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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 900, July 2007--- Screen Reader Version*

Bargaining, Fairness, and Price Rigidity in a DSGE Environment ^*

David M. Arseneau and Sanjay K. Chugh

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

A growing body of evidence suggests that an important reason why firms do not change prices nearly as much as standard theory predicts is out of concern for disrupting ongoing customer relationships because price changes may be viewed as "unfair". Existing models that try to capture this concern regarding price-setting are all based on goods markets that are fundamentally Walrasian. In Walrasian goods markets, transactions are spot, making the idea of ongoing customer relationships somewhat difficult to understand. We develop a simple dynamic general equilibrium model of a search-based goods market to make precise the notion of a customer as a repeat buyer at a particular location. In this environment, the transactions price plays a distributive role as well as an allocative role. We exploit this distributive role of prices to explore how concerns for fairness influence price dynamics. Using pricing schemes with bargaining-theoretic foundations, we show that the particular way in which a "fair" outcome is determined matters for price dynamics. The most stark result we find is that complete price stability can arise endogenously. There are issues about which models based on standard Walrasian goods markets are silent.

Keywords: Sticky prices, fair pricing, customer markets, search models

JEL classification: E20, E30, E31, E32

1.  Introduction

A growing body of evidence suggests that an important reason why firms do not change prices nearly as much as standard theory predicts is out of concern for disrupting ongoing customer relationships because price changes may be viewed as "unfair." Several recently-developed models try to capture this concern regarding price-setting, but in all of them the underlying model of the goods market is Walrasian. In Walrasian goods markets, transactions are spot, making the idea of ongoing customer relationships somewhat difficult to understand. Instead, models that feature explicit bilateral relationships between customers and firms seem to be called for in order to study the interactions between customer relationships and price rigidities. We develop a simple model that embeds a search-based goods market in an otherwise-standard dynamic general equilibrium model, making the notion of a customer as a repeat buyer at a particular location well-defined. In this framework, the transactions price (the terms of trade) plays a distributive role as well as an allocative role. We exploit this distributive role of prices to explore how concerns for fairness influence price dynamics. The most stark result we find is that if pricing is guided by a "fairness norm," complete price stability can arise endogenously. More generally, the consequences of menu costs of price adjustment on the dynamics of prices and allocations depend crucially on the manner in which price and quantity in specific relationships are determined, something about which standard models based on Walrasian goods markets are silent.

The two foundations of our model are a specific notion of customer relationships and menu costs of changing prices in those customer markets. In our model, customer relationships are valuable to both consumers and firms because of search frictions that each must overcome before goods trade can occur. The presence of search frictions leads to a surplus when a customer and a firm meet, and the parties must decide how to share the local monopoly rents. Regarding menu costs, we do not claim we have an explanation any deeper than existing ones for why there may be costs of changing prices; for convenience, such costs can be thought of in the typical fashion of costs associated with recording, reporting, and implementing new prices. By situating menu costs in a clearly-defined concept of a customer relationship, however, we are able to show that the consequences of price rigidities as typically formulated may depend critically on how prices and quantities traded are determined in specific relationships.

We focus on bargaining schemes as the mechanism by which prices and quantities are determined in customer markets. Despite using the formalisms of bargaining theory, we do not need to take literally the idea that customers and firms haggle over prices in every meeting. Rather, it seems to us that the idea that customers wield some "bargaining power" accords with the evidence of Blinder et al (1998) and others that firms often try to avoid upsetting their existing customers.

Given this interpretation, we adopt three bargaining protocols to pin down terms of trade between customers and firms: Nash bargaining, proportional bargaining, and a pricing system that we refer to as fair bargaining. We adopt Nash bargaining as a benchmark because of its familiarity: it has recently become relatively well-understood in macroeconomics due to the ongoing explosion of quantitative labor search models that employ it. Both proportional bargaining and fair bargaining implement an idea of a fairness norm under which parties always split the surplus in a fixed proportion, a feature that Nash bargaining does not always respect. The main difference between proportional bargaining and fair bargaining is the manner in which parties arrive at the fair outcome. Proportional bargaining assumes that the fair outcome is achieved through a mechanical sharing rule. In contrast, our notion of fair bargaining, although not axiomatic like the Nash and proportional outcomes, attempts to retain Nash bargaining's strategic foundation by imposing fairness as a constraint on the standard Nash optimization problem.

Akerlof (2007) makes the case that incorporating norms in macroeconomic models may be an evolutionary step for the field. We view our fairness norm, whether captured through proportional bargaining or fair bargaining, as in this spirit. We also view our idea as complementary to Rotemberg (2005, 2006), who has also stressed the notion that modeling fairness in pricing may be important. The crucial way in which our models of fair-pricing differ from Rotemberg (2005, 2006) is that we embed fairness as a feature of the trading structure of the environment, rather than altering preferences to account for it.

In our model, if there are no menu costs, the Nash-bargaining outcome, the proportional-bargaining outcome, and the fair-bargaining outcome all coincide. In the presence of menu costs, however, the three bargaining protocols imply quite different dynamics of prices and allocations. Under proportional bargaining, menu costs turn out to be completely irrelevant for both quantity and price dynamics, which seems to accord with survey evidence, such as Blinder et al (1998) and Fabiani et al (2006), that menu costs are not a very important friction in practice. Under fair bargaining, prices always remain at their steady-state values and dynamic allocations are completely unaffected by price rigidity. The key to understanding the dynamics under both proportional bargaining and fair bargaining is the fact that under Nash bargaining, price movements cause a time-varying wedge between short-run and long-run shares of the surplus accruing to customers and firms. With a fairness norm, parties eliminate such wedges, but the way in which the wedges are eliminated matters. If the wedges are eliminated according to proportional bargaining's mechanical sharing rule, customers and firms efficiently and equally share the consequences of menu costs and are able to engineer the zero-menu-cost Nash outcome. On the other hand, if the wedges are eliminated with the strategic considerations of fair bargaining in the background, menu costs are borne entirely by firms and eliminating the wedges requires complete price stability.

Due to the presence of local monopoly rents, our model also has something to say about markup behavior irrespective of menu costs. There lately has been a surge of interest in developing models in which markups are endogenously time-varying for reasons other than the presence of price rigidities.¹ Our flexible-price models deliver a time-varying markup; moreover, the flexible-price markup is countercyclical with respect to demand shocks, which seems to accord with empirical evidence and which has been the attention of much modeling effort. However, with fair bargaining and the presence of menu costs, the markup is constant, which is simply a reflection of the fact that both prices and marginal cost are time-invariant under fair bargaining.

In terms of bringing to bear data on our model, we exploit a central idea captured by our model: firms and consumers expend resources looking for trading partners. In our model, firms direct resources towards advertising in order to attract customers, and shoppers spend time looking for and purchasing goods from firms. Empirical evidence shows that the resources expended in such search activities are not negligible. Firm expenditures on advertising constitute over two percent of GDP, and time-use surveys show that individuals spend an average of about one hour per day shopping. Using such evidence, we can calibrate two deep features of our model. A by-product of our structure is that our model reproduces remarkably well the cyclical dynamics of aggregate advertising behavior in U.S. data.

We articulate our ideas in a non-monetary model, meaning the price dynamics on which we focus are those of real (relative) prices. It is apparent that much interest would lie in whether and to what extent our results carry over to monetary environments. We have reason to believe that the crucial aspects of our results -- namely, that the consequences of menu costs depend critically on how customers and firms determine prices and quantities and, in particular, the conclusion that some trading arrangements would lead to the endogenous emergence of price stability -- would carry over to monetary economies because the core mechanisms at work in our environment do not depend on things being cast in nominal or real terms. However, adding a monetary dimension to our model may not be as straightforward as imposing an ad-hoc cash-in-advance constraint or other typical monetary formulation used in the literature because once customer relationships are modeled explicitly, we may have to be careful about issues such as which consumers carry cash, which firms require payment in cash, etc. A monetary extension seems a logical next step; here, though, we concentrate on understanding some basic principles of the interactions between price rigidity and customer relationships.

Hall (2007) takes a very similar view of product markets as we take here and does use it to think about monetary policy issues. As in our model, the price at which goods change hands in Hall's (2007) model plays a distributive role in additional to the standard allocational role. Different prices inside a customer relationship achieve different distributions of the surplus between the consumer and the firm, but this does not affect the underlying efficiency of a trade. This admits the possibility of, in Hall's (2007) language, equilibrium sticky prices in customer markets. Given what we perceive as a growing sense of frustration with pricing models currently used in macroeconomic models, stemming from the growing body of micro pricing facts that challenge standard time-dependent or state-dependent pricing rules, allowing a distributional role for prices, as both our model and Hall's (2007) model do, may be a useful new direction for macroeconomic models. It seems to us that allowing this additional role for prices accords better with the idea that firms do not re-set prices out of worry for upsetting customers than do standard views of price rigidity. Besides the difference in focus on monetary versus non-monetary issues, we think another aspect of our model that sets it apart from Hall's (2007) is that we embed it from the start in a fully-articulated, quantitative DSGE environment, making comparisons with predictions of existing DSGE models straightforward.

Indeed, there lately has been a general surge of interest in developing simple structures of customer-firm interactions that can be tractably incorporated into state-of-the-art quantitative macroeconomic models. Our work here also falls into this broad category. A few recent examples of work in this broadly-defined area are the deep habits models of Ravn, Schmitt-Grohe, and Uribe (2006) and Nakamura and Steinsson (2007) and the switching-cost model of Kleschelski and Vincent (2007). As we mentioned, in terms of some basic motivation -- the idea that fairness norms may interact with or may lead to price rigidity -- the studies by Rotemberg (2005, 2006) are the closest in spirit to what we set out to achieve here. We view our work as complementary to all these recent efforts because we model customer relationships and fairness concerns through trading arrangements rather than essentially just through preferences. One could of course go back much further and tie ideas to the rich customer-markets literature, which includes studies by, among many others, Diamond (1971) and Klemperer (1995). Okun (1981) voices some of the ideas -- namely, that search frictions in goods markets may have important consequences for aggregate phenomena -- that we formalize through a modern search and matching framework. We certainly cannot do justice to the rich history of thought on these topics.

The rest of our paper is organized as follows. In Section 2, we lay out our baseline model with standard Nash bargaining, proportional bargaining, and fair bargaining. In the baseline model, only an extensive margin of consumption exists, and bargaining occurs just over the price paid by the customer. We provide some partial equilibrium analytics in Section 3, which illustrate the core forces at work in our environment. We examine the fully general equilibrium baseline model's numerical properties in Section 4, including a discussion of how and why fair bargaining leads to complete price rigidity in the face of arbitrarily small menu costs of price adjustment. In Section 5, we enrich our environment to allow for bargaining over both price and quantity in a given customer relationship, thus allowing for extensive and intensive margins of consumption. Finally, in Section 6, we briefly extend our model to include exogenous government purchases to demonstrate that our basic results carry over to an environment with demand shocks. Section 7 concludes and offers some ideas for continuing work. Most of the derivations of key relationships in our model are relegated to the appendices.

2.  Baseline Model

Our key point of departure from standard macro models is that, for some goods trades, households and firms each have to expend resources finding individuals on the other side of the market with whom to trade. A fraction of goods market exchange is thus explicitly bilateral, in contrast to all trades happening against the anonymous Walrasian auctioneer. Our model also does feature standard Walrasian goods for which search frictions are absent. We believe this view of goods markets is quite natural -- some goods require effort to find and some goods do not. For search goods, households spend time looking for firms from which to purchase goods, while firms direct part of their revenues towards trying to attract customers. We think of these two search activities as shopping and advertising, respectively.

Because we want to avoid taking a specific stand on the details of why it is that goods-market trading is costly -- there are probably a great many reasons -- we adopt the modeling device of an aggregate matching function from the labor search literature. Hall (2007) also takes this route. We describe more fully this matching mechanism below.² For our purposes, the important consequence of these search and matching frictions is that once a customer relationship is formed, each party has an incentive to keep the match intact because dissolving the relationship would mean each has to re-enter the costly search process. The existence of a surplus to be shared in a customer relationship means that we must think beyond standard Walrasian marginal pricing conditions because prices play both distributional and allocative roles. In the rest of this section, we describe in detail the households and firms in our model as well as the rest of the economic environment.

2.1  Households

There is a measure one of identical households, with a measure one of individuals that live within each household. In a given period, an individual member of the representative household can be engaged in one of four activities: purchasing goods (shopping) at a firm, working, searching for goods, or leisure. More specifically, $l_{t}$ members

There is a measure one of identical households, with a measure one of individuals that live within each household. In a given period, an individual member of the representative household can be engaged in one of four activities: purchasing goods (shopping) at a firm, working, searching for goods, or leisure. More specifically, $l_{t}$ members of the household are working in a given period; $s_{t}$ members are searching for firms from which to buy goods; $N^{h}_{t}$ members are shopping at firms with which they previously formed relationships; and $1 - l_{t} - s_{t} - N^{h}_{t}$ members are enjoying leisure. Note our distinction between shopping and searching for goods. Individuals who are searching are looking to form relationships with firms, which takes time. Individuals who are shopping were previously successful in forming customer relationships, but the act of acquiring and bringing home goods itself takes time.³ We assume that the members of a household share equally the consumption that shoppers acquire.

With this atomistic structure, we assume that lifetime discounted household utility is

$$E_{0} \sum_{t=0}^{\infty} \beta^{t} \left[ u(x_{t}) + \vartheta v\left( \int_{0}^{N^{h}_{t}} c_{it} di \right) + g(1-l_{t} -s_{t}-N^{h}_{t}) \right]$$ (1)

where $x$ is consumption of a standard Walrasian good and $c_{i}$ is the quantity of the search good that shopper $i$ brings back to the household. The household costlessly (aside from the direct purchase price) and instantaneously purchases the good $x$, which is of course what it means for the good to be traded in a Walrasian market. Total consumption $\int _{0}^{N^{h}} c_{i} di$ of the search goods obtained by shoppers is pooled by the household and divided equally amongst all family members. Instantaneous utility over leisure is $g(.)$, and the parameter $\vartheta$ governs how the household prefers to divide its total consumption between search and non-search goods.

Note that consumption of the search good potentially has two dimensions in our model: an extensive margin (the number of shoppers who buy goods) and an intensive margin (the number of goods each shopper buys). For the results in this section, we shut down adjustment at the intensive margin by setting $c_{i} = \bar{c}$. We begin by closing down the intensive margin for two reasons: doing so emphasizes the extensive margin, which is the most novel aspect of our model, and it also allows us some flexibility in calibrating our model. We discuss this issue further below when we present the calibration of the baseline model. In Section 5, we endogenize adjustment at the intensive margin of consumption. In the remainder of this section, then, we specialize to the case $c_{it} = \bar{c}$, and our notation reflects this. Finally, we point out that searching and shopping each detract from household leisure in the same, linear, manner.⁴

Whether or not we allow intensive adjustment, note that the aggregator inside $v(.)$ is linear in the total amount of search goods, meaning what we have in mind is a world in which all search goods are perfect substitutes in utility. Our baseline model focuses on this polar case because it means that any "monopoly markups" that arise in our model are due solely to search and matching frictions that create temporary bilateral monopolies, rather than to any ex-ante differentiation of products that create pure monopolies. That is, beginning with this assumption again allows us to isolate effects stemming from the search frictions in goods markets. As we will see in Section 5, in order to endogenize the intensive quantity traded, we must add some curvature to the consumption aggregator inside $v(.)$. The economic content of such curvature is that goods obtained from distinct matches are imperfect substitutes. We defer further discussion on this point until we encounter the full model in Section 5.

Using $c_{i} = \bar{c}$ for now, then, the flow budget constraint the household faces is

$$x_{t} + \int_{0}^{N^{h}_{t}} p_{it} \bar{c} di + b_{t} = w_{t} l_{t} + R_{t} b_{t-1} + d_{t},$$ (2)

where $b_{t-1}$ is holdings of a state-contingent one-period real private bond at the end of period $t-1$, which has gross payoff $R_{t}$ at the beginning of period $t$, $w_{t}$ is the real wage, and $d_{t}$ is firm dividends received lump-sum by the household. The Walrasian good $x$ serves as the numeraire, hence the price $p_{i}$ of a given search good is measured in units of $x$. With this structure so far, the household's first-order conditions with respect to Walrasian consumption $x_{t}$, labor $l_{t}$, and bond holdings $b_{t}$ are, respectively,

$$u^{\prime}(x_{t}) - \lambda_{t} = 0,$$ (3)

$$-g^{\prime}(1-l_{t}-s_{t}-N^{h}_{t}) + \lambda_{t} w_{t} = 0,$$ (4)

$$-\lambda_{t} + \beta E_{t} \left\{ \lambda_{t+1} R_{t+1} \right\} = 0,$$ (5)

where $\lambda_{t}$ is the Lagrange multiplier associated with the time-$t$ flow budget constraint and measures, in equilibrium, the marginal value of wealth to the household. Conditions (3), (4), and (5) are completely standard and imply the usual consumption-leisure optimality condition

$$\frac{g^{\prime}(1-l_{t}-s_{t}-N^{h}_{t} )}{u^{\prime}(x_{t})} = w_{t}%$$ (6)

and consumption-savings optimality condition

$$u^{\prime}(x_{t}) = \beta E_{t} \left\{ u^{\prime}(x_{t+1}) R_{t+1} \right\} .$$ (7)

The household must also choose how much effort to devote to searching and a desired number of future shoppers; $N^{h}_{t}$ is not free to be chosen at the beginning of period $t$ because that depends on how many searchers were previously successful in forming customer relationships. The household faces a perceived law of motion for the number of active customer relationships in which it is engaged,

$$N^{h}_{t+1} = (1-\rho^{x}) (N^{h}_{t} + s_{t} k^{h} (\theta_{t})),$$ (8)

where $k^{h}$ is the probability that a searcher finds a good. This matching probability depends on $\theta\equiv a / s$, which measures the tightness of the goods market -- how many advertisements there are per searcher -- and is taken as given by the household. With fixed probability $\rho^{x}$, which is known to both households and firms, an existing customer relationship dissolves at the beginning of a period. The dissolution of a customer relationship may occur for any of a number of reasons: the customer may move away, the firm may close shop, the customer may simply choose to stop visiting the same store for some reason, and so on. A natural potential future extension would be to endogenize the rate at which customer-firm relationships break up.

Finally, then, the household first-order conditions with respect to $s_{t}$ and $N^{h}_{t+1}$ are

$$-g^{\prime}(1-l_{t}-s_{t}-N^{h}_{t}) + (1-\rho^{x}) \mu ^{h}_{t} k^{h}(\theta_{t}) = 0$$ (9)

and

$${\small -\mu^{h}_{t} + \beta(1-\rho^{x}) E_{t} \mu^{h}_{t+1} - \beta E_{t} \left\{ \lambda_{t+1} p_{Nt+1} \bar{c} \right\} + \beta E_{t} \left\{ \vartheta v^{\prime}\left( \int_{0}^{N^{h}_{t+1}} \bar{c} di \right) \bar{c} - g^{\prime}(1-l_{t+1}-s_{t+1}-N^{h}_{t+1}) \right\} = 0,}%$$ (10)

where $\mu^{h}_{t}$ is the Lagrange multiplier on the law of motion for shoppers, $p_{Nt+1}$ is the relative price of the $N$-th good at time $t+1$, and $\bar{c}$ is the fixed quantity consumed of the $N$-th good at time $t+1$. As we present below, the price $p_{i}$ is determined in bargaining. From here on, we conserve on notation by using $v^{\prime}_{t}$ to stand for $v^{\prime }\left( \int_{0}^{N^{h}_{t}} \bar{c} di \right)$, $g^{\prime}_{t}$ to stand for $g^{\prime}(1-l_{t}-s_{t}-N^{h}_{t})$, and $u^{\prime}_{t}$ to stand for $u^{\prime}(x_{t})$.

Having taken first-order conditions and given that we restrict attention to symmetric equilibria in which $p_{i} = p_{j}$ for all $i \neq j$, the first-order condition on $N^{h}_{t+1}$ becomes

$$-\mu^{h}_{t} + \beta(1-\rho^{x}) E_{t} \mu^{h}_{t+1} - \beta E_{t} \left\{ \lambda_{t+1} p_{t+1} \bar{c} \right\} + \beta E_{t} \left\{ \vartheta v^{\prime}_{t+1} \bar{c} - g^{\prime}_{t+1} \right\} = 0,$$ (11)

where $p_{t+1}$ now stands for the real (measured in units of $x$) price of any given good for which there exists a customer-firm relationship. Condensing this expression with the household first-order condition on $s$, we have

$$\frac{g^{\prime}_{t}}{k^{h}(\theta_{t})} = \beta(1-\rho^{x}) E_{t} \left\{ \vartheta v^{\prime}_{t+1} \bar{c} - g^{\prime}_{t+1} - \lambda_{t+1} p_{t+1} \bar{c} + \frac{g^{\prime}_{t+1}}{k^{h}(\theta_{t+1})} \right\} .$$ (12)

Using (3) and re-grouping terms, we have

$$\frac{g^{\prime}_{t}}{ k^{h}(\theta_{t})} = \beta (1-\rho^{x}) E_{t} \left\{ \bar{c} \left[ \vartheta v^{\prime}_{t+1} - p_{t+1} u^{\prime}(x_{t+1}) \right] - g^{\prime}_{t+1} + \frac{g^{\prime }_{t+1}}{k^{h}(\theta_{t+1})} \right\} ,$$ (13)

which we refer to as the household's shopping condition. The shopping condition simply states that at the optimum, the household should send a number of individuals out to search for goods such that the expected marginal cost of shopping (the left-hand-side of (13)) equals the expected marginal benefit of shopping (the right-hand-side of (13)). The expected marginal benefit of shopping is composed of two parts: the utility gain from obtaining $\bar{c}$ more goods via the search market rather than via the Walrasian market (net of the direct disutility $g^{\prime}$ of shopping) and the benefit to the household of having one additional pre-existing customer relationship entering period $t+1$. If all trades were frictionless, household optimal choices would imply $\vartheta v^{\prime}_{t} = p_{t} u^{\prime}(x_{t})$. With frictions, in order to engage in costly search, it must be that on the margin, the household expects $\vartheta v^{\prime}_{t+1} > p_{t+1} u^{\prime}(x_{t+1} )$.⁵ This positive flow return ensures that the household finds it worthwhile to send some of its members shopping.

2.2  Walrasian Firms

To make pricing labor simple, we assume that there is a representative firm that buys labor in and sells the Walrasian good $x$ in competitive spot markets. The firm operates a linear production technology that is subject to aggregate TFP fluctuations, $y_{t} = z_{t} l^{W}_{t}$, where $l^{W}$ denotes the labor hired by Walrasian firms. Profit-maximization yields the standard result that

$$w_{t} = z_{t},$$ (14)

which all participants in the economy, including the non-Walrasian firms described next, take as given.⁶

2.3  Non-Walrasian Firms

We also assume that there is a representative firm that sells a large number of goods in bilateral trades. For each good that it sells, the representative search firm must first attract customers. To attract customers, the firm must advertise, and how any given level of advertisements it posts maps into how many customers it finds is governed by a matching technology to be described below. Owing to frictions associated with finding customers, the firm views existing customers as assets. Its total stock of customers evolves according to the perceived law of motion

$$N^{f}_{t+1} = (1-\rho^{x}) (N^{f}_{t} + a_{t} k^{f}(\theta_{t})),$$ (15)

where $a_{t}$ is the number of advertisements the firm posts in period $t$, and $k^{f}$ is the probability that one of the firm's advertisements attracts a customer, which depends on goods market tightness $\theta$; $\theta$ is taken as given by the firm.

As with competitive firms, the production technology is linear in labor and subject to an exogenous aggregate productivity shock. Total output of the non-Walrasian firm is thus $y_{t} = z_{t} l^{A}_{t}$, where $l^{A}$ denotes the labor hired by the non-Walrasian firm. Because we assume a constant-returns production technology with no fixed costs of production (there is a fixed cost of advertising, but no fixed cost of producing), its real marginal cost of production is constant and coincides with average cost.⁷ Denoting marginal production cost by $mc_{t}$ in period $t$, we can express the firm's total production costs as the sum of production costs across all of its customer relationships, $\int_{0}^{N^{f}_{t}} mc_{t} c_{it} di$.

The firm also faces a menu cost of adjusting the per-unit price of each good it sells to a given customer. Specifically, we use a Rotemberg-type quadratic cost of price adjustment, which is a fairly conventional way of modeling menu costs. As we mentioned earlier, our goal is not to provide a compelling micro-foundation for why price adjustment may entail costs; rather, adopting a typical reduced-form specification is just a tractable way to get at our ultimate objective.

With this structure in place, total profits of the representative search firm in a given period $t$ are

$$\int_{0}^{N^{f}_{t}} p_{it} c_{it} di - \int_{0}^{N^{f}_{t}} mc_{t} c_{it} di - \int_{0}^{N^{f}_{t}} \frac{\kappa}{2} \left( \frac{p_{it}}{p_{it-1}} - 1\right) ^{2} - \gamma a_{t},$$ (16)

where $p_{it}$ is the relative price (which is, recall again, measured in units of $x$) of good $i$, $\gamma$ is the flow cost of an advertisement, thus $\gamma a_{t}$ is the total flow advertising cost the firm incurs. We again specialize right away to the case $c_{i} = \bar{c}$ and, as we have already mentioned, defer considering intensive adjustment until Section 5. The parameter $\kappa$ measures how large menu costs are; setting $\kappa=0$ of course means there are no menu costs. The firm's customer base $N^{f}_{t}$ is pre-determined entering period $t$. Discounted lifetime profits of the firm are thus

$$E_{0} \sum_{t=0}^{\infty} \Xi_{t\vert}\left[ \int _{0}^{N^{f}_{t}} p_{it} \bar{c} di - \int_{0}^{N^{f}_{t}} mc_{t} \bar{c} di - \int_{0}^{N^{f}_{t}} \frac{\kappa}{2} \left( \frac{p_{it}}{p_{it-1}<tex2html_comment_mark>138 }-1\right) ^{2} - \gamma a_{t} \right] ,$$ (17)

where $\Xi_{t\vert}$ is the period-0 value to the household of period-$t$ goods, which we assume the firm uses to discount profit flows because the households are the ultimate owners of firms.⁸

The problem of the firm is thus to maximize (17) subject to the evolution of its customer base (15) by choosing $\{a_{t}, N^{f}_{t+1}\}$. In the firm's pursuit of customers, it takes $p_{i}$ and $\bar{c}$ as given because those will be determined in the trading protocols to be described below. Note an important point of departure from standard macro models of goods markets: the firm is not a unilateral price-setter here.⁹ The first-order conditions with respect to $\{a_{t}, N^{f}_{t+1}\}$ thus are

$$-\gamma+ (1-\rho^{x}) \mu^{f}_{t} k^{f}(\theta_{t}) = 0$$ (18)

and

$$- \Xi_{t\vert} \mu^{f}_{t} + (1-\rho^{x}) E_{t} \left\{ \Xi_{t+1\vert} \mu ^{f}_{t+1} \right\} + E_{t} \left\{ \Xi_{t+1\vert} \left[ p_{N,t+1} \bar{c} - mc_{t+1} \bar{c} - \frac{\kappa}{2} \left( \frac{p_{N,t+1}}{p_{N,t}} - 1\right) ^{2} \right] \right\} = 0,$$ (19)

where $\mu^{f}_{t}$ is the Lagrange multiplier on the firm's customer constraint. Condensing these first-order conditions, we arrive at the firm's optimal advertising condition,

$$\frac{\gamma}{k^{f}(\theta_{t})} = (1-\rho ^{x}) E_{t} \left\{ \Xi_{t+1\vert t} \left[ p_{t+1} \bar{c} - mc_{t+1} \bar{c} - \frac{\kappa}{2} \left( \frac{p_{N,t+1}}{p_{N,t}} - 1\right) ^{2} + \frac{\gamma}{k^{f}(\theta_{t+1})} \right] \right\} ,$$ (20)

where $\Xi_{t+1\vert t} \equiv\Xi_{t+1\vert} / \Xi_{t\vert}$ is the household discount factor (again, technically, the real interest rate) between period $t$ and $t+1$. In equilibrium, $\Xi_{t+1\vert t} = \frac{\beta\lambda_{t+1}}{\lambda_{t}}$, which in turn, by the household optimality condition (3), is $\Xi_{t+1\vert t} = \frac{\beta u^{\prime}(x_{t+1})}{u^{\prime}(x_{t})}$ . In writing (20), we have also imposed symmetric equilibrium, in which $p_{it} = p_{jt} = p_{t}$ for any $i \neq j$.

The advertising condition states that at the optimum, the expected marginal cost of posting an ad (the left-hand-side of (20)) equals the expected marginal benefit of forming a relationship with a new customer (the right-hand-side of (20)). The expected marginal benefit takes into account the revenue from selling to one extra customer, the production costs incurred for producing to sell those extra units, future menu costs in that customer relationship, and the cost savings of finding another customer in the future due to the pre-existing (in time $t+1$) customer relationship. Condition (20) is a free-entry condition in advertising. The fact that an entry decision must be made before a firm can enjoy any profit flows means that profit flows from sales of goods are not pure rents as they are in commonly-employed formulations of goods markets.

2.4  Price Determination

With bilateral relationships between customers and firms, there is an array of ways to think about how prices are determined. We focus on three bargaining schemes, two that are axiomatic and one that, although it is not axiomatic, seems to us to capture important intuitive elements underlying each of the two axiomatic schemes. In considering bargaining, we do not need to take literally the idea that customers and firms haggle over prices in every meeting, even though that is the formalism we use. Rather, it seems to us that the idea that customers wield some "bargaining power" accords with the evidence that firms often try to avoid "upsetting their existing customers."

To make progress with this idea, then, we consider three bargaining protocols. The first protocol is Nash bargaining, the second is proportional bargaining, and the third is a modified Nash problem in which the two parties always divide the match surplus in a fixed proportion. We refer to this third trading protocol as fair bargaining, although we recognize that bargaining theorists would not accord this trading protocol "bargaining" status. Conceding this point, we nonetheless use the term fair bargaining to make the discussions symmetric.

Proportional bargaining and fair bargaining implement the idea of "fairness" in trading outcomes in similar, but distinct, ways. Our specific notion of fairness is one in which the surplus accruing to customers is always a fixed ratio of the surplus accruing to firms. Clearly, as discussed by Binmore (2007) and many others, there are a great many ways to operationalize the concept of fairness. Given our environment of explicit relationships between consumers and firms, we think ours is at least one natural definition. As we show below, and is well-known in bargaining applications (see Aruoba, Rocheteau, and Waller (2007) for a particularly recent application), the Nash solution does not generally satisfy this definition of fairness.

It is well-understood that Nash bargaining has explicit strategic foundations. Specifically, Rubinstein (1982) shows that the Nash bargaining solution is the limiting solution of a strategic alternating-offers bargaining environment. In contrast, proportional bargaining, while it does, as we show below, capture our definition of fairness, does not have as clear a strategic foundation. Instead, it accords better with the concept of a focal-point equilibrium, an outcome that, perhaps for some evolutionary reason, simply is the accepted social norm. Experimental evidence on bilateral games people play, summarized by, among others, Binmore (2007, Chapter 6), suggests that both strategic and focal-point elements are typically at work. This motivates us to construct our fair-bargaining scheme in an attempt to retain the strategic element inherent in Nash bargaining as well as the fairness/focal-point element inherent in proportional bargaining.

2.4.1  Nash Bargaining

In Nash bargaining over the $i$-th product, the firm and the customer jointly choose $p_{it}$ to maximize the Nash product

$$(\mathbf{M_{t}} - \mathbf{S_{t}})^{\eta} \mathbf{A_{t} }^{1-\eta},$$ (21)

where $\mathbf{M_{t}}$ is the value to a household of having a member engaged in a relationship with a firm, $\mathbf{S_{t}}$ is the value to a household of having a member searching for goods, $\mathbf{A_{t}}$ is the value to a firm of being engaged in a relationship with a customer, and $\eta$ is a standard time-invariant Nash bargaining weight. The value to a firm of an advertisement that failed to attract any customers is normalized to zero. As we show in Appendix B (where we present the definitions of $\mathbf{M}$, $\mathbf{S}$, and $\mathbf{A}$), the price $p_{it}$ that emerges from Nash bargaining in any particular customer-firm relationship satisfies the sharing rule

$$(1-\omega_{t}) (\mathbf{M_{t}} - \mathbf{S_{t}}) = \omega_{t} \mathbf{A_{t}},$$ (22)

in which $\omega_{t}$ is the effective bargaining power of the customer and $1-\omega_{t}$ is the effective bargaining power of the firm. Specifically,

$$\omega_{t} \equiv\frac{\eta}{\eta+ (1-\eta) \Delta^{F}_{t} / \Delta^{H}_{t}},$$ (23)

where $\Delta^{F}_{t}$ and $\Delta^{H}_{t}$ measure marginal changes in the value of a customer relationship for the firm and the household, respectively.

We provide more details in Appendix B, but three points are worth mentioning here. First, time-variation in $\omega_{t}$ means that the household's surplus $\mathbf{M_{t}} - \mathbf{S_{t}}$ from an active customer relationship is not a fixed ratio of the firm's surplus $\mathbf{A_{t}}$. The Nash solution thus does not generally satisfy our definition of fairness. Second, with zero costs of price adjustment in period $t$ (which may arise for two reasons: either $\kappa=0$ or $\kappa> 0$ but $p_{it} = p_{it-1}$), $\omega_{t} = \eta$ (because in that case $\Delta ^{F}_{t} / \Delta^{H}_{t} = 1$). It is thus variation in prices coupled with the presence of menu costs that drives a time-varying wedge between effective bargaining power and the Nash bargaining weights. Indeed, if the Nash product the customer and firm maximized were $(\mathbf{M_{t}}-\mathbf{S_{t}})^{\omega_{t}} \mathbf{A_{t}}^{1-\omega_{t}}$ rather than (21), the outcome would be the sharing rule (22). Third, in the long run (i.e., the deterministic steady state), $\omega= \eta$ because $p_{it} = p_{it-1} = \bar{p}_{i}$ (recall these are real prices). Thus, the wedge between $\eta$ and $\omega _{t}$ is a business-cycle phenomenon.

With time-varying effective bargaining power, the price solves

$$\lefteqn{\frac{\omega_{t}}{1-\omega_{t}} \left[ p_{it} \bar{c} - mc_{t} \bar{c} - \frac{\kappa}{2} \left( \frac{p_{it}}{p_{it-1} }-1\right) ^{2} + \frac{\gamma}{k^{f}(\theta_{t})}\right] = \frac{\tilde {u}(\bar{c})}{\lambda_{t}} - p_{it} \bar{c} + }$$

$$+ (1-\theta_{t} k^{f}(\theta_{t})) E_{t}\left[ \Xi_{t+1\vert t} \left( \frac{\omega_{t+1}}{1-\omega_{t+1}}\right) (1-\rho^{x}) \left[ p_{it+1} \bar{c} - mc_{t+1} \bar{c} - \frac{\kappa}{2}\left( \frac{p_{it+1}}{p_{it} }-1\right) ^{2} + \frac{\gamma}{k^{f}(\theta_{t})} \right] \right] .$$ (24)

where $\tilde{u}(\bar{c})$ is the utility function defined over the quantity consumed of a good obtained from a given customer relationship (rather than the utility $v(.)$ defined over the household's aggregate consumption of search goods $\int_{0}^{N} \bar{c} di$.).¹⁰ The pricing condition (24) shows that price-setting is forward-looking for two distinct reasons. One reason is a standard sticky-price reason: with costs of price adjustment, a setting for $p_{it}$ has ramifications for future setting of $p_{it+1}$. But note that even with $\kappa=0$, $p_{it}$ is affected by expectations regarding $p_{it+1}$. This forward-looking aspect of pricing has to do with the long-lived customer relationship: with probability $1-\rho^{x}$, the customer and firm will bargain over the same good again in the future. In typical models, price-setting is static in the absence of menu costs; in our framework, it is dynamic even in the absence of menu costs.¹¹

Finally, we note that with $\kappa=0$, the Nash sharing rule (22) reduces to the more standard $(1-\eta )(\mathbf{M_{t}}-\mathbf{S_{t}}) = \eta\mathbf{A_{t}}$, and the pricing equation (24) simplifies dramatically to

$$p_{it} = (1-\eta) \left( \frac{\tilde{u}(\bar {c})}{\lambda_{t}} \right) + \eta(mc_{t} - \gamma\theta_{t}),$$ (25)

which one can obtain by working through the derivations in Appendix B.

2.4.2  Proportional Bargaining

Condition (22) shows that in the presence of menu costs, the surplus is split between consumers and firms according to time-varying shares. Suppose instead that some fairness norm in pricing were in place in which the customer and firm ensure that they always split the total surplus in a time-invariant ratio. An axiomatic solution (see Kalai (1977) for the original exposition) that guarantees constant splits is the proportional bargaining solution. Under proportional bargaining, the price $p_{it}$ solves

$$(1-\eta) (\mathbf{M_{t}} - \mathbf{S_{t}}) = \eta\mathbf{A_{t}},$$ (26)

without any reference to an underlying maximization problem.¹² Substituting the definitions of $\mathbf{M_{t}}$, $\mathbf{S_{t}}$, and $\mathbf{A_{t}}$ presented in Appendix B, the proportional-bargaining price solves

$$\lefteqn{\frac{\eta}{1-\eta} \left[ p_{it} \bar{c} - mc_{t} \bar{c} - \frac{\kappa}{2} \left( \frac{p_{it} }{p_{it-1}}-1\right) ^{2} + \frac{\gamma}{k^{f}(\theta_{t})}\right] = \frac{\tilde{u}(\bar{c})}{\lambda_{t}} - p_{it} \bar{c} + }$$

$$+ (1-\theta_{t} k^{f}(\theta_{t})) \left( \frac{\eta}{1-\eta}\right) E_{t}\left[ \Xi_{t+1\vert t} (1-\rho^{x}) \left[ p_{it+1} \bar{c} - mc_{t+1} \bar{c} - \frac{\kappa}{2}\left( \frac{p_{it+1}}{p_{it}}-1\right) ^{2} + \frac{\gamma}{k^{f}(\theta_{t})} \right] \right] ,$$ (27)

which differs from (24) only in that $\eta/(1-\eta)$ replaces $\omega_{t} / (1-\omega_{t})$.

2.4.3  Fair Bargaining

As we discussed above, Nash bargaining has clear strategic foundations, while those underlying proportional bargaining are less clear. To try to capture both the strategic element underlying Nash bargaining and the focal-point element underlying proportional bargaining -- both elements that the evidence of Binmore (2007) suggests are important for understanding bilateral interactions -- we construct a pricing scheme that draws on both.

The way in which we implement this idea is to continue using the Nash product as the objective the parties seek to maximize; however, constant shares are now enforced. Specifically, $p_{it}$ is chosen to maximize

$$(\mathbf{M_{t}} - \mathbf{S_{t}})^{\eta} \mathbf{A_{t}}^{1-\eta}%$$ (28)

subject to a constant-split rule

$$(1-\varphi) \left( \mathbf{M_{t}}-\mathbf{S_{t}}\right) = \varphi \mathbf{A_{t}}.$$ (29)

The most straightforward constant-split rule to understand is the case $\varphi= \eta$, which amounts to enforcing the standard Nash sharing condition despite the presence of menu costs. We focus just on this case.¹³

We provide the details behind this problem in Appendix C, but the outcome of fair bargaining over price is

$$\kappa(\pi_{it}-1) \pi_{it} - (1-\rho^{x}) \kappa E_{t} \left[ \Xi_{t+1\vert t} (\pi_{it+1}-1) \pi_{it+1}\right] = 0,$$ (30)

in which $\pi_{it}$ is defined as the gross rate of price change between period $t-1$ and $t$, $\pi_{it} \equiv p_{it} / p_{it-1}$. If this were a monetary model, we would of course call $\pi$ inflation and thus may be tempted to refer to condition (30) as a modern Phillips curve because it links period-$t$ price growth (inflation) to expected future price growth (inflation).

What prevents condition (30) from being a Phillips curve (aside from the fact that our model is not cast in nominal terms) is that it contains nothing at all about allocations. To show how tantalizingly close (30) is to a standard New Keynesian Phillips curve, compare it with the standard pricing equation that emerges from New Keynesian models; for example, the analogous condition in Chugh (2006, equation 7) is

$$f(mc_{t}) + \kappa(\pi_{t} - 1) \pi_{t} - \kappa E_{t} \left[ \Xi_{t+1\vert t} (\pi_{t+1}-1) \pi_{t+1}\right] = 0,$$ (31)

where $f$ is some function that depends on the marginal cost of production.¹⁴ The fact that marginal cost, which reflects something about allocations, appears in a standard New Keynesian Phillips curve of course provides the linkage between real activity and price movements in that class of models.

Our fair-pricing condition (30), missing marginal costs, is thus of course not a Phillips curve because it does not link price changes to real activity. Indeed, it is quite the opposite of the spirit of a Phillips Curve: condition (30) describes only the dynamics of prices and demonstrates that the dynamics of allocations are divorced from the dynamics of prices. Thus, price rigidity coupled with a fairness norm in bargaining effectively decouples prices from quantities in our model.

Finally, we point out that our notion of fairness in pricing is an assertion about the bargaining protocol. We are not modeling deeper-rooted reasons that may underlie why fair bargaining (or proportional bargaining, for that matter) may be adopted in the first place. One potential candidate explanation is the recent model of Rotemberg (2006), in which customer "anger" over prices that are perceived to be exploitative may lead to a fairness norm. More broadly, Akerlof (2007), in his overture to macroeconomics to adopt more "norm-based" behavior into standard models, discusses why incorporating norms and, at least as a first step, ignoring endogeneity of norm adoption may be an important evolutionary step for the field. Such issues are quite interesting to consider, and if our model proves to be useful in thinking about some aspects of goods-market relationships, one may want to embed such mechanisms in our model in future work. For now, our focus is on the consequences of menu costs of price adjustment given a fairness norm in pricing.

2.5  Goods Market Matching

The number of new customer-firm relationships that are formed in any period $t$ is described by an aggregate matching function $m(s_{t}, a_{t})$. We assume the matching technology is Cobb-Douglas, $m(s_{t}, a_{t})$. With Cobb-Douglas matching, the probabilities that shoppers and firms, respectively, find partners is

$$k^{h}(\theta) = \frac{m(s, a)}{s} = m\left( 1, \frac{a} {s}\right) = m(1, \theta)$$ (32)

$$k^{f}(\theta) = \frac{m(s, a)}{a} = m\left( \frac{s}{a}, 1\right) = m(\theta^{-1}, 1),$$ (33)

with $\theta\equiv a/s$ a measure of how thick (the ratio of firms searching for customers to individuals searching for goods) the goods market is.

As in the labor search literature, the matching function is meant to be a reduced-form way of capturing the idea that it takes time for parties on opposite sides of the market to meet. Rogerson, Shimer, and Wright (2005, p. 968) note that the ability to be agnostic about the actual mechanics of the process by which parties make contact with each other may be a virtue. Our modeling motivation is very much in line with this idea.

With the matching function describing the flow of new customer relationships, the aggregate number of active customer relationships evolves according to

$$N_{t+1} = (1-\rho^{x}) (N_{t} + m(s_{t}, a_{t})).$$ (34)

2.6  Resource Constraint

Total output of the economy is absorbed by Walrasian consumption, non-Walrasian consumption, price adjustment costs, and advertising costs. The resource constraint is thus

$$x_{t} + \int_{0}^{N_{t}} c_{it} di + \int_{0}^{N_{t}} \frac{\kappa}{2}(\pi_{t} - 1)^{2} + \gamma a_{t} = z_{t} l_{t}.$$ (35)

2.7  Equilibrium

We restrict attention to a symmetric equilibrium in which the price $p$ is identical across all active customer relationships. Because (part of) goods trade is carried out in non-Walrasian markets, the core notion of equilibrium in our model is that of a search equilibrium, rather than a competitive equilibrium. This motivates the definition of a symmetric search equilibrium for our model.

A symmetric search equilibrium is endogenous processes for the household's choice $\{x_{t}, l_{t}, s_{t}, N^{h}_{t+1}, b_{t}\}_{t=0}^{\infty}$, the Walrasian firm's choice $\{l^{W}_{t}\}_{t=0}^{\infty}$, the non-Walrasian firm's choice $\{a_{t}, N^{f}_{t+1}, l^{A}_{t}\}_{t=0}^{\infty}$, the real wage $\{w_{t}\}_{t=0}^{\infty}$, bond returns $\{R_{t}\}_{t=0}^{\infty}$, dividends $\{d_{t}\}_{t=0}^{\infty}$, prices in the non-Walrasian goods market $\{p_{t}\}_{t=0}^{\infty}$, and matching probabilities $\{k^{h}_{t}, k^{f} _{t}\}_{t=0}^{\infty}$ such that

• given $\{p_{t}, w_{t}, R_{t}, d_{t}, k^{h}_{t}, k^{f}_{t}\}_{t=0} ^{\infty}$ , the household maximizes (1) subject to (2) and (8);
• given $\{w_{t}\}_{t=0}^{\infty}$, the Walrasian firm chooses labor to maximize profit;
• given $\{p_{t}, w_{t}, k^{h}_{t}, k^{f}_{t}\}_{t=0}^{\infty}$, the non-Walrasian firm maximizes (17) subject to (15);
• $\{p_{t}\}_{t=0}^{\infty}$ satisfies either (24) (Nash bargaining) or (30) (fair bargaining);
• the resource constraint (35) holds for $t = 0, 1,....$;
• the labor market clears, $\{l_{t} = l^{W}_{t} + l^{A}_{t}\}_{t=0} ^{\infty}$;
• the customer market clears, $\{N^{h}_{t} = N^{f}_{t}\}_{t=0}^{\infty}$;
• the aggregate law of motion for active customer relationships is given by (34) for $t = 0, 1,....$;
• the bond market clears, $\{b_{t} = 0\}_{t=0}^{\infty}$;
• dividends $\{d_{t}\}_{t=0}^{\infty}$ are determined residually from the non-Walrasian firm's profit function;
• $\{k^{h}_{t}, k^{f}_{t}\}_{t=0}^{\infty}$ are given by (32) and (33).

Collecting conditions that summarize the equilibrium, equilibrium is endogenous processes $\{x_{t}, N_{t}, p_{t}, s_{t}, a_{t}, l_{t}, w_{t}, R_{t}\}_{t=0}^{\infty}$ that satisfy (6), (7), (13), (14), (20), either (24) or (30), (34), and (35), for given exogenous process $\{z_{t}\}_{t=0}^{\infty}$.

3.  Partial Equilibrium Analytics

Our main interest lies in some general equilibrium business cycle consequences of search frictions in goods trade and price rigidity. However, we can gain a lot of intuition for the economic forces at work in our model by examining both analytically and numerically its steady-state equilibrium. The triple $(p, \theta, s)$ are the most important endogenous variables describing our frictional goods market.¹⁵ Features such as the intensive quantity $c$, presence of a production technology, and endogenous labor force participation are all present to make our model quantitatively realistic and readily comparable to other quantitative macroeconomic models. We again emphasize that the fundamental notion of goods-market equilibrium here is that of a search equilibrium, not a Walrasian (or Walrasian-based) equilibrium.¹⁶

To make some analytical progress, then, for the moment suppose the general equilibrium features of our model were shut down. Specifically, suppose $c=1$ and labor is not needed ($l = 0$).¹⁷ In this partial equilibrium version of our model, then, imposing steady-state on the firm advertising condition gives us

$$\frac{\gamma}{k^{f}(\theta)} = \frac{\beta(1-\rho^{x})}{1-\beta(1-\rho^{x})} (p - mc),$$ (36)

in which we have used the fact that in the deterministic steady state, prices do not change (remember, these are real prices), hence there are no menu costs of price adjustment. With Cobb-Douglas matching, the probability a firm that has advertised matches with a customer is $k^{f}(\theta) = \theta^{-\xi}$. Imposing this and rearranging, we have

$$\frac{\beta(1-\rho^{x})}{1-\beta(1-\rho ^{x})} p = \frac{\beta(1-\rho^{x})}{1-\beta(1-\rho^{x})} mc + \gamma \theta^{\xi},$$ (37)

which shows that the markup of price over marginal cost of production is governed by the advertising cost $\gamma$ and customer market tightness $\theta$. If $\gamma= 0$, we have $p=mc$, which makes perfect sense because in that case it is costless for firms to find customers and we have assumed that goods are ex-ante perfect substitutes; in other words, the goods market is (nearly) Walrasian with $\gamma=0$. With $\gamma= 0$, firms make normal profits by charging simply their marginal production cost as in a textbook model. Clearly, then, it is the search friction embodied in $\gamma\theta ^{\xi}$ that drives a wedge between price and marginal production cost. Indeed, because $\gamma\theta^{\xi} \geq0$, $p \geq mc$. Figure 1 shows that the advertising condition is upward-sloping in $(p, \theta)$ space.¹⁸ To preview some of the intuition behind the dynamics of markups we present soon: because $\theta$ in general will vary over time as part of business cycle fluctuations, the markup of price over marginal production cost should be expected to be time-varying, and this markup is endogenous. Time-variation in search costs drives time-variation in markups in our framework.¹⁹

To determine the steady-state $(p,\theta)$, we must also examine (the steady-state version of) the pricing condition (24), which is the relevant pricing condition for both Nash and fair bargaining (because, recall again, both bargaining schemes deliver the same outcome when there is no cost of price adjustment, which is true in the steady-state of all our models). Again assuming $c=1$ and after also using the steady-state version of the advertising condition, we can express the steady-state pricing condition as

$$p = (1-\eta) \left( \frac{\tilde{u}(1)}{\lambda}\right) + \eta(mc - \gamma\theta),$$ (38)

where we have used the fact that in steady state, $\omega= \eta$. The first term on the right-hand-side of this expression, because it itself depends on $\theta$ in general equilibrium (because the marginal utility of wealth $\lambda$ is a general equilibrium object), makes analyzing this expression prohibitively more complicated than analyzing the steady-state advertising condition. To focus ideas, though, suppose it were simply a constant, $A$. In that case, we have the very simple pricing equation

$$p = (1-\eta) A + \eta(mc - \gamma\theta),$$ (39)

which states that the price lies inside an interval bounded above by the firm's production cost net of savings on search costs and bounded below by the household's utility of consumption $A$. The convex weights are simply the Nash bargaining weights.²⁰ The locus (39) is downward-sloping in $(p, \theta)$ space, as illustrated in Figure 1; the steady-state equilibrium $(p, \theta)$ is determined where it and the pricing condition cross in Figure 1.

Conditions (37) and (39) characterize $(p, \theta)$ in terms of deep parameters. It is fairly straightforward to show (we provide the details in Appendix D) that the steady-state price is strictly decreasing in customer bargaining power $\eta$. This result makes a lot of intuitive sense: the more bargaining power customers wield, the less rents the firm can extract out of the match surplus, meaning the smaller is the markup of unit price over unit marginal production cost.

So far, we have characterized the equilibrium pair $(p, \theta)$. To complete the description of the steady-state equilibrium core of our model, it remains to characterize $s$ (or, equivalently, $a$, because $\theta\equiv a/s$) for a given $(p,\theta)$. To do this, we begin by noting that in a steady-state equilibrium, the flow of individuals from search into customer relationships equals the flow of individuals from customer relationships (gone sour) back into search. Equating these flows, $k^{h}(\theta)s = \rho^{x} N = \rho^{x} (1-s)$.²¹ Rearranging,

$$s = \frac{\rho^{x}}{\rho^{x} + k^{h}(\theta)}.$$ (40)

Cobb-Douglas matching implies $k^{h}(\theta) = \theta k^{f}(\theta) = \theta^{1-\xi}$. In $(a,s)$ space, then, the flow condition (40), which characterizes activity on the household side of the goods market, defines the downward-sloping locus shown in Figure 2.

Finally, to describe equilibrium in $(a,s)$ space, we need a description of activity on the firm side of the goods market; such a description comes from combining (37) and (39). Specifically, solving (37) for $p$, substituting in (39), and using the implicit function theorem to compute the slope of $a$ with respect to $s$ (we again relegate the details to Appendix D), we can show that the resulting form of the advertising condition is the upward-sloping ray in $(a,s)$ space shown in Figure 2; the steady-state equilibrium $(a,s)$ is determined where it and the locus (40) cross in Figure 2. In combination, then, Figures 1 and 2 fully characterize the steady-state (partial) equilibrium triple $(p,\theta,a)$ that describes goods-market trade.

4.  Quantitative Results in Baseline Model

With these analytics in mind, we turn to characterizing the full deterministic steady state of our model numerically (i.e., fully endogenizing $\tilde{u} / \lambda$ by re-introducing elastic labor supply). We begin by describing the calibration we use.

4.1  Calibration

For instantaneous utility, we choose the common functional forms

$$u(x) = \log x,$$ (41)

$$v(y) = \log y,$$ (42)

and

$$g(z) = \frac{\zeta}{1-\nu} z^{1-\nu}.$$ (43)

The time unit of our model is meant to be one quarter, so we set $\beta= 0.99$, in line with an average annual real interest rate of about four percent. We fix the curvature parameter for the subutility function over leisure to $\nu= 0.4$. We set the preference parameter $\vartheta= 1$ as a baseline. With this baseline setting and given the rest of the calibration described below, the fraction of equilibrium total consumption that is comprised of consumption obtained through search is 43 percent. That is, $\vartheta= 1$ delivers $\frac{Nc}{Nc + x} = 0.43$, which does not seem unreasonable. Varying $\vartheta$ varies this share; in the limit, of course, $\vartheta= 0$ collapses our model to one in which all goods are exchanged via Walrasian trade.

As we mentioned earlier, we choose a Cobb-Douglas specification for the matching function,

$$m(s, a) = \psi s^{\xi_{s}} a^{1-\xi_{s}},$$ (44)

and set the elasticity to $\xi_{s} = 0.5$. We choose Cobb-Douglas because of its convenient properties -- in particular, the fact that matching probabilities depend only on goods-market tightness $\theta\equiv a/s$, as we already mentioned -- but recognize that it would be desirable to test how empirically useful the Cobb-Douglas description is for goods-market frictions.²² For the Nash bargaining weight $\eta$, we choose a middle-of-the-road calibration $\eta= 0.50$ for most of the results we report, but do vary it in some of our experiments. One virtue of setting $\eta= \xi_{s}$, well-known to search theorists, is that, as Hosios (1990) first showed in the context of labor search models, the underlying search equilibrium is socially efficient. We of course do not know if an efficient search equilibrium in the goods market is the best description of the data, but it seems useful as a starting point.

We calibrate a number of other parameters in the version of our model in which there are zero costs of price adjustment ($\kappa=0$). In this flexible-price version of our model, we set $\zeta= 4.3$ so that the household spends 30 percent of its time working (equivalently, the household sends 30 percent of its family members to work) and then hold this value fixed as we move to other versions of our model. We also calibrate $\gamma$, $\bar{c}$, and $\psi$, all of which we discuss in more detail immediately below, in the zero-menu-cost version of our model and hold the resulting values constant throughout all versions of our model.

Two novel properties of our model about which we can obtain some empirical evidence are the amount of time consumers spend shopping and firms' expenditures on advertising. According to the American Time Use Survey (ATUS), conducted annually by the U.S. Bureau of Labor Statistics, the average American consumer spends just under one hour per day shopping, which is roughly one-fourth as much time spent working.²³ The questionnaire that is the basis for the survey does not distinguish between, in the terminology of our model, "searching" for goods and "shopping" for goods. Thus, we calibrate our model so that, in the deterministic steady state, $(N + s)/l = 0.25$ and allow the model to endogenously determine $N$ and $s$ separately. Hitting this target requires setting the intensive quantity traded in a customer relationship to $\bar{c} = 1.4$.

Regarding advertising expenditures, we use data from the BLS and Universal McCann, an advertising agency that tracks and projects developments in the industry.²⁴ According to these data, total nominal advertising expenditures in the U.S. in 2005 were about $276 billion and are estimated to have been about $290 billion in 2006, putting the share of advertising in total nominal GDP around 2.25 percent. This figure strikes us as non-neglible: firm spend quite a lot attracting and retaining customers. Going back to 1950, this share has generally fluctuated within the range 2 percent to 2.5 percent. Because the notion of advertising in our model is likely a bit more general than activities typically expensed as advertising, we use the upper limit of 2.5 percent as our guidepost.²⁵ We thus calibrate $\gamma$ so that $\gamma a$ in our model is 2.5 percent of GDP in steady state. In Appendix F, we provide for interested researchers annual aggregate advertising data.

Unfortunately, neither the shopping data nor the advertising data gives us any guidance (at least not that we have been able to discern) about how to calibrate our model's matching and separation probabilities. Lacking solid evidence, we calibrate the matching function parameter $\psi$ in the flexible-price version of our model so that $k^{h} = 0.4$. The resulting value is $\psi= 0.45$, which we hold fixed as we move to other versions of our model. We simply set the parameter that governs the breakup of a customer relationship at $\rho^{x} = 0.10$, which states that a firm loses ten percent of its existing customers in any given period. Equivalently, this parameter setting means that a newly-formed customer-firm relationship is expected to last for $1/\rho^{x} = 10$ periods (quarters), which we think does not seem implausible.

We also face a problem in terms of calibrating the price-adjustment parameter $\kappa$. Ideally, we would like to set it so that in the deterministic steady state, resources devoted to price adjustment absorb some empirically-relevant portion of output. However, because all prices are real in our model, price-adjustment costs are always zero in any deterministic steady state, making $\kappa$ irrelevant for the steady state. Hence, we report results for several values of $\kappa$, corresponding to no, small, moderate, and large menu costs. We think this strategy is sufficient because the main results we wish to convey are conceptual rather than quantitative.

Finally, the exogenous TFP process follows an AR(1) in logs,

$$\log z_{t+1} = \rho_{z} \log z_{t} + \epsilon^{z}_{t+1},$$ (45)

with $\epsilon^{z} \sim iid N(0, \sigma_{z})$. We choose $\rho_{z} = 0.95$ and $\sigma_{z} = 0.007$ in keeping with the RBC literature -- see, for example, King and Rebelo (1999, p. 955). As our results show, this setting for the volatility of the shock to TFP delivers volatility of total output in our model of about 1.6 percent, in line with empirical evidence. Thus, the amplification of TFP shocks to GDP fluctuations in our model is no different than in a standard RBC model.

4.2  Steady State Numerical Results

Steady-state prices and quantities are reported in Table 1 for our baseline calibration. At our benchmark value $\eta= 0.50$, the markup in the search goods sector is just above eight percent, in line with empirical evidence and with the settings for product-market markups employed by many quantitative macroeconomic models.

The basic motivation of our work is that customer bargaining power may be important in pricing (and other) decisions. As such, one would want to know the predictions of our framework regarding key endogenous variables as customer bargaining power changes. Figure 4 illustrates how a number of steady-state variables vary with customer bargaining power $\eta$ regardless of whether or not there are menu costs of price adjustment. The results in Figure 4 are invariant to both the menu cost parameter $\kappa$ and the specific bargaining protocol because in the steady state, $\pi= 1$ (i.e., prices are unchanging) because the interesting price in our model is a real price -- in the steady state, real prices are of course constant.

As shown in Figure 4, an increase in consumer bargaining power depresses the bargained price (the top left panel), confirming our analytical results. Because marginal production cost is constant at unity by construction, the markup declines as $\eta$ rises, as well. A lower price leaves the household with a larger gain $\mathbf{M} - \mathbf{S}$ from forming a match (the top right panel), which also induces it to put more effort into search. The firm, on the other hand, loses from a rise in $\eta$, as the fall in $\mathbf{A}$ shows. The firm reduces its advertising because lower prices eat directly into profits, making customer relationships less valuable to it. With lower $a$ and higher $s$, market thickness $\theta$ $(\equiv a/s)$ unambiguously falls.

Regarding the flow of new relationships formed, the reduction in advertising expenditures dominates the increase in search activity, and the number of customer-firm relationships ($N$) falls. For our calibration, even though $s$ rises, the total amount of time that households spend engaged in shopping-related activities ($N+s$) declines. Households optimally reallocate this additional time between labor and leisure according to the consumption-leisure optimality condition (expression (6)), and it turns out that total labor declines, which, in turn, leads to lower total output.

The latter result highlights an interesting point of contrast with the standard Dixit-Stiglitz model of monopolistic competition. In the standard model, reducing the degree of firms' pricing power (in the form of a higher elasticity of demand for its output) pushes price closer to marginal cost, causing output to rise. Figure 4 shows that in our model, reducing firms' pricing power (here, in the form of lower bargaining power for the firm) causes households to devote less time to production activities and enjoy more leisure, causing output to decline.

The reason for the difference in the response of output as the markup of price over marginal cost falls is of course a direct result of the non-Walrasian features of our model. Advertising activities open up a supply-side channel missing in a standard model. When the firm has less to gain from trading in the non-Walrasian market, it simply chooses to reduce its activity there by cutting back on advertising. The resulting number of transactions in the non-Walrasian market declines. Shopping becomes less of a burden for households, who enjoy more leisure and devote less time to production activities.²⁶

4.3  Dynamics

To study dynamics, we approximate our model by linearizing in levels the equilibrium conditions of the model around the non-stochastic steady-state. Our numerical method is our own implementation of the perturbation algorithm described by Schmitt-Grohe and Uribe (2004b). We conduct 5000 simulations, each 100 periods long. For each simulation, we compute first and second moments and report the medians of these moments across the 5000 simulations. To make the comparisons meaningful as we vary model parameters, the same realizations for productivity shocks are used across versions of our model. Because by construction marginal production cost is always $mc_{t} = 1$ in our model, we will speak interchangeably about the dynamics of prices and markups. Depending on the issue at hand, one or the other language will typically be more convenient to use, which we think is clear in the ensuing discussion.

4.3.1.  Nash Bargaining with No Menu Costs

Table 2 presents simulation-based first and second moments for key variables under Nash bargaining and zero menu costs. Examining basic business cycle statistics reveals that the search goods market is a source of consumption smoothing, in the sense that the volatility of total consumption is lower than the volatility of GDP.²⁷ In our model, total consumption is $Nc + x$. The volatility of total search consumption $Nc$, at 1.13 percentage points, is noticeably lower than the volatility of Walrasian consumption $x$. In our baseline calibration, search consumption makes up 43 percent of total consumption in the steady-state $Nc + x$. The volatility of total consumption, at 1.55 percent (not shown in the table), is thus a bit lower than volatility of total output. The consumption smoothing, although obviously not as quantitatively strong as in an RBC model with capital, arises because active customer-firm relationships are a state variable in our environment, making part of total consumption pre-determined. At roughly 0.94% ( = 1.55/1.66), the relative volatility of aggregate consumption in our model is clearly not as low as that which obtains in an RBC model with capital (see, for example, Table 3 in King and Rebelo (1999, p. 957)). Thus, we are clearly not claiming that search consumption induces as quantitatively strong a smoothing effect as does capital. The fact that consumption smoothing can arise in a model with completely perishable goods and no capital, though, is novel.

Nevertheless, in terms of developing intuition for the basic dynamics of the model, we find it useful to think of the formation of customer-firm relationships as an investment decision for participants on both sides of the market. Households invest time searching for retailers, and firms invest advertising dollars in order to attract shoppers. A comparison of the volatility of advertising expenditures in our model to the volatility of investment in a standard RBC model with capital supports this analogy. We have more to say about the business cycle properties of advertising expenditures below.

The final point worth emphasizing is that even with flexible prices, our model delivers a time-varying markup, a result that is of some interest on its own. Our baseline model predicts that markups are procyclical, as the fifth column in Table 2 documents. However, most empirical evidence -- for example, Rotemberg and Woodford (1999) or, more recently, Jaimovich (2006) -- suggests that markups are countercyclical. We do not view this as a shortcoming of our model for two reasons. First, our basic goal was not to model markup behavior per se. Second, our baseline model features only TFP shocks. Our reading of the empirical evidence is that it is not even clear whether countercyclicality of markups observed in the data is due to demand shocks or supply shocks. In Section 6, we briefly extend our model to include demand shocks and demonstrate that our model is in fact capable of delivering countercyclical markups.

4.3.2  Nash Bargaining with Menu Costs

We now turn on menu costs. Table 3 presents simulation-based first and second moments for key variables under Nash bargaining and for several values of $\kappa$. Comparing results across models, one feature that stands out is that the volatility and correlation properties of the typical quantity variables GDP, (Walrasian) consumption $x$, and time spent working $l$ are virtually invariant to the size of the menu cost $\kappa$. The same is true of the correlation and volatility properties of time spent shopping $N$ and time spent searching $s$. Quantity variables as a whole are thus largely unaffected by the magnitude of menu costs. An exception is goods-market tightness