Elsevier

Journal of Economic Theory

Volume 150, March 2014, Pages 740-777
Journal of Economic Theory

Money and price posting under private information

https://doi.org/10.1016/j.jet.2013.12.005Get rights and content

Abstract

We study price posting with undirected search in a search-theoretic monetary model with divisible money and divisible goods. Ex ante homogeneous buyers experience match-specific preference shocks in bilateral trades. The shocks follow a continuous uniform distribution, and the realizations of the shocks are private information. We show that there exists a unique monetary equilibrium for generic values of the inflation rate. In equilibrium, each seller posts a continuous pricing schedule that exhibits quantity discounts. Buyers may spend nothing, a fraction or all of their money holdings, depending on their preference-shock realizations. Inflation reduces the extent of non-linear pricing. The model captures the hot-potato effect of inflation along both the extensive margin, as an increase in the trading probability, and the intensive margin, as higher fractions of money being spent.

Introduction

In this paper, we develop a monetary search model with private information and use the model to study how inflation affects the sellerʼs pricing decision and the buyerʼs spending pattern. The model has three main features: buyers are randomly matched with sellers; buyers experience match-specific preference shocks and have private information about the realizations of the shocks; and the terms of trade are determined by sellers making take-it-or-leave-it offers. This trading protocol is termed price posting with undirected search in the monetary search literature. “Undirected” captures the feature that buyers cannot choose which sellers to go to (this is in contrast with competitive or directed search, where sellers post and commit to prices, and buyers observe the prices and direct their search toward sellers who offer the highest expected trading surplus). During a match, the seller makes a take-it-or-leave-it offer through monopolistic pricing and offers a menu of price-quantity pairs from which the buyer can choose. The model has a unique monetary equilibrium in which sellers post a non-linear pricing schedule that exhibits quantity discounts, and buyers spend a fraction of or their entire money holdings contingent on the realizations of their preference shocks. There are two major implications about the effects of inflation on trading behaviors: first, inflation reduces the extent of non-linear pricing; and second, inflation induces buyers to speed up spending, so the model is able to capture the hot-potato effect of inflation.

One of our main contributions is to complement the existing search-theoretic monetary literature on price posting with undirected search. Under this trading protocol, monetary equilibria do not exist with public information. When the nominal interest rate is positive, buyers incur a costly ex ante investment when they decide to carry money. The existence of monetary equilibria hinges critically on the condition that buyers extract some trading surpluses during the monetary exchange to cover the investment cost. With public information, sellers would propose terms of trade to extract the entire trading surplus. Monetary equilibria would unravel as a result. In the literature, private information about match-specific preference shocks has been introduced to restore monetary equilibria in this class of models. The presence of private information forces sellers to share some trading surpluses with buyers that could cover the cost of holding money.

Three other papers study price posting with undirected search in monetary economies. Jafarey and Masters [9] and Curtis and Wright [3] use the indivisible-money framework of Trejos and Wright [23]. Curtis and Wright [3] study the case where there are multiple (⩾2) discrete realizations of the preference shock. Jafarey and Masters [9] assume that the preference shock follows a uniform distribution. More recently, Ennis [4] extends price posting with undirected search to a divisible-money framework, as in Lagos and Wright [13], with the preference shock following a two-point distribution. In all three papers, each seller posts a single price (if money is indivisible) or a single price-quantity pair (if money is divisible). At the aggregate level, at most two prices are observed in equilibrium, a result labelled as the “law of two prices” by Curtis and Wright [3]. As for buyers, they make binary choices, spending either nothing or their entire money holdings. Our study complements the existing literature by considering a model with divisible money and a continuous uniform distribution of the preference shock. The new environment leads to a different trading pattern. Instead of offering a single price, sellers post a continuum of price-quantity pairs that exhibits quantity discounts. For buyers, the dichotomous decision to spend nothing or all is replaced by a continuous choice, with the fraction of spending ranging from 0 to 100%.

The other main contribution of our paper is to provide a useful framework to study private information in monetary economies. As described in the previous paragraph, introducing a continuous distribution of the preference shock to a model with divisible money generates more variation in the trading pattern. This feature enables us to analyze how private information shapes the trading pattern, and how inflation affects the trading behavior in the presence of private information. In this paper we are particularly interested in two aspects of the trading behavior: the sellerʼs pricing choice and the buyerʼs spending decision.

About the pricing decision, our model shows that sellers use quantity discounts to screen buyers who have private information about their willingness to pay. The result has been generated in non-monetary models, including Maskin and Riley [16], Che and Gale [2], Thomas [22] and Faig and Jerez [6]. We show that the result continues to hold in a monetary economy. Furthermore, the model generates a new and testable implication that inflation induces more linear pricing. When buyers have private information about their preference shocks, sellers use quantity discounts to align buyersʼ incentives, encouraging those with higher preference-shock realizations to spend more. The cost of the scheme is that buyers with lower preference shocks are charged higher unit prices, so they buy little (or not at all). Sellers weigh the benefits and costs to determine the optimal level of quantity discounts. When inflation is low, buyers choose to hold more real money balances. The benefit from catering to buyers with higher preference shocks is relatively large. Sellers respond by offering more quantity discounts to increase trading with buyers with higher preference shocks. When inflation is high and buyers hold less real money balances, sellers have limited capacity to extract surpluses from buyers with higher preference shocks. Instead, the optimal strategy is to turn to buyers with lower preference shocks and expand trading with them. To induce buyers with lower preference-shock realizations to spend more, sellers reduce the extent of price discrimination (or quantity discounts). Faig and Jerez [7] show that quantity discounts are also observed in a competitive-search monetary model. However, their paper does not explore how inflation affects the extent of non-linear pricing.

About the buyerʼs spending pattern, our model predicts the hot-potato effect of inflation as an indirect result of the buyerʼs choice of money holdings and the sellerʼs optimal pricing strategy. As inflation rises, buyers choose to carry less money in real terms. Sellers respond by reducing the extent of quantity discounts to induce buyers with lower preference shocks to either start trading or trade more. As a result, the model captures the hot-potato effect along both the extensive margin (as a higher trading probability) and the intensive margin (as increased fractions of money being spent). The hot-potato effect exists at all levels of inflation.

There are several alternative approaches to generating the hot-potato effect in the literature. In Peterson and Shi [18], the economy consists of large households, which send a large number of buyers to purchase goods of varying quality. As inflation rises, buyers hold less money in real terms, which reduces the amount of high-quality goods being purchased. In response, households choose as a substitute, and speed up spending on, low-quality goods. Lagos and Rocheteau [12] show that, under competitive search, moderate inflation may increase the trading surpluses of buyers, thereby inducing them to search harder and trade more frequently. Faig and Jerez [7] analyze a competitive-search economy where buyers hold private information about their match-specific preference shocks. As inflation rises, sellers compete to offer buyers flatter pricing schedules that reduce the need for precautionary balances. Therefore, buyers with low preference-shock realizations consume and spend more. Ennis [4] presents a model in which inflation induces buyers who are less likely to consume to exit the market. Those who stay have a higher tendency to spend. A critical feature in Ennis [5] is that sellers are better able to avoid inflation taxes than buyers. Higher inflation makes buyers rush to dump their money to sellers. Liu et al. [14] assume that buyers have to pay an entry fee to participate in monetary exchange. Inflation reduces buyersʼ trading surpluses, so fewer buyers enter the market, which increases the trading probabilities of those who enter. Nosal [17] assumes that accepting a current trade reduces the probability of future trading. In the presence of preference shocks, buyers trade only when their preference-shock realizations are above a reservation shock level. As inflation reduces the value of future trading, the reservation shock level falls and buyers are more likely to accept a current trade.

The rest of this paper is organized as follows. Section 2 describes the environment. In Section 3, we characterize the monetary equilibrium. Section 4 examines the sellerʼs pricing behavior and studies how private information and inflation affect non-linear pricing. We investigate the effect of inflation on the speed of spending in Section 5. Section 6 discusses the efficiency properties of the monetary equilibrium, the modelʼs results under an alternative price-posting mechanism – competitive search, and an extension of the model with a more general distribution of the preference shock. Section 7 concludes. Appendix A provides the technical proofs of Lemma 1, Lemma 2. Appendix B discusses the situation where money holdings are unobservable. Appendix C solves the model under competitive search. Appendix D illustrates the role of private information in generating non-linear pricing in models with Nash bargaining and competitive search. Appendix E discusses the modelʼs implications with a more general distribution of the preference shock.

Section snippets

Environment

The model is based on Rocheteau and Wright [20]. Time is discrete and runs from 0 to ∞. A decentralized market (DM) and a centralized market (CM) open sequentially in each period. There is one non-storable good in each market: a DM good (q) and a CM good (x). The discount factor between two periods is β(0,1). There are two permanent types of agents: buyers and sellers, distinguished by their roles in the DM. Each type is of measure 1.

In the DM, agents are anonymous. Buyers are those who want

Equilibrium

To solve for the monetary equilibrium, we begin by analyzing choices in the CM and then consider decisions in the DM.

Non-linear pricing

As long as γ<, buyers choose to hold a positive amount of money and sellers post a continuum of price-quantity pairs (qe,ze)e[e0,eˆ], which may imply different unit prices. In this section, we examine the pricing schedule in more detail. We begin by investigating whether the pricing schedule involves non-linear pricing. If there is non-linear pricing, we analyze the role played by private information in generating the result. Finally, we explore how inflation affects the pricing schedule in

Hot-potato effect of inflation

The model has the feature that buyers hold precautionary money balances and may spend nothing, part or all of their money during exchange. This allows us to study how inflation affects the speed at which buyers spend their money. The results are stated in Proposition 4, Proposition 5.

Proposition 4

Effects of inflation on (e0,eˆ) and qe for e(e0,eˆ): for generic values of γ, de0/dγ<0, deˆ/dγ<0, and dqe/dγ>0.

Proof

From (9),de0dzˆ=(1eˆ)deˆdzˆ>0. Recall from Proposition 1 that dzˆ/dγ<0. Using (12) and (16), we can

Discussion

In this section, we analyze the efficiency properties of the monetary equilibrium under price posting and undirected search. We also discuss our main results under an alternative price-posting mechanism where buyers engage in directed search. Finally, we consider a more general distribution of the preference shock.

Conclusion

In this paper, we study trading behaviors in a monetary model of price posting and undirected search where buyers have private information about their match-specific preference shocks. This paper contributes to the literature on price posting with undirected search by considering a continuous uniform distribution of the preference shock in a divisible-money framework. The model has a unique monetary equilibrium where the seller posts a continuous pricing schedule. Our study overturns the result

Acknowledgments

For their comments and suggestions, we thank three anonymous referees, Aleksander Berentsen, Jose Dorich, Huberto Ennis, Ben Fung, Ricardo Lagos, Peter Norman, Brian Peterson, Tsz-Nga Wong, Randy Wright and seminar participants at the Aarhus University, Bank of Canada, Brock University, Ryerson University, University of Guelph, University of New South Wales, University of Technology Sydney, Wilfrid Laurier University, the 2009 Canadian Economic Association Meetings, the 2009 Chicago Fed Summer

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