Monetary neutrality under evolutionary dominance of bounded rationality.

Lima, Gilberto Tadeu ; Silveira, Jaylson Jair

I. INTRODUCTION

A key property of the dynamic relationship between money and output is that due to price stickiness monetary shocks have persistent, although not permanent, impacts on the real output (see, e.g., Christiano, Eichenbaum, and Evans 1999). The nominal adjustment of the aggregate price level in response to a monetary shock is, therefore, temporarily incomplete, and considerable supporting empirical evidence for price rigidity has been found in several microprice studies (see, e.g., Nakamura and Steinsson 2013, and Klenow and Malin 2011, for comprehensive discussions of these studies).

Mankiw and Reis (2002) develop a novel model of temporarily incomplete nominal adjustment based on the assumption that costly information disseminates slowly. In each period, a random fraction of the price setters updates itself on the state of the economy so as to compute and set the optimal price, while the remaining fraction keeps setting prices based on outdated information. They admit that micro-foundations for price stickiness may require a better understanding of bounded rationality, and that information processing is more complex than the time-contingent adjustment assumed in their paper. In fact, as added in Mankiw and Reis (2011), while the cost of acquiring information may be small, and may even have fallen in this information age, the costs of absorbing and processing information may be rather large and actually higher today than those in the past.

This paper contributes to the intuition on price setting under bounded rationality by developing a model in which the share of firms that have up-to-date information on relative prices to establish the optimal price follows evolutionary dynamics (with and without mutation), and not a random dynamics a la Calvo (1983), as in Mankiw and Reis (2002), and the related literature. Our aim is to devise evolutionary game-theoretic dynamics to provide alternative microfoundations for a temporarily incomplete nominal adjustment following a monetary shock. We investigate the behavior of the general price level in an economy in which an individual firm can either pay a cost (featuring a random component) to update its information set and establish the optimal price (Nash strategy) or freely use full information from the previous period and set a one-period lagged optimal price (bounded rationality strategy). Indeed, detailed empirical studies such as the investigation for the Euro zone by Alvarez et al. (2006) have found persistent coexistence of forward looking and backward looking price-setting behavior, and there is also robust empirical evidence showing persistent and time-varying heterogeneity both in the frequency and in the extent of information updating (see, e.g., Branch 2007; Coibion and Gorodnichenko 2012). We then devise evolutionary game-theoretic microdynamics in discrete time that, by interacting with the dynamics of the macroeconomic variables, affects the coevolution of the frequency distribution of information-updating strategies across firms and the completeness of the nominal adjustment of the general price level to a monetary shock.

Our model economy is composed of monopolistically competitive firms whose individual optimal price depends positively upon the prices set by others, implying the existence of strategic complementarity. In a given period, there is a fraction of firms that set prices without perfect knowledge of the structure of relative prices, and hence of the prices that other firms will be setting, as they decide not to pay the information cost involved in knowing it. We refer to these firms as boundedly rational firms. The remaining fraction is composed of firms that decide to incur the information-updating cost to obtain that perfect knowledge, and we follow Droste, Hommes, and Tuinstra (2002) in referring to them as Nash firms, as they become perfect foresight firms. Meanwhile, the optimal price is assumed to be given as in Ball and Romer (1991), varying positively with the actual general price level and nominal stock of money, which is publicly known. Therefore, drawing again on Droste, Hommes, and Tuinstra (2002), the price-setting strategy of Nash firms is conceived of as like a Nash equilibrium in a game contaminated with boundedly rational firms.

While the distribution of information-updating strategies is predetermined in a given period, it varies over time according to an evolutionary dynamics. Firms' choice between the two available information-updating strategies is based on expected payoffs, with a loss (having a random component to it) being faced by firms that do not carry out the updating of the information set required to establish the optimal price. The flows of firms between the two strategies are driven by expected payoffs through an evolutionary dynamics (with and without mutation), and are derived following Weibull (1995) and Samuelson (1997). This evolutionary approach to information updating is indeed fitting given the robust empirical evidence that fluctuations in the degree of (in)attention to new information are persistent and not purely random (see, e.g., Andrade and Le Bihan 2013). As it turns out in the model developed herein, in the unique evolutionary equilibrium that emerges in the long run, and which features either most or all firms playing the boundedly rational strategy as local attractor, monetary shocks have nonetheless only temporary real effects. Besides, the extent of the temporary output response to a monetary shock depends both on the frequency distribution of information-updating strategies across firms and on the heterogeneity in the cognitive abilities of the firms playing a given information-updating strategy.

The remainder of this paper is organized as follows. The next section presents the motivating issues and briefly discusses some related literature, while Section III lays out the structure of the model. Meanwhile, Section IV evaluates the existence and stability of an evolutionary equilibrium and discusses the corresponding implications for the response of the general price level and real output to a monetary shock. A final section with concluding remarks closes this paper.

II. RELATED LITERATURE AND CONTRIBUTION OF THIS PAPER

Contributions by Sims (1998, 2003) and Mankiw and Reis (2002) recently revived the interest in the importance of information imperfections for price and output dynamics, giving rise to a now voluminous literature (see, e.g., Mankiw and Reis 2011, and Sims 2011, for surveys of this literature, and Veldkamp 2011, for a book-length discussion of other recent applications of models with information frictions). This literature is related one way or another to the early research on price decisions under imperfect information dating back to the prominent work of Phelps (1969) and Lucas (1972).

Mankiw and Reis (2011) describe the recent literature as being split between partial and delayed information models, with some contributions combining elements of both. Drawing on Sims (1998, 2003), in partial information models, firms observe economic conditions continuously but subject to a noise. Following Mankiw and Reis (2002), meanwhile, delayed information models describe firms as observing conditions perfectly but subject to a lag, as a result of fixed costs to information updating. Nonetheless, both families of models assume that agents form expectations optimally but under some form of incomplete information. Delayed information models usually assume that only a proportion of firms have up-to-date information, with the remaining having old information from previous periods. Meanwhile, partial information models typically assume that firms observe a noisy signal with some relative precision. Essentially, both families of models introduce just one new parameter, measuring either the proportion of firms with up-to-date information or the relative precision with which a noisy signal is observed, which can be interpreted as an index of informational rigidities. While the notion of rational inattention as originally developed by Sims (1998) has been used to justify the assumption of partial information, models of rational inattentiveness as pioneered by Reis (2006) have provided micro-foundations for delayed information models.

In the delayed information model of Mankiw and Reis (2002), firms update their information set infrequently, but nonetheless acquire full-information rational expectations when the update is performed. As information disseminates slowly, in every period, only a fraction of firms obtains perfect information concerning all current conditions, while all other firms continue to set prices using old information. While the probability of updating information is taken to be exogenously given in Mankiw and Reis (2002), Reis (2006) considers instead the endogenous determination of the time interval between information updates, finding that a time-dependent schedule a la Calvo (1983) is optimal: the firm rationally chooses to be inattentive to news, only sporadically updating its information set.

Meanwhile, Sims (1998) argues that continuous-time optimizing behavior with adjustment costs may sometimes, and under some specific conditions, approximate well behavior that is actually based on limited information-processing capacity. In fact, Sims (2003) elaborates on the intuition about the relevance of information-processing constraints by showing that a capacity constraint can substitute for adjustment costs in a dynamic optimization problem. Woodford (2003) builds on Sims' (1998, 2003) suggestion to model firms as having a limited capacity for absorbing and processing information. Given that price setters learn about monetary policy via a limited-information channel, it is as if they observe monetary policy with a random error and hence have to solve a signal-extraction problem. As a result, real effects of monetary policy may be persistent. Mackowiack and Wiederholt (2009) also model firms as continuously updating their information set but never fully observing the true state, having to form and update beliefs about the underlying fundamentals through a signal-extraction problem. Meanwhile, Woodford (2009) assumes that firms can pay a fixed information cost at discrete times to perform a price review, and when they do so, they obtain complete information about the current state of the economy, just as in delayed information models. The model also assumes that between these adjustment dates, firms obtain imperfect signals as in partial information models.

In the model of this paper, just as in the delayed information model of Mankiw and Reis (2002), in a given period, only a share of firms have up-to-date information, but this share is not an exogenously given constant. As in the canonical delayed information model, all information-updating firms obtain perfect information and hence derive the same optimal price. However, as we assume that there is heterogeneity in the cognitive abilities of firms, individual information-updating costs differ across firms. In fact, it is also because of the same heterogeneity that firms not paying the information-updating cost to compute the optimal price have imperfect perception of the resulting losses. Besides, while Reis (2006) derives an optimal frequency of information updating for a representative producer making optimal output and price decisions, we devise evolutionary microdynamics that interact with the macrodynamics of the price level and output to explore the coevolution of the frequency distribution of information-updating strategies across firms and the adjustment of the general price level to a one-time monetary shock. As it turns out, we contribute to the research on temporary real effects of monetary shocks by offering an alternative imperfect information model based on a novel analytical notion that we refer to as boundedly rational inattentiveness.

Nonetheless, our contribution differs from Carroll's (2006) apparently similarly motivated one mostly in that we do not use epidemiological models of spread of disease to provide a microfoundation to the assumption of slow dissemination of information adopted in Mankiw and Reis (2002). Carroll (2006) proposes an approach to expectations formation where only a small set of agents (professional forecasters) form their own expectations, which spread through the population via the news media in a way analogous to the spread of a disease. The paper provides an epidemiological micro-foundation for a simple, aggregate expectations model that is mathematically very similar to that proposed by Mankiw and Reis (2002). Carroll's (2006) baseline model is indeed mathematically very similar to the Calvo (1983) model in which firms change their prices with a given probability. While in the model of Mankiw and Reis (2002), the updating agents construct their own rational forecast of the future course of the economy, in the Carroll (2006) model, the updating agents learn about the experts' forecast from the news media.

Our contribution also differs from Saint-Paul's (2005) interesting evolutionary microfoundation to nominal adjustment first in that we do not assume from the outset that there are only imperfectly rational strategies being played, although we consider only one of these coexisting with a fully rational one. Second, while in Saint-Paul (2005), the rational expectations equilibrium is characterized by a simple pricing rule that firms can easily adopt, although there is no convergence to that equilibrium for all parameter values, in our model, an individual firm has to pay an information-updating cost to establish the optimal price. In our model, however, evolutionary learning dynamics take the information-updating process to a long-run equilibrium where, albeit either most or even all firms do not pay the information-updating cost, the general price level is the symmetric Nash equilibrium price. Third, in Saint-Paul (2005), firms' payoffs depend on aggregate demand, their own price, and their neighbor's price; while in our model, the local interaction coming through the latter is absent, but individual payoffs feature a random component. Fourth, in both models, firms drop a rule either because it yields a lower payoff than an alternative rule or simply because they want to experiment with another rule; but Saint-Paul (2005) considers a more complex experimentation process, which even includes the adoption of an entirely new rule.

Our contribution is also related to the model set forth in Sethi and Franke (1995), in which firms make output decisions in a stochastic environment (because of shocks to costs of production) featuring strategic complementarity. While naive firms follow a costless adaptive rule to form expectations about returns of production projects, sophisticated firms incur a common (and deterministic) optimization cost (treated as both fixed and varying with the share of sophisticated firms) to form rational expectations. The composition of the population of firms then evolves over time under pressure of differential payoffs, with sophisticated agents being favored if optimization is cheap or the stochastic environment highly variable, but naive agents usually survive. In fact, with positive optimization cost in the special case of a macroeconomic deterministic environment, sophisticated firms do not survive in the long run and naive expectations converge to rational expectations.

Our contribution differs from the interesting one proposed in Sethi and Franke (1995) in several respects. First, our underlying economic model is a different one, featuring firms that make interdependent price and output decisions in a deterministic macroeconomic environment. However, in our model, firms' individual payoffs have a random component, a feature that plays an important role both in the evolutionary dynamics of strategy revision and in the convergence dynamics of the macroeconomic variables. Second, in our model, firms' behavior differs regarding whether or not to update the information set (at a cost) to establish the optimal price. Third, although we also consider that firms choose to behave as either fully rational at a cost or boundedly rational at no similar cost, in our model, the individual cost of full rationality features a random component (but the average cost of full rationality is fixed) and boundedly rational firms' adaptive behavior (which yields losses differing across firms) is faster than that in the model by Sethi and Franke (1995). In fact, while in the latter, naive firms revise their expected return by less than the discrepancy between the expected and the actual return, in our model, boundedly rational firms revise their price by the full discrepancy between their price and the optimal one, and therefore set a one-period lagged optimal price. Although the evolutionary dynamics driving the composition of the population of firms is to some extent similar in both models, the fact that adopted strategies are also assumed to be revised every period in both models implies that the faster adaptive behavior assumed in this paper is more reasonable.

Fourth, our evolutionary dynamics of strategy revision feature an asymmetry implied by a key assumption about information possessed by firms. While a firm that pays the information-updating cost is able to both establish the optimal price and compute the average payoff of firms that do not pay such cost, a firm of the latter type has to rely on a random pairwise matching process to know the payoff of a firm of the former type. Moreover, in our model, firms' heterogeneous cognitive abilities, by creating heterogeneity in individual payoffs, have important implications for the evolutionary dynamics. Fifth, the evolutionary dynamics set forth herein can operate subject to a perturbation analogous to mutation in natural environments. As a result, and in contrast to Sethi and Franke's (1995) special case of a positive optimization cost in a deterministic macroeconomic context, in the continuous presence of mutant firms, the boundedly rational strategy is not the only survivor in the long-run equilibrium (albeit it is the most played strategy).

This paper is also related to the literature on predictor choice following the discrete-decision, multinomial logit model elaborated in Brock and Honimes (1997). It is worth pointing out several similarities and differences between the two models. First, as in Brock and Flommes (1997), in the model herein, firms choose a predictor from a set of alternative predictors whose cost ordering increases in a predictor's precision, while the share of firms in the population using a certain predictor varies positively with its relative net benefit. In our model, however, there is heterogeneity in the cognitive abilities of firms, so that both the individual cost paid to have a perfect foresight and the individual loss associated with forming costless adaptive foresight vary across firms. As in Brock and Flommes (1997), perfect foresight firms also have perfect knowledge about the frequency distribution of all other foresight strategies. However, unlike in Brock and Hommes (1997), in our model, the individual (and hence average) performance of each preditor is not publicly available to all firms. While firms which pay the cost to have perfect foresight know the average payoff of the adaptive foresight strategy, and hence know the relative net benefit of playing the perfect foresight strategy, firms which play the adaptive foresight strategy have to rely on a pairwise random matching process to learn the relative net benefit of so playing. In fact, while the predictor switching mechanism in Brock and Hommes (1997) is derived from a random utility model under very specific assumptions about the underlying stochastic process, the strategy switching in our paper is driven by an asymmetric evolutionary mechanism (only firms with adaptive foresight rely on pairwise comparisons of losses) without and with mutation (some firms choose a predictor refraining from comparing payoffs and rather change predictor at random). Second, Brock and Hommes (1997) assume that predictors' costs are exogenously determined, and although we assume that the average cost of the perfect foresight strategy is exogenously given, individual costs are assumed to be normally distributed around such average. However, the main qualitative results found in our paper are robust to the consideration of the average cost of the perfect foresight strategy as varying with the frequency distribution of foresight strategies rather than fixed. Third, in Brock and Hommes (1997), firms use a discrete choice model to pick a predictor where the deterministic part of the utility of the predictor is the performance measure. While the standard discrete choice model features deterministic and random individual-specific characteristics, in our model, the payoff of individual firms playing each available foresight strategy features both a deterministic component and random firm-specific cognitive abilities. In our model, such a random component does not, therefore, correspond to random shocks experienced by the firm, but to random firm-specific cognitive abilities, although in both cases, the random component influences the payoff of each of the possible choices.

Fourth, as in Brock and Hommes (1997), firms base decisions upon predictions of future values of endogenous variables whose actual values are determined by equilibrium equations. In both models, there is a dynamics across predictor choice which is coupled to the equilibrium dynamics of the endogenous variables. In the cobweb model in Brock and Hommes (1997), the predictor choice feeds into the market equilibrium dynamics, which in turn feeds into predictor choice, whereas in our model, the predictor choice feeds into the equilibrium dynamics of the general price level and real output, which in turn feeds into predictor choice. Fifth, while the selection model in Brock and Hommes (1997), for a finite intensity of choice (or intensity with which firms react to increases in relative net benefit) and information cost, yields a strictly positive share of firms using the strictly dominated perfect foresight strategy even in the long-run equilibrium, in our model, without mutation, strictly dominated foresight strategies vanish asymptotically. In our model, the long-run, evolutionary equilibrium is characterized either by all firms playing the adaptive foresight strategy (in the absence of mutation) or by most firms playing such strategy (in the presence of mutation). In our model, therefore, it takes an exogenous perturbation in the form of mutation for the perfect foresight strategy to survive asymptotically, whereas Brock and Hommes (1997) obtain coexistence of both foresight strategies in the long-run equilibrium by devising a predictor dynamics whose underlying selection mechanism is based on a random utility model. When there is mutation, the long-run equilibrium share of firms playing the perfect foresight strategy in our model depends, among other parameters, on the distribution of cognitive abilities across firms, which is random. (1)

III. STRUCTURE OF THE MODEL

Ball and Romer (1991) explore price stickiness arising from coordination failures (because of strategic complementarities in price setting) in a monopolistically competitive economy populated by a continuum of producers. This setting is borrowed from Ball and Romer (1989), in which there is a finite population of producers. Even though these two contributions are based on Blanchard and Kiyotaki (1987), they differ from the latter by abstracting from the labor market (the economy is populated by "yeoman farmers" who sell differentiated goods produced with their own labor and purchase the products of all other farmers). We draw on the model developed in Ball and Romer (1991) because of not only its focus on the goods market but also its assumption of a continuum of producers, which is a more convenient structure for our evolutionary game-theoretic modeling. As we clarify shortly, however, there are significant differences between our modeling strategy and that adopted in Ball and Romer (1989, 1991).

As in Ball and Romer (1991), in a given period t. each farmer i [member of] [0, 1] [subset] R produces [Y.sub.i], units of a differentiated good i by employing L, units of his own labor. The respective production function is given by:

(1) [Y.sub.i] = [L.sub.i].

Farmer i sets the unit price [P.sub.i] of his product and sells it taking other farmers' prices as given, thus obtaining an amount of nominal income [I.sub.i]. Meanwhile, this nominal income is allocated to the purchase of a continuum of differentiated goods whose associated utility is given by the following constant elasticity of substitution consumption index:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [C.sub.ij] stands for farmer z's consumption of the product of farmer j and [epsilon] > 1 is the elasticity of substitution between any two goods. As it turns out, farmer z's utility function is given by:

(3) [U.sub.i] = [C.sub.i] - [[epsilon] - 1/[lambda][epsilon]] - [L.sub.[lambda].sub.i],

where [lambda] > 1 is a parameter measuring the extent of increasing marginal disutility of labor, and the coefficient on [L.sub.[lambda].sub.i] in (3) is chosen for convenience. (2)

Farmer i's decision problem can be solved in two stages. The first stage consists in computing the consumption basket that maximizes his consumption index [C.sub.i], taking as given his nominal income, the price of his product, and the other farmers' prices. The second stage, in turn, consists in finding the price of product i that maximizes farmer i's utility and, by extension, gives the production of good i, given the vector of prices of the other goods. Therefore, farmer i first solves the following constrained maximization problem:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The solution to this constrained maximization problem yields the following Marshallian demand function for each good j [member of] [0, 1] [subset] R by farmer i's:

(5) [C.sub.ij] = (P/[[P.sub.j].sup.[epsilon]]) [[I.sup.i]/p],

where P is the general price level given by (3):

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Plugging the Marshallian demand function in

(5) into (2), we obtain the optimal consumption index (indirect utility function):

(7) [C.sub.i] = [[I.sub.i]/P].

Let us now turn to the second stage of farmer i's decision problem, which consists in finding the price of product i that maximizes his utility, given the vector of prices of the other goods. First, let us compute the aggregate demand for good which is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] dj. Using (5) and (7), aggregate demand for good i can be expressed as follows:

(8) [Y.sup.D.sub.i] = C ([[P.sup.i]/P]).sup.-[epsilon]]).

3. Given that farmer i must spend P to obtain one unit of [C.sub.i], (Ball and Romer 1989, 510), (2) implies that the minimum expenditure to obtain C, = 1 is given by the following expenditure minimization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In fact, the solution to this problem is the Hicksian demand function for each good j = 1,2, ..., N, which is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Plugging this Hicksian demand function into the objective function of the expenditure minimization problem above, we obtain the expression for the general price level in (6).

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is aggregate consumption.

Farmer i's nominal income is given by [I.sub.i] = [P.sub.i][Y.sub.i]. Assuming equality between supply and demand, [Y.sub.i], = [Y.sup.D.sub.i], for all goods I [member of] [0,1] [subset] R, it follows from (7) and (8) that:

(9) Ci = C ([[P.sub.i]/P).sup.1-[epsilon]]).

As in Ball and Romer (1991), money is introduced in the model by assuming that it is required for transactions, which conveniently allows taking the nominal money supply, M, as a proxy for the nominal aggregate demand. Therefore, aggregate nominal spending equals the money stock:

(10) PC = M.

The production function in (1) and the assumption of market clearing for all goods jointly imply that [L.sub.i], = [Y.sub.i] = [Y.sup.D.sub.i]. As it turns out, the utility function in (3) can be expressed as [U.sub.i] = [C.sub.i] - [([epsilon] - 1)/([epsilon][lambda])] [([Y.sup.D.sub.i]).sup.[lambda]]. Plugging (10) into (9) and (8) and then the resulting expressions into the utility function just reexpressed, we obtain the maximum utility of farmer i as a function of the nominal aggregate demand, the price of good i, and the general price level resulting from the simultaneous decisions of all farmers (and hence farmer i's relative price, [P.sub.i]/P) (see Equation (10) in Ball and Romer 1991, 541):

(11) [U.sub.i] = M/P ([P.sub.i]/P).sup.1-[epsilon]] - [[epsilon] - 1/[lambda][epsilon] [([M/P).sup.[lambda]] (([P.sub.i]/P).sup.[lambda][epsilon]]).

Therefore, farmer i's optimal price is the maximizer of (11), which is given by:

(12) [P.sup.*.sub.i] = [P.sup.phi][M.sup.1-[phi]],

where [phi] = 1 -([lambda] - 1)/([lambda][epsilon] - [[epsilon] - 1) [member of] (0, 1) [subset] R is a constant denoting the elasticity of each individual price with respect to the observed general price level. Therefore, from profit maximization, farmer i chooses a relative price that varies positively with the real stock of money. The reason is that an increase in the real stock of money, by increasing aggregate demand, shifts out the demand faced by farmer i and hence leads him to increase his relative price.

The symmetric Nash equilibrium of this price-setting game occurs when [P.sub.i] = P for all i [subset] [0,1][subset] R, in which case it follows that P = M and Y [equivalent to] C = 1. Unlike in Ball and Romer (1991), however, in this paper, the general price level is not common knowledge. More precisely, at each period t e {0, 1, 2, ...}, there is a fraction [[lambda].sub.t] of the population of firms (as we refer to farmers from now on) that pay a cost to update its information set and establish the optimal price, and we refer to them as Nash firms. Meanwhile, the remaining fraction of firms, 1 - [[lambda].sub.t], set their price without fully knowing the structure of relative prices, as they decide not to pay the cost to update the relevant information set (we refer to these firms as bounded rationality firms). This heterogeneity in price setting can be alternatively interpreted as a result of firms having to form expectations about the general price level. (4) In this alternative interpretation, while Nash firms pay an information cost to form rational expectations about the general price level, boundedly rational firms forecast the latter using a non-updated information set. (5)

Based on the best-reply response in (12), the optimal price established by Nash firms in a given period t is given by (6):

(13) [P.sub.n,t] = [P.sup.[phi].sub.t][M.sup.1-[phi].sub.t].

Boundedly rational firms, meanwhile, having decided not to pay the cost to update their information set, establish the following price:

(14) [p.sub.b,t] = [p.sup.[phi].sub.t-1] [M.sup.1-[phi].sub.t] Therefore, the price set at each period t by Nash firms, [P.sub.n,t], is the standard optimal price, that is, the optimal price choice given the current general price level, [P.sub.t]. The price set at each period t by boundedly rational firms, [P.sub.bt], in turn, is the suboptimal price formed using the one-period lagged general price level, [P.sub.t-1]. Note that at each period t, all firms know the current nominal stock of money, that is, we assume that [M.sub.t] is publicly known for all t [member of] {0, 1, 2, ...}. The reason for this specific assumption about the nominal stock of money is our purpose to investigate the implications of boundedly rational inattentiveness in a context of strategic complementarity where the optimal price varies positively with the general price level, which in turn depends on the frequency distribution of information-updating strategies across firms. We, therefore, assume that M, is publicly known in order to focus attention on the interplay between the evolutionary dynamics of the distribution of information-updating strategies and the adjustment dynamics of the general price level and real output to a monetary shock.

We approximate the current general price level by the geometric average between the current prices set by Nash firms and bounded rationality firms (7):

(15)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given (13)--(15), we can express the general price level and the price set by Nash firms as:

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

7. As in a given period t there is a fraction X, of Nash firms and a fraction 1 - X, of bounded rationality firms, Equation (6) can be used to express the general price level as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For e sufficiently close to one, which we assume, it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A similar approximation is rather assumed in Blanchard and Fischer (1989, section 8.2) in a simplified version of the monopolistically competitive model set forth in Blanchard and Kiyotaki (1987) then extended to incorporate staggered price decisions in the spirit of Fischer (1977).

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [xi][([[lambda].sub.t]) = 1/(1 -[psi] [[lambda].sub.t]). For a given distribution of information-updating strategies at period t, that is, for a given [[lambda].sub.t] the price set by each type of firm and the general price level are determined by (14), (16), and (17). Hence (17) can be seen as the best-reply function of Nash firms in a repeated game in which a fraction 1 - [[lambda].sub.t], of bounded rationality firms contaminates the game (in the sense used by Droste, Hommes, and Tuinstra 2002, 244) by not updating their information set. In fact, Nash firms not only best reply to other firms, but also take into account that other firms do likewise. Therefore, between themselves, the Nash firms coordinate on a Nash equilibrium.

Firms not setting the optimal price face losses whose average value at each period t, [L.sub.bt], is given by a quadratic function of the discrepancy between the price [P.sub.b,t] and the optimal price [P.sub.n,t]. As shown in Appendix A, this loss function can be obtained as a second-order Taylor approximation around the point [P.sub.b,t] = [P.sub.n,t]. Using (14) and (17), the average loss of bounded rationality firms can then be expressed as follows:

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Appendix A. To facilitate later usage of the average loss given by (18), it is convenient to reexpress it as a function of [P.sub.t] rather than [P.sub.t-1]. This can be done by isolating the latter in (16) and substituting the resulting expression in (18), which yields:

(18a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We assume that the actual losses of bounded rationality firms are normally distributed around the average value given by (18a), so that the loss of the kth boundedly rational firm can be expressed as:

(19) [L.sup.k.sub.b,t] = [L.sub.b] ([[lambda].sub.t], [P.sup.t])+ [[mu].sup.k.sub.t],

where [[mu].sup.k.sub.t] denotes a random variable having a normal probability distribution with mean zero and variance [[sigma].sup.2], that is, [[mu].sup.k.sub.t] ~ N (0, [[sigma].sup.2]). This random variable captures the dispersion in the cognitive abilities of firms when playing the boundedly rational strategy. These cognitive idiosyncrasies are assumed to be independent across firms and time, and from the variables in (18a). (8) Given that E [[mu].sup.k.sub.t]] = 0, it follows that E [[L.sup.k.sub.b,t] = [L.sub.b] ([lambda].sub.t], [P.sub.t]) at every t [member of] {0, 1, 2, ...}. Therefore, using (A4) in Appendix A, the average payoff of the subpopulation of firms adopting the strategy of not updating the relevant information set (to set the current optimal price) at period t can be written as E [U.sup.k.sub.b,t] = [U.sup.k *.sub.b,t] + [L.sub.b] ([[lambda].sub.t], [P.sub.t], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Meanwhile, Nash firms, by always setting the current optimal price, do not face the loss given by (18a), although they face an information-updating cost. As this perfect foresight requires knowing the distribution of information-updating strategies in the population of firms, firms' potentially heterogeneous information-updating behavior imposes an information cost on the subpopulation of Nash firms. We suppose that the average value of such information cost associated with computing the current optimal price is an exogenously given strictly positive constant, c > 0. As a result, such information cost can be interpreted as representing the average loss of Nash firms at a given period t, with the corresponding individual losses being normally distributed around it. Formally, the loss of the lth Nash firm can then be expressed as:

(20) [L.sup.l.sub.n,t] = -c + [[mu].sup.l.sub.t].

where [[mu].sup.l.sub.t] if denotes a random variable having a normal probability distribution with mean zero and variance [[sigma].sup.2], that is, [[mu].sup.l.sub.t] ~ N (0, [[sigma].sup.2]). This random variable captures the dispersion in the cognitive abilities of firms when playing the Nash information-updating strategy. Analogously to the case where firms are playing the boundedly rational strategy, these cognitive idiosyncrasies are independent across firms and time, and from the variables in (18a). (9) As we assume that [[mu].sup.l.sub.t] N (0, [[sigma].sup.2]), it follows that for any t [member of] {0, 1, 2, ...}, the expected loss of a (randomly chosen) firm playing the costly strategy of updating its relevant information set is given by E [L.sup.l.sub.t] = -c. Using (A4) in Appendix A, the average payoff of the subpopulation of firms adopting the strategy of updating the relevant information set (to establish the optimal price) at period t can, therefore, be written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The flows of firms between the two information-updating strategies depend on the rate of strategy revision per period and the corresponding choice probabilities (Weibull 1995, 152). We assume that all firms reevaluate their information-updating strategy in the transition between any two contiguous periods, so that the rate of strategy revision per period for all firms is equal to one. However, while the rate of strategy revision is time dependent, the probability and the rate of individual information-updating (and hence price change) are state dependent, as described in what follows.

Let us assume that in every period, each boundedly rational firm compares its own loss with the loss incurred by a randomly chosen firm. As the probability that the loss of a boundedly rational firm is compared with the loss of a Nash firm is then equal to the number (measure) of boundedly rational firms that can potentially switch to the Nash strategy at period t + 1 is given by (1 - [[lambda].sub.t])[[lambda].sub.t].

Now, a k boundedly rational firm randomly comparing its actual payoff [U.sup.k.sub.b,t] = [U.sup.k *.sub.b,t] [L.sub.b]([[lambda].sub.t], [P.sub.t]) + [[mu].sup.k.sub.t] with the payoff of a l Nash firm [U.sup.l.sub.n,t] = [U.sup.l *.sub.n,t] - c + [[mu].sup.l.sub.t] will switch to the information-updating strategy (and hence ultimately to the price strategy) of the latter if, and only if, [U.sup.l.sub.n,t] > [U.sup.k.sub.b,t] As [U.sup.l *.sub.n,t] = [U.sup.k *.sub.n,t] for any k and l, the inequality [U.sup.l.sub.n,t] > [U.sup.k.sub.b,t] (will hold if, and only if, [L.sup.k.sub.b,t] = [L.sub.b] + ([lambda].sub.t], [P.sup.t]) + [[mu].sup.k.sub.b,t] < - c + [[mu].sup.l.sub.t] = [L.sup.l.sub.n,t], which is equivalent to [[mu].sup.k.sub.t] - [[mu].sup.l.sub.t] < - c - [L.sub.b] ([lambda].sub.t], [P.sub.t]). As [[mu].sup.k.sub.t] - [[mu].sup.k.sub.t] and [[mu].sup.l.sub.t] are independent and identically normally distributed random variables with zero mean and constant variance, the random variable [[mu].sup.k.sub.t] - [[mu].sup.l.sub.t] is also normally distributed with zero mean and constant variance: [[[mu].sup.k.sub.t] - [[mu].sup.l.sub.t]] ~ N (0, 2 [[sigma].sup.2]). Let G: R [right arrow] [0,1] [subset] R be the cumulative distribution function of the random variable [[mu].sup.k.sub.t] - [[mu].sup.l.sub.t]. The probability with which a k boundedly rational firm will switch to the alternative information-updating strategy (and hence ultimately to the Nash price strategy) is then given by G (-c - [L.sub.b] ([[lambda].sub.t], [P.sub.t])).

Therefore, while the probability with which a boundedly rational firm becomes a potentially revising firm is given by [[lambda].sub.t] the probability with which a potentially revising boundedly rational firm switches to the Nash price strategy is given by G{-c-[L.sub.b]([lambda].sub.t], [P.sub.t])}. Assuming that these two events are statistically independent, the product of their respective probabilities yields the probability with which a boundedly rational firm becomes a Nash firm, namely, [[lambda].sub.t]G(-c -[L.sub.b]([[lambda].sub.t], [P.sub.t])). As there are 1 - [[lambda].sub.t], boundedly rational firms at period t, the expected number (measure) of boundedly rational firms becoming Nash firm at period t + 1 is then given by:

(21) (1 - [lambda].sub.t]) ([lambda].sub.t])G (-c - [L.sub.b] ([lambda].sub.t], [P.sub.t])).

Unlike a boundedly rational firm, a Nash firm comes to know (at a cost) the average loss of the alternative information-updating strategy at any period t, so that it does not have to rely on a random pairwise comparison of losses. Consequently, a l Nash firm switches to the alternative strategy of refraining to pay the information-updating cost in the next period if, and only if, [L.sup.l.sub.n,t] = -c + [[mu].sup.l.sub.t] < [L.sub.b] ([[lambda].sub.t], [P.sub.t]), which is equivalent to [[mu].sup.k.sub.t] < [subset] + [L.sup.b] ([lambda].sub.t], [P.sub.t]. Meanwhile, let F: R [right arrow] [0,1] [subset] R be the cumulative distribution function of the random variable [[mu].sup.l.sub.t]). It follows that the probability of a strategy switch of a l Nash firm is then given by F(c + [L.sub.b] ([[lambda].sub.t], [P.sub.t])). As all Nash firms are potentially revising players in every period, the number (measure) of Nash firms that decide not to pay the information-updating cost in the next period is simply:

(22) [[lambda].sub.t]F (c + [[L.sub.b]([lambda].sub.t], [[P.sub.t])).

Therefore, note that Nash firms are not intertemporally rational. (10) In fact, Nash firms will immediately switch to being boundedly rational in the next period if the average payoffs for playing the boundedly rational strategy are higher than payoffs for playing the Nash strategy in the current period. Yet, such strategy choice is not necessarily intertemporally rational, for in the next period, Nash firms may perform better than bounded rationality firms. Thus, as regards strategy choice, Nash firms are also boundedly rational.

The difference between the influx described by (21) and the efflux described by (22) yields the rate of change of the proportion of Nash firms in the population between periods t and t + 1:

(23) [[lambda].sub.t+1] - [[lambda].sub.t] = [[lambda].sub.t] [(1 - [[lambda].sub.t] G (-c - [L.sub.b] ([[lambda].sub.t], [P.sub.t])) - F (c + [L.sub.b] ([[lambda].sub.t], [P.sup.t]))].

As Appendix B shows, [[lambda].sub.t+1] [epsilon](0, 1) [subset] R for all states ([lambda].sub.t], [P.sub.t]) [member of][THETA] = {[R.sup.2.sub.+]: 0 [less than or equal to] [[lambda].sub.t] < 1, [P.sup.t] > 0}, vector of parameters (M, c, [sigma]) [epsilon] [R.sup.3.sub.++] and ([phi] [epsilon] (0,1) [subset] R. The intuition for why our model is not defined for [lambda] = 1 is that, given the nature of the information imperfection embedded in it, [lambda] = 1 means that all firms are paying the cost to learn the measure of the heterogeneity in the choice of information-updating strategy when there is no such cost to be paid.

To gain in generality, we consider the possibility that the evolutionary dynamics in (23) operates in the presence of a noise term, analogous to mutation in natural environments. In a biological setting, mutation is interpreted literally as consisting of random changes in genetic codes. In economic settings, as pointed out by Samuelson (1997, ch. 7), mutation refers to a situation in which a player refrains from comparing payoffs and changes strategy at random. Therefore, the present extension features mutation as exogenous noise in the evolutionary dynamics (23) leading some firms to choose a price setting strategy (Nash or bounded rationality) at random. This disturbance component is intended to capture the possibility, as sugested by Kandori, Mailath, and Rob (1993), either that a firm exits the economy with some (fixed) probability and is replaced with a new firm who knows nothing about (or is inexperienced in) the respective decision-making process or that each firm simply "experiments" occasionally with exogenously fixed probability.

Drawing on Gale, Binmore, and Samuelson (1995), mutation can be incorporated into (23) as follows. Let [theta][epsilon] (0,1) [subset] R be the measure of mutant firms that choose an information-updating strategy in a given revision period independently of the respective payoffs. Consequently, there are [theta][[lambda].sub.t], Nash firms and [theta](1 - [[lambda].sub.t]) bounded rationality firms behaving as mutants. We assume that mutant firms choose either one or the other of the two information-updating strategies with equal probability, so that there are [[lambda].sub.t],(l/2) Nash mutant firms and [theta](1 - [[lambda].sub.t])(l/2) boundedly rational mutant firms changing information-updating strategy. The net flow of mutant firms becoming Nash firms in a given revision period, which can be either positive or negative, is then the following:

(24) [theta](l - [[lambda].sub.t])1/2 - [theta][[lambda].sub.t] 1/2 = [theta](1/2 - [[lambda].sub.t]).

Following Gale, Binmore, and Samuelson (1995), this perturbation can be added to the evolutionary dynamics (23) to yield the following noisy evolutionary dynamics'.

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or, in a compact form,

(25a) [[lambda].sub.t + 1] = [psi] > {[[lambda].sub.t], [P.sub.t]},

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As shown in Appendix B, [[lambda].sub.t+1][epsilon](0,1) [subset] R for all state ([[lambda].sub.t+1], [P.sup.t] [epsilon] [THETA] = {R.sup.2.sub.+] : 0 [less than or equal to] [[lambda].sub.t] < 1, [P.sub.t] > 0}, vector of parameters (M, c, [sigma]) [epsilon] [R.sup.3.sub.++], [psi] [epsilon](0, 1) [subset] R and [theta] [epsilon] [0, 1) [subset] R.

Meanwhile, we can use (16) and (25a) to obtain the difference equation associated with the general price level:

(26) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The state transition of the economy is, therefore, determined by the system (25a)-(26), whose state space is defined by [THETA] = {([[lambda].sub.t], [P.sub.t]) [epsilon] [[R.sup.2.sub.+]: 0 [less than or equal to] [[lambda].sub.t] < 1, [P.sup.t], > 0}.

IV. EXISTENCE AND STABILITY OF AN EVOLUTIONARY EQUILIBRIUM

Throughout the ensuing analysis, we suppose that [M.sup.t] = M > 0 for all t [epsilon] {0,1,2, ...}. In the following proposition, we establish the existence and uniqueness of an evolutionary equilibrium.

PROPOSITION 1. For a given constant nominal stock of money, the dynamic system (25a) - (26) has a unique evolutionary equilibrium given by ([[lambda].sub.*], M) [member of] [THETA], with [[lambda].sub.*] [member of] (0,1/2) [subset] R if [theta][epsilon] (0,l) [subset] R and [[lambda].sup.*] = 0 if [theta] = 0.

Proof. See Appendix C.

Note that the general price level in the long-run equilibrium is the symmetric Nash equilibrium price. This general price level is nonetheless achieved in a long-run equilibrium configuration in which the boundedly rational information-updating strategy is played either by all firms (when mutation is absent) or by the majority of the firms (when there is mutation).

Now, a question arises regarding whether the evolutionary dynamics in (25a)-(26) takes the economy to the long-run equilibrium configuration whose existence is established in Proposition 1. Or, to phrase it differently, do boundedly rational firms come to learn to play the Nash strategy without ever having to pay the information-updating cost? The answer is yes, as formally established in the following proposition.

PROPOSITION 2. (i) The evolutionary equilibrium ([[lambda].sub.*], M) [member of] [THETA] of the dynamic system (25a) - (26) is locally asymptotically stable; (ii) For any initial condition given by ([[lambda].sub.0], [P.sub.0]) [member of] [THETA], the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is positively invariant; (iii) For any initial condition given by ([[lambda].sub.0], [P.sub.0]) [epsilon] [THETA], the general price level converges monotonically toward a given constant nominal stock of money M; and (iv) If the economy is initially at any state (0, [P.sup.0]) [member of] [THETA], the proportion of Nash firms rises. Meanwhile, for any [[lambda].sub.t] sufficiently close to one, the proportion of Nash firms falls.

Proof. See Appendix D.

Therefore, the bounded rationality information-updating strategy is either the only or the most played strategy in the long-run equilibrium despite the asymmetric nature of the evolutionary mechanism of strategy revision.

In fact, in the long-run equilibrium, the Nash strategy is either extinct or played by the minority of firms even though all firms playing it are potentially revising firms at each period, while the probability that a bounded rationality firm becomes a potentially revising firm at each period, given by [[lambda].sub.t] < 1 (cf. Appendix B), is decreasing in the proportion of bounded rationality firms. As noted earlier, given the nature of the information imperfection embedded in our model, X - 1 is not admissible because it means that all firms are paying the cost to learn the measure of a heterogeneity that does not exist. Meanwhile, when there is no mutation, X -- 0 is an equilibrium because it means that no firm is paying the cost of an inexistent heterogeneity, and when there is mutation, X* e [0,1/2) is an equilibrium because there is a heterogeneity cost to be paid.

Furthermore, it is shown in Appendix E that, when there is mutation, so that the two information-updating strategies survive in the evolutionary equilibrium (yet the boundedly rational strategy predominates), such long-run equilibrium heterogeneity varies expectedly with some parameters. Intuitively, the long-run equilibrium proportion of firms playing the Nash strategy varies negatively with the average information-updating cost (see Equation (A27)) and positively with the rate of mutation (see Equation (A26), and recall that mutant firms choose either one or the other of the two information-updating strategies with equal probability). Meanwhile, the long-run equilibrium proportion of firms that play the bounded rationality strategy varies negatively with the standard deviation of the cognitive abilities of firms. To grasp the intuition for this result, recall that the evolutionary mechanism of strategy revision is asymmetric (bounded rationality firms have to rely on random pairwise comparisons of losses, while Nash firms do not). Hence, a more dispersed distribution of cognitive abilities across firms, with everything else constant, including a strictly positive mutation rate (recall again that mutant firms select one or the other of the two available strategies with equal probability), lowers the probability with which a firm playing the bounded rationality strategy switches to the Nash strategy, which is given by G(x) in (21). Interestingly, then, when there is mutation, the rate of survival of the Nash information-updating strategy in the evolutionary equilibrium varies positively with the dispersion in the cognitive abilities of firms.

Hence, while the model in Brock and Homines (1997) can exhibit local instability of the long-run equilibrium and complicated global equilibrium dynamics, in our model (with and without mutation), the long-run, evolutionary equilibrium is unique and locally stable. In a binary prediction choice setting, Brock and Hommes (1997) show that a rational choice between a cheap destabilizing (adaptive foresight) predictor and a costly (perfect foresight) stabilizing one does lead to the existence of a very complicated dynamics when the intensity of choice to switch predictors is high. In other words, such a model shows that with information costs it may be rational for firms to select methods other than perfect foresight, with the conflict between cheap free riding and costly sophisticated prediction being a potential source of instability and complicated global dynamics. In our model, meanwhile, the conflict between free riding and costly sophisticated prediction is rather an actual source of stability even if there is mutation, so that the assertion by Brock and Hommes (1997) that instability is inherent in such situations when more sophisticated prediction methods are more expensive is not confirmed in our model.

In fact, the information-updating game investigated in this paper is subject to evolutionary social learning dynamics, which takes it to a long-run equilibrium in which, albeit either the majority or even all of the firms play the boundedly rational information-updating strategy, the general price level is the symmetric Nash equilibrium price (and therefore the nominal adjustment of the general price level to a monetary shock is only temporarily incomplete). In response to a one-time permanent monetary shock, temporarily incomplete nominal adjustment (and hence monetary non-neutrality) obtains not only if all firms are boundedly rational, but also if both information-updating strategies coexist as a long-run equilibrium configuration. In both cases, a permanent rise (fall) in the nominal stock of money, by temporarily leading to a less than one-to-one increase (decrease) in the general price level, causes a temporary rise (fall) in real output. As a result, there is long-run monetary neutrality even if either the majority or all of the firms playing the boundedly rational information-updating strategy are the only (and stable) evolutionary equilibrium.

In fact, it follows from (16) that the elasticity of the general price level with respect to the nominal money stock is (d ln [P.sup.t]/d ln [M.sup.t]) = (1 - [psi])) [xi] ([[lambda].sub.t]), from which it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Moreover, as such elasticity is strictly increasing in [[lambda].sub.t], the extent of the incomplete nominal adjustment of the general price level varies positively with the proportion of boundedly rational firms. Or, using the novel notion introduced in this paper, the real impact of a one-time permanent monetary shock varies positively with the proportion of firms behaving as boundedly rational inattentive.

Therefore, as aptly argued by Samuelson (1997), an equilibrium configuration may be reached through an evolutionary dynamics when decision making is guided by adaptive behavior rather than by unbounded rationality. As Samuelson (1997, 3) describes an equilibrium configuration emerging from rule-of-thumb behavior through an evolutionary dynamics: "The behavior that persists in equilibrium then looks as if it is rational, even though the motivations behind it may be quite different. An equilibrium does not appear because agents are rational, but rather agents appear rational because an equilibrium has been reached." In the present model, in which either only the bounded rationality strategy (when mutation is absent) or both strategies (in the presence of mutation) survive in the evolutionary equilibrium, we can say that the symmetric Nash equilibrium price emerges (and the nominal adjustment of the general price level to a monetary shock is only temporarily incomplete) not because all firms play the Nash strategy, but rather firms appear unboundedly rational because an evolutionary equilibrium has been reached.

It is also worth analyzing the dynamics of the convergence to the evolutionary equilibrium. Per the Jacobian matrix (A29) in Appendix D, around the evolutionary equilibrium represented by ([[lambda].sub.*], [P.sup.*]), the trajectories of the state variables defined by the nonlinear system (25a)--(26a) can be approximated by:

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(28) [P.sub.t] = [P.sub.0] [[[THETA](1 - [[lambda].sub.*])/1 - [theta] [[lambda].sub.*]].sup.t].

Equation (27) shows that without mutation ([theta] = 0), in which case [[lambda].sub.t] = 0, the pace of extinction of the Nash strategy around the evolutionary equilibrium depends essentially on the dispersion in the cognitive abilities of firms. Meanwhile, still without mutation, per (28), the dynamics of the incomplete nominal adjustment of the general price level sufficiently near the long-run equilibrium depends essentially on the elasticity of the individual prices with respect to the general price level, given by [phi]). The higher this parameter, the higher the strategic complementarity involved and the larger the real impact of a monetary shock. When there is mutation ([theta] > 0), (27) reveals that the pace of convergence around the evolutionary equilibrium given by [[lambda].sub.*] [member of] (0,1/2) R depends also on the mutation rate and hence on the specific value of [[lambda].sup.*]. Besides, still with mutation, per (28), the extent of the incomplete nominal adjustment sufficiently near the long-run equilibrium depends on the specific value of [[lambda].sub.*] as well. (11)

In order to gain further intuition about the main qualitative results derived along the way, and some resulting implications for the conduct of monetary policy, we report results from several simulation exercises. (12)

Figure 1 presents simulation results for a situation where mutation is absent ([theta] = 0) and the evolutionary equilibrium is, therefore, characterized by the survival only of the bounded rationality information-updating strategy ([[lambda].sub.*] = 0). In period 1, the general price level and output are in the long-run equilibrium ([P.sub.1] = [P.sub.*] = [M.sup.1] = 1 and [Y.sub.1], = [Y.sub.*] = [M.sub.1]/[P.sub.1] = 1), while the frequency distribution of information-updating strategies is in one of two possible disequilibrium positions, [[lambda].sub.1], = 0.01 (left column) and [[lambda].sub.1] = 0.99 (right column). Moreover, two different values for the degree of strategic complementarity are considered, [psi] = 0.3 and [psi] = 0.7. In period 2, the economy is hit by a one-time permanent monetary shock that raises the nominal money stock by 5% permanently, so that the long-run equilibrium value of the general price level rises by the same proportion.

Several features of this first set of simulations are worth noting. First, the monotonic convergence of [lambda] to the evolutionary equilibrium ([[lambda].sup.*] = 0) obtains for the two levels of strategic complementarity even if the initial evolutionary disequilibrium is quite substantial (as shown in the first line of panels; see also (27)) and in the very next period, the monetary shock generates a disequilibrium in the macroeconomic variables as well (as shown in the second and third lines of panels). Second, the model replicates the hump-shaped temporary response of real output (and inflation, as it can be easily checked) to such a monetary shock, which is a well-established stylized fact. In fact, a large literature in monetary macroeconomics has produced a considerable body of evidence supporting the notion that demand shocks have sizeable and persistent (although not permanent) effects on real output. One strand of this large literature has focused on the effects of monetary shocks, documenting evidence for substantial monetary non-neutrality in the short run (see, e.g., Christiano, Eichenbaum, and Evans 1999; Romer and Romer 2004). Third, as shown in the second and third lines of panels, a higher strategic complementarity generates a larger (smaller) temporary output (price level) response to such a nominal shock. As the model features heterogeneity in information updating, there are some firms that are better informed than others. The higher the degree of strategic complementarity, therefore, the less willing are the better-informed firms to change their prices by much to keep them in line with the less-informed firms, and hence the more significant is the temporary monetary non-neutrality.

Fourth, and as also shown in the second and third lines of panels, the larger initial disequilibrium in [lambda] intuitively generates the smaller (larger) temporary output (price level) response to such a nominal shock. The reason is that the immediate impact on the general price level of such a monetary stock varies positively with the proportion of Nash firms. Fifth, the monotonic convergence of the general price level and real output is faster than the convergence of the frequency distribution of information-updating strategies. This means that a (falling) proportion of firms will keep playing the (costly) Nash strategy even after the general price level has achieved its new long-run equilibrium level (which is equal to new nominal stock of money) and the average loss of the (costless) bounded rationality strategy has, therefore, become zero (see Equation (18a)). Bear in mind, though, that the loss of an individual boundedly rational firm is normally distributed around the respective average loss (see Equation (19)), and that the loss of an individual Nash firm is normally distributed around the positive average cost c (see Equation (20)). Recall, further, that the evolutionary mechanism of strategy revision is asymmetric (boundedly rational firms have to rely on random pairwise comparisons of losses, whereas Nash firms do not), and the intuition for the sluggish extinction of the Nash strategy (even from a very low initial measure of firms playing it, as in the left column of Figure 1) will become clear (see Equation (23)). Sixth, the boundedly rational information-updating strategy is the only survivor in the long-run equilibrium despite the asymmetry in the mechanism of strategy revision. In other words, the long-run equilibrium is characterized by extinction of the Nash strategy even though all firms playing it are potentially revising firms at each period, while the probability that a bounded rationality firm becomes a potentially revising firm at each period, given by [[lambda].sub.t], is therefore decreasing in the proportion of bounded rationality firms.

[FIGURE 1 OMITTED]

Figure 2 reports simulations for a situation where there are mutant firms and the evolutionary equilibrium is, therefore, characterized by the survival of both information-updating strategies, with dominance of the boundedly rational strategy (per Proposition 1). This mixed-strategy evolutionary equilibrium is consistent with robust empirical evidence showing persistent and time-varying heterogeneity in both the frequency and the extent of information updating (see, e.g., Andrade and Le Bihan 2013; Branch 2007; Coibion and Gorodnichenko 2012). In period 1, the general price level and the real output are both in the long-run equilibrium ([P.sub.1] = [P.sup.*] = [M.sub.1] = 1 and [Y.sub.1] = [Y.sub.*] = [M.sub.1]/[P.sub.1] - 1), whereas the distribution of information-updating strategies is in one of two possible long-run equilibrium positions, each one corresponding to a given mutation rate, Ql = l% (left column) and [[theta].sub.1] = 5% (right column). These initial values for the proportion of information-updating firms are indeed empirically plausible. (13) In the empirical literature, the frequency of updating an inflation (and sometimes output) forecast is seen as providing a reasonable measure of the extent with which agents are (in)attentive to new information by incorporating it in their forecasts. As noted earlier, the present model can be alternatively interpreted as assuming that firms form expectations about the current general price level (and hence current inflation) at the beginning of the current period (or in the very end of the previous one). (14)

[FIGURE 2 OMITTED]

In period 2, a one-time monetary shock raises the nominal money supply by 5% permanently, so that the long-run equilibrium value of the general price level rises in the same proportion. Several features of this second set of simulations are worthy of highlight. First, the temporary response of real output (and inflation, as it is easy to check) to such a monetary shock is hump-shaped, a robust stylized fact. Second, convergence to the new long-run equilibrium (which features the same frequency distribution of information-updating strategies, real output, and inflation rate as those of the initial ones, but a higher general price level) is accompanied by a temporary (and slight) hump-shaped response of both the modulus of the loss of bounded rationality firms (as shown in the first line of panels) and the proportion of Nash firms (as shown in the second line of panels). Third, while the response of both the modulus of the loss of bounded rationality firms and the real output peak right at period 2, the response of the proportion of Nash firms peaks a few periods later. In fact, given the evolutionary dynamics in (25), the temporary fall in the proportion of bounded rationality firms could not start contemporaneously with the occurrence of the monetary shock. Fourth, the mutation rate, and then the corresponding long-run equilibrium level of the distribution of information-updating strategies, matters for the disequilibrium dynamics of the latter (as shown in the second line of panels; see also Equation (27)). As it turns out, given the hypothesized parameter values, the higher the mutation rate, the greater the height of (but closer in time relatively to the monetary shock) the peak of the temporary increase in the proportion of Nash firms. In fact, and relatedly, the higher the mutation rate, the greater the height of the peak of the temporary rise in the modulus of the loss of boundedly rational firms. Fifth, the mutation rate also matters for the disequilibrium dynamics of the macroeconomic variables, even if less significantly than it matters for the disequilibrium dynamics of k (as shown in the third line of panels; but see Equation (28)). Meanwhile, as in the simulations reported in Figure 1, and for the same qualitative reasons, the convergence of the general price level and real output is, however, faster than the convergence of the frequency distribution of information-updating strategies. Sixth, the bounded rationality strategy is played by most firms in the long-run equilibrium in spite of the asymmetric nature of the evolutionary mechanism of strategy revision.

Meanwhile, Figure 3 presents simulations for a context where there are mutant firms and the long-run equilibrium is characterized by the survival of both information-updating strategies with dominance of the boundedly rational strategy. In period 1, the general price level and output are in long-run equilibrium ([P.sub.1] = [P.sup.*] - [M.sub.1] = 1 and [Y.sub.1] = [Y.sup.*] = [M.sup.1]/[P.sub.1] = 1), whereas the frequency distribution of information-updating strategies is in one of three possible positions, namely, one long-run equilibrium position [[lambda].sub.*] = 0.18 corresponding to a mutation rate of 5% and two disequilibrium positions, [[lambda].sub.1], = 0.01 and [[lambda].sub.1] =0.99.

In period 2, a one-time contractionary monetary shock lowers the nominal stock of money by 10% permanently, which leads the long-run equilibrium value of the general price level to fall proportionately. Several interesting qualitative results stand out. First, the fall of real output is only temporary. Second, convergence to the corresponding new long-run equilibrium (featuring the same distribution of information-updating strategies and real output as the initial ones, but a lower general price level) is accompanied by a temporary rise in the proportion of Nash firms only if the latter is initially in the long-run equilibrium. In this case, however, the negative response of the real output peaks contemporaneously with the occurrence of the contractionary monetary shock, whereas the positive response of the proportion of Nash firms intuitively peaks a couple of periods later.

Third, whether the frequency distribution of information-updating strategies is initially in the long-run equilibrium does matter for the convergence dynamics of the general price level and real output. In fact, if the frequency distribution of information-updating strategies is initially in disequilibrium, how large is such disequilibrium similarly matters for the convergence dynamics of the macroeconomic variables. Intuitively, the temporary output cost of disinflation is lower for the higher the initial proportion of Nash firms. The output cost of disinflation is thus minimized when the distribution of information-updating strategies is in the largest possible disequilibrium between the hypothesized initial values. Note that this result shows that whether the evolutionary dynamics operates in the presence of mutation matters for whether a larger initial disequilibrium in [lambda] yields a more complete or less complete nominal adjustment of the price level after a monetary shock. The reason is that both in Figure 1 (where there is no mutation) and in Figure 3 (where mutation is present), the temporary output (price level) response to a permanent monetary shock is smaller (larger) for the larger disequilibrium in [lambda]. The intuition is clear: in both sets of simulations, the larger disequilibrium is characterized by a huge dominance of Nash firms. Fourth, as in the two preceding sets of simulations, and for the same qualitative reasons, the convergence of the general price level and real output is faster than the convergence of the distribution of information-updating strategies.

[FIGURE 3 OMITTED]

Finally, Figure 4 reports other simulations for a situation where there is mutation. In period 1, the price level and real output are both in the long-run equilibrium (Pl=P*=Ml = 1 and [Y.sub.1] = [Y.sup.*] = [M.sub.1]/[P.sub.1] =1), while the distribution of information-updating strategies is in a disequilibrium position [[lambda].sub.1] = 0.99. Meanwhile, two different values for the heterogeneity in the cognitive abilities of firms ([sigma] = 1 and [sigma] = 0.01) and three different values for the average information-updating cost (c = 0.05, c = 0.005 and c = 0.0005) are considered. Recall from Appendix E that, when there is mutation, the long-run equilibrium proportion of Nash firms varies negatively with the average information-updating cost and positively with the rate of mutation and the standard deviation of the cognitive abilities of firms. Consequently, given the rate of mutation, each pair of standard deviation and average cost formed with the values above yields a given long-run equilibrium value for the distribution of information-updating strategies.

In period 2, a one-time contractionary monetary shock lowers the nominal money stock by 10% permanently, so that the long-run equilibrium general price level falls in the same proportion. Several other interesting qualitative results stand out. First, the average information-updating cost matters for the convergence dynamics of the proportion of Nash firms. Given the dispersion in the cognitive abilities of firms, the long-run equilibrium distribution of strategies and the respective disequilibrium dynamics vary with such cost; and, given the considered parameter values, they vary more for the lower dispersion in the cognitive abilities of firms, as shown in the first line of panels. Intuitively, the initial drop in the share of Nash firms occurring in period 2 is larger (and it is large enough to ensure immediate convergence to the evolutionary equilibrium) for the higher average information cost and the lower heterogeneity in the cognitive abilities of firms. Second, and relatedly, the average information-updating cost also matters for the convergence dynamics of the general price level, mattering more for the lower dispersion in the cognitive abilities of firms. In fact, for the lower dispersion in the cognitive abilities, the initial drop in the general price level in period 2 is larger (but not as larger as the accompanying fall in the proportion of Nash firms) for the lower average information cost. Third, and consequently, the average information-updating cost also matters for the convergence dynamics of the real output; and it also matters more for the lower dispersion in the cognitive abilities of firms. Intuitively, given the large initial disequilibrium in [lambda], the output cost of disinflation is minimized for a combination of lower information-updating cost and lower heterogeneity in the cognitive abilities of firms. In fact, the output cost of disinflation is larger when the hypothesized parameter values are such that the initial drop in the proportion of Nash firms is large enough to ensure immediate convergence of such proportion to the evolutionary equilibrium. Fourth, and relatedly, while in the preceding simulations, the convergence of the general price level and real output is faster than the convergence of the distribution of information-updating strategies, the converse is obtained when the dispersion in the cognitive abilities of firms is sufficiently low.

[FIGURE 4 OMITTED]

V. CONCLUDING REMARKS

This paper provides evolutionary game-theoretic microfoundations to the adjustment dynamics of the general price level and real output in response to a monetary shock. We investigate the behavior of the general price level in an economy in which an individual firm can either pay a cost (featuring a random component) to update its information set and establish the current optimal price or freely use information from the previous period and set a lagged optimal price. We devise evolutionary social learning dynamics that, by interacting with the dynamics of the macroeconomic variables, determine the coevolution of the distribution of information-updating strategies across firms and the adjustment of the general price level and real output to a monetary shock. Interestingly, although either the bounded rationality information-updating strategy is the only survivor (when mutation is absent) or both information-updating strategies survive with predominance of the bounded rationality strategy (when there is mutation) in the evolutionary equilibrium, monetary non-neutrality is only temporary. In fact, the evolutionary social learning dynamics succeed in taking the information-updating process to a long-run equilibrium configuration where, despite either the majority or even all of the firms playing the bounded rationality information-updating strategy, the general price level is the symmetric Nash equilibrium price. Further, the extent to which monetary shocks have persistent, although not permanent, impacts on real output depends both on the frequency distribution of information-updating strategies across firms and on the dispersion in the cognitive abilities of firms playing a given information-updating strategy. As regards the conduct of monetary policy, therefore, a fundamental implication of our analysis is that there are two different (albeit interlinked) ways through which heterogeneity in firms' cognitive abilities and boundedly rational inattentiveness matter. Indeed, as the management of expectations has become a central element of the theory of monetary policy, a better understanding of the persistent and endogenously time-varying heterogeneity in expectations about macroeconomic variables is essential.

Consequently, we contribute to the research on real effects of monetary shocks by developing an alternative approach based on a novel analytical notion to which we refer as boundedly rational inattentiveness. Although we introduce this notion in the context of a given macroeconomic model and apply it to the exploration of a specific, even if perennial, issue (i.e., how the general price level reacts to a monetary shock), the seemingly likely prospect that other kinds of behavior are also subject to boundedly rational inattentiveness deserves future investigation.

doi: 10.1111/ecin.12195

APPENDIX A: THE AVERAGE LOSS OF BOUNDED RATIONALITY FIRMS

Using (12), the relative price established by farmer i can be expressed as follows:

(A1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using (A1), the utility function in (11) can be re-expressed in the following way:

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Given that [phi] = 1 -([lambda] - 1)/([lambda][epsilon] - [epsilon] + 1), it follows that [lambda][1 - (1 - [phi])[epsilon]] = 1 + (1 - [psi]))(1 - [psi]) = [lambda]/([lambda][epsilon] - [epsilon] + 1). As a result, (A2) becomes (see Equation (12) in Ball and Romer 1991):

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For a given macroeconomic state (Af, P), a Nash-type farmer i will establish the optimal price, thus obtaining the maximum utility [U.sup.*.sub.n,i], which can be computed by applying [P.sub.i] = [P.sup.*.sub.i] in (A3):

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Meanwhile, a boundedly rational farmer i, by establishing Pi [not equal to] [P.sup.*.sub.i], will incur in a welfare loss given by [U.sub.i] - [U.sup.*.sub.i]. For analytical convenience, in this paper, we work with an approximation of this welfare loss (recall the quadratic loss function in (17)) resulting from a second-order Taylor approximation of (A3) around the point [P.sub.i] = [P.sup.*.sub.i]. This approximation can be obtained as follows.

First, let us compute the first derivative of (A3) with respect to the price of good i:

(A5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Of course, there are no first-order impacts, given that: dU:

(A6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, the second derivative of (A3) with respect to the price of good i is given by:

(A7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Evaluating this derivative at [P.sub.i] = [P.sup.*.sub.i], we obtain:

(A8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the negative sign in (18) is obtained because [epsilon] > 1 and [lambda] > 1. The second-order Taylor approximation around [P.sub.i] = [P.sup.*.sub.i] is, therefore, given by:

(A9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, given (A6) and (A8), the welfare loss of farmer i resulting from establishing a price which differs from the optimal one can be approximated by:

(A10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Recall the distinction made in this paper between Nash firms and boundedly rational firms, with only the former establishing the optimal price. Therefore, the welfare loss in (A10) is incurred solely by boundedly rational firms and can be rewritten (adding the subscript t as well) as [L.sup.b,t] = [U.sup.b,t] - [U.sup.N,t] [congruent to] - [beta] ([[P.sub.bt] - [P.sub.n,t]).sup.2], from which it follows the expression for the loss in (18), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

APPENDIX B: PROOF THAT [[lambda].sub.t + 1] [member of] [0, 1) [subset] R FOR ALL STATE ([[lambda].sub.t], [P.sub.t]) [member of] [THETA], VECTOR (M, c, [sigma]) [member of] [R.sup.3.sub.++], [psi] [member of] (0, 1) [subset] R, AND [theta] [[O, 1) [subset] R

First, let us shown that X,+ ]>0 if 0 < A., <l. Given that F(-) is a cumulative distribution function, we have F(c + [L.sup.b] (X,,P,))< l, which implies that:

(A11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Given that 1 - [[lambda].sub.t] > 0 and G (-c- [L.sup.b] ([[lambda].sub.t], [P.sub.t])) [greater than or equal to] 0, and because G(x) is a cumulative distribution function, it follows that:

(A12) (1 - [[lambda].sub.t]) G (-c-[L.sup.b] ([[lambda].sub.t], [P.sub.t])) [greater than or equal to] 0.

Given (All) and (A 12), it follows that:

(A13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By multiplying both sides of (A 13) by [[lambda].sub.t] > 0 we obtain:

(A 14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because [theta] [member of] [0,1) [subset] R, we can use (A14) to write:

(A15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can add 0/2 to the left-hand side of the above expression to obtain:

(A16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, given (A16) and (25), we can establish that [[lambda].sub.t + 1] - [[lambda].sub.t], [greater than or equal to] - [[lambda].sub.t], from which it follows that [[lambda].sub.t + 1] [greater than or equal to] 0 for all state ([[lambda].sub.t] [P.sub.t]) [member of] [THETA], vector (M, c, [sigma]) [member of] [R.sup.3.sub.++], [psi] [member of] (0, 1) [subset] R, and [THETA] [member of] (0, 1) [subset] R.

Let us now show that [[lambda].sub.t + 1] < 1 if 0 [less than or equal to] [[lambda].sub.t + 1] < 1. Given that G(x) is a cumulative distribution function, it follows that:

(A17) (-c - [[L.sub.b]([[lambda].sub.t], [[P.sub.t]) [less than or equal to] 1.

Given that 1 - [[lambda].sub.t] > 0, it follows from (A17) that:

(A18) (1 - [[lambda].sub.t] G (-c - [[L.sub.b] ([lambda].sub.t], [[P.sub.t])) [less than or equal to] 1 - [[lambda].sub.t].

Meanwhile, given that F(c + [L.sup.b] (Pt)) > 0, and because F(-) is a cumulative distribution function, it follows from (A 18) that:

(A 19) (1 - [[lambda].sub.t] G (-c - [L.sub.b] ([lambda].sub.t], [[P.sub.t])) [less than or equal to] 1 - [[lambda].sub.t]

Because [theta] [epsilon] [0,1) [epsilon] R, we can use (A19) to write:

(A20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can subtract 0/2 from the left-hand side of the above expression to obtain:

(A21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given that 0 [less than or equal to] [[lambda].sub.t + 1] < 1, we can use (A21) to establish that:

(A22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, given (A21), (A22), and (25), we can establish that [[lambda].sub.t + 1] - [[lambda].sub.t] < 1 [[lambda].sub.t], from which it follows that [[lambda].sub.t + 1] < 1 for all state ([[lambda].sub.t], [[P.sup.t]) [member of] [THETA], vector (M, c, [sigma]) [epsilon] [R.sup.3.sub.++] [psi] [member of] (0, 1) [subset] R, and [theta][member of][0,1) [subset] R.

APPENDIX C: PROOF OF PROPOSITION 1

Suppose that [M.sub.t] = M > 0 for all t [member of] {0, 1, 2, ...}. Let [[lambda].sub.t + 1] = [[lambda].sub.t] = [[lambda].sub.* and [P.sub.t + 1] = [P.sup.t] = [P.sup.*] for any re {0, 1, 2, ...}. It then follows from (25a) that [[lambda].sub.*] = [psi] ([[lambda].sub.*], [P.sup.*]). Substituting the previous identities in (26), we find that [P.sup.*] = M, which is the symmetric Nash equilibrium price. As in the long-run equilibrium with [P.sup.*] = M (18a) yields [L.sub.b] ([[lambda].sub.*], [P.sup.*]) = [L.sup.b] {[[lambda].sub.*], M) = 0, the condition [[lambda].sub.*] = [psi]([[lambda].sub.*], M) is satisfied if: (A23)

(1 - [theta]) [[lambda].sup.*] [(l-[[lambda].sup.*]) G (-c)-F(c)] + [theta] (1/2 - [[lambda].sup.*]) =0.

After some algebraic manipulation, this condition can be rewritten as a quadratic equation:

(A24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This equation has two distinct real roots given by:

(A25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us show that ([[lambda].sup.*.sub.1], M) is not a long-run equilibrium in the economically relevant state space. As c>0, we have F(c) > G(-c) > 0. For all [theta] [member of] [0,1) [subset] R, therefore, the numerator in (A25) is strictly positive and the denominator in (A25) is strictly negative. Hence, the root X* < 0 is such that ([[lambda].sup.*.sub.1], M) [??] [THETA].

Let us now prove that ([[lambda].sup.*.sub.2], M) is the economically relevant long-run equilibrium for all [THETA] [member of] [0,1) [subset] R. Firstly, let us assume that [theta] = 0 (there is no mutation). In this case, based on (A26), we obtain [[lambda].sup.*.sub.2] = 0, that is, the Nash information-updating strategy "becomes extinct when mutation is absent.

Now we will assume that there is mutation, that is, [theta][member of](0, 1) [subset] R. First, let us prove that [[lambda].sup.*.sub.2] > 0 for all

[theta] [epsilon] (0, 1) [subset] R. Based on (A14), this will be true if the following condition holds:

(A27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As F(c) > G(-c), we have (1 - [theta])(F(c) - G(-c)) + 0 > 0. Therefore, we can square both sides of (A27) and simplify the resulting expression to obtain the equivalent condition:

(A27a) 2[theta] (1 - [theta])G(-c) > 0.

Because [theta] [epsilon] (0, 1) [subset] R and G(-c)>0, the inequality (A27a) holds, so that the condition (A27) is also satisfied. It follows from (A27) that the numerator in (A26) is strictly negative. As the denominator in (A26) is strictly negative, it then follows that [[lambda].sup.*.sub.2] > 0 for all [theta] [member of] (0, 1) [subset] R.

Finally, we will show that [[lambda].sup.*.sub.2] < 1 /2 for all [theta] [member of] (0, 1) [subset] R. Using (A26), this will be true if the following condition holds:

(A28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can rewrite this inequality as follows:

(A28a) [[(1 - [theta]) F (c) + 0].sup.2] > [[(1 - [theta]) (F (c) - G (-c)) + [theta]].sup.2] + 2[theta] (1 - [theta])G(-c).

Finally, after some algebraic simplification, we obtain the following equivalent inequality:

(A28b) (1 - [theta])G(-c)[1/2 - F(c)] < 0.

This inequality holds given that c > 0, G(-c) > 0, and F (c) > F (0) = 1/2. As a result, it follows that [[lambda].sup.*.sub.2] < 1/2 for all [theta] [member of] (0, 1) [subset] R.

Therefore, we have shown that there is one, and only one, economically relevant long-run equilibrium (X*, M) G 0, with [[lambda].sup.*] = [[lambda].sup.*.sub.2] [theta] [member of] (0, 1) [subset] R

APPENDIX D: PROOF OF PROPOSITION 2

Throughout the ensuing demonstrations, we suppose that [M.sub.1] = M > 0 for all t [member of] (0,1,2, ...). Unless specified otherwise, these demonstrations are valid for any point in the parameter space [OMEGA] = ([psi] [member of] R: 0 < [psi] < 1} x {(M, c, [sigma]) [member of] [R.sup.3.sub.++]} x {[theta] [member of] R : 0 [less than or equal to] [theta] < 1).

(i) The equilibrium ([[lambda].sup.*], M) [member of] [theta] of the system (25a) - (26) is locally asymptotically stable.

Consider the Jacobian matrix of the linearization around the long-run equilibrium of the system (25a)--(26), which is given by ([[lambda].sup.*], A/):

(A29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whose eigenvalues are [[alpha].sub.1] = (1 - [theta])[1 - F(c) + (1 -2[[lambda].sup.*])G(-c)] and [[alpha].sub.1] = [psi] (1 - [[lambda].sup.*]/(1 - [psi] [[lambda].sup.*]). Given that F(c)<l, it follows that 1-F(c)>0. Moreover, given that [[lambda].sup.*] [member of] (0, 1/2) [subset] R (per Proposition 1) and G(-c)> 0, it follows that 1 - F(c) + (1 - 2[[lambda].sup.*])G(-c) > 0. Therefore, because [theta] [member of] (0, 1) [subset] R, we obtain [[alpha].sup.1] > 0. Furthermore, given that c > 0, it follows that G(-c) < F(c). Therefore, it follows that (1 - 2[[lambda].sup.*]))G(-c) < F(c) for any [[lambda].sup.*] [theta] [member of] (0, 1/2) [subset] R. From the preceding inequality, meanwhile, we obtain 1 -F(c) + (1 -2[[lambda].sup.*])G(-c)< 1. As a result, because [theta] [member of] (0, 1) [subset] R, it follows that [[alpha].sup.1] < 1.

Now, as [theta] [member of] (0, 1) [subset] R and [[lambda].sup.*] [theta] [member of] (0, 1/2) [subset] R, we obtain [psi] (1 - [[lambda].sup.*]) > 0 and 1 - [psi] [[lambda].sup.*] > 0, so that [[alpha].sup.2] > 0. Besides, we obtain [psi](1 -[[lambda].sup.*]) - (1 -[psi] [[lambda].sup.*]) = [psi] - 1 < 0, so that [[alpha].sup.2] < 1.

Therefore, as we have shown that [[alpha].sup.1]] [member of] (0, 1/2) [subset] R and [theta] [member of] (0, 1) [subset] R, it then follows that the long-run equilibrium configuration given by ([[lambda].sup.*], M) is a local attractor.

(ii) The set {([[lambda].sup.t], [P.sub.t]) [member of] [R.sup.2.sub.+] : 0 [less than or equal to] [[lambda].sup.t] < 1, Min {[P.sub.0], M} [less than or equal to] [P.sub.t] [less than or equal to] Max {[P.sub.0], M}} [subset] [subset] R is positively invariant for any initial condition ([[lambda].sup.0], [P.sup.0]) [epsilon] [THETA].

We show in Appendix B that [[lambda].sup.t+1] = [psi] [member of] (0, 1) [subset] R for all state ([[lambda].sup.0], [P.sub.t]) [member of][theta]. Given that such result holds for any [[lambda].sup.0] [member of] (0, 1) [subset] R, it follows by induction that [[lambda].sup.1] [member of] (0, 1) [subset] R for all r [member of] {0,1,2, ... and initial condition ([[lambda].sup.0], [P.sup.0]) [member of] [THETA].

We still have to show that [P.sup.t], [member of] [Min {[P.sup.0], M}, Max {{[P.sup.0], M})] [subset]R for all t [member of] (0, 1, 2, ... 1 and initial condition (Xq,Po)G0. By dividing both sides in (26) by M and applying the logarithmic operator to the resulting expression, we can rewrite (26) as follows:

(A30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Given that [[lambda].sup.t+1] = [[psi] ([[lambda].sup.t], [P.sub.t]) [member of] (0, 1) [subset] R for all state ([[lambda].sup.t], [P.sub.t]) [member of][THETA]

X(+1 =t|/(X,, P,) [theta] [member of] (0, 1) [subset] R for all state (X,, P,) E 0, it follows that:

(A31) [[[phi][1 - [psi] ([[lambda].sup.t], [P.sub.t])]/ 1 - [phi][psi] ([[lambda].sup.t] [P.sub.t])]/ [member of] (0, 1) [subset] R

Equation (A30) in conjunction with (A31) implies that lln ([P.sub.t+1]/M) < lln ([P.sup.t]/M)| for all t [member of] (0,1,2, ...} and initial condition (([[lambda].sup.t], [P.sub.0]) [member of] [THETA].

(iii) For any initial condition ([[lambda].sup.0], [P.sub.0]) [member of] [THETA], the general price level converges monotonically toward a given constant nominal stock of money M.

If [P.sub.t] [member of] (M, [infinity]) [subset] R, it follows that ln ([P.sub.1]/M) > 0. Given (A30) and (A31), it follows that 0 < ln (([P.sub.t]/M)) < ln([P.sub.t+l]/M) < ln ([P.sub.t+1,/M) <0 for all r [member of] {0,1,2, ...). Consequently, the general price level converges monotonically toward M for any [P.sub.0] [member of] (M, [infinity]) [subset] R.

If [P.sub.t] [member of] (M, [infinity]) [subset] R, it follows that ln ([P.sub.t]/M) > 0. Given (A30) and (A31), it follows that 0< ln ([P.sub.t+1]/M) < ln([P.sub.t]/M) for all t [member of] {0,1,2, ...}. Therefore, the general price level converges monotonically toward M for any [P.sup.0] [member of] (M, [infinity]) [subset] R.

(iv) If the economy is initially at any state (0, [P.sup.0]) [member of] [theta], the proportion of Nash firms rises. Moreover, for any ([[lambda].sup.t], sufficiently close to one, the proportion of Nash firms falls.

When there is mutation, so that (per Proposition 1) the Nash information-updating strategy is played by the minority of firms in the long-run equilibrium, the boundedly rational information-updating strategy is not evolutionarily stable. In fact, when ([[lambda].sup.0] = 0, it follows that ([[lambda].sup.1] - ([[lambda].sup.0] = [theta]/2 > 0 for all [theta] [member of] (0, 1) [member of] R.

Meanwhile, when the Nash information-updating strategy is played by the vast majority of firms, the proportion of firms playing the boundedly rational information-updating strategy rises. This result can be demonstrated as follows. If we add ([[theta].sup.0] (1/2 - ([[lambda].sup.t]) to both sides of the inequality in (A20), we can establish the following inequality for any X, e (1/2,1) [subset] R:

(A32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, for any ([[lambda].sup.1][member of] (1 - [theta]/2,1) [subset] R with [theta] [member of] (0,1) [subset] R, we obtain ([[lambda].sup.t+1] + ([[lambda].sup.t] < 0.

APPENDIX E: COMPARATIVE STATICS FOR THE MIXED-STRATEGY EVOLUTIONARY EQUILIBRIUM

Applying the implicit function theorem to the long-run equilibrium condition (A23), we obtain the following comparative static results:

(A33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The common denominator in (A33) and (A34) is strictly positive. Because c [member of] [R.sub.++], it follows that F(c)>G(-c). Therefore, we have (1 - [[lambda].sup.*])G(-c) - F(c) < 0 for all c [member of] [R.sub.++] and [[lambda].sup.*] e [0,1/2) c R. Besides, because G(-c)>0, we have --[[lambda].sup.*]G(-c)<0. Hence, we obtain [theta] - (1 - [theta]) {[(1 - [[lambda].sup.*])G(-c) - [[lambda].sup.*]G(c)] - [[lambda].sup.*]G(-c)} < 0 for all c [member of] [R.sup.++], [theta] [member of] (0,1) [subset] R, and [[lambda].sup.*] [member of] [0,1/2) [subset] R.

Meanwhile, the numerator in (A33) is strictly positive. It is immediate that 1/2 - [[lambda].sup.*] > 0 for all [[lambda].sup.*] [member of] [0,1/2) [subset] R. As shown above, we have (1 -[[lambda].sup.*])G(-c) -F(c) <0 for all c [member of] [R.sub.++] and [[lambda].sup.*] e [0,1/2) c R. Hence, it follows that (1/2-[[lambda].sup.*])- [[lambda].sup.*] [(1 - [[lambda].sup.*])G(-c)-F(c)]>0 for all c [member of] [R.sub.++] and [[lambda].sup.*] [member of] [0,1/2) c R, which completes the proof that [delta] [[lambda].sup.*]/[delta][theta] > 0.

Now, it can be shown that the numerator in (A34) is strictly negative. Given that F'(c)> 0 and G'(-c)>0, it follows that -(1-0)[(1-X*)G'(-c) + F'(c)] <0 for all c [member of] [R.sub.++], [theta](0,1)[subset]R, and [[lambda].sup.*] [member of] [0,1/2) [subset] R. This completes the proof that [[lambda].sup.*] < 0.

In order to evaluate how [[lambda].sup.*] varies with the standard deviation of firms' cognitive abilities [sigma], it is convenient to express the values of the cumulative distribution functions F and G in the long-run equilibrium as follows:

(A35) F (c) = 1/2 [1 + erf (x)],

(A36) G(-c) = 1/2 [l + erf (-x/2)] = 1/2 [l - erf (x/2)] ,

where x = c/([sigma][square root of 2]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the error

function (Gautschi 1972; Zelen and Severo 1972). Therefore, we can use (A35) and (A36) to obtain the following derivatives:

(A37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(A38) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Applying again the implicit function theorem to the long-run equilibrium condition (A23), we obtain:

(A39) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have already demonstrated above that the denominator in (A39) is strictly positive. Moreover, given that (A37) and (A38) jointly imply that the numerator in (A39) is also strictly positive, the demonstration that [delta] [[lambda].sup.*]/[delta][theta] > 0. is completed.

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(1.) Branch and McGough (2008) compare the dynamics across predictor choice in Brock and Hommes (1997) with an alternative evolutionary dynamics (the replicator dynamic) to the one set forth in our model. In fact, they extend the dynamics in Brock and Hommes (1997) by introducing a generalized version of the replicator dynamic having more than two predictors. The main reason why our evolutionary dynamics is not a replicator dynamic is our purpose to treat the individual access to the net benefit of each alternative foresight strategy as asymmetric. While firms playing the adaptive foresight strategy rely on a pairwise random matching process to access the relative net benefit of so playing, perfect foresight firms access the same relative net benefit as a further gain from paying the information cost. A small (but representative) sample of macroeconomic models with heterogeneous behavior embedding the dynamics across predictor choice in Brock and Hommes (1997) include Branch and McGough (2010), Lines and Westerhoff (2010), and Anufriev et al. (2013).

(2.) A first significant difference with respect to the basic model set forth in Ball and Romer (1989, 1991) is that the utility function in (3) does not feature a utility-reducing (constant) menu cost a producer has to pay to change his nominal price. As specified later, in this paper, a producer decides about whether or not to pay an information-updating cost to obtain perfect knowledge of the structure of relative prices in an environment characterized by strategic complementarity in price setting.

(4.) This interpretation will be convenient later on when we compare our simulation results to the empirical evidence. The reason is that the empirical literature sees the frequency of updating a forecast about a variable such as inflation as furnishing a reasonable measure of the extent with which forecasters are (in)attentive to new information.

(5.) Mankiw and Reis (2002) note that microfoundations for the Phillips curve may ultimately require a better understanding of bounded rationality. Their conclusion is suggestive: "Yet we must admit that information processing is more complex than the time-contingent adjustment assumed here ... [developing better models of how quickly people incorporate information about monetary policy into their plans, and why their response is faster at some times than others, may prove a fruitful avenue for future research on inflation-output dynamics" (1319). Indeed, an essential feature of the model set forth herein is that firms do not follow an exogenous rule in updating the information set, as in Mankiw and Reis (2002), in which, following Calvo (1983), firms do so randomly. In this paper, firms decide continuously on the convenience of updating the information set relevant for setting the optimal price based on net (of costs) payoffs featuring a random term. Moreover, the resulting evolutionary dynamics may operate in the presence of a noise component, analogous to mutation in natural environments.

(6.) As we intend to follow the behavior of this imperfectly competitive economy over time, thereafter we attach a sub script t to variables.

(8.) As firms have heterogeneous cognitive abilities, it is conceivable that the perceived losses associated with the bounded rationality information-updating strategy differ across firms playing it. For instance, firms with lower cognitive ability may make greater miscalculations of the (say) fundamental value of such losses (given by 18a) by (say) misprocessing the relevant available information. These heterogeneous cognitive abilities are represented in (19) through individual losses of bounded rationality firms featuring a random component. The essential role played by the firm's cognitive abilities in its strategic decision making is established, for instance, in Gavetti (2005) and Gavetti and Rivkin (2007).

(9.) As conjectured in Branch (2004), actual prediction costs may be different across individuals if some people have higher calculation costs. For instance, some people may have greater cognitive ability that allows them to form a sophisticated forecast at a lower cost. The specification in (20) can, therefore, be seen as somehow carrying out the suggestion in Branch (2004) that heterogeneous cognitive abilities could be introduced via stochastic costs.

(10.) We thank one of the referees for pointing this out to us.

(11.) It can be shown that the main qualitative results found in this paper (especially the existence and stability of an evolutionary equilibrium in which monetary policy shocks have persistent, but not permanent, real effects) are robust to the specification of the average information-updating cost as varying (positively or negatively) with the distribution of information-updating strategies rather than being fixed (when the average optimization cost is treated as variable in Sethi and Franke 1995, it varies only positively with the proportion of sophisticated firms). Predictors' costs are also exogenously given in Brock and Hommes (1997). Further research is needed to find sound theoretical support for information-updating costs, which vary with the proportion of updating agents, but it is reassuring that our qualitative results are robust to this specification. Admittedly, however, information-updating costs may be found to vary with variables unrelated to the distribution of information-updating strategies.

(12.) While in the evolutionary microfoundation to nominal adjustment in Saint-Paul (2005), the results are obtained through simulations, our results are derived both as explicit analytical solutions and using simulations. Also, while in Saint-Paul (2005), money is roughly neutral in the long run if the autocorrelation of money shocks is high, we obtain longrun monetary neutrality without assuming a specific process for the exogenous money supply. Meanwhile, the evolutionary contribution in Sethi and Franke (1995) does not include any policy exercise, explores the response (solely) of aggregate output to shocks to production costs arriving in every period (and only) through simulation, and investigates analytically the evolutionary dynamics (only) in the degenerate case when the production cost shock is constant. In this paper, we explore the response of the price level and output to a one-time permanent monetary policy shock in a deterministic macroeconomic context both analytically and through simulation.

(13.) Coibion and Gorodnichenko (2012), using survey forecast quarterly data from the U.S. professional forecasters, consumers, firms, and central bankers, obtain an average estimate across all specifications (and with a small dispersion across agent types) of X = 0.18. Meanwhile, Mankiw and Reis (2003) and Branch (2007) perform survey-based empirical studies for the U.S. setting [lambda] = 0.1.

(14.) In their simulations, Mankiw and Reis (2002) set [lambda] = 0.25, which implies a frequency of information updating [[lambda].sup.-1] of four quarterly periods (i.e., information sets are updated once a year). Other studies reporting values of X estimated for the United States or the European countries directly from surveys or using proxies (and also using quarterly periods) include: Mankiw and Reis (2003), 0.25; Carroll (2003), 0.27; Khan and Zhu (2006), 0.14 (for long forecasting horizon) and 0.33 (for short forecasting horizon); Kiley (2007), an average of about 0.56; Dopke et al. (2008a, 2008b), an average of about 0.25; and Andrade and Le Bihan (2013), an average not much higher than 0.50. Meanwhile, Pfajfar and Santoro (2010) find an average minimum updating period of 7 months, and Branch (2007) finds that, on average, the highest proportion of agents update their information sets every 3-6 months, a lower proportion of agents do so every period, and few agents update their information sets at periods of 9 months or more, with these proportions varying over time.

GILBERTO TADEU LIMA and JAYLSON JAIR SILVEIRA *

* We are grateful to three anonymous referees and the Co-Editor, Bruce McGough, for helpful comments and suggestions. Any remaining errors are our own. We are also grateful to CNPq (Brazil) for providing us with the research funding. JJ.S. gratefully acknowledges the Department of Economics of the University of Massachusetts Amherst, whose hospitality assisted his work on the completion of this paper, and the CAPES Foundation, Ministry of Education of Brazil, for the grant (Proc. BEX 18175/12-0), which funded his visit to UMass-Amherst. Lima: Professor, Department of Economics, University of Sao Paulo, Sao Paulo, Brazil. Phone +55 11 30915907, E-mail [email protected]

Silveira: Associate Professor, Department of Economics and International Relations, Federal University of Santa Catarina, Florianopolis, Brazil. Phone +554837219458, Email [email protected]