An Evolutionary Theory of Inflation Inertia by Alexis Anagnostopoulos^*, Omar Licandro^** Italo Bove^*** and Karl Schlag DOCUMENTO DE TRABAJO 2006-25

October 2006

* Stony Brook.

** European University Institute.

*** Universidad de la República.

Alexis Anagnostopoulos Stony Brook

Omar Licandro European University Institute

Italo Bove Universidad de la Rep˙blica

Karl Schlag European University Institute

September 2006

Abstract

We provide a simple theory of ináation inertia in a staggered price setting framework a la Calvo (1983). Contrary to Calvoís formulation, the frequency of price changes is allowed to vary according to an evolutionary criterion. Inertia is the direct result of gradual adjustment in this frequency following a permanent change in the rate of money growth.

1 Introduction

The recent literature on monetary policy has widely used the sticky price model proposed by Calvo (1983) as a simple way of generating the so-called (New Keynesian) Phillips curve, a negative relation between ináation and the output gap ñsee Woodford (1994) or Clarida, Gali and Gertler (1999), amongst others. However, as pointed out by Fuhrer and Moore (1995), the Calvo model does not succeed in reproducing the observed persistence of ináation. In Calvo (1983), even if only a few Örms change prices at any period, the assumption of rational expectations implies that these Örms fully understand the price e§ects of changes in the environment. This in turn has important implications for the behavior of ináation. In particular, when the economy faces an unexpected change in the rate of money growth, the forward-looking behavior of those few Örms currently changing prices is enough for the economy to anticipate the e§ect of the shock on ináation. In other words, Örms currently setting prices will choose their prices in such a way that the overall level of ináation will immediately jump to the new steady state.

In this paper, we use an evolutionary game approach to solve a Calvo (1983) economy with a cash-in-advance constraint as a simple way of generating ináation inertia. The description of the economy is taken directly from Calvo (1983), but money is introduced following Clower (1967) instead of Sidrauski (1967). When a rational expectations equilibrium is computed for this economy, the standard result that ináation automatically adjusts to a permanent, unexpected shock in the growth rate of money supply holds. However, when Örms play an evolutionary game in prices, instead of behaving consistently with rational expectations, ináation responds slowly and gradually to exogenous shocks to the money supply growth rate. In the evolutionary economy, a Örm receiving a price-change signal follows a simple rule of thumb relating its price to the observed money supply. The probability of observing a price-change signal, and thus the frequency with which Örms change prices, evolves depending on the performance of this price rule relative to any other price observed in the economy. When Örms changing prices perform better (worse) than those that keep posting their old prices, price change signals become more (less) frequent. We refer to this dynamic adjustment as the Darwinian dynamics, to highlight the underlying idea that successful behavior tends to spread in the population. Within this setup, the response of ináation to a permanent negative shock on the rate of money growth is gradual. Even though price-changing Örms keep playing the same price rule as before, ináation will tend towards the new rate of money growth due to the Darwinian dynamics. That is the dynamic adjustment of ináation will work through the adjustment in the probability of observing a price-change signal. That adjustment will be slow and gradual, delivering in a simple, intuitive way ináation inertia.

Anagnostopoulos and Licandro would like to thank the Önancial support of the European Commission under the MAPMU network . A Örst draft of this paper was written while Licandro was visiting the London Business School during August 2006. Comments by Gilles Saint-Paul on a previous version are gratefully acknowledged.

Our approach is not unique in delivering the result of ináation inertia. A growing recent literature seeks ways of obtaining ináation persistence by modifying the sticky price model that originates in Calvoís staggered price setting formulation. The most notable of such attempts is due to Mankiw and Reis (2002). They stress the role of sticky information as an alternative to sticky prices. Under sticky information, Örms revise prices every period but their decisions are not always based on current information. Inertia results from the fact that some price setters decide on price changes based on past information. A similar reasoning explains why a sticky price framework augmented with price indexation delivers the same result.¹² In an evolutionary framework, information spreads across the economy through evolution, i.e. the replication of high performing behavior. In this paper, changes in the environment a§ect price-pioneersíproÖts making its size in population adjust. Inertia results because news are not observed and, more important, agents donít adapt their behavior to them, but the economy learns slowly through the replication of those behaviors that are most adapted to the new environment.

2 The Economy

The economy in this paper is very close to the model of Calvo (1983), with the di§erence that money is introduced following Clower (1967) instead of Sidrauski (1967). The fundamentals are described as in the general equilibrium tradition. However, instead of using an equilibrium concept to formalize the behavior of agents and markets, an evolutionary game in prices is assumed and its outcome is computed.

¹ For a detailed analysis of the similarities and di§erences between the two models see Trabandt (2006).

² Yet another interesting recent development on the issue is due to Mackowiak and Wiederholt (2006) who derive ináation and price setting implications of rational inattention, an idea originally suggested by Chris Sims (2003).

Time is continuous. There is a sole, perishable and divisible good. The economy is populated by a continuum of Örms in the interval [0; 1]. At every instant, each Örm receives, as manna from heaven, a strictly positive endowment of size Therefore, the total amount of goods available at t is q. There is a continuum of individuals; the representative individual has inÖnite life, time additive preferences with constant, strictly positive discount rate, owns Örms and holds money. Money is issued by a central bank and is permanently increasing at the instantaneous rate

\[M _ {t} = M _ {0} e ^ {\mu t}.\]

Any increase in money supply is distributed across individuals as a lump-sum transfer.

In the spirit of Clower (1967), money is required for transactions ñany form of barter is forbidden ñmeaning that money plays the role of both a unit of account and a medium of exchange. Individuals hold money and use their money holdings to buy goods. Firms collect proÖts in the form of money and give the money back to individuals as dividends.

What can we learn from equilibrium theory? Under standard conditions on preferences, an equilibrium path with binding cash-in-advance constraint exists. Such an equilibrium is consistent with the quantity theory of money , where is the equilibrium price of the physical good and the velocity of money is one.³ In the evolutionary economy described below the equilibrium price plays no role, since agents do not use it to take any economic decision. However, we will use the equilibrium outcome as a benchmark to analyze the performance of the evolutionary economy, by comparing evolutionary prices to the equilibrium price.

The evolutionary game we propose is extremely simple and very close to the price setting process in Calvo (1983). Firms keep announcing the same price until they receive a price-change signal. Firms receiving such a signal, instead of computing an optimal price consistent with rational expectations as in Calvo, set a price proportional to the observed money supply. This simple rule of thumb allows prices to follow money. We call these Örms price-pioneers; they understand that prices must increase since money is growing and they implement it in a simple way. They play a similar role as Calvo players in the Calvo model. The price-change signal arrives with an instantaneous rate that adjusts upwards or downwards depending on whether the price-pioneer is making the largest proÖts. This is a simple application of the Darwinian principle that successful behavior tends to replicate. Let us now be more precise on the deÖnition of the evolutionary game.

At any time t, Örms announce prices and engage on satisfying demand at these prices until they run out of stock. As a consequence, it may be that some Örms keep unsold units of the perishable good, which cannot be transferred to the next period. In such a case, these Örms are said to be quantity constrained by a demand shortage. The representative individual uses her money holdings to buy as much of the physica good as she can. She is assumed to observe all current prices, order Örms by prices and buy following the order of prices, the lowest price Örst. As a result, either individuals spend the total available amount of money or all Örms run out of stock. In order to ensure individuals are identical, we make the assumption that any excess demand is proportionally distributed across individuals. We say that the representative individual is quantity constrained when she cannot buy as many units as she would like, because of a shortage of supply.⁴

³ See Lucas and Stokey (1987) and Woodford (1994).

Firmsí prices are set according to the following rules. At the initial time Örms carry prices from the past. The relevant information on the initial distribution of prices can be summarized by , the average price announced at . At any time , Örms keep announcing their previous prices until they receive a price-change signal, in which case they set a price proportional to the money supply, We refer to the Örms that receive a price change signal as price-pioneers. In order to follow this price rule, the only information requirement is that price-pioneers observe the money supply . Note that this rule is equivalent to setting a price proportional to the equilibrium price , which was deÖned previously as : That is, one can equivalently express the pioneersíprice setting rule as , with . We use this notation for convenience from now on, but it is important to clarify that we do not require price pioneers to observe the equilibrium price observing is enough. Since the pioneer always sets the largest price let us assume that , otherwise the average announced price would be smaller than the equilibrium price and the economy would (by construction) permanently be in a situation of excess demand.

The probability density of receiving a price-change signal during a period of length k from time t is assumed to be

\[v _ {t + k} \mathrm{e} ^ {- \int_ {0} ^ {k} v _ {t + z} \mathrm{d} z},\]

with representing the density of price-pioneers in the distribution of Örms at time t. This is similar to Calvo (1983), the di§erence amounting to the fact that in Calvo is constant.

At any time t, let us denote by the price set by price-pioneers at By deÖnition of the pioneerís price rule, , since the equilibrium price follows money by deÖnition. Let be the density of the pioneers still announcing at time which is . Lastly, the average price announced by Örms at time , which we denote by , can be expressed as

\[\mathcal {P} _ {t} = \mathcal {P} _ {0} \mathrm{e} ^ {- \int_ {0} ^ {t} v _ {z} \mathrm{d} z} + \int_ {k} p _ {k t} h _ {k t} \mathrm{d} k = \mathcal {P} _ {0} \mathrm{e} ^ {- \int_ {0} ^ {t} v _ {s} \mathrm{d} s} + (1 + \eta) \mathcal {A} _ {t} P _ {t},\tag{1}\]

where

\[\mathcal {A} _ {t} = \int_ {0} ^ {t} e ^ {- \mu (t - z)} v _ {z} \mathrm{e} ^ {- \int_ {z} ^ {t} v _ {s} \mathrm{d} s} d z.\]

We have used the variable change to obtain . When t goes to inÖnity, initial conditions vanish and the ratio of the average to the equilibrium price converges to

\[\frac {\mathcal {P} _ {t}}{P _ {t}} = (1 + \eta) \frac {v}{v + \mu}\]

if converges to a constant value .

⁴ A large literature in the seventies and eighties has developed equilibrium concepts to deal with situations where markets are not cleared by prices. See Benassy (1982) and Dreze (1974).

From the description of the evolutionary game above, the economy is in excess supply or excess demand depending on

\[M _ {t} \lessgtr q \mathcal {P} _ {t} \qquad \text {or equivalently} \qquad \frac {P _ {t}}{\mathcal {P} _ {t}} \lessgtr 1.\]

The second relation follows directly from the Örst after substituting the deÖnition of the equilibrium price . The following Lemma shows that there is a one-to-one relationship between excess supply and excess demand on the one hand and the pioneers performance on the other hand.

Lemma 1 then the pioneer is making the largest proÖts, otherwise the pioneer is not making the largest proÖts.

Proof. Let and denote goods sold and proÖts made by a Örm that was a price pioneer at and has not changed its prices since. Thus for a current price-pioneer, goods sold are denoted by and proÖts by . When , the economy is in excess demand, implying that and for all . When , the economy is in excess supply. Since pioneers are setting the largest price and have zero measure in the distribution of prices, and , implying that price-pioneers are not making the largest proÖts. That is because prices are strictly positive and some Örms are selling therefore making strictly positive proÖts.

In the Örst situation, the average price is lower than the equilibrium price, implying that the economy is in excess demand, i.e. consumers cannot spend all their money holdings. In this case, the pioneer is setting the largest price, running out of stock, and making the largest proÖts. In the other case, the economy is facing an excess supply. Since pioneers are charging the largest price, they receive no demand and make zero proÖts.

Lastly, let us specify the Darwinian dynamics as

\[\dot {v} _ {t} = \lambda \frac {P _ {t} - \mathcal {P} _ {t}}{\mathcal {P} _ {t}}\tag{2}\]

where is the velocity of the evolutionary process. From the previous Lemma, is an indicator of the price-pioneersíperformance. The Darwinian dynamics assumes that the probability density of being a pioneer increases (decreases) when pioneers are (not) making the largest proÖts. This is an application of the Darwinian principle saying that successful behavior tends to replicate in the population. Note that the absolute size of both the equilibrium and the evolutionary price grows with money holdings. The observed absolute distances between the two will therefore also increase with time. To ensure the evolutionary adjustment is not a§ected by this artiÖcial increase, we divide the right-hand-side by i.e. we relate the rate of change in to relative distance from the equilibrium price.

3 Ináation inertia

In this section, we study the dynamics of the evolutionary economy and compare them to the dynamics of the Calvo model. The main objective is the analysis of ináation inertia in the case of a permanent change in the rate of money growth. We show that, contrary to the behavior of the Calvo model where ináation jumps instantaneously to the new rate of money growth, in the evolutionary economy ináation moves gradually from its past level to the new one. That is, there is no ináation inertia in the Calvo model, but the evolutionary economy generates it since there is high persistence in past behavior.

3.1 Evolutionary Economy

By di§erentiating (1) with respect to we get

\[\dot {\mathcal {P}} _ {t} = v _ {t} \left(\left(1 + \eta\right) P _ {t} - \mathcal {P} _ {t}\right).\tag{3}\]

Note that only pioneers change prices and they do so with probability density . On average, they move from the average announced price to the new price

It is easy to see that (3) is a particular version of the Phillips curve. Let us deÖne the output gap, a measure of excess supply, as

\[y _ {t} \equiv \frac {q - \frac {M _ {t}}{\mathcal {P} _ {t}}}{q}\]

where is the amount of goods individuals would like to buy at the announced prices and q is the total endowment, a measure of capacity. Using this deÖnition, we can write (3) as

\[\pi_ {t} \equiv \frac {\dot {\mathcal {P}} _ {t}}{\mathcal {P} _ {t}} = v _ {t} (\eta - (1 + \eta) y _ {t}),\]

which implies that ináation depends negatively on the output gap.

Let us now rewrite the dynamic system. From the deÖnition of the equilibrium price , the output gap can be written as , with . After straightforward algebra (3) becomes

\[\frac {\dot {p} _ {t}}{p _ {t}} = \mu - v _ {t} ((1 + \eta) p _ {t} - 1).\tag{4}\]

From the deÖnition of , the Darwinian dynamics (2) can be written as

\[\dot {v} _ {t} = \lambda (p _ {t} - 1).\tag{5}\]

The economy has now been reduced to equations (4) and (5), an ODE system in and . Since the system is purely backward looking, both and are state variables, implying that initial conditions and need to be speciÖed. From (4) and (5), p = 1 and at steady state. At the steady state of the evolutionary economy, the average price announced by Örms is equal to the equilibrium price, implying that the economy is in equilibrium. The instantaneous probability of being a pioneer depends positively on the growth rate of money supply and negatively on the constant in the pioneerís price setting rule. When the money supply is growing at a large rate, nominal aggregate demand is growing at a large rate too. In a sticky price framework, pioneers are making large proÖts since other Örms are not changing their prices. Evolution increases the number of pioneers, which increases ináation and reduces pioneerís proÖts up to the point where the evolutionary economy is at steady state. This is very di§erent from the Calvo economy, where the probability of receiving a price-change signal is constant and the economy adjusts through changes in Calvo playersíprices. In the evolutionary economy, the probability of receiving a price-change signal is endogenous and positively related to the growth rate of money, which implies the desirable prediction that in a high ináation environment Örms change prices more frequently.⁵

The eigenvalues of the linearized system (4)-(5) are

\[- \frac {(1 + \eta) \mu}{\eta} \pm \sqrt {\left(\frac {(1 + \eta) \mu}{\eta}\right) ^ {2} - 4 \lambda \eta}.\]

Both eigenvalues are negative real numbers if , otherwise they are complex numbers with negative real part. The solution does converge, but it may converge by oscillations if the Darwinian dynamics adjust at high speed.

We study ináation inertia by analyzing the reaction of (4)-(5) to a permanent, negative shock to the rate of money growth, under the assumption that the economy was initially at steady state. The unit of time is a quarter. The quarterly rate of money growth was 2.5% before the shock and the new one is 2%, a 2 percentage points reduction in the annual growth rate. The instantaneous rate of price-change signals decreases from 0.30 to 0.24 in the new steady state. As it can be observed in Figure 1 ináation is persistent, with an impulse response function close to the one generated by sticky information.⁶ Pioneers keep playing the same price rule as before the shock and ináation reduces gradually at the speed with which the Darwinian dynamics reduces the size of price-pioneers in the total population.⁷

3.2 Calvo (1983)

To understand better the novelty of our result, we compare it to the cash-in-advance version of Calvo (1983). Equilibrium in the Calvo model is described by an ODE system for ináation and real money balances

\[\frac {\dot {m} _ {t}}{m _ {t}} = \mu - \pi_ {t}\tag{6}\]

\[\dot {\pi} _ {t} = b (q - m _ {t}).\tag{7}\]

⁵ A similar dynamic system emerges from an economy with constant probability density and a simple learning process in of the type , with , where the pioneers rule adjusts upwards or downwards depending on pioneersí proÖts. For example, under excess demand , pioneers can increase proÖts by increasing .

⁶ See Mankiw and Reis (2002), the bottom picture of Figure II on page 1305.

⁷ Inertia depends crucially on the velocity of the replicator dynamics process. We set implying that eigenvalues are complex. The eigenvalues are real for in which case ináation converges monotonically in less than 20 quarters. For very large values of , ináation convergences by oscillations moving at a very high frequency.

(20b,c)

c = m

⁸ See equations (20b,c) in Calvo (1983), with c = m because of the cash-in-advance constraint.

Figure 1: Reaction of quarterly ináation to a permanent reduction in the quarterly growth rate of money from 0.025 to 0.02

Since nominal money balances are a stock and prices are sticky, real money balances are a predetermined variable, with given initial condition . Ináation is not predetermined, even if the average price level is. Equation (6) comes directly from the deÖnition of real balances, which stop moving when ináation is equal to the growth rate of money. Equation (7) results from the price setting process and says that ináation adjusts to the output gap, i.e., the di§erence between aggregate capacity and aggregate demand m. Parameter b represents the speed of ináation adjustment, which depends negatively on the average time before a price change occurs (the inverse of the rate of price-change signals ). Remember that in Calvo the average price is a state variable. When the economy is in excess supply, prices have to increase; since they cannot jump instantaneously, ináation has to increase to reduce the output gap. At steady state, ináation is equal to the rate of money growth and the endowment is fully consumed The system is saddle path stable, with both m and increasing monotonically if the economy is initially in excess supply, i.e. if

In order to study ináation inertia, as in the case of the evolutionary economy, let us assume the Calvo economy is initially at steady state. It is easy to see that it adjusts to an unexpected permanent shock on the growth rate of money supply by jumping to the new steady state. Remember that a change in only a§ects the steady state value of ináation, implying that the steady state value of m remains unchanged. Since at the time of the shock real balances are assumed to be at steady state, from the saddlepath properties of the model, ináation has to directly jump to the new steady state. Consequently, the Calvo model shows no ináation inertia.

At the stationary solution of the Calvo model, the economy consumes the whole endowment meaning that the average price is equal to the equilibrium price. At steady state, given and Örms receiving a price-change signal set a price equal to where as in the evolutionary economy. However, this equilibrium outcome results from the forward-looking behavior of Calvo players. It is important to notice that, at the time of the shock, Calvo players automatically adjust to its new steady state value. This is a direct implication of rational expectations. They know the economy is jumping to the new steady state and use this information to perfectly forecast present and future ináation. Importantly, they incorporate this information in their price rule, making it jump to the new steady state too. In contrast, our price pioneers are backward looking and follow a simple rule of thumb. The ináation adjustment in the evolutionary economy takes place gradually because it relies on the Darwinian dynamics.

4 Conclusion

Calvoís (1983) method of incorporating staggered price setting in a utility maximizing framework is by now standard in the literature on New Keynesian Macroeconomics. One of the main weaknesses of the original formulation was a failure to deliver the empirically observed levels of persistence in ináation. Prominent remedies of this shortcoming include the introduction of price indexation in an otherwise standard sticky price model and the replacement of the assumption of sticky prices with sticky information. Here we have provided a simple alternative to those remedies. We have shown that, if prices are determined using an evolutionary principle, ináation inertia arises naturally even in a sticky price framework without price indexation.

In Calvoís original formulation, a fraction of Örms sets prices optimally, using all available information and forming expectations rationally. This has the undesirable e§ect that, even if the optimizing Örms are a tiny minority, their choices will be such that the overall ináation level will jump immediately to the new steady state in response to a permanent change in the money growth rate. Our evolutionary economy eventually converges to a stationary state where the average price (and ináation) is equal to the rational expectations equilibrium price (and ináation). However, in response to a change in the fundamentals, its convergence to the rational expectations ináation is only gradual. This is because no single Örm realizes the changes in the economy, it is rather the economy as a whole that learns about the new environment through a process of Darwinian selection. Firms receiving a price change signal keep playing the same price rule as before. But the frequency of the price change signal (or equivalently the number of Örms that receive it) evolves as it becomes more or less proÖtable for Örms to change their price and eventually converges to a stationary level consistent with rational expectation ináation, but only after a signiÖcant period of adjustment.

A numerical example provides an indication as to the strength of ináation inertia, more detailed quantitative prediction will be presented in a follow up article.

References

1. [1] Benassy, Jean-Pascal (1982), The Theory ofMarket Disequilibrium, Academic Press, New York.
2. [2] Calvo, Guillermo (1983), ìStaggered Prices in a Utility-Maximizing Framework, Journal of Monetary Economics 12(3), 983-998.
3. [3] Clarida, Richard, Jordi Gali and Mark Gertler (1999), "The Science of Monetary Policy: A New Keynesian Perspective", Journal of Economic Literature XXXVII,1661-1707.
4. [4] Clower, Robert (1967), ìA Reconsideration of the Microfoundations of Monetary Theory,îWestern Economic Journal 6, 1-9.
5. [5] Dreze, Jacques (1975), ìExistence of an Exchange Equilibrium under Price rigiditiesî, International Economic Review 16(2), 301-320.
6. [6] Fuhrer, Je§rey C. and George R. Moore (1005), "Ináation Persistence", Quarterly Journal of Economics 110(1), 127-159.
7. [7] Lucas, Robert and Nancy Stokey (1987), ìMoney and Interest in a Cash-in-Advance Economy,îEconometrica 55(3), 491-513.#
8. [8] Mackowiak, Bartosz and Mirko Wiederholt (2006), "Optimal Sticky Prices under Rational Inattention", manuscript:
9. [9] Mankiw, Gregory and Ricardo Reis (2002), ìSticky Information versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve,î The Quarterly Journal of Economics 117, 1295-1328.
10. [10] Saint-Paul, Gilles (2005), ìSome Evolutionary Foundations for Price Level Rigidity,îAmerican Economic Review, forthcoming.
11. [11] Samuelson, Larry (2002), ìEvolution and Game Theory,îJournal of Economic Perspective 16(2), 47-66.
12. [12] Sidrauski, Miguel (1967), ìRational Choice and Patterns of Growth in a Monetary Economy,îAmerican Economy Review 57(2), 534-544.
13. [13] Sims, Christofer (2003), "Implications of Rational Inattention", Journal of Monetary Economics, 50 (3), 665-690.
14. [14] Woodford, Michael (1994), ìMonetary policy and price level determinacy in a cashin-advance economy,îEconomic Theory 4(3), 345-380.