Sticky Prices, Coordination and Enforcement

[Forthcoming, Topics in Macroeconomics] Sticky Prices, Coordination and Enforcement John C. Driscoll Harumi Ito 1 Federal Reserve Board Brown University and NBER Abstract Price-setting models with monopolistic competition and costs of changing prices exhibit coordination failure: in response to a monetary policy shock, individual agents lack incentives to change prices even when it would be Pareto-improving if all agents did so. The potential welfare gains are in part evaluated relative to a benchmark equilibrium of perfect, costless coordination; in practice, since agents will still have incentives to deviate from the benchmark equilibrium, coordination is likely to require enforcement. We consider an alternative benchmark equilibrium in which coordination is enforced by punishing deviators. This is formally equivalent to modeling agents as a cartel playing a punishment game. We show that this new benchmark implies that the welfare losses from coordination failure are smaller. Moreover, at the new benchmark equilibrium, prices are upwardsflexible but downwards-sticky. These last results suggest that the dynamic behavior of sticky-price models may more generally depend on the kind of imperfect competition assumed. JEL Classi(cid:12)cation Numbers: D43, E12, E30, L13 Keywords: coordination failure, menu costs, monopolistic competition, cartel. 1FederalReserveBoard,MailStop75,20thStreetandConstitutionAvenueNW,Washington, DC 20009 and Department of Economics, Brown University, Box B, Providence RI 02912. email: John Driscoll@alum.mit.edu, harumi@itosun.pstc.brown.edu. We thank PeterIreland,JohnLeahy,theeditorandtwoanonymousrefereesforveryhelpfulremarks. A previous version of this paper circulated under the title \Sticky Prices, Coordination and Collusion." The views expressed here are those of the authors and not necessarily those of the Federal Reserve Board or its sta(cid:11).

1 Introduction Price-setting models with monopolistic competition and costs of changing prices can exhibit coordination failure. In response to a monetary policy shock, individual agents maylackincentives tochange prices even ifallwould be better o(cid:11) by doing so, implying that price stickiness is a symmetric Nash equilibrium(SNE). Authors typically evaluate the welfare losses from coordination failure in part with reference to a benchmark equilibrium (called the symmetric cooperative equilibrium (SCE) by Cooper and John, 1988) in which coordination is implicitly assumed to be both costless and perfectly enforced. The latter assumption is necessary because individual agents have incentives to deviate from this equilibrium. In this paper, we consider an alternative benchmark equilibrium in which coordination must be enforced by threat of punishment of the deviators. As one might expect, welfare losses are smaller when measured relative to this equilibrium (which we refer to as the symmetric enforced equilibrium, or SEE). Our method of modeling enforcement closely resembles the analysis of implicit cartels in which the incentive to cheat is contained by threat of punishment (applied to macroeconomic contexts by Rotemberg and Saloner, 1986 and Rotemberg and Woodford, 1991, 1992). Hence our paper also generalizes the study of the macroeconomic e(cid:11)ects of menu costs to forms of imperfect competition other than monopolistic competition. We (cid:12)nd that at the SEE, prices are upwards-flexible but downwards-sticky (for small monetary shocks). This result suggests that the behavior of prices and welfare in models with sticky prices may more generally depend on the kinds of imperfect competition assumed. The rest of the paper proceeds as follows. Section 2 reviews the results from Ball and Romer (1991)’s \yeoman farmer" model of price setting under monopolistic competition and evaluates the costs of coordination failure assuming perfect coordination is possible. Section 3 considers an equilibrium assuming that coordination must be enforced by threatening to punish deviators. Section 4 looks at the aggregate supply properties of this equilibrium. Section 5 concludes. 1

2 Review: Monopolistic Competition and Perfect Coordination 2.1 Flexible Prices We follow Ball and Romer (1991)’s \yeoman farmer" model of price-setting. Assume there are N di(cid:11)erentiated products indexed by j produced by N producer-consumers ((cid:12)rms) indexed by i, where N is large. Producerconsumer i consumes C of product j (j = 1;2;:::N), receiving utility of ij C i = ( P N j=1 C i (cid:15) j − (cid:15) 1 )(cid:15)− (cid:15) 1. With labor supply L i , her utility function is given by: (cid:15)−1 U = C − L γ : (1) i i i γ(cid:15) There is increasing marginal disutility of labor, so that γ > 1, and the elasticity of substitution across goods is (cid:15) > 1. The production function is linear in labor, so that Y = L . i i P N 1−(cid:15) 1 De(cid:12)ne the aggregate price level as P = [ j=1 P j ]1−(cid:15) where P j is the price of good j. We assume the quantity theory of money holds with unit velocity, so that Y = M .1 Then one can easily show that the demand for P each good is: (cid:18) (cid:19) −(cid:15) M P i Y = : (2) i P P We may then rewrite each agent’s utility as a function of the price of his or her good P , the aggregate price level P and the money stock M as: i (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) M P 1−(cid:15) (cid:15)−1 M γ P −γ(cid:15) U (P ;P;M) = i − i : (3) i i P P γ(cid:15) P P Now consider a price competition game in which each agent chooses its price P . Let P(cid:3)(P;M) be the maximizer of the above individual utility i i function when the aggregate price level is P and the money stock is M. We can show that: P i (cid:3)(P;M) (cid:18) M (cid:19) (cid:15)(γ γ − − 1 1 )+1 = : (4) P P 1AsnotedinBallandRomer(1990,1991),wecanobtainthisstandardresulteitherby imposing a cash-in-advance constraint and assuming as in Rotemberg (1987) that money is distributed at the beginning of the period and must be spent within the period, or by assuming money enters the utility function. 2

We de(cid:12)ne the symmetric Nash equilibrium (SNE) to be the price level P that solves the following equation: SNE P = P (cid:3) (P ;M) 8i = 1;2;:::;N: (5) SNE i SNE AsinBallandRomer(1991),thesolutiontotheabovesymmetricequilibrium (cid:15)(γ−1)+1 condition yields P = M and individual utility of U = . SNE SNE γ(cid:15) This equilibrium is not joint-utility maximizing. Following Cooper and John(1988),wecan(cid:12)ndthesymmetriccooperativeequilibrium(SCE),which is Pareto-superior to the SNE. By imposing symmetry of actions (P = P), (cid:16) (cid:17) i γ the individual utility function in (3) becomes U = M − (cid:15)−1 M . P i P γ(cid:15) P SCE jointly maximizes the sum of individual utilities: (cid:18) (cid:19) ! M (cid:15)−1 M γ P = argmaxN − : (6) SCE P P γ(cid:15) P The closed form solution to the above maximization problem is P = SCE (cid:15)−1 1 ( (cid:15) )γ−1M, which is lower than P SNE derived above. The utility level at (cid:16) (cid:17)(cid:16) (cid:17) − 1 this price is U = γ−1 (cid:15)−1 γ−1, which is greater than U . Not SCE γ (cid:15) SNE coincidentally, this solution fully internalize all the externalities. By letting P = P (M) in equation (4), we have: SCE ! γ−1 P(cid:3)(P (M);M) M (cid:15)(γ−1)+1 i SCE = : (7) P (M) P (M) SCE SCE Since P < P = M and γ > 1, the above right-hand side ratio is SCE SNE greater than 1, implying P(cid:3)(P (M);M) is larger than P (M). Therei SCE SCE fore, the SCE is not a Nash equilibrium because individual producers have incentive to deviate by raising prices from P . SCE The Pareto suboptimality result derived above is not a form of coordination failure in the sense of Cooper and John (1988), since the SNE is unique, and the SCE is not a Nash equilibrium. It is worth noting, however, that if the producer-consumers were able to coordinate, they could internalize the ine(cid:14)ciency arising fromimperfect competition2 ; thiswould yield the socially optimal solution and maximize utility of the producer-consumers. 2One can also think of this ine(cid:14)ciency as arising from an aggregate demand externality (Blanchard and Kiyotaki, 1987), in which each producer-consumers’s choice of price, throughitse(cid:11)ectsonthepricelevelandthusaggregatedemand,a(cid:11)ectsallotherproducerconsumers. 3

2.2 Sticky Prices In the presence of a cost of changing prices (or menu cost) z, there is true coordination failure in the sense of Cooper and John (1988). Consider a change in nominal money M by an amount !. As shown by many authors (including Mankiw (1985)andBall andRomer (1990, 1991)), there is a range of values of! for which all producers would be better o(cid:11) if allchanged prices, but no individual producer-consumer has an incentive to do so. Hence for given values of M and ! within this range, there will be a sticky-price SNE 0 as well as a flexible-price SNE. Let M denote the initial value of the money stock. Assuming that 0 the economy is already at the P (M ) = M , we introduce a mone- SNE 0 0 tary shock of size !. If no producers move, an individual producer receives U (M ;M ;M +!) by remaining at the original price level.3 Assuming that i 0 0 0 no other producers move their prices from M , the best that an individ- 0 ual producer can achieve is U (P(cid:3)(M ;M +!);M ;M +!). If the largest i i 0 0 0 0 achievable gain from moving is smaller than the menu cost, (cid:12)rms will not move, generating a hysteresis band. The width of this band is given by the values of !+ and !− which solve the producer-consumer’s decision whether changing prices would increase its utility net of the menu cost: U (P (cid:3) (M ;M (cid:6)! (cid:6) );M ;M (cid:6)! (cid:6) )−U (M ;M ;M (cid:6)! (cid:6) ) = z; (8) i i 0 0 0 0 i 0 0 0 where P(cid:3) is de(cid:12)ned as above and we have used the result that at the SNE, i P = M. The criterion for determining whether all (cid:12)rms would be better o(cid:11) if they all changed prices is given by: U (M (cid:6)! (cid:6) ;M (cid:6)! (cid:6) ;M (cid:6)! (cid:6) )−U (M ;M ;M (cid:6)! (cid:6) ) (cid:21) z: (9) i 0 0 0 i 0 0 0 We can compute the welfare costs associated with economic fluctuations in this model in two ways. The (cid:12)rst way is to compare utility levels under the flexible-price SNE with that under the sticky-price SNE. Mankiw (1985) and Ball and Romer (1990, 1991) show that for small (second-order) values of z, this welfare loss is much greater than the loss su(cid:11)ered by an individual 3For the rest of this paper, we assume that the initial price level is set without any anticipation of future price chance. Thus, these monetary shocks can be considered as unanticipated shocks. 4

producer by not adjusting.4 The second way is to compare utility levels with those of the SCE. This will also be large, for the same reason: imperfect competition implies that all of the SNE are far from the SCE.5 3 Enforced Cooperation Inevaluatingthewelfarelossfromcoordinationfailure,much oftheliterature surveyed by and following on from Cooper and John (1988) has used the same dual approach as in the previous section: comparing the Pareto-inferior SNEs with the Pareto-superior SNE and with the SCE. In the absence of enforcement, the SCE is not a Nash equilibrium, as individual producerconsumers have an incentive to deviate. Hence in practice any attempt to enhance welfare by imposing an equilibrium other than the SNE will require some kind of enforcement. There might be a variety of possible institutional solutions to this problem. For any enforcement mechanism to be successful, however, it needs to curtail the individual incentive to deviate, either by arti(cid:12)cially reducing the bene(cid:12)ts or by increasing the costs of deviating. So, instead of de(cid:12)ning the social optimum over all price levels, we focus on the set of ‘enforced’ price levels and (cid:12)nd the optimum price level within. We will re-evaluate the welfare properties of the coordination failure using our new ‘enforced’ benchmark. Our problemresembles that ofthedynamic modelof aprice cartel. There is collective gain from cooperating, but the individual incentive to deviate needs to be contained by the threat of possible punishment. Although the entire economy is modeled as a single cartel, it is not our intention to argue that this is a good description of the current real-world economy. Instead, this is an assumption about the benchmark economy against which we calculate the welfare properties of other equilibria.6 We use a cartel because it 4For large (i.e. (cid:12)rst-order) values of z, the welfare loss will be small; but such large costs of changing prices seem implausible. Ball and Romer (1990) show how the ratio of these two quantities varies with di(cid:11)erent kinds of real rigidities. 5It should also be noted that in the presence of menu costs, there will in general also be hysteresis bands around the SCE in response to monetary shocks. 6Note also that unlike traditional colluders, e.g. in the models of Rotemberg and Saloner(1986)andRotemberg and Woodford(1991,1992),producershere will collude to lowerprices,notraisethem. Thisarisesfromthefactthatloweringindividualpriceslowers theaggregatepricelevelandthusraiseaggregatedemand;henceitisaconsequenceofany generalequilibriumimperfectly competitivemodelinwhichdemandforgoodsdependson 5

is simply the most generic and best-understood form of self-enforcing mechanism of cooperation under negative spillovers and strategic complementarity. It allows us to illustrate the constraint and the consequences of departing from Nash equilibrium for welfare gain. We would also like to stress that punishment-based enforcement mechanisms have broad generality in the discussion of cooperative equilibria, beyond the cartel model we discuss in this paper. For example, an alternative institutional solution such as price controls also su(cid:11)ers from the same individual incentive problem of the SCE. Unless the government is able to gather information about all transaction prices, any price control is subject to individual manipulation such as black market transactions. Thus, some enforcement mechanism that curtails such individual incentives through punishment (such as legal sanctions) will be necessary. So, we focus on a cartel as one of such punishment-based enforcement mechanisms. 3.1 The Symmetric Enforced Equilibrium (Under Flexible Prices) We de(cid:12)ne a punishment-enforced equilibrium in a game theoretic framework. We consider an in(cid:12)nitely repeated game in which the stage game takes the form of the static model in section 2.1. This will take the form of a dynamic game for a price cartel. Agents’ actions are assumed to be observable to everyone. Again, we focus on symmetric strategies and equilibria. In this section, we de(cid:12)ne and analyze the properties of the punishment enforced equilibrium for a (cid:12)xed money level assuming no menu costs. Later in section 3.2, we re-introduce menu costs and analyze its influence on the price adjustment to money shocks in section 4. Given a money stock M, consider a price P 2 (P (M);P (M)). SCE SNE Then we have: U (P;P;M) > U (P (M);P (M);M): (10) i i SNE SNE Even if this price P cannot be sustained as a Nash equilibrium in a stage game, it can possibly be a Nash equilibrium in an in(cid:12)nitely repeated game. Consider the incentive problem of agent i’s pricing decision while all other agents are charging the above-mentioned price P. Agent i considers conforming to the aggregate price level P or defecting and charging P(cid:3)(P;M) aggregate demand. 6

which maximizes its stage payo(cid:11). Subsequently, the cartel will punish such a defection by imposing P , the worst possible outcome in the price game. SNE We de(cid:12)ne the punishment K to be a linear function of the missed pro(cid:12)t opportunities, U (P;P;M)−U : i SNE K(P;M) = (cid:27)(U (P;P;M)−U ); (11) i SNE where (cid:27) parameterizes the harshness of the punishment. Theoretically, the worst possible punishment in a price game, as in Abreu et al. (1986), is (cid:14) in(cid:12)nitely repeating P . In this case, we can let (cid:27) = where (cid:14) is the SNE 1−(cid:14) discount rate. However, such a harsh and prolonged punishment may not be institutionally feasible in the real world. Since our purpose is to be realistic about the enforcement mechanism, we accommodate such imperfect punishment in our model by varying the value of (cid:27). For example, the cartel may be able to impose P for only a (cid:12)nite punishment period, T, then it SNE (cid:14)(1−(cid:14)T) will revert to P. In that case, we have (cid:27) = . From this construction, 1−(cid:14) we expect (cid:27) to increase with the length of the punishment period and the relative patience of the individuals. Later, we will examine how di(cid:11)erences in the severity of punishment (di(cid:11)erent values of (cid:27)) influence the equilibrium. P can be imposed as an equilibrium of this dynamic game if the following condition is satis(cid:12)ed: U (P (cid:3) (P;M);P;M)−U (P;P;M) (cid:20) K(P;M): (12) i i We denote the left-hand side of this inequality as (cid:1)U(P;M), which represents the maximum gain from one-shot deviation. Hereafter, we refer to this inequality astheequilibrium condition (constraint).7 Wedroptheisubscript because both (cid:1)U(P;M) and K(P;M) are symmetric across producers. AsinRotemberg andSaloner(1986),thecloser thecooperativepricelevel P is to P , the smaller the size of the punishment K. The gain from the SNE cheating, on the other hand, is larger if the cartel’s price P is farther away from P . SNE With this punishment-enforcement mechanism at work, we de(cid:12)ne \the symmetric enforced equilibrium (SEE)" to be the price level P (M) at SEE 7Chari and Kehoe (1990) characterize equilibria in in(cid:12)nite dynamic games in which one player is much larger than the others. They develop the concept of \sustainable" equilibria; in their most prominent example of such an equilibrium, the incentive for a one-shot deviation is contained by threat of in(cid:12)nite punishment. This condition, given in their equation (6), is analogous to the above equilibrium condition (12). 7

which the producers’ joint utilities are maximized subject to the equilibrium condition: (cid:18) (cid:19) ! M (cid:15)−1 M γ P (M) = argmaxN − (13) SEE P P γ(cid:15) P s:t:(cid:1)U(P;M) (cid:20) K(P;M): (14) For some parameter values, the equilibrium condition will not bind for any value of M, and an e(cid:14)cient cartel will reach the P as it was de(cid:12)ned SCE in the previous static game. In that case, there is no problem of enforcement and P will be the same as P . (P > P = P ). SEE SCE SNE SEE SCE If, on the other hand, the equilibrium condition is binding, P will lie SEE between the socially optimal price (P ) and the monopolistically competi- SCE tive price P (P > P > P ). Similarly, we can show that welfare SNE SNE SEE SCE at the SEE as measured by the representative producer-consumer utility lies in between welfare of the other two equilibria: i.e. U < U < U .8 SNE SEE SCE Inthiscase, thewelfarelossoftheSNEissmaller whenitiscomparedagainst the SEE, than it is against the SCE. Moreover, as long as the equilibrium constraint binds, the more severe the punishment (the larger the value of (cid:27)), the greater the value of K and the closer the price P to the socially optimal P . This is a standard SEE SCE result in the literature on dynamic pricing games. 3.2 Sticky Prices Now suppose that there is a small cost of changing prices, z, for each producer. We rede(cid:12)ne the SEE under this new setup. In the following discussion, we assume that the producers see the collective gains in coordinating their prices and enforce their cooperative strategy through the threat of punishment. That is, when there is collective gain from moving prices, but an individual producer does not conform, that is considered as a deviation and calls for punishment. In this way, we can examine the enforcement mechanism that achieves the cooperative equilibrium endogenously. The presence of the menu costs requires some modi(cid:12)cation in the equilibrium condition (12) (or equivalently (14)). Since cheating involves changing 8As shown in Appendix A. 8

prices, the cheater has to pay z when cheating. So, the gain from cheating should be modi(cid:12)ed to be: (cid:1)U(P;M) = U (P (cid:3) (P;M);P;M)−U (P;P;M)−z: (15) i i Let K(cid:22) be the punishment modi(cid:12)ed accordingly. For example, if the punishment P lasts for T periods, K(cid:22)(P;M) can be expressed as (cid:14)z +(cid:14)T+1z + P SNE T (cid:14)t[U (P;P;M)−U ]. (cid:14)z and (cid:14)T+1z account for menu costs incurred t=1 i SNE by moving in and out of the punishment phase. More generally, the punishment K is an increasing function of (U (P;P;M) − U ) as it was in i SNE de(cid:12)nition (11) and it co-varies with the patience of individual producers and menu costs z. Thus, the symmetric enforced equilibrium price is the solution to the maximization problem in (13) subject to the following modi(cid:12)ed equilibrium condition: (cid:1)U(P;M) (cid:20) K (cid:22) (P;M): (16) As in the flexible price case, for some values of (cid:15);γ and (cid:14), this constraint may not bind if the punishment is harsh (i.e. if (cid:27) is high), for any level of the money stock. We again focus our attention on parameter values for which (16) can bind. As above, P > P > P , and it is also immediate that U < SNE SEE SCE SNE U < U 9 Thus the absolute welfare loss from being at the SNE when SEE SCE there are menu costs is smaller relative to the SEE than it was relative to the SEE - though of course the welfare loss from being at a sticky-price SNE relative to being at a flexible-price SNE remains the same as before. 4 Enforced Cooperation and Aggregate Supply We now describe the response of (cid:12)rms under the SEE and menu costs to monetary shocks. Although we (still) do not think that the current macroeconomy can be described as a giant implicit cartel, we believe determining the properties of the SEE to be of interest for two reasons. First, were the 9More precisely, with menu costs z there will be a range of prices corresponded to the SNE, SEE and SCE. For small menu costs and reasonable choices of the other parameter values, these ranges of prices will not intersect, allowing these inequalities to hold. 9

SEE actually implemented as a benchmark equilibrium, we would like to know its properties. More important, while the behavior of (cid:12)rms under menu costs is well understood for the case of monopolistic competition, that is only one form of imperfect competition. Understanding how (cid:12)rms behave under a cartel may help shed light on how (cid:12)rms would behave more generally under other forms of imperfect competition; in particular, if (cid:12)rm responses to monetary shocks were quite di(cid:11)erent, that would suggest that the results under monopolistic competition may not fully generalize. Assume that the initial level of the money stock is M and the economy is 0 already at P (M ). We investigate the behavior of the cartel when there is SEE 0 a permanent shock that changes the money stock from M to M (cid:6)!(cid:6)(!+ > 0 0 0;!− > 0). This is an unanticipated monetary shock to the extent that the initial price level P (M ) was set without any anticipation of future SEE 0 price changes. We investigate if the cartel is willing to adjust to the new SEE price level, P (M (cid:6) !(cid:6)) and if individual producers are willing to SEE 0 conform to whatever the cartel decides to do. The purpose of this exercise is to identify the case in which the incentive of the cartel and individual (cid:12)rms prevent the price of adjusting to the new SEE, generating a hysteresis band. Moreover, in the discussion through section 4.2, we focus on cases in which the equilibrium constraint (16) binds at the initial money stock M and the 0 initial price level P (M ). 10 SEE 0 Inthefollowingdiscussion, weexploitseveralpropertiesoftheequilibrium constraint (16). Both the incentive to cheat (cid:1)U and the punishment K(cid:22) are M increasing functions of . If the constraint (16) binds for some value of P M=P, they have a single-crossing property where (cid:1)U eventually exceeds K(cid:22) M for arbitrarily large values of . When the equilibrium constraint (16) is P binding, the SEE is at the intersection of (cid:1)U and K(cid:22). As (M=P) rises above (cid:22) the SEE level, we have (cid:1)U > K. And, as (M=P) falls below the SEE level, we have (cid:1)U < K(cid:22). Finally, as long as the equilibrium constraint (16) is binding, the ratio M=P (M) is invariant to M. SEE Since positive and negative changes in the money stock a(cid:11)ect the constraint (16) in opposite ways, we consider each case separately. In each scenario, we look at the cartel’s incentive to change the enforced price and 10If the equilibrium constraint (16) does not bind, the economy achieves the SCE, so there is no issue of coordination or enforcement. Moreover, this model generates a hysteresis in which the economy is not at the SEE.Section 4.3 discusses how the economy in such initial states respond to monetary shocks. 10

an individual (cid:12)rm’s incentive to conform or deviate- implying four cases in all. 4.1 Positive Monetary Shocks Since the economy is already at the SEE, an increase in the money stock will push M=P above the intersection of (cid:1)U and K(cid:22) and break the equilibrium constraint (16). The natural prediction is that the cartel will raise prices to P (M + !+), which keeps M=P constant (this logic follows that of SEE RotembergandSaloner,1986,whoarguedthatacartellowersitspriceduring a boom). In this section, we check to see if this outcome is possible along with other possibilities. In the following discussion, we carefully check if the cartel’s action that pursue the collective gain is compatible with the incentives of individual producers. Once the cartel’s preferred decision, to more or not to move, is decided, the cartel can impose punishment K on any producers who do not conform to its decision. We check the sustainablity of such an equilibrium by examining the incentive of a single producer to conform to the cartel’s decision when all other producers are already conforming. If the incentive of this single non-conforming producer outweigh the punishment K, then we assume that all other (cid:12)rms will also deviate from the cartel’s decision, thus that particular equilibrium is not sustainable. On the other hand, if the punishment outweigh the incentive to deviate, the cartel’s decision can be sustained as an equilibrium. First, we check the cartel’s incentive to move its price. By moving, they will maintain U , but they have to pay the menu costs. Alternatively, SEE they may have an option of maintaining the original price P (M ) and SEE 0 receiving utility level U (P (M );P (M );M + !+) on the condition i SEE 0 SEE 0 0 that individual (cid:12)rms conform (we check individual incentives in a moment). The cartel would like to move if the following condition holds: 1 1 U − U (P (M );P (M );M +! + ) > z: (17) 1−(cid:14) SEE 1−(cid:14) i SEE 0 SEE 0 0 This condition may or may not hold, depending on parameter values and the size of the money stock change. So, we consider both cases along with individual (cid:12)rms’ incentives to move or not to move their prices. 1. The cartel does not want to move. Firms do not want to move. 11

If(17)doesnot hold, thecartel hasnoincentive to move. Since individ- M ual producers’ utilities are increasing in , this is a possible outcome. P Even if the cartel does not want to move, would individual (cid:12)rms conform to that decision? We consider the incentive of an individual (cid:12)rm to cheat assuming that all other (cid:12)rms conform to the cartel. Here, cheating means individually optimizing while all other (cid:12)rms are charging P (M ). The gainfrom this cheating is (cid:1)U(P (M );M +!+) SEE 0 SEE 0 0 while the punishment that the cartel can impose is K(cid:22)(P (M );M + SEE 0 0 !+). Then, this (cid:12)rm conforms if the following inequality holds: + (cid:22) + (cid:1)U(P (M );M +! ) < K(P (M );M +! ): (18) SEE 0 0 SEE 0 0 Recall that (cid:1)U(P;M) < K(cid:22)(P;M) binds at M = M 0 . Since P P (M ) SEE 0 (M + !+)=P (M ) > M =P (M ), this inequality (18) will not 0 SEE 0 0 SEE 0 hold. Thus, individual (cid:12)rms will deviate. This implies that not moving is not an option available for the cartel since individual (cid:12)rms would like to move. We can exclude this case. 2. The cartel does not want to move. Firms want to move. Our analysis of the previous case indicates that this can happen. Positive money shocks imply that the following is true. (cid:1)U(P (M );M +! + ) > K(cid:22)(P (M );M +! + ) (19) SEE 0 0 SEE 0 0 As a result, (cid:12)rms will deviate by changing prices, the cartel will break down, and we will return to the SNE. But this outcome implies that the cartel’s choice was not between moving to the SEE or staying at the current price, as was assumed in equation (17) above, but between moving to a new SEE, if that is feasible, or moving to the SNE. We thus check below whether a new SEE is feasible, and, if so, whether the cartel would prefer to move to it instead of letting the cartel dissolve. 3. The cartel wants to move. Firms want to move. Now, we consider a case where (17) holds. The cartel wants to move to P (M +!+) which also keep the equilibrium constraint (16) intact. SEE 0 Will the (cid:12)rm conform? It will if its incentive to cheat is smaller than the punishment: 12

+ + U (P (M );P (M +! );M +! ) i SEE 0 SEE 0 0 −U +z < K (cid:22) (P (M +! + );M +! + ): (20) SEE SEE 0 0 Again,theleft-handsideofthisinequalityissmallerthan(cid:1)U(P (M + SEE 0 !+);M + !+). The equilibrium constraint (16) implies that this in- 0 equality is also satis(cid:12)ed. Thus, if the cartel wants to move, individual (cid:12)rms will move. 4. The cartel wants to move. Firms do not want to move. From our discussion of the previous case, we can rule out this case. The binding equilibrium condition (16) implies that (cid:12)rms would like to move. These outcomes are summarized in Table 1 below. Even if the cartel has an incentive to remain at P (M ), individual (cid:12)rms do not have incentives SEE 0 to conform to that decision. The consequence of the cartel not moving is the breakdown of the cartel and return to the SNE. On the other hand, as shown in case 3, by moving to the new SEE, the cartel achieves U . By SEE comparing these two outcome, we have: 1 1 U < U (21) 1−(cid:14) SNE 1−(cid:14) SEE For the cartel, moving in response to the positive money shock dominates not moving regardless of inequality (17).11 Thus, we rule out the SNE outcome. When there is a positive money shock, the economy will move to the new SEE, and therefore there will be no hysteresis band for a positive monetary shock when the cartel starts out at the lowest price consistent with an SEE. Table 1: Positive Monetary Shocks Cartel Does Not Want to Move Wants to Move Firms Do Not Want to Move Ruled Out Ruled Out Want to Move Ruled Out New SEE 11Unless the di(cid:11)erence inutilities is less thanthe menucost. But, in this case,the SEE is already nearly indistinguishable from the SNE. 13

4.2 Negative Monetary Shocks The equilibrium constraint (16) will become slack as a negative money shock lowers the value of M=P. The cartel has an incentive to lower its price to reachP (M −!−). Thecartelwouldliketomoveifthefollowingcondition SEE 0 holds: 1 1 U − U (P (M );P (M );M −! − ) > z: (22) 1−(cid:14) SEE 1−(cid:14) i SEE 0 SEE 0 Again, we check the four possible outcomes, interacting the cartel’s incentive to move and an individual (cid:12)rm’s incentive to move. 1. The cartel does not want to move. Firms do not want to move. Suppose that the above inequality (22) does not hold, implying that the cartel does not want to move. This may happen when P (M − SEE 0 !−) is so close to P (M ) that the gain from moving is smaller SEE 0 than the menu costs involved. Next, we look at the individual incentive to conform this cartel’s decision. Assuming that all other (cid:12)rms are charging P (M ), an individual (cid:12)rm’s incentive to cheat is SEE 0 (cid:1)U(P (M );M −!−)whilethepossiblepunishmentisK(cid:22)(P (M );M − SEE 0 0 SEE 0 0 !−). Anindividual(cid:12)rmwouldconformifthefollowinginequalityholds: (cid:1)U(P (M );M −! − ) < K(cid:22)(P (M );M −! − ): (23) SEE 0 0 SEE 0 0 Since (cid:1)U(P;M) < K (cid:22) (P;M) binds at M = M 0 , for M 0 −!− P P (M ) P (M ) SEE 0 SEE 0 M (< 0 ), the above inequality will hold. Thus individual (cid:12)rms P (M ) SEE 0 would conform to the cartel’s decision not to move. There must be !(cid:3)(> 0) such that the following equality holds: 1 1 U −z = U (P (M );P (M );M −! (cid:3) ): (24) 1−(cid:14) SEE 1−(cid:14) i SEE 0 SEE 0 0 This implies a hysteresis band of [M −!(cid:3);M ]. 0 0 2. The cartel does not want to move. Firms want to move. Fromourdiscussionofthepreviouscase, wecanexcludethispossibility. 14

3. The cartel wants to move. Firms do not want to move. Suppose now that the inequality (22) holds. Thus, the cartel would like to move. Would the individual (cid:12)rms conform? Assuming that all other (cid:12)rmsconform,weinvestigateanindividual(cid:12)rm’sincentivenottomove. By not moving, this cheating (cid:12)rm will gain U (P (M );P (M − i SEE 0 SEE 0 !−);M − !−) while moving will yields U − z. If the gain from 0 SEE cheating is larger than the possible punishments as shown below, the (cid:12)rms would not move. U (P (M );P (M −! − );M −! − )−U +z i SEE 0 SEE 0 0 SEE > K(cid:22)(P (M −! − );M −! − ) (25) SEE 0 0 However, the left-hand side of the above inequality should be smaller than(cid:1)U(P (M −!−);M −!−)since(cid:1)U(P (M −!−);M −!−) SEE 0 0 SEE 0 0 is the maximum obtainable gain from cheating. Since (cid:1)U(P (M − !−);M − !−) = K(cid:22)(P (M − !−);M − SEE 0 0 SEE 0 0 !−), the left-hand side of the inequality (25) should be smaller than K(cid:22)(P (M −!−);M −!−) This implies that the inequality (25) does SEE 0 0 not hold and individual (cid:12)rms have incentive to move, thus we can rule out this case. 4. The cartel wants to move. Firms want to move. When the inequality (22) does hold, the cartel would like to move. From our previous discussion, (cid:12)rms would conform. The economy will smoothly moves to the new SEE. Table 2 summarizes the outcomes when the money stock decreases. When the cartelwants to move, (cid:12)rms will also want to move, and a new SEE will be reached. When the cartel does not want to move, the economy will exhibit hysteresis. Table 2: Negative Monetary Shocks Cartel Does Not Want to Move Wants to Move Firms Do Not Want to Move Hysteresis [M −!(cid:3);M ] Ruled Out 0 0 Want to Move Ruled Out New SEE 15

The asymmetric response of the price to positive and negative money shocks implies that the welfare loss of coordination failure, as we discussed in section 2.2, is larger for a positive money shock and smaller for a negative money shock when the SEE is used as the benchmark economy than when the SCE is used. However, this di(cid:11)erence in welfare between positive and negative shocks is much smaller than the di(cid:11)erence in welfare between being at an SNE and being at the SEE. 4.3 Implications for Aggregate Supply Wehave shown thatthe presence of menu costs creates asymmetric responses to money shocks under the SEE. When the money stock increases, the economy will smoothly transit to the new SEE. When the money stock decreases, the SEE will exhibit hysteresis: prices will be sticky downwards. The above analysis supposes the cartel begins at the lowest price consistent with the SEE- i.e. at the point where the equilibrium constraint (16) is binding. However, it is likely that after a series of monetary shocks, the cartel will be at some point in the interior of the hysteresis band. At such a point, both positive and negative monetary shocks, if su(cid:14)ciently small, will leave the cartel within the band. Hence in practice observed hysteresis bands may be asymmetric in either direction, or even symmetric, depend on the past sequence of shocks. But note that since a large money shock, either positive or negative, will produce a return to the SEE, it is likely that most observed hysteresis bands will be larger downwards than upwards. The hysteresis band will be larger than those under monopolistic competition; this can be seen by comparing equation (8) with equation (24). In equation (8), which determines the size of the monopolistically competitive band, the size of the band is determined by the utility gain from a single (cid:12)rm deviating from the aggregate price level. In equation (24), which determines the size of the band under SEE, the size of the band is determined by the utility gain from all (cid:12)rms changing their prices. The aggregate demand externality implies that the utility gain will be greater in the latter case than in the former. So, a larger menu cost will be needed to produce the same size hysteresis band, or, equivalently, having the same size menu cost produces a larger hysteresis band. If the punishment is not very harsh (i.e. (cid:27) is small), so that the equilibrium is near the SNE, the size of the hysteresis bands is small. As (cid:27) decreases, the width of the hysteresis band diminishes 16

and approaches those of the SNE. Figure1plotsaggregatesupply, whichishorizontalattheSEE.P (M ) SEE 0 denotes the price level for the SEE at the initial money stock M . Y is 0 SEE the level ofoutput corresponding to the lowest price consistent with the SEE. Note that this level of output is independent of the money stock. Asnotedabove, atthelowestpriceconsistent withanSEE(corresponding to aggregate demand curve AD , at equilibrium point A) there is only a 1 downwards hysteresis band; positive monetary shocks will cause prices to increase with no change in output, while small negative monetary shocks will cause output to decrease with no change in prices. Suppose that sequences of monetary shocks push aggregate demand into the interior of the hysteresis band (aggregate demand curve AD and equi- 2 librium point B). Subsequent small monetary shocks can push output in either direction, while the price level remains unchanged A single large negative monetary shock or a series of negative monetary shocks will push aggregate demand to the lower end of the hysteresis band (aggregate demand curve AD and equilibrium point C). At this point, 3 a further negative monetary shock (to aggregate demand curve AD and 4 equilibrium point D) will cause the price level to jump down to the new SEE level, and output up to the level implied by the SEE. A single large positive monetary shock or a series of positive monetary shocks will pushaggregatedemand toa pointwhere it willno longer intersect the old hysteresis band. At such a point (aggregate demand curve AD and 5 equilibrium point E), the price level will jump upwards to the new SEE level, while output will remain at its original level. 4.4 Idiosyncratic Shocks The previous subsection assumes the existence of only one shock, an aggregate one to the money stock. Caplin and Spulber (1987) and Caplin and Leahy (1991) have shown that the neutrality of money can depend on assumptions about the money supply process, idiosyncratic shocks and the initial distribution of prices. Caplin and Leahy (1997) present a dynamic model of coordination failure with idiosyncratic shocks; they show that there is only one equilibrium, in which some producers adjust even though they know no other producer will do so. John and Wolman (1999) generalize Ball and Romer’s model to a dynamic setting; they (cid:12)nd that doing so weakens the degree of strategic complentarity to the degree that there may be no 17

multiplicity of equilibria. We may be able to extend our model to allow for idiosyncratic shocks to individualproducers. However, thecurrentprice-coordinationschemecannot take account of such idiosyncratic price shocks and will simply impose the same pricing rules for all producer-consumers, yielding a degenerate crosssectional distribution of prices.12 5 Conclusion Welfare implications of New Keynesian pricing models are commonly drawn in part by comparing an equilibrium under perfect e(cid:11)ortless coordination with equilbria under coordination failure. In practice, since individual producershaveanincentivetodeviatefromtheperfectcoordinationequilibrium, any coordination must be enforced. In this paper, we propose an alternative benchmark for evaluating the welfare loss from the coordination failure by focusing on the set of ‘enforced’ price levels. In order to (cid:12)nd the optimum within such a set, we employ a model of an implicit cartel, as in Rotemberg and Saloner (1986). Our new benchmark, the symmetric enforced equilibrium (SEE) is a punishment-enforced price coordination equilibrium that lies between the monopolistically competitive, symmetric Nash equilibrium (SNE) and the symmetric cooperative equilibrium (SCE). We re-evaluate the welfare loss of coordination failure under our benchmark. It remains true that di(cid:11)erence in utility between the sticky-price SNE and the flexible price SNE is large relative to the utility loss su(cid:11)ered by an individual producer-consumer by not adjusting. However, the absolute loss in utility is smaller relative to the SEE than to the SCE, since the SEE’s utility level is lower than the SCE. Depending on how harsh the punishment may be, the absolute loss may itself be quite small (i.e. zero to (cid:12)rst-order). We also study the aggregate supply curve implied by the SEE, not only because the latter is the constrained social optimum, but also because doing so may shed light on how aggregate supply behaves under other forms of imperfect competition than monopolistic competition. Aggregate sup- 12The recent paper by Athey et. al. (2003) has drawn a rich set of a dynamic model implications into static cartel models. It shows that the degree of patience influences the price rigidity. An impatient cartel will move prices in order to attenuate the incentive to cheatasinRotembergandSaloner(1986),whileapatientcartelwillfollowarigid-pricing scheme. 18

ply is horizontal at the minimum price consistent with the SEE. Prices are upwardly-flexible, but downwardly rigid for small monetary shocks. A large negative monetary shock will cause aggregate supply to shift downwards, with prices moving downwards and output returning to the SEE level. The coordination failure problem is unlikely to be solved by (cid:12)at, but ratherbydevelopingmechanismsorinstitutionsdesignedtoaddressit. Moreover, the problem itself in the context of price-setting has so far only been studied under monopolistic competition. This paper takes (cid:12)rst steps in the directions of widening the kinds of imperfectly competitive models under which the problem is considered and in thinking more concretely about how coordination might be implemented. 19

A Theorems on Symmetric Enforced Equilibrium (SEE) Recall that utility is de(cid:12)ned as: (cid:18) (cid:19) M (cid:15)−1 M γ U = − : (26) i P γ(cid:15) P and denote the price under the SNE as P = M and under the SCE as SNE (cid:16) (cid:17) 1 P = 1− 1 γ−1 M < P . SCE (cid:15) SNE Note that this implies that: 1 1 U = 1− − (27) SNE γ γ(cid:15) and !(cid:18) (cid:19) − 1 1 1 γ−1 U = 1− 1− > U : (28) SCE SCE γ (cid:15) P is de(cid:12)ned as in equation 14. Note that at the SNE, the incentive SEE @U to cheat (cid:1)U = 0 and at the SCE, (cid:1)U > 0. Also, i < 0 at the SCE and @P C @U i < 0 = 0 at the SNE. @P C Theorem 1 P (cid:20) P (cid:20) P . SCE SEE SNE Proof: Suppose K is smaller than the value K(cid:3) = (cid:1)U(P ), but larger SCE than zero. Suppose the resulting P < P . Then (cid:1)U(P > (cid:1)U(P ) > K. SEE SCE SEE SCE But this contradicts the de(cid:12)nition of P . SEE Suppose the resulting P > P . Then U < U , but SNE is SEE SNE SEE SNE feasible. This again contradicts the de(cid:12)nition of P . SEE Suppose the resulting P = P . Then there is a P(cid:3) which still SEE SNE satis(cid:12)es the equilibrium constraint but has higher utility that P or P . SEE SNE This again contradicts the de(cid:12)nition of P . SEE So, P (cid:20) P (cid:20) P . SCE SEE SNE Theorem 2 U (cid:21) U (cid:21) U . SCE SEE SNE @U Proof: Follows immediately from (cid:12)rst proposition and fact that < 0. @P 20

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Figure1 AggregateSupplyfortheSymmetricEnforcedEquilibrium(SEE) P P C B A SEE AD AD 1 AD 2 D AD 3 AD 4 Y Y SEE . E (M) 0 5