxes 11 yi eld the aggregate resource equilibrium condition Ct +dt +9t :;;; f(nt)+w. Equation 12 is the outcome rule for bargaining over tbe net flow of funds between the lender and the borrower. Equation 13 imposes subgame perCection on the government. Solutions to Program 1 show that in punishment phases oC pa' ths arising in subgame per· Cect equilibria, tbe government must move taxes up and down with fluctuating expenditure. In high welfare equilibria, the government borrows and repays to allow smoothing of taxes. Tbese results are graphed and discussed in greater detail in Section 4. affect consumption and labor supply. Equations 2, 9 and 4.2 Program 2: the government can save In this program, the government has a.ccess to a storage technology. This makes the lender's penalty oC cutting off future credit much less severe. The government can smooth without creditors, but it must save in advance. The government now solves the problem W(Yo, Ao) = 00 ",ax {....T.,A.+I} (l-ó)¿;ó'U(c" I,) t=O (8) (14) subject to: -20- U,(c"I,)/U,(c" 1,) e, = = (1- T,)f' (n,) (1 - T,)f(n,) if if (10) (11 ) TI and T, are of the same sigo (12) TI and Tt are oí the opposite signo 00 (l-ó)Ló'U(c" I,)::> W(g"A,) f=. (13) Households' and the lender's problems are unchanged from Program lo Granting the government a storage technology causes lending to break clown. This is because in the first period the government owes the lender a. repayment it can instead save the resources that were to be used to repay. In the next period the government has need of resources in arder to smooth taxes, it draws clown ¡ts savings. The lender, in anticipation of this, never ¡ends. These results are sensitive to tbe assumptions: T" f(C(T")¡) ;::: 91 is criticaL It is also important that the sequence {gtl� is known with certainty. If {gr} �o was stochastic, a. no-lending result would depend the on distribution and support of gl' With a storage technology and no lending the government is worse off at the best equilibria oí Program 2 than it would be at the best equilibria oí Program 1: in those states oí the world where the government has low savings and high expenditures, it must raise taxes and deíer a plan oí smooth taxes until the next period oC low government expenditure. If it could borrow, it could begin smoothing right away. These findings are sensitive to the efficiency oC the government's storage technology. [f it can save and earn interest equal to or greater than the lender's discount rate, there will be no lending whatsoever. As interest earned by the government approaches zero, or becomes negative, small amounts oC debt can be supported. As the storage technology becomes less efficient, lending nears levels for Program 1. This confirms results of ehari .nd Kehoe (1993b) .nd Bulow and Rogoff (1989.) in tbis environment. 4.3 Computational results For the computation oC all programs, we assume the Collowing íunctional forms: ( 15) - 21 - f(n,) = n,'. ( 16) The period ayer which the experiments will be run will be about three or four years. We choose this period length because we are modeling expenditure fluctuations and this seerns a reasonable duration to posit for a typical cause oí expenditure fluctuation, war. Therefore 6 0.9 ror most experiments. = 4.3.1 Computations for Program 1 To run the first computation we set 0") = 0.6, 0"2 = 0.2. While these elasticities are oot particularly "realistic", Cor the additive separable specification oí utility and the households' environment it yields negative labor supply response to the tax rateo (For lag preferentes, ¡.e. 0'"1 = (72 = 1, labor supply is inelastic; Cor 0"1 > 1 and 0'2 > 1, labor supply rises in response to higher taxes.) The production parameter "1 is set at 0.65. The process for exogenous government expenditure is of period two: one period of high expenditure, one oc. low (four years of wa.r, four years of pea.ce). {gl}�o {O, 0.24, O, 0.24, ...}. The endowment for the lender, w = 0.24. The lender is always capable of funding the government's full need for finance. Utility functions were scaled up or down by affine transformation so that payofrs lie in the interval [O, IOJ. Calling the government player 1, the lender player 2, and ·their respective payoffs IT1(a) and Ih(a), we have = 11,(a) =�, + 6U(c" 1,) and 11,(a) = (, + (,d,. The values � = (-26,15.5) and ( (0,20.8) map government and lender payoffs into ¡O, 10J x ¡O, IOJ. Finally, we, discretize both action and state spaces oí tbe government and bankers. Dis­ cretization is not necessary for varia.bles ct, and lt, for they are determined as a result of the government's choice of T and can be solved exactly without increasing computational complexity. For the government's actions, = TI r, = lJ,-R,B,_, e {-0.24, E {O.O, 0.2, 0.46, 0.715}; -0.18, -0.12, -0.06, 0.0, 0.06, 0.12, 0.18, 0.24}. -0.18, -0.12, -0.06, 0.0, 0.06, 0.12, 0.18, 0.24}. For the lender's a.ction:. r, = lJ,-R,B,_, e {-0.24, Negative transfers go from government to lender; positive transfers go írom lender to govern­ mento We have chosen the grids on gt, Tt and TI so that the accounting oí the government's -22- 6.2 r------�--__, 6.4 6.2 6 o 6 5.8 �5.8 i .2 5.6 '" �5.6 � = o � .. Q. o:: .E �5.4 5.4 � 5.2 5.2 5 5 4.85 • 5.5 6 6.5 4.84L- --o- -o5--: 6 --: - --7-: 7 - - Payoff tar Player 1 Payoff tar Player 1 Figure 1: Value sets, Program 1 budget ¡eaves only small amounts oí taxed resources unaUocated. However, the government still has the choice to give these inter·grid rounding errcrs back lo consumers lump sum, or lo give thero to the lender, as a small adjustment to the bargained transfer in the lender's favor. The value sets for Program 1 are shown in Figure 1 for tbe two values oC the state variable 9c. The axes oC tbe graphs represent the average oC the present discounted values each pIayer receives ftom playing a subgame perfect equilibrium. The value sel corresponding to peace (9t = O) appears on the left, war (ge = .24 ) on the right. There are many possible equilibria, and by extension, many equilibrium values in each state of the world. The punishment values on the southwest border of the value sets arise in two types of equilibria: the first are autarkic-the government.never borrows, and the lenders never lend. The second are "carrot and stick" punishments-in these equilibria, it is not best to play non.cooperatively, but rather to go along with your own punisbment. In such equilibria, the player that is known as the deviator deliberately chooses an action that results in a low one-shot payoff. - 23 - Of course the inducement is that the continuation promise associated with going along with one's punishment is high. As a con.sequence, carrot and stick paths can generate severe punishment5, yet they still allow pIayers to emerge from punishment, or non-cooperative play. It is possible to generate outcome allocations along �he paths that alise in equilibrlum from V(x) using optimal penal codes, in the language oC Abreu (1988). The evolution oC the value of playing a particular equilibrium is shown in Figure 5. The value of the outcome • ) in the state g, .24. ' can move all over the value sets. O, at the point (5.2, 5.4). Since gt = .24 in tbe subsequent perlod, hand value set: the star al (4.4, 5.44) i. the value oC Ihe equilibrium path i. marked by circl.. (o) in the otate g, = O and by staro ( = Firstly note that the value of playing a given equilibri� Play originates in g, = .kip over to the right in the second period. The value of play evolves rising to nearly the top of the value sets, altemating from one set to the other. Then, play moves the position of continuation values down, in southeast directioo, along the Pareto frootier. The Cour ploto in Figure top row (the govermnent bottom row. In periods 00 • graph the on...shot payoffs Cor each player through time on the the left, the lender on the right), the evolution of value on the 1-11, perlod payoft'a for tbe govemmeot are volatile, reftecting the fact that it is in punishment phase4 and that it ia not being allowed to smooth by tbe tender. Tbe government also mues small tranafers to tbe tender out of revenues not allocated to 9,. Trua is part of "going along with its own punishment." In perlod 10 the government begins lo mue larger IransCers lo the lender during periods oC peace (declining peaks in Ihe Iop left plot). With that show oC "good faith", the lender slarts lending lo Ihe governmenl. Govemment ta.xes, and perlod-payoft's are tben smooth for tbe remainder of play. The lender now aborbs the volatility of the sequence g, through financial flows. The four plots in Figure 3 graph the underlying economic allocations through time. Note thal tbis path i. only one arnong literally infinile palhs lhal arise in equilbrium. We were able to see allocations during a punishment because we chose to start the path at a punishment value. Paths moving along the northeast frontiers of the equilibrlum value sets always uve the govemment smootbing taxes. 4.3.2 Computations for Program 2 In Program 2, the government has the capacity to save. As with Program 1, parameters are al = 0.6, 002 = 0.2, and -¡ = 0.65. The process for exogenous government expenditure remains {gt}:'o = {O, 0.24, O, 0.24, . . . }, and tbe endowment for tbe lender w = 0.24. The values e (-26,15.5) and ( (0, 20.8). set as � � ·Once on a cooperative path, punishments do not arise in equilibrium as can happen in a game with imperCect inCormation. However by initiating tbe. game at a punisbment value, we are able to trace out punisbment patbs tbat would otherwise líe "off path. n - 24 - - i rt. 10 10 8 8 6 .;¡ 6 � Q. 4 ii: w 4 2 2 O O - 10 20 Periodo 30 O O 40 10 10 8 8 6 .;¡ 6 2 2 � Q. 4 :> O O 10 20 30 40 20 Periodo 30 40 Periodo f4 :> 10 20 Periodo Figure 6: 30 O O 40 10 Evolution of Value, Payoff's ayer time, Program 1 - 25 - 0.8 0.8 � 0.6 10.6 .. 0.4 � ° 0.2 "l;: 0.4 Ir o ° 0.2 O 10 30 20 O 40 jj 20 30 40 0.8 ;:. "R:O.6 <il .8 0.4 0.5 � � 10 Periad: Periad: !I 0.2 -0.5 O �--�4O 0�--�17 0--� 2O �--7 30 -1 � � -=--��� 40 0 --10 -- 20 30 Periad: Period: Figure 7; Evolution oí allocations over time, Program 1 - 26 - For the government's action space, T. E T {O.O, is discretized as 0.2, 0.46, The gross rate of interest for the government, R, 0.715}. is constant at 1/6 or 1.111. The govern­ ment's assets lie in the grid A. E {O.O, 0.06, 0.12}. The government's bargaining position for transfers Hes in T. = B.-R.BI-l E {-0.24, E {-0.24, -0.18, -0.12, -0.06, 0.0, 0.06, 0.12, 0.18, 0.24}. -0.12, -0.06. 0.0, 0.06, 0.12, 0.18, 0.24}. The lender's transfer offers lie in t. = B.-R.BI-l -0.18, There are six possible states in the garne, one for eac� combination of gt and The value sets for Program (g" A.) = {(O,O), (0,0.12), At. 2 are shown in Figure 8.. The only four states are shown, (.24,0), (.24,.12)}. The value set in e&eh st.te has col­ lapsed to a single point. TOOs is because lending breaks down, and there is a unique equilib­ rium action (and hence a unique subsequent path oí play and a unique value) in each state. The lenders never lend, and the government smooths using its own assets. The path shown is marked by a + and a 6 (gt = .24, At = x: .12), in state 1 (gt = O. it raises revenues oí At = .12, O), the government saves 0.12. In state draws down savings, and gives interest back to the households as a lump-sum payrnent. The penod payoffs and evolution oí values through time are shown in Figure �, and Figures 1 0 and l 1 show the economy's allocations through time. 5. Conc1usion In this paper we have described and implemented a computer algorthm to solve discounted sugergarnes with perfect monitoring and state variables. Key assumptions are restriction to pure strategies, the inclusion of public randomization and a finite state space. We represent the key object in the "dynamic prograrnrrung approach to supergames", the value set, by its boundary. This parsimonious representa.tion of the value set allows us to compute a.n inner approxima.tion oí the set rapidly, far more ra.pidly tnan would be possible using "brute force" methods such as exhaustive search. We demonstrate the algonthm for a. fiscal policy game played by a government and its lender. The level of government expenditures to be financed is the state variable. Com­ puta.tions venfy theoretical results of Cronshaw and Luenberger and Abreu, Pearce and Stacchetti, Bulow and 'Rogoff and Chari and Kehoe. Tbis computa.tiooal approach is gener­ alizable to games of imperfect information, stochastic garues, a.nd games oí asymmetnc in­ íormation. Areas oí research we are exploriog are wa.ys to produce an "outer-approxima.tion" oí the value set, and other a.pplications. - 27 - 6 6 '" '" I!i 5.5 � 5.5 ... .. l .e 5 ¡;: .e 5 + "' o ... � .f 4.5 4 6 ¡f 4.5 6.5 7 7.5 Payoff for Player 4 6 8 6.5 7 7.5 Payoff for Player 1 1 8 6 ,------, 6 ,------, '" � 5.5 '" :s 5.5 � l ¡;: .2 5 x .2 5 "' � ¡f 4.5 � .f 4.5 4 �--�------------� 8 6 7.5 6.5 7 --��--�--��� 4� 7 7.5 6 8 6.5 Payoff for Player 1 Payoff for Player 1 Figure 8 : Value sets, Program 2 - 28 - 10 10 8 8 .;.; 6 N 6 2IL 4 a: 2 � iL 4 2 O O O O 10 5 Penad: � 10 10 8 8 6 N 6 4 ¡¡: 4 >- :> 2 2 O O 10 !;. 2- IL 5 Penad: O O 10 5 Figure 9: 5 10 Periad: Perlad: Evolution oí Value. Payoffs over time, Program - 29 - 2 0.8 0.8 0.6 1 >< ';'! 0.4 ¡,; !' 0.6 .. a: :r- � CI 0.2 O O 0.4 iS CI 0.2 5 O lO O Period: � lO 0.8 0.5 � ! 5 Period: O -0 5 . O � ---5 -0L----10 -oL-------� 5 lO Period: Period: Figure l O: Evolution oC allocations over time, Program - 30 - 2 0.8 ,;: z el 0.8 0.6 .!i 0.6 0.4 a: � O.4 0.2 0.2 O O .. 5 O O 10 B O en .8 0.4 -0.5 � 0.2 -1 O 10 0.8 >. 8:0.6 0.5 ¡,; 5 Periodo Periodo " 5 O O 10 5 Periodo Periodo Figure 11: Evolution oí allocations over time, Program 2 - 31 - 10 References [1] Abreu, Dilip. (1988): "On the Theory oC Infinitely Repeated Games with Discounting," Economelrica 56, 383-396. [2J Abreu, Dilip, David Pearce, and Eneio Stacchetti (1986): "Optimal Cartel Equilibria with Imperfect Monitoriog," Joumal oi Economic Theory 39, 251-269. [3] Abreu, Dilip, David Pearce, and Enoio Stacchetti ( 1990): "Toward a Theory oC Dis­ counted Repeated Games with Imperfect Monitoring," Econometrica 58, 1041-1063. [4] Bulow, Jererny and Kennetb Rogoff. 1989a. "Is to Forgive lo Forget?" nomic Review 12: 497-513. American Eco­ [5) Bulow, Jeremy and Kenneth Rogoff. 1989b. '"A Constant Recontracting Model oC Sovereign Debt," Journal 01 Political Economy 97(1): 155-178. [6J Chari, V.V. and Patrick J. Kehoe. 1990. "Sustainable Plans," omy 98, 783-802. [7J Chari, V.V. Joumal 01 Political Econ­ and Patrick J. Kehoe. 1993a. "Sustainable Plans and Debt," 230-261. Journal ol Economic Theory 61: [8J Chari, V.V. and Patrick J. Kehoe. 1993b. "Sustainable Debt and Mutual DefauIt," 60: 175-195. Review 01 Economic Studies [9J Judd, Kenneth and Conklin, James. (199.5 ). "Cornputing Supergarne Equilibria," . rnirneo, Hoover Institution. [lOJ Cronshaw, Mark and David C . Luenberger ( 1994): "Strongly Syrnmetric Subgarne Per­ fect Equilibrium in Infinitely Repeated Carnes with Perfect Monitoring and Discount­ ing," Cames and Economic Behavior 6: 220 - 237. [11] Judd, Kenneth L. (1996): Numerical Methods in Economics, [12] Stokey, Nancy L. (1991): "Credible Public Poliey," and Control 15, 627-656. - 33 - manuscript. 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Sastre de Miguel: Los mercados de operaciones bancarias en España: especialización productiva y competencia. 94/1 Olympia Bover and Ángel Estrada: Durable consumption and house purchases: Evidence from Spanish panel data. 94/2 José Viñals: Building a Monetary Union in Europe: Is it worthwhile, where do we stand, and where are we going? (The Spanish original oí this publication has the same number.) 94/3 Carlos Chuliá: Los sistemas financieros nacionales y el espacio financiero europeo. 94/4 José Luis Escrivá and Andrew G. Haldane: The interest rate transmission mechanism: Sec­ 94/5 M.a de los Llanos Matea y Ana Valentina Regil: Métodos para la extracción de señales y toral estimates for Spain. (The Spanish original of this publication has the same number.) para la trimestralización. Una aplicación: Trimestralización del deflactor del consumo pri­ vado nacional. 94/6 José Antonio Cuenca: Variables para el estudio del sector monetario. Agregados moneta­ rios y crediticios, y tipos de interés sintéticos. 9417 Ángel Estrada y David López-Salido: La relación entre el consumo y la renta en España: un modelo empírico con datos agregados. 9418 9419 José M. González Mínguez: Una aplicación de los indicadores de discrecionalidad de la política fiscal a los países de la UE. Juan Ayuso, María Pérez Jurado and Fernando Restoy: Is exchange rate iisk hjgher in the E.R.M. after the widening oí fluctuation bands? (The Spanish original of this publication has the same number.) 9420 Simon Mitner and David Metcalf: Spanish pay setting institutions and perfonnance outcomes. 942/ Javier Santillán: El SME, los mercados de djvisas y la transición hacia la Unión Monetaria. 9422 Juan Luis Vega: ls lhe ALP long-run demand funclion stable? (The Spanish original of this publication has the same number.) 9423 9424 Gabriel Quirós: El mercado italiano de deuda pública. Isabel Argimón, José Manuel González-Páramo y José María Roldán: Inversión privada, gasto público y efecto expulsión: evidencia para el caso español. 9425 Charles Goodhart and José Viñals: Strategy and tactics of monetary policy: Examples from Europe and the Antipodes. 9426 Carmen Melcón: Estrategias de política monetaria basadas en el seguimiento directo de objetivos de inflación. Las experiencias de Nueva Zelanda, Canadá, Reino Unido y Suecia. 9427 Olympia Bover and Manuel ArellaRo: Female ¡abour force participation in the 1980s: the case of Spain. 9428 Juan María Peñalosa: The Spanish catching-up process: General deterrninants and contribution oi the manufacturing industry. 9429 Susana Núñez: Perspectivas de los sistemas de pagos: una reflexión crítica. 9430 José Viñals: ¿Es posible la convergencia en España?: En busca del tiempo perdido. 9501 Jorge Blázquez y Miguel Sebastián: Capital público y restricción presupuestaria guberna­ mental. 9502 9503 Ana Buisán: Principales determinantes de los ingresos por turismo. Ana Buisán y Esther Gordo: La protección nominal como factor determinante de las im­ portaciones de bienes. 9504 Ricardo Mestre: A macroeconomic evaluation oi the Spanish monetary policy transmis­ sion mechanism. 9505 Fernando Restoy and Ana Revenga: Optimal exchange rate flexibility in an economy with intersectoral rigidities and nontraded goods. 9506 Ángel Estrada and Javier Vallés: Investment and financial costs: Spanish evidence with pa­ 9507 Francisco Alonso: La modelización de la volatilidad del mercado bursátil español. nel data. (The Spanish original oi this publication has the same number.) 9508 Francisco Alonso y Fernando Restoy: La remuneración de la volatilidad en el mercado es­ pañol de renta variable. 9509 Fernando C. BaUabriga, Miguel Sebastián y Javier Vallés: España en Europa: asimetrías reales y nominales. 9510 Juan Carlos Casado, Juan Alberto Campoy y Carlos Chuliá: La regulación financiera espa­ 9511 Juan LuiS Díaz del Hoyo y A. Javier Prado Domínguez: Los FRAs como guías de las expec­ ñola desde la adhesión a la Unión Europea. tativas del mercado sobre tipos de interés. 9512 José M: Sánchez Sáez y Teresa Sastre de Miguel: ¿Es el tamaño un factor explicativo de 9513 Juan Ayuso y Soledad Núñez: ¿Desestabilizan los activos derivados el mercado al conta­ las diferencias entre entidades bancarias? do?: La experiencia española en el mercado de deuda pública. 9514 M: Cruz Manzano Frías y M" Teresa Sastre de Miguel: Factores relevantes en la determinación del margen de explotación de bancos y cajas de ahorros. 9515 Fernando Restoy and Philippe Weil: Approximate equilibrium asset prices. 9516 Gabriel Quirós: El mercado francés de deuda pública. 9517 Ana L. Revenga and Samuel Bentolila: What affects the employrnent rate intensity of growth? 9518 Ignacio Iglesias Araúzo y Jaime Esteban Velasco: Repos y operaciones simultáneas: estu- 9519 Ignacio Fuentes: Las instituciones bancarias españolas y el Mercado Único. 9520 Ignacio Hernando: Política monetaria y estructura financiera de las empresas. dio de la normativa. 9521 Luis Julián Álvarez y Miguel Sebastián: La inflación latente en España: una perspectiva macroeconómica. 9522 Soledad Núñez Ramos: Estimación de la estructura temporal de los tipos de interés en España: elección entre métodos alternativos. 9523 Isabel Argimón, José M. González·Páramo y José M: Roldán Alegre: Does public spen­ ding crowd out private investment? Evidence from a panel of 14 OECD countries. 9524 Luis Julián Álvarez, Fernando C. Ballabriga y Javier Jareño: Un modelo macroeconomé­ trico trimestral para la economía española. 9525 Aurora Alejano y Juan M.a Peñalosa: La integración financiera de la economía española: efectos sobre los mercados financieros y la política monetaria. 9526 Ramón Gómez Salvador y Juan J. Doladó: Creación y destrucción de empleo en España: un análisis descriptivo con datos de la CBBE. 9527 Santiago Femández de Lis y Javier Santillán: Regímenes cambiaríos e integración moneta­ ria en Europa. 9528 Gabriel Quirós: Mercados financieros alemanes. 9529 Juan Aruso Huertas: Is there a trade-off between exchange rate risk and interest rate risk? (The Spanish original of this publication has the same number.) 9530 Fernando Restoy: Determinantes de la curva de rendimientos: hipótesis expectacional y primas de riesgo. 9531 Juan Ayuso and María Pérez Jurado: Devaluations and depreciation expectations in the EMS. 9532 Panl Schulstad and Ángel Senat: An Empirical Examination of a Multilateral Target Zone Model. 9601 Juan Ayuso, Soledad Núüez and María Pércz-Jurado: Volatility in Spanish financial markets: The recent experience. 9602 Javier Andrés e Ignacio Hernando: ¿Cómo afecta la inflación al crecimiento económico? Evidencia para los países de la OCDE. 9603 Barbara Dluhosch: On the fate of newcomers in the European Union: Lessons from the Spanish experience. 9604 Santiago Femández de Lis: Classifications of Central Banks by Autonomy: A comparative analysis. 9605 M: Cruz Manzano Frías y Sofía Galmés Belmonte: Credit Institutions' Price Policies and Type of Customer: lmpact on the Monetary Transmission Mechanism. (The Spanish original of this publication has the same number.) 9606 Malte Krüger: Speculation, Hedging and Intermediation in the Foreign Exchange Market. 9607 Agustín Maravall: Short-Term Analysis of Macroeconomic Time Series. 9608 Agustín MaravaU and Christophe Planas: Estimation Error and the Specification of Un­ observed Component Models. 9609 9610 Agustín Maravall: Unobserved Components in Economic Time Series. Matthew B. Canzoneri, Behzad Diba and Gwen Eudey: Trends in European Productivity and Real Exchange Rates. 9611 Francisco Alonso, Jorge Martínez Pagés y María Pérez Jurado: Weighted Monetary Aggregates: an Empirical Approach. (The Spanish original of this publication has the same number.) 9612 Agustín Maravall and Daniel Peña: Missing Observations and Additive Outliers in Time Series Models. 9613 Juan Ayuso and Juan L. Vega: An empirical analysis of the peseta's exchange rate dynamics. 9614 Juan Ayuso Huertas: Un análisis empírico de los tipos de n i terés reales ex-ante en España. 9615 Enrique Alberola lIa: Optimal exchange rate targets and macroeconomic stabilization. 9616 A. Jorge Padilla. Samuel Bentolila and Juan J. Dolado: Wage bargaining in industries with market power. 96/7 Juan J. Dolado and Francesc Marmol: Efficient estimation of cointegrating relationships 9618 Juan J. Dolado y Ramón Gómez: La relación entre vacantes y desempleo en España: per­ among higher order and fractionally integrated processes. turbaciones agregadas y de reasignación. 9619 Alberto Cabrero y Juan Carlos Delrieu: Elaboración de un índice sintético para predecir la inflación en España. 9620 Una�Louise BeU: Adjustment costs, uncertainty and employment inertia. 9621 M.a de los Llanos Matea y Ana Valentina RegiI: Indicadores de inflación a corto plazo. 9622 James Conklin: Computing value correspondences fOT repeated games with state variables. (1) Previously published Working Papers are Usted in the Banco de España publications catalogue. Queries should be addressed to: Banco de España Sección de Publicaciones. Negociado de Distribución y Gestión Telephone: 338 51 80 Alcalá, 50. 28014 Madrid