Pricing under Fairness Concerns
PPricing under Fairness Concerns
Erik Eyster, Kristof Madarasz, Pascal MichaillatMarch 2020
This paper proposes a theory of pricing based on two facts: customers care about thefairness of prices, and firms take these concerns into account when setting prices.The theory assumes that customers dislike unfair prices, namely those marked upsteeply over cost. Since costs are unobservable, customers must extract them fromprices. The theory assumes that customers infer less than rationally: when a pricerises after a cost increase, customers partially misattribute the higher price to ahigher markup—which they find unfair. Firms anticipate this response and trim theirprice increases, which drives the passthrough of costs into prices below one: pricesare somewhat rigid. Embedded in a New Keynesian model as a replacement for theusual pricing frictions, our theory produces monetary-policy nonneutrality: whenmonetary policy loosens and inflation rises, customers misperceive markups ashigher and feel unfairly treated; firms mitigate this perceived unfairness by reducingtheir markups; in general equilibrium, employment rises. The New Keynesian modelalso features a hybrid short-run Phillips curve, realistic impulse responses of outputand employment to monetary and technology shocks, and an upward-sloping long-run Phillips curve.
Eyster: London School of Economics & University of California–Santa Barbara. Madarasz: London School of Eco-nomics. Michaillat: Brown University. We thank George Akerlof, Roland Benabou, Daniel Benjamin, Joaquin Blaum,Arpita Chatterjee, Varanya Chaubey, Olivier Coibion, Stephane Dupraz, Gauti Eggertsson, John Friedman, XavierGabaix, Nicola Gennaioli, Yuriy Gorodnichenko, David Laibson, John Leahy, Bobak Pakzad-Hurson, Matthew Rabin,Ricardo Reis, David Romer, Emmanuel Saez, Klaus Schmidt, Jesse Shapiro, Andrei Shleifer, Joel Sobel, and SilvanaTenreyro for helpful discussions and comments. This work was supported by the Institute for Advanced Study. a r X i v : . [ ec on . T H ] M a r . Introduction Prices are neither exactly fixed nor fully responsive to cost shocks (Carlsson and Skans 2012; DeLoecker et al. 2016; Caselli, Chatterjee, and Woodland 2017; Ganapati, Shapiro, and Walker 2019).Such price rigidity is of first-order importance, as it determines the transmission of shocks andgovernment policies to the economy.Asked why they show such restraint when setting prices, firm managers explain that theyavoid alienating customers, who balk at paying prices that they regard as unfair (Blinder et al.1998). Yet theories of price rigidity almost never include fairness considerations (Blanchard 1990;Mankiw and Reis 2010). The notable exception is Rotemberg (2005), which calls attention to therole of fairness in pricing. Due to its innovative nature, however, the theory itself is somewhatdifficult to analyze or use in other frameworks (see section 2).This paper therefore develops a pricing theory that incorporates the fairness concerns ob-served among firms and their customers and uses such concerns to generate the price rigidityobserved in the data. The theory is designed to be easy to analyze, permitting closed-form ex-pressions for price markups and passthroughs, as well as a set of comparative statics. It is alsodesigned for easy transferal to other frameworks: here, we port it to a New Keynesian model tostudy its macroeconomic implications.The first element of our theory is that customers dislike paying prices marked up heavilyover marginal costs because they find these prices unfair, and that firms understand this. Thisassumption draws upon evidence from numerous surveys of consumers and firms, our ownsurvey of French bakers, and religious and legal texts (section 3). We formalize this assumption byweighting each unit of consumption in the utility function by a fairness factor that is a function ofthe markup that customers perceive firms to charge for the consumed good: the fairness factor isdecreasing in the perceived markup (since higher markups seem less fair) and concave (sincepeople tend to respond more strongly to increases in markups than to decreases).Because customers do not observe firms’ costs but need them to assess markups, their fairnessperceptions depend upon their cost estimates. The second element of our theory is that customersupdate their beliefs about marginal costs less than rationally. First, customers underinfer marginalcost from price: they form beliefs that depend upon some anchor, which may be their priorexpectation of marginal cost. Second, insofar as customers do update their beliefs about marginalcost from price, they engage in a form of proportional thinking by estimating marginal costs that Fairness has received more attention in other contexts: Akerlof (1982), Akerlof and Yellen (1990), and Benjamin(2015) add fairness to labor-market models; Rabin (1993), Fehr and Schmidt (1999), and Charness and Rabin (2002) togame-theoretic models; and Fehr, Klein, and Schmidt (2007) to contract-theoretic models. For surveys of the fairnessliterature, see Fehr and Gachter (2000), Sobel (2005), and Fehr, Goette, and Zehnder (2009). x %, customers correctly infer thatmarginal cost has increased by x %, and therefore that the markup has not changed. Since theprice change does not change the perceived markup, the price elasticity of demand does notchange, and neither does the markup.Once fairness concerns and subproportional inference are combined, however, pricingchanges. Demand decreases in price not only through the standard channel, but also througha fairness channel. When customers see a higher price, they attribute it partially to a highermarginal cost and partially to a higher markup—which they find unfair. Thus the higher pricelowers their marginal utility of consumption, which further decreases demand. This rendersdemand more elastic than it would be otherwise, leading the monopoly to lower its markup.Second, fairness concerns and subproportional inference create price rigidity. After a priceincrease spurred by a cost increase, customers underappreciate the increase in marginal cost andpartially misattribute the higher price to higher markup. Since the fairness factor is decreasingand concave in the perceived markup, it is more elastic at higher perceived markups. This propertytranslates to demand, which is more price elastic at higher perceived markups. Hence, afterthe cost increase, the monopoly reduces its markup. As a result, the price increases less thanproportionally to the underlying marginal cost: the passthrough of marginal costs into pricesfalls short of one. This mild form of price rigidity is consistent with the response of prices tomarginal-cost shocks estimated in empirical studies.Theories of price rigidity are central to macroeconomic models. To illustrate how our theorycan be embedded into such a model, and develop its implications, we substitute it for the usualpricing frictions in a New Keynesian model (section 5). Again we assume that customers inferless than they should about marginal cost from price. In the dynamic model, subproportionalinference means that in each period t , customers average their period- t −
2. Related literature
Rotemberg (2005) developed the first theory of price rigidity based on fairness considerations. Customers in his model care about firms’ altruism, which they re-evaluate following every pricechange. Customers buy a normal amount from a firm unless they can reject the hypothesis thatthe firm is altruistic, in which case they withhold all demand in order to lower the firm’s profits.Firms react to the discontinuity in demand by refraining from passing on small cost increases,creating price stickiness.We depart from Rotemberg’s discontinuous, buy-normally-or-buy-nothing formulation to onein which customers continuously reduce demand as the unfairness of the transaction increases.Our continuous formulation seems more realistic and offers greater tractability. Its tractabilityallows us to obtain closed-form expressions for the markup and passthrough, and thus to performa range of comparative-statics exercises. The tractability also allows us to embed our pricing Rotemberg (2011) explores other implications of fairness for pricing, such as price discrimination.
3. Microevidence supporting the assumptions
This section presents microevidence in support of the assumptions underlying our theory. First,we show that people care about the fairness of prices, and that they assess a price to be fair whenit carries a low markup over marginal cost. Second, we document that people erroneously infermarginal costs from prices and thus misperceive markups. Finally, we show that firms accountfor customers’ fairness concerns when they set prices.
Our theory assumes that customers deem a price to be unfair when it entails a high markupover marginal cost, and that they dislike such prices. Here we review evidence supporting thisassumption.
Price increases due to higher demand.
Our assumption implies that people will find price in-creases unjustified by cost increases to be unfair. In a survey of Canadian residents, Kahneman,Knetsch, and Thaler (1986, p. 729) document this pattern. They describe the following situation:“A hardware store has been selling snow shovels for $15. The morning after a large snowstorm, thestore raises the price to $20.” Among 107 respondents, only 18% regard this pricing as acceptable,whereas 82% regard it as unfair.Subsequent studies confirm and refine Kahneman, Knetsch, and Thaler’s results. For example,in a survey of 1,750 households in Switzerland and Germany, Frey and Pommerehne (1993, pp. 297–298) confirm that customers dislike a price increase that involves an increase in markup; so too do4hiller, Boycko, and Korobov (1991, p. 389) in a comparative survey of 391 respondents in Russiaand 361 in the United States.One concern about the snow-shovel evidence is that people may find the price increase unfairsimply because it occurs during a period of hardship. To address this question, Maxwell (1995)asks 72 students at a Florida university about price increases following an ordinary increase indemand as well as those following a hardship-driven increase in demand. While more find priceincreases in the hardship environment unfair (86% versus 69%), a substantial majority in eachcase perceive the price increase as unfair.
Price increases due to higher costs.
Conversely, our fairness assumption suggests that customerstolerate price increases following cost increases so long as the markup remains constant. Kah-neman, Knetsch, and Thaler (1986, pp. 732–733) identify this pattern: “Suppose that, due to atransportation mixup, there is a local shortage of lettuce and the wholesale price has increased.A local grocer has bought the usual quantity of lettuce at a price that is 30 cents per head higherthan normal. The grocer raises the price of lettuce to customers by 30 cents per head.” Among 101respondents, 79% regard the pricing as acceptable, and only 21% find it unfair. In a survey of 307Dutch individuals, Gielissen, Dutilh, and Graafland (2008, table 2) also find that price increasesfollowing cost increases are fair, while those following demand increases are not.
Price decreases allowed by lower costs.
Our assumption equally implies that it is unfair for firmsnot to pass along cost decreases. Kahneman, Knetsch, and Thaler (1986, p. 734) find mildersupport for this reaction. They describe the following situation: “A small factory produces tablesand sells all that it can make at $200 each. Because of changes in the price of materials, the cost ofmaking each table has recently decreased by $20. The factory does not change its price of tables.”Only 47% of respondents find this unfair, despite the elevated markup.Subsequent studies, however, find that people do expect the price to fall after a cost reduction.Kalapurakal, Dickson, and Urbany (1991) conduct a survey of 189 business students in the UnitedStates, and asked them to consider the following scenario: “A department store has been buyingan oriental floor rug for $100. The standard pricing practice used by department stores is to pricefloor rugs at double their cost so the selling price of the rug is $200. This covers all the sellingcosts, overheads and includes profit. The department store can sell all of the rugs that it can buy.Suppose because of exchange rate changes the cost of the rug rises from $100 to $120 and theselling price is increased to $220. As a result of another change in currency exchange rates, thecost of the rug falls by $20 back to $100.” Then two alternative scenarios were evaluated: “Thedepartment store continues to sell the rug for $220” compared to “The department store reduces5he price of the rug to $200.” The scenario in which the department store reduces the price inresponse to the decrease in cost was considered significantly more fair: the fairness rating was + . − . − + Norms about markups.
Religious and legal texts written over the ages display a long history ofnorms regarding markups—which suggests that people deeply care about markups. For example,Talmudic law specifies that the highest fair and allowable markup when trading essential itemsis 20% of the production cost, or one-sixth of the final price (Friedman 1984, p. 198).Another example comes from 18th-century France, where local authorities fixed bread pricesby publishing “fair” prices in official decrees. In the city of Rouen, for instance, the official breadprices took the costs of grain, rent, milling, wood, and labor into account, and granted a “modestprofit” to the baker (Miller 1999, p. 36). Thus, officials fixed the markup that bakers could charge.Even today, French bakers attach such importance to convincing their customers of fair markupsthat their trade union decomposes the cost of bread and the rationale for any price rise intominute detail (https://perma.cc/GQ28-JMFC).Two more examples come from regulation in the United States. First, return-on-cost regulationfor public utilities, which limits the markups charged by utilities, has been justified not onlyon efficiency grounds but also on fairness grounds (Zajac 1985; Jones and Mann 2001). Second,most US states have anti-price-gouging legislation that limits the prices that firms can chargein periods of upheaval (such as a hurricane or an epidemic). But by exempting price increasesjustified by higher costs, the legislation only outlaws price increases caused by higher markups(Rotemberg 2009, pp. 74–77).
Fairness and willingness to pay.
We assume that customers who purchase a good at an unfairprice derive less utility from consuming it; as a result, unfair pricing reduces willingness to pay.Substantial evidence documents that unfair prices make customers angry, and more generallythat unfair outcomes trigger feelings of anger (Rotemberg 2009, pp. 60–64). A small body ofevidence also suggests that customers reduce purchases when they feel unfairly treated. In atelephone survey of 40 US consumers, Urbany, Madden, and Dickson (1989) explore—by lookingat a 25-cent ATM surcharge—whether a price increase justified by a cost increase is perceived asmore fair than an unjustified one, and whether fairness perceptions affect customers’ propensityto buy. While 58% of respondents judge the introduction of the surcharge fair when justifiedby a cost increase, only 29% judge it fair when not justified (table 1, panel B). Moreover, those6eople who find the surcharge unfair are indeed more likely to switch banks (52% versus 35%, seetable 1, panel C). Similarly, Piron and Fernandez (1995) present survey and laboratory evidencethat customers who find a firm’s actions unfair tend to reduce their purchases with that firm.
Customers do not observe firms’ marginal costs. Consequently, their perception of the fairnessof firms’ prices depends upon their estimates of these costs. Since customers cannot easilylearn about hidden costs, however, they are prone to develop mistaken beliefs. To describe suchmisperceptions, we assume subproportional inference. First, consumers underinfer marginalcost from price: they form beliefs that depend upon some anchor. Second, insofar as consumersdo update their beliefs about cost from price, they engage in a form of proportional thinking byestimating marginal costs that are proportional to price. We now review evidence that supportsthis pair of assumptions.
Underinference in general.
Numerous experimental studies establish that people underinferother people’s information from those other people’s actions (Eyster 2019). In the context ofbilateral bargaining with asymmetric information, Samuelson and Bazerman (1985), Holt andSherman (1994), Carillo and Palfrey (2011), and others show that bargainers underappreciate theadverse selection in trade. The papers collected in Kagel and Levin (2002) present evidence thatbidders underattend to the winner’s curse in common-value auctions. In a metastudy of social-learning experiments, Weizsacker (2010) finds that subjects behave as if they underinfer theirpredecessors’ private information from their actions. In a voting experiment, Esponda and Vespa(2014) show that people underinfer others’ private information from their votes. Subproportionalinference encompasses such underinference.
Underinference from prices.
Shafir, Diamond, and Tversky (1997) report survey evidence thatpoints at underinference in the context of pricing. They presented 362 people in New Jersey withthe following thought experiment: “Changes in the economy often have an effect on people’sfinancial decisions. Imagine that the US experienced unusually high inflation which affected allsectors of the economy. Imagine that within a six-month period all benefits and salaries, as wellas the prices of all goods and services, went up by approximately 25%. You now earn and spend25% more than before. Six months ago, you were planning to buy a leather armchair whose priceduring the 6-month period went up from $400 to $500. Would you be more or less likely to buythe armchair now?” The higher prices were distinctly aversive: while 55% of respondents were aslikely to buy as before and 7% were more likely to buy as before, 38% of respondents were less7 able 1.
Opinions about price movements in Japan, 2001–2017 likely to buy then before (p. 355). Our model makes this prediction. While consumers who updatesubproportionally recognize that higher prices signal higher marginal costs, they stop short ofrational inference. Consequently, consumers perceive markups to be higher when prices arehigher. These consumers deem today’s transaction less fair, so they have a lower willingness topay for the armchair.A survey conducted by Shiller (1997) confirms that when consumers see higher prices, theysystematically believe that markups are higher. Among 120 respondents in the United States,85% report that they dislike inflation because when they “go to the store and see that prices arehigher,” they “feel a little angry at someone” (p. 21). The most common perceived culprits are“manufacturers,” “store owners,” and “businesses,” whose transgressions include “greed” and“corporate profits” (p. 25). In the presence of higher prices, many people indeed infer that firmshave increased their profit margins, which angers them.
Underinference from inflation and deflation.
In our model, customers dislike inflation becauseit leads them to perceive higher markups; conversely, they enjoy deflation because it leads themto perceive lower markups. An opinion poll conducted by the Bank of Japan between 2001 and8 able 2.
Description of firm surveys about pricing
Survey Country Period Number of firmsBlinder et al. (1998) United States (US) 1990–1992 200Hall, Walsh, and Yates (2000) United Kingdom (GB) 1995 654Apel, Friberg, and Hallsten (2005) Sweden (SE) 2000 626Nakagawa, Hattori, and Takagawa (2000) Japan (JP) 2000 630Amirault, Kwan, and Wilkinson (2006) Canada (CA) 2002–2003 170Kwapil, Baumgartner, and Scharler (2005) Austria (AT) 2004 873Aucremanne and Druant (2005) Belgium (BE) 2004 1,979Loupias and Ricart (2004) France (FR) 2004 1,662Lunnemann and Matha (2006) Luxembourg (LU) 2004 367Hoeberichts and Stokman (2006) Netherlands (NL) 2004 1,246Martins (2005) Portugal (PT) 2004 1,370Alvarez and Hernando (2005) Spain (ES) 2004 2,008Langbraaten, Nordbo, and Wulfsberg (2008) Norway (NO) 2007 725Olafsson, Petursdottir, and Vignisdottir (2011) Iceland (IS) 2008 262
Proportional thinking.
Finally, a small body of evidence documents that people think proportion-ally, even in settings that do not call for proportional thinking (Bushong, Rabin, and Schwartzstein2017). In particular, Thaler (1980) and Tversky and Kahneman (1981) demonstrate that people’swillingness to invest time in lowering the price of a good by a fixed dollar amount dependsnegatively upon the good’s price. Rather than care about the absolute savings, people appear tocare about the proportional savings. Someone who thinks about a price discount not in absoluteterms but as a proportion of the purchase price may think about marginal cost not in absoluteterms but rather as a percentage of price. If so, then the simplest assumption is that, insofar asthe person infers marginal cost from price, she infers a marginal cost proportional to price.
In our model, in response to their customers’ fairness concerns, firms pay great attention tofairness when setting prices. This seems to hold true in the real world: firms identify fairness tobe a major concern in price-setting. 9 urveys of firms.
Following Blinder et al. (1998), researchers have surveyed more than 12,000firms across developed economies about their pricing practices (table 2). The typical study asksmanagers to evaluate the relevance of different pricing theories from the economics literature toexplain their own pricing, in particular price rigidity. Amongst the theories that the managersdeem most important, some version of fairness invariably appears, often called “implicit con-tracts” and described as follows: “firms tacitly agree to stabilize prices, perhaps out of fairness tocustomers.” Indeed, fairness appeals to firms more than any other theory, with a median rankof 1 and a mean rank of 1.9 (table 3). The second most popular explanation for price rigiditytakes the form of nominal contracts—prices do not change because they are fixed by contracts: ithas a median rank of 3 and a mean rank of 2.6. Two common macroeconomic theories of pricerigidity—menu costs and information delays—do not resonate at all with firms, who rank themamongst the least popular theories, with mean and median ranks above 9.Firms also understand that customers bristle at unfair markups. According to Blinder et al.(1998, pp. 153–157), 64% of firms say that customers do not tolerate price increases after demandincreases, while 71% of firms say that customers do tolerate price increase after cost increases.Firms seem to agree that the norm for fair pricing revolves around a constant markup overmarginal cost. Based on a survey of businessmen in the United Kingdom, Hall and Hitch (1939,p. 19) report that the “the ‘right’ price, the one which ‘ought’ to be charged” is widely perceived tobe a markup (generally, 10%) over average cost. Okun (1975, p. 362) also observes in discussionswith business people that “empirically, the typical standard of fairness involves cost-orientedpricing with a markup.”
Survey of French bakers.
To better understand how firms incorporate fairness into their pricingdecisions, we interviewed 31 bakers in France in 2007. The French bread market makes a goodcase study because the market is large, bakers set their prices freely, and French people careenormously about bread. We sampled bakeries in cities and villages around Grenoble, Aix-en-Provence, Paimpol, and Paris. The interviews show that bakers are guided by norms of fairnesswhen they adjust prices in order to preserve customer loyalty. In particular, cost-based pricingis widely used. Bakers raise the price of bread only in response to increases in the cost of flour, In 2005, bakeries employed 148,000 workers, for a yearly turnover of 3.2 billion euros (Fraichard 2006). Since1978, French bakers have been free to set their own prices, except during the inflationary period 1979–1987 whenprice ceilings and growth caps were imposed. For centuries, bread prices caused major social upheaval in France.Miller (1999, p. 35) explains that before the French Revolution, “affordable bread prices underlay any hopes forurban tranquility.” During the Flour War of 1775, mobs chanted “if the price of bread does not go down, we willexterminate the king and the blood of the Bourbons”; following these riots, “under intense pressure from irate andnervous demonstrators, the young governor of Versailles had ceded and fixed the price ‘in the King’s name’ at twosous per pound, the mythohistoric just price inscribed in the memory of the century” (Kaplan 1996, p. 12). a b l e . R a n k i n g o f p r i c i n g t h e o r i e s i n fi r m s u r v e y s C o u n t r y o f s u r v e y O v e r a ll r a n k T h e o r y U S G B S E J P C AA T B E F R L U N L P T E S N O I S M e d i a n M e a n I m p li c i t c o n t r a c t s . N o m i n a l c o n t r a c t s . C oo r d i n a t i o n f a il u r e . . P r i c i n g p o i n t s – . M e n u c o s t s . I n f o r m a t i o n d e l a y s – – – – – . S u r v e y r e s p o n d e n t s r a t e d t h e r e l e v a n c e o f s e v e r a l p r i c i n g t h e o r i e s t o e x p l a i n p r i c e r i g i d i t y a tt h e i r o w n fi r m . T h e t a b l e r a n k s c o mm o n t h e o r i e s a m o n g s tt h e a l t e r n a t i v e s . B li n d e r e t a l . ( , t a b l e . ) d e s c r i b e s t h e t h e o r i e s a s f o ll o w s ( w i t h w o r d i n g v a r y i n g s li g h t l y a c r o sss u r v e y s ) : “ i m p li c i t c o n t r a c t s ” s t a n d s f o r “ fi r m s t a c i t l y a g r ee t o s t a b ili z e p r i c e s , p e r h a p s o u t o ff a i r n e ss t o c u s t o m e r s ” ; “ n o m i n a l c o n t r a c t s ” s t a n d s f o r “ p r i c e s a r e fi x e d b y c o n t r a c t s ” ; “ c oo r d i n a t i o n f a il u r e ” s t a n d s f o r t w o c l o s e l y r e l a t e d t h e o r i e s , w h i c h a r e i n v e s t i g a t e d i n s e p a r a t e s u r v e y s : “ fi r m s h o l d b a c k o n p r i c e c h a n g e s , w a i t i n g f o r o t h e r fi r m s t o g o fi r s t ” a n d “ t h e p r i c e i s s t i c k y b e c a u s e t h e c o m p a n y l o s e s m a n y c u s t o m e r s w h e n i t i s r a i s e d , b u t g a i n s o n l y a f e w n e w o n e s w h e n t h e p r i c e i s r e d u c e d ” ( w h i c h i s l a b e l e d “ k i n k e dd e m a n d c u r v e ” ) ; “ p r i c i n g p o i n t s ” s t a n d s f o r “ c e r t a i n p r i c e s ( li k e $ . ) h a v e s p e c i a l p s y c h o l o g i c a l s i g n i fi c a n c e ” ; “ m e n u c o s t s ” s t a n d s f o r “ fi r m s i n c u r c o s t s o f c h a n g i n g p r i c e s ” ; “ i n f o r m a t i o n d e l a y s ” s t a n d s f o r t w o c l o s e l y r e l a t e d t h e o r i e s , w h i c h a r e i n v e s t i g a t e d i n s e p a r a t e s u r v e y s : “ h i e r a r c h i c a l d e l a y ss l o w d o w n d e c i s i o n s ” a n d “ t h e i n f o r m a t i o n u s e d t o r e v i e w p r i c e s i s a v a il a b l e i n f r e q u e n t l y . ” T h e r a n k i n g s o f t h e t h e o r i e s a r e r e p o r t e d i n t a b l e . i n B li n d e r e t a l . ( ) ; t a b l e i n H a ll , W a l s h , a n d Y a t e s ( ) ; t a b l e i n A p e l , F r i b e r g , a n d H a ll s t e n ( ) ; c h a r t i n N a k a g a w a , H a tt o r i , a n d T a k a g a w a ( ) ; t a b l e i n A m i r a u l t , K w a n , a n d W il k i n s o n ( ) ; t a b l e i n K w a p il , B a u m g a r t n e r , a n d S c h a r l e r ( ) ; t a b l e i n A u c r e m a nn e a n d D r u a n t ( ) ; t a b l e . i n L o up i a s a n d R i c a r t ( ) ; t a b l e i n L u nn e m a nn a n d M a t h a ( ) ; t a b l e i n H o e b e r i c h t s a n d S t o k m a n ( ) ; t a b l e i n M a r t i n s ( ) ; t a b l e i n A l v a r e z a n d H e r n a n d o ( ) ; c h a r t i n L a n g b r aa t e n , N o r d b o , a n d W u l f s b e r g ( ) ; a n d t a b l e i n O l a f ss o n , P e t u r s d o tt i r , a n d V i g n i s d o tt i r ( ) .
4. Monopoly model
We extend a simple model of monopoly pricing to include fairness concerns and subproportionalinference, along the lines described in section 3. In this extended model, the markup charged bythe monopoly is lower. Furthermore, the markup responds to marginal-cost shocks, generatingsome price rigidity: prices are not completely fixed, but they respond less than one-for-one tomarginal costs.
A monopoly sells a good to a representative customer. The monopoly cannot price-discriminate,so each unit of good sells at the same price P . The customer cares about fairness and appraisestransactional fairness by assessing the markup charged by the monopoly. Since the customerdoes not observe the marginal cost of production, she needs to infer it from the price. We assumethat the perceived marginal cost at price P is given by a belief function C p ( P ) . For simplicity, werestrict C p ( P ) to be deterministic. Having inferred the marginal cost, the customer deduces thatthe markup charged by the monopoly is M p ( P ) = PC p ( P ) . The perceived markup determines the fairness of the transaction through a fairness function F ( M p ) >
0. Both functions C p ( P ) and F ( M p ) are assumed to be twice differentiable.A customer who buys a quantity Y of the good at price P experiences the fairness-adjustedconsumption Z = F ( M p ( P )) · Y . The customer also faces a budget constraint: P · Y + B = W , where W > B designates remaining money balances. Fairness-12djusted consumption and money balances enter a quasilinear utility function ϵϵ − · Z ( ϵ − )/ ϵ + B , where the parameter ϵ > F and price P , the customer chooses purchases Y and money balances B to maximize utility subjectto the budget constraint.Finally, the monopoly has constant marginal cost C >
0. It chooses price P and output Y tomaximize profits ( P − C ) · Y subject to customers’ demand for its good. We begin by determining customers’ demand for the monopoly good. The customer choosespurchases Y to maximize utility ϵϵ − ( F · Y ) ( ϵ − )/ ϵ + W − P · Y . The customer’s utility function is strictly concave so the following first-order condition gives itsglobal maximum: F ( ϵ − )/ ϵ · Y − / ϵ = P . This first-order condition yields the demand curve(1) Y d ( P ) = P − ϵ · F ( M p ( P )) ϵ − . The price affects demand through two channels: the typical substitution effect, captured by P − ϵ ;and the fairness channel, captured by F ( M p ( P )) ϵ − . The fairness channel appears because theprice influences the perceived markup and thus the fairness of the transaction; this in turn affectsthe marginal utility of consumption and demand.We turn to the monopoly’s pricing. The monopoly chooses price P to maximize profits ( P − C ) · Y d ( P ) . The first-order condition is Y d ( P ) + ( P − C ) dY d dP = . We introduce the price elasticity of demand, normalized to be positive: E = − d ln (cid:0) Y d (cid:1) d ln ( P ) = − PY d · dY d dP . P = EE − · C ;that is, the monopoly optimally sets its price at a markup M = E /( E − ) over marginal cost. To learn more about the monopoly’s markup, we compute the elasticity E . Using (1), we find(2) E = ϵ + ( ϵ − ) · ϕ · (cid:20) − d ln ( C p ) d ln ( P ) (cid:21) , where ϕ = − d ln ( F )/ d ln ( M p ) is the elasticity of the fairness function with respect to the perceivedmarkup, normalized to be positive. The first term, ϵ , describes the standard substitution effect.The second term, ( ϵ − ) · ϕ · [ − d ln ( C p )/ d ln ( P )] , represents the fairness channel and splits intotwo subterms. The first subterm, ( ϵ − ) · ϕ , appears because a higher price mechanically raises theperceived markup and thus reduces fairness. The second subterm, −( ϵ − ) · ϕ · [ d ln ( C p )/ d ln ( P )] ,appears because a higher price conveys information about the marginal cost and thus influencesperceived markup and fairness. We now use (2) to compute the markup in various situations. Before studying the more realistic case with fairness concerns, we examine the benchmark casein which customers do not care about fairness.
Definition 1.
Customers who do not care about fairness have a fairness function F ( M p ) = . Without fairness concerns, the fairness function is constant, so its elasticity is ϕ =
0. Ac-cording to (2), the price elasticity of demand is therefore constant: E = ϵ . This implies that theoptimal markup for the monopoly takes a standard value of ϵ /( ϵ − ) .Since the markup is independent of costs, changes in marginal cost are fully passed throughinto the price; that is, prices are flexible. Formally, the cost passthrough is β = d ln ( P ) d ln ( C ) , which measures the percentage change in price when the marginal cost increases by 1%. Thepassthrough takes the value of one because P = ϵ · C /( ϵ − ) .The following lemma summarizes the results: At this stage, we cannot guarantee that the first-order condition identifies the maximum of the monopoly’s profitfunction. But in appendix A.1, we use the assumptions introduced in the next sections to verify that in all cases thefirst-order condition indeed gives the maximum of the profit function. emma 1. When customers do not care about fairness, the monopoly sets the markup to M = ϵ /( ϵ − ) , and the cost passthrough is β = . We now introduce fairness concerns. As a preliminary step to the analysis with unobservablecosts, we explore pricing when costs are observable.To describe fairness concerns, we impose some structure on the fairness function.
Definition 2.
Customers who care about fairness have a fairness function F ( M p ) that is positive,strictly decreasing, and weakly concave on [ , M h ] , where F ( M h ) = and M h > ϵ /( ϵ − ) . The assumption that the fairness function strictly decreases in the perceived markup capturesthe pattern that customers find higher markups less fair and resent unfair transactions. Theassumption that the fairness function is weakly concave means that an increase in perceivedmarkup causes a utility loss of equal magnitude (if F is linear) or of greater magnitude (if F isstrictly concave) than the utility gain caused by an equal-sized decrease in perceived markup.We could not find evidence on this assumption, but it seems natural that people are at least asoutraged over a price increase as they are happy about a price decrease of the same magnitude.The assumptions about the fairness function lead to the following properties: Lemma 2.
When customers care about fairness, the elasticity of the fairness function ϕ ( M p ) = − d ln ( F ) d ln ( M p ) is strictly positive and strictly increasing on ( , M h ) , with lim M p → ϕ ( M p ) = and lim M p → M h ϕ ( M p ) =+ ∞ . As an implication, the superelasticity of the fairness function σ = d ln ( ϕ ) d ln ( M p ) is strictly positive on ( , M h ) . Proof . By definition, ϕ ( M p ) = − M p · F (cid:48) ( M p )/ F ( M p ) . Using the properties of the fairness functionlisted in definition 2, F ( M p ) > F (cid:48) ( M p ) <
0, so ϕ ( M p ) >
0. The properties also indicate that F > M p , and that F (cid:48) < M p (as F is concave in M p ). Thus,both 1 / F > − F (cid:48) > M p , which implies that ϕ is strictly increasing in M p .The properties also indicate that F ( ) > F (cid:48) ( ) is finite, so lim M p → ϕ ( M p ) =
0. Last, theproperties indicate that F ( M h ) = M h > F (cid:48) ( M h ) <
0, so that lim M p → M h ϕ ( M p ) = + ∞ .The final result immediately follows, as σ = M p · ϕ (cid:48) ( M p )/ ϕ ( M p ) , ϕ (cid:48) ( M p ) >
0, and ϕ ( M p ) > (cid:4)
15 key property in the lemma is that the superelasticity of the fairness function is positive—meaning that the fairness function is more elastic at higher perceived markups. This propertyfollows from the assumptions in definition 2 because a positive, decreasing, and weakly concavefunction always has positive superelasticity. It will play a central role in the analysis. Since the marginal cost is assumed to be observable, customers correctly perceive marginalcost ( C p = C ), so the perceived markup equals the true markup ( M p = M ). From (2), we see thatthe price elasticity of demand is E = ϵ + ( ϵ − ) · ϕ ( M ) > ϵ ; therefore, the markup charged bythe monopoly satisfies(3) M = + ϵ − · + ϕ ( M ) . Since ϕ ( M ) is strictly increasing from 0 to + ∞ when M increases from 0 to M h (lemma 2), theright-hand side of the equation is strictly decreasing from ϵ /( ϵ − ) to 1 when M increases from 0to M h > ϵ /( ϵ − ) >
1. We infer that the fixed-point equation (3) admits a unique solution, locatedbetween 1 and ϵ /( ϵ − M is well-defined and M ∈ ( , ϵ /( ϵ − )) .The next lemma records the results: Lemma 3.
When customers care about fairness and observe costs, the monopoly’s markup M isimplicitly defined by (3) . This implies that M ∈ ( , ϵ /( ϵ − )) and the cost passthrough is β = .Hence, the markup is lower than without fairness concerns, but the passthrough is identical. Without fairness concerns, the price affects demand solely through customers’ budget sets.With fairness concerns and observable marginal costs, the price also influences the perceivedfairness of the transaction: when the price is high relative to marginal cost, customers deemthe transaction to be less fair, which reduces the marginal utility from consuming the good andhence demand. Consequently, the monopoly’s demand is more price elastic than without fairnessconcerns, which forces the monopoly to charge a lower markup.However, (3) shows that with fairness concerns and observable costs, the markup does notdepend on costs, as in the absence of fairness concerns. Since changes in marginal cost do notaffect the markup, they are completely passed through into price: prices remain flexible. The concavity of the fairness function is not a necessary condition for any of the results in the paper. Thenecessary conditions are that the fairness function is decreasing in the perceived markup, and that its elasticity isincreasing in the perceived markup. The elasticity is increasing with weakly concave functions but also with other,not-too-convex functions. For example, the logistic function F ( M p ) = /[ + ( M p ) θ ] with θ > ϕ ( M p ) = − d ln ( F )/ d ln ( M p ) = θ /[ + ( M p ) − θ ] . All the results wouldcarry over with such logistic fairness function. We limit ourselves to concave functions instead of allowing for anyfunction with an increasing elasticity because we find such restriction more natural (it is a natural extension of alinear function) and easier to interpret. .5. Fairness concerns and rational inference of costs Next, we combine fairness concerns with unobservable marginal costs. We study a last prelimi-nary case: we assume that customers rationally invert the price to uncover the hidden marginalcost. In this case, the model takes the form of a simple signaling game in which the monopolylearns its marginal cost and chooses a price, before customers observe the monopoly’s price—butnot its marginal cost—and formulate demand. Let [ , C h ] ⊂ (cid:82) + be the set of all possible marginalcosts for the monopoly. The monopoly knows its marginal cost C ∈ [ , C h ] , but customers do not;instead, customers have non-atomistic prior beliefs over [ , C h ] .A pure-strategy perfect Bayesian equilibrium (PBE) of this game comprises three elements:a pure strategy for the monopolist, which is a mapping P : [ , C h ] → (cid:82) + that selects a pricefor every possible value of marginal cost; a belief function for customers, which is a mapping C p : (cid:82) + → [ , C h ] that determines a marginal cost for every possible price; and a pure strategyfor customers, which is a mapping Y d : (cid:82) + → (cid:82) + that selects a quantity purchased for everypossible price. We look for a PBE that is fully separating: the monopoly chooses different prices for differentmarginal costs, which allows a rational customer who knows the monopoly’s equilibrium strategyand observes the price to deduce marginal cost.We now demonstrate the existence of a PBE in which the monopolist uses the strategy P ( C ) = ϵ · C /( ϵ − ) ; customers believe C p ( P ) = ( ϵ − ) P / ϵ if P ∈ P ≡ (cid:2) , ϵC h /( ϵ − ) (cid:3) , and C p ( P ) = Y d ( P ) = P − ϵ · F ( P / C p ( P )) ϵ − . In such a PBE, customerscorrectly infer marginal costs from prices on the equilibrium path ( P ∈ P ), and they infer theworst when they observe a price off the equilibrium path ( P (cid:60) P )—namely that the firm has zeromarginal cost and thus infinitely high markup.The argument proceeds in three steps. First, given their beliefs, customers’ strategy is indeedoptimal, as we have shown in (1). Second, given the monopolist’s strategy, customers’ beliefsare indeed correct for any price on the equilibrium path. Third, given customers’ beliefs andstrategy, the monopolist’s strategy is optimal. Indeed, given customers’ beliefs for P ∈ P , we have d ln ( C p )/ d ln ( P ) =
1. Then, according to (2) (which is implied by customers’ strategy), the priceelasticity of demand for any price on P is E = ϵ . Hence, it is optimal for the monopolist to chargea price P = ϵC /( ϵ − ) . It remains to show that the monopoly has no incentive to charge someprice not belonging to P . This is straightforward: if it does, customers infer that the markup isinfinite, which brings fairness factor, demand, and thus profits to zero. Thus, the monopolist hasno incentive to deviate from the equilibrium markup ϵ /( ϵ − ) , regardless of its marginal cost. Strictly speaking, C p should allow the consumer to hold probabilistic beliefs about the firm’s marginal cost givenprice, but we sidestep this subtlety because it does not affect our analysis. Lemma 4.
When customers care about fairness and rationally infer costs, there is a PBE in whichthe monopoly uses the markup M = ϵ /( ϵ − ) , and customers learn marginal cost from price. Inthis PBE, the cost passthrough is β = . Hence, in this PBE, the markup and passthrough are thesame as without fairness concerns. The lemma shows that when customers care about fairness and rationally infer costs, there isa PBE in which fairness does not play a role. The intuition is the following. Without fairness con-cerns, the price affects demand only by changing customers’ budget sets. With fairness concerns,the price affects demand through a second channel, by changing the perceived markup. In thisequilibrium, however, after observing any price chosen by the monopoly, rational customersperceive the same markup ϵ /( ϵ − ) . The second channel closes, so the monopoly indeed setsthe standard markup ϵ /( ϵ − ) . Since the markup does not depend on marginal cost, changes inmarginal cost are fully passed through into prices—prices are flexible again.Of course, there may exist other equilibria beside the one described in lemma 4. An exampleof a pooling PBE is one in which all types of the firm charge the same price P > C h , and consumersbelieve that a firm who prices otherwise has zero marginal cost. However, this and other non-fully-separating PBEs fail standard signaling refinements. Because the linear PBE in lemma 4 isso simple and robust, it is more plausible than any alternative, which suggests that fairness isunlikely to matter when customers rationally infer costs.
We turn to the case of interest: customers care about fairness and subproportionally infer costsfrom prices. In this case, the fairness function satisfies definition 2, and the belief function takesthe following form:
Definition 3.
Customers who update subproportionally use the belief-updating rule (4) C p ( P ) = (cid:16) C b (cid:17) γ (cid:18) ϵ − ϵ P (cid:19) − γ , Only a separating PBE satisfies the D1 Criterion from Cho and Kreps (1987). Intuitively, consumers ought tointerpret a price P (cid:48) > P as coming from type C = C h rather than type C =
0, which undermines the poolingequilibrium. Indeed, if consumers demand no less at P (cid:48) than in equilibrium, then all types of firm benefit fromdeviating; if consumers demand less at P (cid:48) than in equilibrium, then the highest-cost firm strictly benefits wheneverany other type of firm weakly benefits. On these grounds, the D1 Criterion suggests that consumers should interpret P (cid:48) > P as coming from the highest marginal-cost firm. here C b > ( ϵ − ) · ( M h ) − / γ · C / ϵ is a prior point belief about marginal cost, and γ ∈ ( , ] governs the extent to which beliefs anchor on that prior belief. We have seen evidence that people do not sufficiently introspect about the relationshipbetween price and marginal cost, which leads them to underinfer the information conveyedby the price, and that they tend to think proportionally. The inference rule (4) geometricallyaverages no inference with proportional inference, so it encompasses these two types of error. First, customers underinfer marginal costs from price by clinging to their prior belief C b . Theparameter γ ∈ ( , ] measures the degree of such underinference. When γ =
1, customers donot update at all about marginal cost based on price; they naively maintain their prior belief C b ,irrespective of the price they observe. When γ ∈ ( , ) , customers do infer something from theprice, but not enough.Moreover, insofar as they infer something, they infer that marginal cost is proportional toprice, given by ( ϵ − ) P / ϵ . Such proportional inference represents a second error: underinferencepertains to how much customers infer, whereas proportional inference describes what customersinfer in as much as they do infer. The updating rule has the property that in the limit as γ = γ =
0, the monopoly optimally sets the markup ϵ /( ϵ − ) ,which makes ( ϵ − ) P / ϵ the marginal cost at price P , and proportional inference agrees withrational inference. When γ ∈ ( , ) , however, the monopoly does not find it optimal to mark upproportionally, and proportional inference becomes an error.Last, we impose a constraint on the parameter C b such that the perceived markup falls below M h when the firm prices at marginal cost; this is necessary for the equilibrium to exist.Despite its apparent arbitrary nature, the assumption of subproportional inference has closeties to game-theoretic models of failure of contingent thinking. It is related to the concept ofcursed equilibrium, developed by Eyster and Rabin (2005), and to the concept of analogy-basedexpectation equilibrium, developed by Jehiel (2005) and extended to Bayesian games by Jehiel andKoessler (2008). Both concepts propose mechanisms that can be used to explain why people mightfail to account for the information that equilibrium prices reveal about marginal costs. Subpro-portional inference is also related to the cursed-expectation equilibrium developed by Eyster,Rabin, and Vayanos (2019) as an alternative to rational-expectations equilibrium in markets. We use a geometric average instead of an arithmetic average because it is much more tractable. In fact, with γ =
1, the beliefs about marginal cost given by (4) resemble those in a fully cursed equilibrium andthe coarsest analogy-based-expectation equilibrium, when recasting our model as a Bayesian game, as in section 4.5.In these equilibrium concepts, an unsophisticated household infers nothing about marginal cost from any economicvariable. Consequently, a consumer with average prior beliefs about marginal cost equal to C b would continue toperceive marginal costs with mean C b given any price. In a cursed-expectation equilibrium of a model in which traders endowed with private information trade a riskyasset, each trader forms an expectation about the value of the asset equal to a geometric average of her expectation
Analytical results.
Plugging the belief-updating rule (4) into M p = P / C p , we obtain the follow-ing: Lemma 5.
When customers update subproportionally, they perceive the monopoly’s markup to be M p ( P ) = (cid:16) ϵϵ − (cid:17) − γ (cid:18) PC b (cid:19) γ , which is a strictly increasing function of the observed price P . Customers appreciate that higher prices signal higher marginal costs. But by underappre-ciating the strength of the relationship between price and marginal cost, customers partiallymisattribute higher prices to higher markups. Consequently, they regard higher prices as lessfair. As the functions M p ( P ) and F ( M p ) are differentiable, customers enjoy an infinitesimal pricereduction as much as they dislike an infinitesimal price increase; therefore, the monopoly’sdemand curve has no kinks, unlike in pricing theories based on loss aversion (Heidhues andKoszegi 2008).Combining (2) and (4), we then find that the price elasticity of demand satisfies E = ϵ + ( ϵ − ) · γ · ϕ ( M p ) . conditional upon her private signal alone and her expectation conditional upon both her private signal and themarket price. Traders’ expectations therefore take the form of a weighted average of naive beliefs and correctbeliefs. The two rules differ in that consumers in our model average naive beliefs with a particular form of incorrectbeliefs (proportional inference); to include rational updating as a limit case, we calibrate the updating rule to matchcorrect equilibrium beliefs for the case in which all consumers are rational. We adopt this approach for its analytictractability and suspect that the main results of the paper would go through if people averaged their prior beliefswith rational beliefs about cost.
20e have seen that without fairness concerns ( ϕ = γ = ϵ . That result changes here. Since γ >
0, the priceelasticity of demand is always greater than ϵ . Moreover, since ϕ ( M p ) is increasing in M p and M p ( P ) in P , the price elasticity of demand is increasing in P . These properties have implicationsfor the markup charged by the monopoly, M = E /( − E ) . Proposition 1.
When customers care about fairness and update subproportionally, the monopoly’smarkup is implicitly defined by (5) M = + ϵ − · + γ ϕ ( M p ( M · C )) , which implies that M ∈ ( , ϵ /( ϵ − )) . Furthermore, the cost passthrough is given by β = (cid:30) (cid:26) + γ ϕσ ( + γ ϕ ) [ ϵ + ( ϵ − ) γ ϕ ] (cid:27) , which implies that β ∈ ( , ) . Hence, the markup is lower than without fairness concerns orwith rational inference; and unlike without fairness concerns or with rational inference, the costpassthrough is incomplete. The proof is relegated to appendix A.2, but the intuition is simple. First, when customerscare about fairness but underinfer marginal costs, they become more price-sensitive. Indeed,an increase in the price increases the opportunity cost of consumption—as in the case withoutfairness—and also increases the perceived markup, which reduces the marginal utility of con-sumption and therefore demand. This heightened price-sensitivity raises the price elasticity ofdemand above ϵ and pushes the markup below ϵ /( ϵ − ) .Second, after an increase in marginal cost, the monopoly optimally lowers its markup. Thisoccurs because customers underappreciate the increase in marginal cost that accompanies ahigher price. Since the perceived markup increases, the price elasticity of demand increases. Inresponse, the monopoly reduces its markup, which mitigates the price increase. Thus, our modelgenerates incomplete passthrough of marginal cost into price—a mild form of price rigidity.Furthermore, customers err in believing that transactions are less fair when the marginal costincreases: transactions actually become more fair. Comparison with microevidence.
The result that prices do not fully respond to marginal-costshocks accords well with evidence on real firm behavior. First, using matched data on prod-uct prices and producers’ unit labor cost in Sweden, Carlsson and Skans (2012) find a limited21assthrough of idiosyncratic marginal-cost changes into prices of about 0.3. Second, using pro-duction data for Indian manufacturing firms, De Loecker et al. (2016, table 7) find that followingtrade liberalization in India, marginal costs fell significantly due to the import tariff reduction,yet prices failed to fall in step: they estimate passthroughs between 0.3 and 0.4. Third, usingproduction and cost data for Mexican manufacturing firms, Caselli, Chatterjee, and Woodland(2017, table 7) also find a modest passthrough of idiosyncratic marginal-cost changes into prices:between 0.2 and 0.4. Last, combining production data for US manufacturing firms with dataon energy prices and consumption, Ganapati, Shapiro, and Walker (2019, tables 5 and 6) find amoderate passthrough of marginal-cost changes caused by energy-price variations into prices:between 0.5 and 0.7. Taking the midpoint estimates from the four studies, we find an averagepassthrough of 0 . + ( . + . )/ + ( . + . )/ + ( . + . )/ = .
4. Hence, across studies, thecost passthrough is well below 1.Additionally, our theory predicts that when customers care about fairness, the passthroughof marginal costs into prices is markedly different when costs are observable and when they arenot. The passthrough is one when costs are observable (lemma 3) but is strictly below one whencosts are not observable (proposition 1). Kachelmeier, Limberg, and Schadewald (1991a,b) andRenner and Tyran (2004) provide experimental evidence consistent with this result: they find thatafter a cost shock, prices adjust much more when costs are observable than when they are not.
Comparison with the literature.
In our model, price rigidity arises from a nonconstant priceelasticity of demand, which creates variations in markups after cost shocks. In that respect,our model shares similarities to other models in which a variable price elasticity leads to pricerigidity. In international economics, these models have long been used to explain the behaviorof exchange rates and prices (Dornbusch 1985; Bergin and Feenstra 2001; Atkeson and Burstein2008). In macroeconomics, such models have been used to create real rigidities—in the senseof Ball and Romer (1990)—that amplify nominal rigidities (Kimball 1995; Dotsey and King 2005;Eichenbaum and Fisher 2007). Whereas many of these models make reduced-form assumptions(in the utility function or the demand curve) to generate a nonconstant price elasticity of demand,our model provides a microfoundation for this property.
Additional analytical results.
To obtain further results, we introduce a simple fairness functionthat satisfies all the requirements from definition 2:(6) F ( M p ) = − θ · (cid:16) M p − ϵϵ − (cid:17) , θ > θ means that a consumergrows more upset when consuming an overpriced item and more content when consuming anunderpriced item. The fairness function reaches 1 when the perceived markup equals ϵ /( ϵ − ) ;then fairness-adjusted consumption coincides with actual consumption. When the perceivedmarkup exceeds ϵ /( ϵ − ) , the fairness function falls below one; and when the perceived markuplies below ϵ /( ϵ − ) , the fairness function surpasses one.Furthermore, to compare different industries or economies, we focus on a situation in whichcustomers have acclimated to prices by coming to judge firms’ markups as acceptable: C b adjustsso M p = ϵ /( ϵ − ) and F =
1. Acclimation is likely to occur eventually within any industry oreconomy, once customers have faced the same prices for a long time. We then obtain the following corollary:
Corollary 1.
Assume that customers care about fairness according to the fairness function (6) ,infer subproportionally, and are acclimated. Then the monopoly’s markup is given by M = + ( + γ θ ) ϵ − . The markup decreases with the competitiveness of the market ( ϵ ), concern for fairness ( θ ), anddegree of underinference ( γ ). And the cost passthrough is given by β = (cid:30) (cid:26) + γ θ [( + θ ) ϵ − ]( ϵ − ) ( + γ θ ) [( + γ θ ) ϵ − ] (cid:27) . The passthrough increases with the competitiveness of the market ( ϵ ), but decreases with theconcern for fairness ( θ ) and degree of underinference ( γ ). The proof is in appendix A.3; it involves applying proposition 1 to the fairness function (6),under acclimation.
Comparison with additional microevidence.
Our theory predicts that the cost passthrough ishigher in more-competitive markets. This property echoes the finding by Carlton (1986) thatprices are less rigid in less-concentrated industries. It is also consistent with the finding by Amiti, As noted by Kahneman, Knetsch, and Thaler (1986, p. 730), “Psychological studies of adaption suggest that anystable state of affairs tends to become accepted eventually, at least in the sense that alternatives to it no longer come tomind. Terms of exchange that are initially seen as unfair may in time acquire the status of a reference transaction. . . .[People] adapt their views of fairness to the norms of actual behavior.” The belief-updating rule (7) introduced in theNew Keynesian model has the property that for any initial belief, people eventually become acclimated. Fairness operates by reducing the markup below its standard level ϵ /( ϵ − ) and toward 1. As the market becomesperfectly competitive, the markup approaches 1, and prices become flexible (see proposition 1 when ϵ → ∞ ). This more personal relationship could have made the retail sector morefairness-oriented, which would help to explain, according to our theory, greater price rigidityin the past. The property that prices are more rigid in markets that are more fairness-orientedcould also contribute to explain the finding by Nakamura and Steinsson (2008, table 8) that pricesare more rigid in the service sector than elsewhere. Indeed, in the service sector, relationshipsbetween buyers and sellers are more personal, which could make fairness concerns more salientand thus prices more rigid.
5. New Keynesian model
We now explore the macroeconomic implications of the pricing theory developed in section 4. Tothat end, we embed it into a New Keynesian model as a substitute for usual pricing frictions (eitherCalvo 1983 pricing or Rotemberg 1982 pricing). We find that when customers care about fairnessand infer subproportionally, the price markup depends on the rate of inflation; thus, monetarypolicy is nonneutral in both short and long run. (All derivations are relegated to appendix B.)
The economy is composed of a continuum of firms indexed by j ∈ [ , ] and a continuum ofhouseholds indexed by k ∈ [ , ] . Firms use labor services to produce goods. Households supplylabor services, consume goods, and save using riskless nominal bonds. Since goods are imperfectsubstitutes for one another, and labor services are also imperfect substitutes, each firm exercisessome monopoly power on the goods market, and each household exercises some monopolypower on the labor market. Fairness concerns.
Households cannot observe firms’ marginal costs. When a household pur-chases good j at price P j ( t ) in period t , it infers that firm j ’s marginal cost is C pj ( t ) . The dynamic Kackmeister notes: “In 1889–1891 retailing often occurred in small one- or two-person shops, retailers suppliedcredit to the customers, and retailers usually delivered the purchases to the customer’s home at no extra charge.Today retailing occurs in large stores, a third party supplies credit, and the customer takes his own items home. Thesechanges decrease both the business and personal relationship between the retailer and the customer” (p. 2008). j ’s marginal cost at time t is given by(7) C pj ( t ) = (cid:104) C pj ( t − ) (cid:105) γ (cid:20) ϵ − ϵ P j ( t ) (cid:21) − γ , where C pj ( t − ) is last period’s perception of marginal cost, and γ ∈ ( , ) is the degree ofunderinference.Having inferred the marginal cost, the household deduces that the markup charged by firm j is M pj ( t ) = P j ( t )/ C pj ( t ) . This perceived markup determines the fairness of the transaction withfirm j , measured by F j ( M pj ( t )) . The fairness function F j , specific to good j , satisfies the conditionslisted in definition 2. The elasticity of F j with respect to M pj is ϕ j = − d ln ( F j )/ d ln ( M pj ) .An amount Y jk ( t ) of good j bought by household k at a unit price P j ( t ) yields a fairness-adjustedconsumption Z jk ( t ) = F j ( M pj ( P j ( t ))) · Y jk ( t ) . Household k ’s fairness-adjusted consumption of the different goods aggregates into a consump-tion index Z k ( t ) = (cid:20)∫ Z jk ( t ) ( ϵ − )/ ϵ d j (cid:21) ϵ /( ϵ − ) , where ϵ > t is given by the price index X ( t ) = (cid:40)∫ (cid:34) P j ( t ) F j ( M pj ( P j ( t ))) (cid:35) − ϵ d j (cid:41) /( − ϵ ) . Households.
Household k derives utility from consuming goods and disutility from working. Itsexpected lifetime utility takes the form of (cid:69) ∞ (cid:213) t = δ t (cid:20) ln ( Z k ( t )) − N k ( t ) + η + η (cid:21) , where (cid:69) t is the mathematical expectation conditional upon time- t information, N k ( t ) is its laborsupply, δ ∈ ( , ) is its time discount factor, and η > t , house-hold k holds B k ( t ) bonds. Bonds purchased in period t have a price Q ( t ) , mature in period t + k ’s consumption-savings decisions in each period t must obey the constraint ∫ P j ( t ) Y jk ( t ) d j + Q ( t ) B k ( t ) = W k ( t ) N k ( t ) + B k ( t − ) + V k ( t ) , where W k ( t ) is the wage rate for labor service k , and V k ( t ) are dividends from firm ownership. Inaddition, household k satisfies a solvency constraint that prevents Ponzi schemes.Finally, in each period t , household k chooses purchases Y jk ( t ) for each j ∈ [ , ] , labor supply N k ( t ) , bond holdings B k ( t ) , and wage rate W k ( t ) . The household’s objective is to maximize itsexpected utility subject to the budget constraint, to the solvency constraint, and to firms’ demandfor labor service k . The household takes as given its initial endowment of bonds B k (− ) , allfairness factors F j ( t ) , all prices P j ( t ) and Q ( t ) , and dividends V k ( t ) . Firms.
Firm j hires labor to produce output using the production function(8) Y j ( t ) = A j ( t ) N j ( t ) α , where Y j ( t ) is output of good j , A j ( t ) > α ∈ ( , ] is the extent ofdiminishing marginal returns to labor, and N j ( t ) = (cid:20)∫ N jk ( t ) ( ν − )/ ν dk (cid:21) ν /( ν − ) is an employment index. In the index, N jk ( t ) is the quantity of labor service k hired by firm j , and ν > A j ( t ) is stochastic and unobservable to households—making the firm’s marginal cost unobservable.Each period t , firm j chooses output Y j ( t ) , price P j ( t ) , and employment levels N jk ( t ) for all k ∈ [ , ] . The firm’s objective is to maximize the expected present-discounted value of profits (cid:69) ∞ (cid:213) t = Γ ( t ) (cid:20) P j ( t ) Y j ( t ) − ∫ W k ( t ) N jk ( t ) dk (cid:21) , where Γ ( t ) = δ t [ X ( ) Z ( )]/[ X ( t ) Z ( t )] is the stochastic discount factor for period- t nominalpayoffs, subject to the production constraint (8), to the demand for good j , and to the law ofmotion of the perceived marginal cost (7). The firm takes as given the initial belief about itsmarginal cost C pj (− ) , all wage rates W k ( t ) , and discount factors Γ ( t ) . The firm’s profits arerebated to households as dividends. 26 onetary policy. We define the inflation rate between t and t + π ( t + ) = ln ( P ( t + )/ P ( t )) ,the nominal interest between t and t + i ( t ) = − ln ( Q ( t )) , and the real interest rate as r ( t ) = i ( t ) − π ( t ) . The nominal interest rate is determined by a simple monetary-policy rule:(9) i ( t ) = i ( t ) + ψ π ( t ) , where i ( t ) is stochastic, and ψ > Symmetry.
We assume a symmetric economy: all households receive the same initial bondendowment and same dividends; and all firms share a common technology and face the samefairness and belief functions. Hence, all households behave identically, as do all firms.
Notation.
Since the equilibrium is symmetric, we drop subscripts j and k to denote the equilib-rium values taken by the variables. We also denote the steady-state value of any variable H ( t ) by H . And for any variable H ( t ) except the interest and inflation rates, we denote the logarithmicdeviation from steady state by (cid:98) h ( t ) ≡ ln ( H ( t )) − ln ( H ) . For the interest and inflation rates, wedenote the deviation from steady state by (cid:98) π ( t ) ≡ π ( t ) − π , (cid:98) i ( t ) ≡ i ( t ) − i , and (cid:98) r ( t ) ≡ r ( t ) − r . Households and firms behave exactly as in the textbook model, except that fairness concernsmodify consumers’ demand and, consequently, firms’ pricing.The demand for good j from all households is Y dj (cid:16) t , P j ( t ) , C pj ( t − ) (cid:17) = Z ( t ) (cid:20) P j ( t ) X ( t ) (cid:21) − ϵ F j (cid:32) (cid:16) ϵϵ − (cid:17) − γ (cid:34) P j ( t ) C pj ( t − ) (cid:35) γ (cid:33) ϵ − , where Z ( t ) = ∫ Z k ( t ) dk describes the level of aggregate demand. The price of good j appearstwice in the demand curve: as part of the relative price P j / X , as in the textbook model; and aspart of the fairness factor F j (·) . This second element leads to unconventional pricing.Once again, fairness affects pricing through the price elasticity of demand. As in the static The equation can be rewritten in the standard form of this type of model: Z dj ≡ F j · Y dj = Z · [( P j / F j )/ X ] − ϵ . Asthe price of one unit of Z j is P j / F j and the price of one unit of Z is X , the relative price of Z j is ( P j / F j )/ X . Hence,the demand for Z j equals aggregate demand Z times the relative price of Z j to the power of − ϵ . E j ( M pj ( t )) = − ∂ ln (cid:16) Y dj (cid:17) ∂ ln (cid:0) P j (cid:1) = ϵ + ( ϵ − ) γ ϕ j ( M pj ( t )) . Unlike in the static model, however, the profit-maximizing markup does not equal E j ( t )/[ E j ( t ) − ] because the price elasticity of demand does not capture the effect of the current price on futureperceived marginal costs and thus future demands. Instead, in equilibrium, firms set their pricemarkup M ( t ) such that(10) M ( t ) − M ( t ) E ( M p ( t )) = − δγ + δ (cid:69) t (cid:18) M ( t + ) − M ( t + ) (cid:2) E ( M p ( t + )) − ( − γ ) ϵ (cid:3) (cid:19) . The gap between M ( t ) and E ( t )/[ E ( t ) − ] reflects how much today’s price affects future perceivedmarginal costs, demand, and profits. Conversely, if firms do not care about the future ( δ = M ( t ) = E ( t )/[ E ( t ) − ] , as in the static model.The price markup plays an important role because it directly determines employment:(11) N ( t ) = (cid:20) ( ν − ) αν · M ( t ) (cid:21) /( + η ) . This equation shows that in equilibrium employment is strictly decreasing in the price markup.This is because in equilibrium the price markup is the inverse of the real marginal cost, which isitself increasing in employment. Since a lower price markup implies a higher real marginalcost, it also implies higher employment.
Before simulating the model, we calibrate it to US data. To set values of the fairness-related param-eters, we use new evidence on price markups and cost passthroughs. For the other parameters,we rely on standard evidence. The calibrated values of the parameters are summarized in table 4.
Fairness function.
We set the shape of the fairness function F to (6). This simple functionalform has two advantages. First, it introduces only one new parameter, θ >
0, which governsthe concern for fairness. Second, it produces a fairness factor equal to one at the zero-inflation The real marginal cost is the ratio of real wage to marginal product of labor. The marginal product of labor isdecreasing in employment because of diminishing returns. Simultaneously, the real wage is proportional to themarginal rate of substitution between leisure and consumption, which is increasing in employment because theutility function is concave and because more employment means more consumption but less leisure. able 4. Parameter values in simulations
Value Description Source or targetA. Common parameters δ = .
99 Quarterly discount factor Annual rate of return = α = = / η = . ψ = . µ i = / µ a = . ϵ = . = . θ = = . γ = . = . ϵ = = . ξ = / = steady state. Indeed, in such steady state, the perceived price markup is M p = P / C p = ϵ /( ϵ − ) (see equation (7)), and so the fairness factor is F =
1. Thus, with no trend inflation, customersacclimate and are neither happy nor unhappy about markups.
Fairness-related parameters.
We then calibrate the three parameters central to our theory: thefairness parameter θ , the inference parameter γ , and the elasticity of substitution across goods ϵ .These parameters jointly determine the average value of the price markup and its response toshocks—which determines the cost passthrough. Hence, for the calibration, we match evidenceon price markups and cost passthroughs. We target three empirical moments: average pricemarkup, short-run cost passthrough, and long-run cost passthrough.First, using firm-level data, De Loecker and Eeckhout (2017) estimate price markups in theUnited States between 1950 and 2014. They find that the average markup hovers between 1.2 and1.3 in the 1950–1980 period and rises from 1.2 to 1.7 in the 1980–2014 period. Since the averagemarkup since 2000 is about 1.5, we adopt this value as a target. The average markup computed by De Loecker and Eeckhout is commensurate to the markups estimated forspecific industries or goods in the United States. In the automobile industry, Berry, Levinsohn, and Pakes (1995,p. 882) estimate that on average ( P − C )/ P = . M = P / C = /( − . ) = . ( P − C )/ P is 0.372, whichtranslate into a markup of M = P / C = /( − . ) = .
6. In the coffee industry, Nakamura and Zerom (2010, table 6)also estimate a markup of 1.6. For most national-brand items retailed in supermarkets, Barsky et al. (2003, p. 166)discover that markups range between 1.4 and 2.1. Finally, earlier work surveyed by Rotemberg and Woodford (1995,pp. 260–267) estimates similar markups: the industrial-organization literature estimates markups to be between 1.2and 1.7, and the marketing literature estimates a typical markup to be around 2. θ primarily affects the level of the cost passthrough, while the inferenceparameter γ primarily affects its persistence. Based on the simulations, we set ϵ = . θ =
9, and γ = .
8. This calibration allows us to achieve a steady-state price markup of 1.5, an instantaneouscost passthrough of 0.4, and a two-year cost passthrough of 0.7.
Other parameters.
We set the labor-supply parameter to η = .
1, which gives a Frisch elasticityof labor supply of 1 / . = .
9. This value is the median microestimate of the Frisch elasticity foraggregate hours (Chetty et al. 2013, table 2). We then set the quarterly discount factor to δ = . α =
1. This calibration guarantees that the labor share, which equals α / M in steady state, takesits conventional value of 2 /
3. Last, we calibrate the monetary-policy parameter to ψ = .
5, whichis consistent with observed variations in the federal funds rate (Gali 2008, p. 52).
Parameters of the textbook New Keynesian model.
We also calibrate a textbook New Keynesianmodel (described in appendix C), which we will use as a benchmark in simulations. For theparameters common to the two models, we use the same values—except for ϵ . In the textbookmodel, the steady-state price markup is ϵ /( ϵ − ) , so we set ϵ = ξ , which governs pricerigidity. To generate price rigidity, the New Keynesian literature uses either the staggered pricingof Calvo (1983) or the price-adjustment cost of Rotemberg (1982). Both pricing assumptions leadto the same linearized Phillips curve around the zero-inflation steady state, and therefore to the30ame simulations (Roberts 1995). For calibration purposes, however, the Calvo interpretation of ξ is easier to map to the data, so we use it here. The parameter ξ indicates the share of firms thatcannot update their prices each period; it can be calibrated from microevidence on the frequencyof price adjustments. If a share ξ of firms keep their price fixed each period, the average durationof a price spell is 1 /( − ξ ) (Gali 2008, p. 43). In the microdata underlying the US ConsumerPrice Index, the mean duration of price spells is about 3 quarters (Nakamura and Steinsson 2013,table 1). Hence, we set 1 /( − ξ ) =
3, which implies ξ = / Price rigidity is a central concept in macroeconomic theory because it is a source of monetarynonneutrality. Here we explore how our pricing theory produces monetary nonneutrality.At this stage, we focus on the short-run effects of monetary policy. We trace how an unexpectedand transitory shock to monetary policy permeates through the economy.
Analytical results.
The dynamics of the textbook model around the steady state are governed byan IS equation, describing households’ consumption-savings decisions, and a short-run Phillipscurve, describing firms’ pricing decisions. In the model with fairness, the same IS equationremains valid, but the Phillips curve is modified—because firms price differently. The main difference is that the Phillips curve involves not only employment and inflation butalso the perceived price markup, which itself obeys the following law of motion:
Lemma 6.
In the New Keynesian model with fairness, the perceived price markup evolves accord-ing to (12) (cid:99) m p ( t ) = γ (cid:104)(cid:98) π ( t ) + (cid:99) m p ( t − ) (cid:105) . Accordingly, the perceived price markup is a discounted sum of lagged inflation terms: (cid:99) m p ( t ) = ∞ (cid:213) s = γ s + (cid:98) π ( t − s ) . Introducing fairness concerns into the New Keynesian model improves the realism of the Phillips curve but notthat of the IS equation. Yet the IS equation is also problematic. It is notably the source of the many anomalies ofthe New Keynesian model at the zero lower bound on nominal interest rates. Other behavioral elements have beenintroduced into the New Keynesian model to improve the realism of the IS equation. For instance, Gabaix (2016)assumes that households are inattentive to unusual events. Alternatively, Michaillat and Saez (2019) assume thathouseholds derive utility not only from consumption and leisure but also from social status, which is measured byrelative wealth.
Proposition 2.
In the New Keynesian model with fairness, the short-run Phillips curve is (13) ( − δγ ) (cid:99) m p ( t ) − λ (cid:98) n ( t ) = δγ (cid:69) t ( (cid:98) π ( t + )) − λ (cid:69) t ( (cid:98) n ( t + )) , where λ ≡ ( + η ) ϵ + ( ϵ − ) γ ϕγ ϕσ (cid:20) + ( − δ ) γ − δγ ϕ (cid:21) λ ≡ ( + η ) δ ϵ + ( ϵ − ) ϕϕσ (cid:20) + ( − δ ) γ − δγ ϕ (cid:21) . Hence the short-run Phillips curve is hybrid, including both past and future inflation rates: ( − δγ ) ∞ (cid:213) s = γ s + (cid:98) π ( t − s ) − λ (cid:98) n ( t ) = δγ (cid:69) t ( (cid:98) π ( t + )) − λ (cid:69) t ( (cid:98) n ( t + )) . The proof appears in appendix B.4. It is obtained by log-linearizing firms’ pricing equation (10)around the steady state, and combining it with the log-linear version of (11)—to link the pricemarkup to employment—and with (12)—to link the future perceived price markup to inflation.
Simulation results.
Next we simulate the dynamical response of our calibrated model to anunexpected and transitory monetary shock. Following the literature, we simulate dynamicsaround the zero-inflation steady state. We assume that the exogenous component i ( t ) of themonetary-policy rule (9) follows an AR(1) process, such that (cid:98) i ( t ) = µ i · (cid:98) i ( t − ) + ζ i ( t ) , where the disturbance ζ i ( t ) follows a white-noise process with mean zero, and µ i ∈ ( , ) governsthe persistence of shocks. We set µ i = /
4, which corresponds to moderate persistence (Gali 2008,32. 52; Gali 2011, p. 26), and we simulate the response to an initial disturbance of ζ i ( ) = − . ζ i ( t ) is an expansionary monetary shock, leading to a fall in the realinterest rate. Without any inflation response, this shock would lead to a decrease of the annualizedinterest rate by 1 percentage point.Figure 1 depicts the dynamical response to the expansionary monetary shock. The real-interestand inflation rates are expressed as deviations from steady-state values, measured in percentagepoints and annualized (by multiplying by four the variables (cid:98) r ( t ) and (cid:98) π ( t ) ); all other variables areexpressed as percentage deviations from steady-state values.Loosening monetary policy generates a decrease in the real interest rate and an increase ininflation. Inflation is positive for two quarters and close to zero thereafter. Observing higher prices,customers underinfer the underlying increase in nominal marginal costs and thus perceive higherprice markups. Firms respond to higher perceived markups by cutting their actual markups.The price markup falls by 1.4%, which raises output and employment by 0.7%. (Output andemployment respond identically because the production function is calibrated to be linear.) Comparison with microevidence.
The dynamics of the perceived price markup in the modelallow us to make sense of the survey responses collected by Shiller (1997) and the Bank of Japan(table 1). When consumers observe inflation, they mistakenly believe that price markups arehigher and transactions are less fair, which lowers their consumption utility and triggers a feelingof displeasure. Conversely, upon observing deflation, they would believe that markups are lowerand transactions more fair, which would boost their consumption utility and trigger a feeling ofhappiness. Hence, our model naturally explains why Japanese customers have a negative opinionof inflation and a positive opinion of deflation. By the same token, it explains Shiller’s findingthat people are angered by inflation, which they attribute to the greed of businesses.
Comparison with macroevidence.
Monetary policy is nonneutral in the model because mon-etary shocks influence output and employment. The nonneutrality of monetary policy is welldocumented; the evidence is summarized by Christiano, Eichenbaum, and Evans (1999) andRamey (2016, sec. 3). Furthermore, the effect of monetary policy is mediated by a hybrid Phillipscurve, which is realistic as both past inflation and expected future inflation enter significantly inestimated New Keynesian Phillips curve (Mavroeidis, Plagborg-Moller, and Stock 2014, table 2).In fact the response of output to a monetary shock is broadly the same in the model as in USdata. First, the shape of the response is similar, as output is estimated to respond to monetaryshocks in a hump-shaped fashion (Ramey 2016, figs. 1–4). Second, the amplitude of the responseis comparable. After a one-percentage-point decrease of the nominal interest rate, the literature33 R ea l i n t e r e s t r a t e Textbook New Keynesian modelNew Keynesian model with fairness -1.5% -1%-0.5% 0% 0.5% 1% 1.5% I n f l a t i on r a t e -1.5% -1%-0.5% 0% 0.5% 1% 1.5% P e r c e i v ed p r i c e m a r k up -1.5% -1%-0.5% 0% 0.5% 1% 1.5% A c t ua l p r i c e m a r k up Years -1.5% -1%-0.5% 0% 0.5% 1% 1.5% E m p l o y m en t Years -1.5% -1%-0.5% 0% 0.5% 1% 1.5% O u t pu t Figure 1.
Effects of an expansionary monetary shock
This figure describes the response of the New Keynesian model with fairness (solid, blue lines) to a decrease inthe exogenous component of the monetary-policy rule (9) by 1 percentage point (annualized) at time 0. The realinterest rate and inflation rate are deviations from steady state, measured in percentage points and annualized. Theother variables are percentage deviations from steady state. For comparison, the figure also displays the response ofthe textbook New Keynesian model (dashed, orange lines). The log-linearized equilibrium conditions used in thesimulation of the model with fairness are described in appendix B.4; those used in the simulation of the textbookmodel are in appendix C. The calibration of the two models is described in table 4.
Comparison with the textbook New Keynesian model.
In both our model and the textbook model,looser monetary policy leads to higher inflation and lower markups, boosting employment andoutput. Beyond these similarities, the two models differ on several counts.First, the textbook’s short-run Phillips curve is purely forward-looking, so it does not includethe backward-looking elements found in US data and present in the fairness model. Of course,other variations of the textbook model append backward-looking components to the Phillipscurve; for example, having firms index their prices to past inflation in periods when they cannotreset their prices (Christiano, Eichenbaum, and Evans 2005).Second, the textbook model cannot produce the positive correlation between perceived pricemarkup and inflation that occurs in the fairness model, and that rationalizes the survey findingsby Shiller (1997) and the Bank of Japan. This is because households in the textbook model correctlyinfer that price markups are lower when they see higher inflation.Third, the textbook model cannot produce the hump-shaped response of output observedin US data and predicted by the fairness model, since it does not include any backward-lookingelement. The fairness model, by contrast, includes a backward-looking element in the form ofthe perceived price markup (cid:99) m p ( t ) , which enters the Phillips curve (13) and depends on the pastvia (12). It is well understood that backward-looking elements generate hump-shaped impulseresponses. Many authors have obtained such responses by assuming that consumers form habits(Fuhrer 2000; Christiano, Eichenbaum, and Evans 2005). Under this assumption, consumers’35ehavior depends on their past consumption, which then enters the IS curve and generateshump-shaped responses.Fourth, the response of output in the textbook model is about one third the size of that in thefairness model, and much smaller than in US data. Despite both models being calibrated throughmicroevidence on price dynamics, monetary shocks are more amplified in the fairness model. The price rigidity arising from fairness concerns allows the transmission of monetary policy toreal variables, such as employment and output. The price rigidity also affects the transmission ofnonpolicy shocks to the economy. Here we illustrate the effects of a technology shock—the moststudied nonpolicy shock in modern macroeconomics—on the economy under fairness concerns.
Simulation results.
We simulate the dynamical response of our calibrated model to an un-expected and transitory shock to technology. Once again, we simulate dynamics around thezero-inflation steady state. We assume that the logarithm of technology A ( t ) in the productionfunction (8) follows an AR(1) process, such that (cid:98) a ( t ) = µ a · (cid:98) a ( t − ) + ζ a ( t ) , where the disturbance ζ a ( t ) follows a white-noise process with mean zero, and µ a ∈ ( , ) governsthe persistence of shocks. We set µ a = .
9, which is typical (Gali 2008, p. 55), and we simulate theresponse to an initial disturbance of ζ a ( ) = (cid:98) π ( t ) ), whereas all other variables are expressed as percentage deviationsfrom steady-state values.The increase in technology reduces marginal costs, which generates a drop in inflation:inflation is negative for about four quarters and virtually zero thereafter. Observing lower prices,customers underinfer the underlying decrease in marginal costs and thus perceive lower pricemarkups and fairer transactions. The improvement in perceived fairness decreases the priceelasticity of the demand for goods. Firms best respond by raising their markups. The pricemarkup increases by 1.3% at the peak, which depresses employment by 0.7%. Despite the drop inemployment, output initially increases by 0.5% because technology is higher.36 -1%-0.5% 0% 0.5% 1% 1.5% T e c hno l og y Textbook New Keynesian modelNew Keynesian model with fairness -1%-0.5% 0% 0.5% 1% 1.5% I n f l a t i on r a t e -1%-0.5% 0% 0.5% 1% 1.5% P e r c e i v ed p r i c e m a r k up -1%-0.5% 0% 0.5% 1% 1.5% A c t ua l p r i c e m a r k up Years -1%-0.5% 0% 0.5% 1% 1.5% E m p l o y m en t Years -1%-0.5% 0% 0.5% 1% 1.5% O u t pu t Figure 2.
Effects of a positive technology shock
This figure describes the response of the New Keynesian model with fairness (solid, blue lines) to a 1% increasein technology at time 0. The inflation rate is a deviation from steady state, measured in percentage points andannualized. The other variables are percentage deviations from steady state. For comparison, the figure also displaysthe response of the textbook New Keynesian model (dashed, orange lines). The log-linearized equilibrium conditionsused in the simulation of the model with fairness are described in appendix B.4; those used in the simulation of thetextbook model are in appendix C. The calibration of the two models is described in table 4. omparison with macroevidence. In our model, an increase in technology leads to higher outputbut lower employment. This prediction conforms to much of the evidence from US data (Gali andRabanal 2005; Basu, Fernald, and Kimball 2006; Francis and Ramey 2009). Our model also predictsthat inflation falls after the increase in technology, as documented by Basu, Fernald, and Kimball(2006, fig. 4). Finally, in the model, price markups and output are positively correlated undertechnology shocks. Nekarda and Ramey (2013) report evidence consistent with this prediction.
Comparison with the textbook New Keynesian model.
The similarities and differences betweenthe fairness model and textbook model identified under monetary shocks also apply undertechnology shocks. The main similarity is that in response to a technology shock, inflation, pricemarkup, employment, and output move in the same directions in the two models. There are threemain differences. First, the fairness model produces a hump-shaped response of employment tothe technology shock, which the textbook model does not. Second, the fairness model producesa negative correlation between perceived and actual price markups, whereas the two coincide inthe textbook model. Last, in response to a positive technology shock, employment falls muchmore in the fairness model than in the textbook model; as a corollary, output increases muchless in the fairness model than in the textbook model.
Our pricing theory implies that monetary policy is nonneutral in the short run, so that a transi-tory monetary shock affects employment. Here we develop another implication of the theory:monetary policy is nonneutral in the long run, so that different rates of steady-state inflation leadto different levels of steady-state employment. In other words, the theory generates a nonverticallong-run Phillips curve.We study the long-run effects of monetary policy by comparing the steady-state equilibriainduced by different values of the exogenous component i in the monetary-policy rule (9). Insteady state the real interest rate equals the time discount rate ρ ≡ − ln ( δ ) ; therefore, by choosing i , monetary policy perfectly controls steady-state inflation: π = ρ − i ψ − . To obtain zero inflation, it suffices to set i = ρ ; to obtain higher inflation, it suffices to reduce i . Acclimation.
Kahneman, Knetsch, and Thaler (1986, p. 730) have hypothesized that “any stablestate of affairs tends to become accepted eventually”. We adapt this idea to our model by assuming38hat people become partially acclimated to the steady-state inflation rate. Formally, we generalizethe fairness function (6) to(14) F ( M p ) = − θ · ( M p − M f ) , where M f is the fair markup resulting from acclimation. We assume that the fair markup is theweighted average of the standard markup, ϵ /( ϵ − ) , and the steady-state perceived markup M p :(15) M f = χ · M p + ( − χ ) · ϵϵ − . The parameter χ ∈ [ , ] measures acclimation: when χ =
0, there is no acclimation, as in theprevious version of the paper; when χ =
1, there is perfect acclimation, so people do not mindwhatever is happening in steady state; when χ ∈ ( , ) , people may be permanently satisfied ordissatisfied in steady state, but less than when χ = Analytical results.
In steady state, the rate of inflation determines the perceived price markup,fairness factor, and elasticity of the fairness function:
Lemma 7.
In the New Keynesian model with fairness, the steady-state perceived price markup isa strictly increasing function of steady-state inflation: M p ( π ) = ϵϵ − · exp (cid:18) γ − γ π (cid:19) . Hence, the steady-state fairness factor is a weakly decreasing function of steady-state inflation: F ( π ) = − θ · ( − χ ) · (cid:104) M p ( π ) − ϵϵ − (cid:105) . Accordingly, the steady-state elasticity of the fairness function is a strictly increasing function ofsteady-state inflation: ϕ ( π ) = θ · M p ( π ) F ( π ) . The proof is relegated to appendix B.3; it requires to manipulate the inference mechanism (7)to obtain M p , and to use (14) and (15) to obtain F and ϕ .The lemma shows that in steady state households perceive higher price markups wheninflation is higher. Households understand that in steady state nominal marginal costs grow at This specification does not change anything at the zero-inflation steady state. With zero inflation, M p = ϵ /( ϵ − ) ,so M f = ϵ /( ϵ − ) for any χ . Therefore, for any χ , the fairness function (14) simplifies to the function (6). χ = Proposition 3.
In the New Keynesian model with fairness, the steady-state price markup is astrictly decreasing function of steady-state inflation: (16) M ( π ) = + ϵ − · + ( − δ ) γ − δγ ϕ ( π ) . Hence, steady-state employment is a strictly increasing function of steady-state inflation: N = (cid:20) ( ν − ) αν · M ( π ) (cid:21) /( + η ) . Thus, the long-run Phillips curve is not vertical (fixed N ) but upward sloping. The proof is in appendix B.3; the main step is reworking (10) in steady state to obtain M .The proposition shows that the long-run Phillips curve slopes upward for any degree ofacclimation. Hence, monetary policy is nonneutral in the long run. The reason is that in the longrun, higher inflation leads to a lower price markup—and thus higher employment in generalequilibrium. In fact, (16) has the same structure as (5) in the monopoly model, so the two modelsoperate similarly. After an increase in inflation, households underappreciate the increase innominal marginal costs, so they partly attribute the higher prices to higher markups, which theyfind unfair. Since perceived markups are higher, the price elasticity of demand increases, leadingfirms to reduce their markups. Simulation results.
To quantify long-run monetary nonneutrality, we compute the long-runPhillips curve in our calibrated model. Figure 3 displays two versions of the curve: one describesthe relationship between steady-state inflation and steady-state price markup, and the otherthe relationship between steady-state inflation and steady-state employment. In the absence ofmicroevidence on acclimation, we compute the long-run Phillips curve for various degrees ofacclimation, and show how acclimation affects the slope of the Phillips curve.With full acclimation ( χ = .2 1.3 1.4 1.5 1.6 1.7 Price markup -2%-1% 0% 1% 2% I n f l a t i on -4% -2% 0% 2% 4% Employment -2%-1% 0% 1% 2% I n f l a t i on Figure 3.
Long-run Phillips curves for various degrees of acclimation
The left-hand panel gives the relationship between steady-state inflation and steady-state price markup. The right-hand panel gives the relationship between steady-state inflation and steady-state employment (measured as percent-age deviation from employment in the zero-inflation steady state). These long-run Phillips curve are constructedusing the expressions in proposition 3 under the calibration in table 4, for various degrees of acclimation: χ = χ = / χ = /
4, and χ = percentage point raises employment by 0.2%. With less-than-full acclimation, the Phillips curvebecomes flatter. For instance, with an acclimation of χ = /
4, a permanent increase in inflationby 1 percentage point raises employment by 1.3%; and with a lower acclimation of χ = /
2, apermanent increase in inflation by 1 percentage point raises employment by 3%. Finally, with noacclimation ( χ = F , depends less on inflation, because con-sumers adapt to a larger degree to different inflation rates. As a result, the elasticity of the fairnessfunction in steady state, ϕ , depends less on inflation. Through formula (16), this means that thesteady-state price markup responds less to steady-state inflation: the Phillips curve is steeper. Comparison with macroevidence.
The property that higher steady-state inflation leads to highersteady-state employment is consistent with evidence that higher average inflation leads to loweraverage unemployment. King and Watson (1994, table 1) find in US data that a permanent increasein inflation by 1 percentage point reduces the unemployment rate between 0.2 and 1.3 percentagepoints, depending on the period and identification strategy. King and Watson (1997) confirmthese findings, while highlighting the uncertainty surrounding the Phillips curve’s slope.Quantitatively, the findings by King and Watson are consistent with a good amount of accli-41ation. If we neglect the effect of monetary policy on labor force participation, these resultsimply that in the United States a permanent increase in inflation by 1 percentage point increasesemployment by 0.2% to 1.3%. This range corresponds to our model’s predictions for a degree ofacclimation between 3 / χ = /
4, a permanent increase ininflation by 1 percentage point raises employment by 1.3%. With more acclimation, the Phillipscurve becomes steeper, and employment rises less. With χ = Comparison with the literature.
The property that in steady state inflation has an effect onthe price markup and employment also appears in the textbook New Keynesian model. WithRotemberg (1982) pricing, an increase in steady-state inflation leads to a lower price markupand higher output, as in our model (Ascari and Rossi 2012, fig. 1). With Calvo (1983) pricing, theopposite occurs: an increase in steady-state inflation leads to a higher price markup and loweroutput, which appears inconsistent with available evidence (Ascari and Rossi 2012, fig. 2). Our mechanism complements the traditional mechanism for an upward-sloping long-runPhillips curve: that higher steady-state inflation reduces the likelihood that firms experiencingnegative shocks are constrained by downward nominal wage rigidity to lay off workers (Akerlof,Dickens, and Perry 1996; Benigno and Ricci 2011). While our mechanism operates on the goodsmarket instead of the labor market, the psychological origins of the two mechanisms could besimilar, since one source of downward wage rigidity is workers’ fairness concerns (Bewley 2007).
6. Conclusion
This paper develops a theory of pricing to fairness-minded customers. The theory revolves aroundtwo assumptions. First, customers derive more utility from a good priced at a low markup— Although the New Keynesian models with Rotemberg and Calvo pricing are the same around the zero-inflationsteady state, they differ when steady-state inflation is nonzero (Ascari and Rossi 2012).
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Appendix A. Derivations for the monopoly model
We derive several of the monopoly results stated in section 4. In particular, we provide proofs forproposition 1 and corollary 1.
A.1. Properties of the profit function
We show that in all the cases treated in section 4, the monopoly’s profit function is single-peaked(strictly increasing to a peak, then strictly decreasing), so its maximum can be determined byfirst-order condition.The monopoly chooses a price P > C to maximize profits V ( P ) = ( P − C ) · Y d ( P ) . The derivative of the profit function is V (cid:48) ( P ) = Y d + ( P − C ) dY d dP = Y d − ( P − C ) Y d P E ( P ) , where E ( P ) ≡ − d ln ( Y d )/ d ln ( P ) = −( P / Y d )( dY d / dP ) is the price elasticity of demand. Hencethe derivative of the profit function satisfies(A1) V (cid:48) ( P ) = Y d ( P ) (cid:20) − P − CP E ( P ) (cid:21) . We now study the properties of the derivative (A1) in the various cases considered in section 4.50 o fairness concerns.
Without fairness concerns, the price elasticity of demand is E = ϵ (sec-tion 4.3). Hence the derivative (A1) becomes V (cid:48) ( P ) = Y d ( P ) (cid:20) − ϵ P − CP (cid:21) . The function P (cid:55)→ ( P − C )/ P is strictly increasing from 0 to 1 as P increases from C to + ∞ , so theterm in square brackets is strictly decreasing from 1 to 1 − ϵ < P increases from C to + ∞ .Hence, the term in square brackets has a unique root P ∗ on ( C , + ∞) , is positive for P < P ∗ , andis negative for P > P ∗ . Since Y d ( P ) >
0, these properties transfer to the derivative of the profitfunction: V (cid:48) ( P ) > P ∈ ( C , P ∗ ) , V (cid:48) ( P ) = P = P ∗ , and V (cid:48) ( P ) < P ∈ ( P ∗ , + ∞) . Weconclude that the profit function is single-peaked, and its maximum P ∗ is the unique solution tothe first-order condition V (cid:48) ( P ) = Fairness concerns and observable costs.
With fairness concerns and observable costs, the priceelasticity of demand is E = ϵ + ( ϵ − ) ϕ ( P / C ) (section 4.4). The profit function is now defined for P ∈ ( C , M h · C ) . The derivative (A1) becomes V (cid:48) ( P ) = Y d ( P ) (cid:20) − P − CP · { ϵ + ( ϵ − ) ϕ ( P / C )} (cid:21) . Again, the function P (cid:55)→ ( P − C )/ P is strictly increasing from 0 to 1 as P increases from C to + ∞ . The elasticity of the fairness function ϕ ( P / C ) is strictly increasing from ϕ ( ) > + ∞ as P increases from C to M h · C (lemma 2). Hence the term in square brackets is strictly decreasingfrom 1 to −∞ as P increases from C to M h · C . This implies that the term in square brackets has aunique root P ∗ on ( C , M h · C ) , is positive for P < P ∗ , and is negative for P > P ∗ . Following thesame argument as in the previous case, we conclude that the profit function is single-peaked,and its maximum P ∗ is the unique solution to the first-order condition V (cid:48) ( P ) = Fairness concerns and rational inference of costs.
With fairness concerns rational inference ofmarginal costs, the price elasticity of demand is again E = ϵ (section 4.5). Hence, as in the case ofno fairness concerns, the profit function is single-peaked so its maximum is the unique solutionto the first-order condition V (cid:48) ( P ) = Fairness concerns and subproportional inference of costs.
With fairness concerns and subpropor-tional inference of costs, the price elasticity of demand is E = ϵ + ( ϵ − ) γ ϕ ( M p ( P )) (section 4.6).51he profit function is now defined for P ∈ ( C , P b ) , where the upper bound is defined by(A2) P b = ϵϵ − ( M h ) / γ C b . The price P b is such that at P b , the perceived markup reaches the upper bound of the domain ofthe fairness function: M p ( P b ) = M h . The derivative (A1) becomes V (cid:48) ( P ) = Y d ( P ) (cid:20) − P − CP · (cid:8) ϵ + ( ϵ − ) γ ϕ ( M p ( P )) (cid:9)(cid:21) . Again, the function P (cid:55)→ ( P − C )/ P is strictly increasing from 0 to 1 as P increases from C to + ∞ .The perceived markup M p ( P ) is strictly increasing from M p ( C ) > M h as P increases from C to P b (lemma 5). Hence, the elasticity of the fairness function ϕ ( M p ( P )) is strictly increasing from ϕ ( M p ( C )) > + ∞ as P increases from C to P b (lemma 2). Since γ >
0, we infer that the term insquare brackets is strictly decreasing from 1 to −∞ as P increases from C to P b . Thus the term insquare brackets has a unique root P ∗ on ( C , P b ) , is positive for P < P ∗ , and is negative for P > P ∗ .Following the same argument as in the previous cases, we conclude that the profit function issingle-peaked, and its maximum P ∗ is the unique solution to the first-order condition V (cid:48) ( P ) = A.2. Proof of proposition 1
Markup.
Since customers care about fairness and infer subproportionally, the price elasticityof demand is E = ϵ + ( ϵ − ) γ ϕ ( M p ( P )) . Moreover, the monopoly’s optimal markup is M = E /( E − ) = + /( E − ) . Combining these equations yields the markup(A3) M = + ϵ − · + γ ϕ ( M p ( M · C )) . In (A3) we have used the fact that the price is related to the markup by P = M · C .Toward showing that (A3) admits a unique solution, we introduce the price P b defined by(A2) and the markup M b = P b / C >
1. Since P = M · C , P strictly increases from 0 to P b when M increases from 0 to M b . Next, lemma 5 shows that M p ( P ) strictly increases from 0 to M h when P increases from 0 to P b . Last, lemma 2 indicates that ϕ ( M p ) strictly increases from 0 to ∞ when M p increases from 0 to M h . As γ >
0, we conclude that when M increases from 0 to M b >
1, theright-hand side of (A3) strictly decreases from ϵ /( ϵ − ) to 1. Hence, (A3) has a unique solution M ∈ (cid:2) , M b (cid:3) , implying that the markup exists and unique. Given the range of values taken by theright-hand side of (A3), we also infer that M ∈ ( , ϵ /( ϵ − )) . We know that P b > C because C b satisfies C b > ( ϵ − ) · ( M h ) − / γ · C / ϵ (definition 3). assthrough. We now compute the cost passthrough, β = d ln ( P )/ d ln ( C ) . The equilibriumprice is P = M ( M p ( P )) · C , where the markup M ( M p ) is given by (A3). Using this price equation,we obtain β = d ln ( M ) d ln ( M p ) · d ln ( M p ) d ln ( P ) · d ln ( P ) d ln ( C ) + . Since d ln ( M p )/ d ln ( P ) = γ (lemma 5) and d ln ( P )/ d ln ( C ) = β (by definition), we get(A4) β = − γ d ln ( M ) d ln ( M p ) . Our next step is to compute the elasticity of M ( M p ) with respect to M p from (A3): − d ln ( M ) d ln ( M p ) = − M · dMd ln ( M p ) = M · ϵ − · + γ ϕ · + γ ϕ · γ · dϕd ln ( M p ) . Using (A3), we find that ( ϵ − )( + γ ϕ ) M = ϵ + ( ϵ − ) γ ϕ . Moreover, by definition, the superelasticity σ of the fairness function satisfies ϕσ = dϕ / d ln ( M p ) .Combining these three results, we obtain(A5) − d ln ( M ) d ln ( M p ) = γ ϕσ [ ϵ + ( ϵ − ) γ ϕ ]( + γ ϕ ) . Finally, combining (A4) with (A5) yields the cost passthrough(A6) β = + γ ϕσ ( + γϕ )[ ϵ + ( ϵ − ) γϕ ] . Since γ > ϕ > σ > β ∈ ( , ) . A.3. Proof of corollary 1
We apply the results of proposition 1 to a specific fairness function:(A7) F ( M p ) = − θ · (cid:16) M p − ϵϵ − (cid:17) . We also assume that customers are acclimated, so M p = ϵ /( ϵ − ) and F = reliminary results. The elasticity of the fairness function (A7) is ϕ = − M p F · dFdM p = M p F · θ . Accordingly, the superelasticity of the fairness function (A7) satisfies σ = d ln ( ϕ ) d ln ( M p ) = − d ln ( F ) d ln ( M p ) = + ϕ . When M p = ϵ /( ϵ − ) and F =
1, the elasticity and superelasticity simplify to ϕ = ϵθϵ − σ = + ϵθϵ − . (A9) Markup.
Combining (A3) with (A8), we obtain the following markup: M = + ϵ − · + γ ϵθ /( ϵ − ) = + ( + γ θ ) ϵ − . This expression shows that M is lower when ϵ , γ , or θ are higher. Passthrough.
Combining (A6) with (A8) and (A9), we find that the cost passthrough β satisfies1 / β = + γ ϵθ [( ϵ − ) + ϵθ ]( ϵ − ) [( ϵ − ) + γ ϵθ ] ( ϵ + γ ϵθ ) = + γ θ [( + θ ) ϵ − ]( ϵ − ) [( + γ θ ) ϵ − ] ( + γ θ ) . Next we introduce the auxiliary function(A10) ∆ ( γ , θ , ϵ ) = γ θ [( + θ ) ϵ − ]( ϵ − ) [( + γ θ ) ϵ − ] ( + γ θ ) , where γ > θ >
0, and ϵ >
1. Dividing numerator and denominator of ∆ by γ , we get ∆ ( γ , θ , ϵ ) = θ [( + θ ) ϵ − ]( ϵ − ) [ θϵ + ( ϵ − )/ γ ] ( θ + / γ ) . The denominator is decreasing in γ , so ∆ is increasing in γ . Since β = /( + ∆ ) , we concludethat β is decreasing in γ . 54ext, we divide numerator and denominator of ∆ in (A10) by ( ϵ − ) : ∆ ( γ , θ , ϵ ) = γ θ [ + θϵ /( ϵ − )][( + γ θ ) ϵ − ] ( + γ θ ) . Since ϵ /( ϵ − ) is decreasing in ϵ > ( + γ θ ) ϵ − ϵ , ∆ is decreasing in ϵ . As β = /( + ∆ ) , we conclude that β is increasing in ϵ .Last, we divide numerator and denominator of ∆ in (A10) by θ ( ϵθ + ϵ − ) : ∆ ( γ , θ , ϵ ) = γ ( ϵ − ) ( γ + / θ ) γϵθ + ϵ − ϵθ + ϵ − , First, γ + / θ is decreasing in θ >
0. Second, ( γ ϵθ + ϵ − )/( ϵθ + ϵ − ) is decreasing in θ > γ ≤
1. Hence, ∆ is increasing in θ >
0. Since β = /( + ∆ ) , we conclude that β isdecreasing in θ . Appendix B. Derivations for the New Keynesian model
We derive the properties of the New Keynesian model with fairness presented in section 5. Inparticular, we prove lemmas 6 and 7, as well as propositions 2 and 3.
B.1. Household and firm problems
We begin by solving the problems of households and firms.
Household k ’s problem. To solve household k ’s problem, we set up the Lagrangian: L k = (cid:69) ∞ (cid:213) t = δ t (cid:26) ln ( Z k ( t )) − N k ( t ) + η + η + A k ( t ) (cid:20) W k ( t ) N k ( t ) + B k ( t − ) + V k ( t ) − Q ( t ) B k ( t ) − ∫ P j ( t ) Y jk ( t ) d j (cid:21) + B k ( t ) (cid:2) N dk ( t , W k ( t )) − N k ( t ) (cid:3) (cid:27) . In the Lagrangian, A k ( t ) is the Lagrange multiplier on the budget constraint in period t ; B k ( t ) is the Lagrange multiplier on the labor-demand constraint in period t ; and Z k ( t ) is the fairness-55djusted consumption index:(A11) Z k ( t ) = (cid:20)∫ Z jk ( t ) ( ϵ − )/ ϵ d j (cid:21) ϵ /( ϵ − ) . In the consumption index, Z jk ( t ) is the fairness-adjusted consumption of good j :(A12) Z jk ( t ) = F j ( t ) · Y jk ( t ) . First-order conditions with respect to consumption.
We first compute the first-order conditionswith respect to Y jk ( t ) for all goods j ∈ [ , ] : ∂ L k / ∂ Y jk ( t ) =
0. From the definitions of Z k ( t ) and Z jk ( t ) given by (A11) and (A12), we find ∂ Z jk ( t ) ∂ Y jk ( t ) = F j ( t ) and ∂ Z k ( t ) ∂ Z jk ( t ) = (cid:20) Z jk ( t ) Z k ( t ) (cid:21) − / ϵ d j . Hence the first-order conditions imply that for all j ∈ [ , ] ,(A13) (cid:20) Z jk ( t ) Z k ( t ) (cid:21) − / ϵ F j ( t ) Z k ( t ) = A k ( t ) P j ( t ) . Taking (A13) to the power of 1 − ϵ and reshuffling terms, we then obtain1 Z k ( t ) − ϵ · Z k ( t ) ( ϵ − )/ ϵ · Z jk ( t ) ( ϵ − )/ ϵ = A k ( t ) − ϵ (cid:20) P j ( t ) F j ( t ) (cid:21) − ϵ . We integrate this equation over j ∈ [ , ] , use the definition of Z k ( t ) given by (A11), and introducethe price index(A14) X ( t ) = (cid:40)∫ (cid:20) P j ( t ) F j ( t ) (cid:21) − ϵ d j (cid:41) /( − ϵ ) . We obtain the following: 1 Z k ( t ) − ϵ · Z k ( t ) ( ϵ − )/ ϵ Z k ( t ) ( ϵ − )/ ϵ = A k ( t ) − ϵ X ( t ) − ϵ . From this equation we infer(A15) A k ( t ) = X ( t ) Z k ( t ) . j by household k satisfies Z jk ( t ) = Z k ( t ) (cid:20) P j ( t ) X ( t ) (cid:21) − ϵ F j ( t ) ϵ . As consumption and fairness-adjusted consumption of good j are related by Y jk ( t ) = Z jk ( t )/ F j ( t ) ,the optimal consumption of good j by household k satisfies(A16) Y jk ( t ) = Z k ( t ) (cid:20) P j ( t ) X ( t ) (cid:21) − ϵ F j ( t ) ϵ − . Integrating (A16) over all households k ∈ [ , ] yields the output of good j : Y j ( t ) = Z ( t ) (cid:20) P j ( t ) X ( t ) (cid:21) − ϵ F j ( t ) ϵ − . We note that the fairness factor F j ( t ) is a function of the perceived price markup, F j ( t ) = F j ( P j ( t )/ C pj ( t )) , and that the perceived marginal cost C pj ( t ) follows the law of motion (7). Theseobservations allow us to obtain the demand for good j : Y dj ( t , P j ( t ) , C pj ( t − )) = Z ( t ) (cid:20) P j ( t ) X ( t ) (cid:21) − ϵ F j (cid:32) (cid:16) ϵϵ − (cid:17) − γ (cid:34) P j ( t ) C pj ( t − ) (cid:35) γ (cid:33) ϵ − . For future reference, the elasticities of the function Y dj ( t , P j ( t ) , C pj ( t − )) are − ∂ ln (cid:16) Y dj (cid:17) ∂ ln (cid:0) P j (cid:1) = ϵ + ( ϵ − ) γ ϕ j ( M pj ( t )) ≡ E j ( M pj ( t )) (A17) ∂ ln (cid:16) Y dj (cid:17) ∂ ln (cid:16) C pj (cid:17) = ( ϵ − ) γ ϕ j ( M pj ( t )) = E j ( M pj ( t )) − ϵ . (A18)The function E j ( M pj ) gives the price elasticity of the demand for good j , normalized to be positive.Moreover, using (A16) and the definition of the price index X given by (A14), we find that ∫ P j Y jk d j = X ϵ Z k ∫ (cid:18) P j F j (cid:19) − ϵ d j = X Z k . This means that when households optimally allocate their consumption expenditures acrossgoods, the price of one unit of fairness-adjusted consumption index is X .57 irst-order condition with respect to bonds. The first-order condition with respect to B k ( t ) is ∂ L k / ∂ B k ( t ) =
0, which gives Q ( t )A k ( t ) = δ (cid:69) t (A k ( t + )) . Using (A15), we obtain household k ’s consumption Euler equation:(A19) Q ( t ) = δ (cid:69) t (cid:18) X ( t ) Z k ( t ) X ( t + ) Z k ( t + ) (cid:19) . This equation governs how the household smooths fairness-adjusted consumption over time.
Firm j ’s problem. Since the wages set by households depend on firms’ labor demands, we turnto the firms’ problems before finishing the households’ problems. To solve firm j ’s problem, weset up the Lagrangian: L j = (cid:69) ∞ (cid:213) t = Γ ( t ) (cid:26) P j ( t ) Y j ( t ) − ∫ W k ( t ) N jk ( t ) dk + H j ( t ) (cid:104) Y dj ( t , P j ( t ) , C pj ( t − )) − Y j ( t ) (cid:105) + J j ( t ) (cid:2) A j ( t ) N j ( t ) α − Y j ( t ) (cid:3) + K j ( t ) (cid:34) C pj ( t − ) γ (cid:20) ϵ − ϵ P j ( t ) (cid:21) − γ − C pj ( t ) (cid:35) (cid:27) . In the Lagrangian, H j ( t ) is the Lagrange multiplier on the demand constraint in period t ; J j ( t ) isthe Lagrange multiplier on the production constraint in period t ; K j ( t ) is the Lagrange multiplieron the law of motion of the perceived marginal cost in period t ; and N j ( t ) is the employmentindex:(A20) N j ( t ) = (cid:20)∫ N jk ( t ) ( ν − )/ ν dk (cid:21) ν /( ν − ) . First-order conditions with respect to employment.
We compute the first-order conditions withrespect to N jk ( t ) for all labor services k ∈ [ , ] : ∂ L j / ∂ N jk ( t ) =
0. From the definition of N j ( t ) given by (A20), we know that ∂ N j ( t ) ∂ N jk ( t ) = (cid:20) N jk ( t ) N j ( t ) (cid:21) − / ν dk . k ∈ [ , ] ,(A21) W k ( t ) = α J j ( t ) A j ( t ) N j ( t ) α − (cid:20) N jk ( t ) N j ( t ) (cid:21) − / ν . Toward deriving firm j ’s labor demand, we introduce the wage index(A22) W ( t ) = (cid:20)∫ W k ( t ) − ν dk (cid:21) /( − ν ) . Taking (A21) to the power of 1 − ν , we obtain W k ( t ) − ν = (cid:2) α J j ( t ) A j ( t ) N j ( t ) α − (cid:3) − ν N j ( t ) ( ν − )/ ν N jk ( t ) ( ν − )/ ν . Integrating this condition over k ∈ [ , ] and using the definitions of N j and W given by (A20)and (A22), we find W ( t ) − ν = (cid:2) α J j ( t ) A j ( t ) N j ( t ) α − (cid:3) − ν N j ( t ) ( ν − )/ ν N j ( t ) ( ν − )/ ν . From this equation we infer(A23) W ( t ) = α J j ( t ) A j ( t ) N j ( t ) α − . Last, we combine (A21) and (A23) to determine the quantity of labor that firm j hires fromhousehold k :(A24) N jk ( t ) = N j ( t ) (cid:20) W k ( t ) W ( t ) (cid:21) − ν . Integrating (A24) over all firms j ∈ [ , ] yields the demand for labor service k :(A25) N dk ( t , W k ( t )) = N ( t ) (cid:20) W k ( t ) W ( t ) (cid:21) − ν , where N ( t ) = ∫ N j ( t ) d j is aggregate employment.Moreover, (A22) and (A24) imply that ∫ W k N jk dk = W ν N j ∫ W − νk dk = W N j . This means that when firms optimally allocate their wage bill across labor services, the cost of59ne unit of labor index is W . First-order conditions with respect to labor and wage.
We now finish solving household k ’sproblem using labor demand (A25). The first-order conditions with respect to N k ( t ) and W k ( t ) are ∂ L k / ∂ N k ( t ) = ∂ L k / ∂ W k ( t ) =
0; they yield N k ( t ) η = A k ( t ) W k ( t ) − B k ( t ) (A26) A k ( t ) N k ( t ) = −B k ( t ) dN dk dW k . (A27)Since the elasticity of N dk with respect to W k is − ν , we infer from (A27) that(A28) A k ( t ) W k ( t ) = B k ( t ) ν . Plugging this result into (A26), we obtain B k ( t ) = N k ( t ) η ν − . Combining this result with (A28) then yields W k ( t ) = νν − · N k ( t ) η A k ( t ) . Finally, by merging this equation with (A15), we find that household k sets its wage rate at(A29) W k ( t ) X ( t ) = νν − N k ( t ) η Z k ( t ) . This equation shows that households set their real wage at a markup of ν /( ν − ) > First-order condition with respect to output.
We then finish solving firm j ’s problem. The first-order condition with respect to Y j ( t ) is ∂ L j / ∂ Y j ( t ) =
0, which gives P j ( t ) = H j ( t ) + J j ( t ) . Using the value of J j ( t ) given by (A23), we then obtain(A30) H j ( t ) = P j ( t ) (cid:20) − W ( t )/ P j ( t ) αA j ( t ) N j ( t ) α − (cid:21) . j ’s nominal marginal cost is the nominal wage divided by the marginal product oflabor:(A31) C j ( t ) = W ( t ) αA j ( t ) N j ( t ) α − . Hence the first-order condition (A30) can be written(A32) H j ( t ) = P j ( t ) (cid:20) − C j ( t ) P j ( t ) (cid:21) . Given that firm j ’s markup is M j ( t ) = P j ( t )/ C j ( t ) , we rewrite this equation as(A33) H j ( t ) P j ( t ) = M j ( t ) − M j ( t ) . First-order condition with respect to price.
The first-order condition of firm j ’s problem withrespect to P j ( t ) is ∂ L j / ∂ P j ( t ) =
0. It yields(A34) 0 = Y j ( t ) + H j ( t ) ∂ Y dj ∂ P j + ( − γ )K j ( t ) C pj ( t ) P j ( t ) . We divide this condition by Y j ( t ) , and we insert the price elasticity of the demand for good j , E j ( M pj ( t )) = − ∂ ln ( Y dj )/ ∂ ln ( P j ) , as well as the perceived price markup for good j , M pj ( t ) = P j ( t )/ C pj ( t ) . We obtain 0 = − H j ( t ) E j ( M pj ( t )) P j ( t ) + ( − γ ) K j ( t ) Y j ( t ) M pj ( t ) . Using the value of H j ( t ) given by (A33), we finally obtain(A35) ( − γ ) K j ( t ) Y j ( t ) M pj ( t ) = M j ( t ) − M j ( t ) E j ( M pj ( t )) − . First-order condition with respect to perceived marginal cost.
Finally, the first-order conditionof firm j ’s problem with respect to C pj ( t ) is ∂ L j / ∂ C pj ( t ) =
0. It gives0 = (cid:69) t (cid:32) Γ ( t + ) Γ ( t ) H j ( t + ) ∂ Y dj ∂ C pj (cid:33) + γ (cid:69) t (cid:32) Γ ( t + ) Γ ( t ) K j ( t + ) C pj ( t + ) C pj ( t ) (cid:33) − K j ( t ) . K j ( t ) = (cid:69) t (cid:32) Γ ( t + ) Γ ( t ) (cid:40) H j ( t + ) Y j ( t + ) C pj ( t ) [ E j ( M pj ( t + )) − ϵ ] + γ K j ( t + ) C pj ( t + ) C pj ( t ) (cid:41)(cid:33) . We modify this equation in two steps: first, we multiply it by C pj ( t )/[ Y j ( t ) P j ( t )] ; second, we insertthe perceived price markups M pj ( t ) = P j ( t )/ C pj ( t ) and M pj ( t + ) = P j ( t + )/ C pj ( t + ) . We get K j ( t ) M pj ( t ) Y j ( t ) = (cid:69) t (cid:32) Γ ( t + ) Y j ( t + ) P j ( t + ) Γ ( t ) Y j ( t ) P j ( t ) (cid:40) H j ( t + ) P j ( t + ) [ E j ( M pj ( t + )) − ϵ ] + γ K j ( t + ) M pj ( t + ) Y j ( t + ) (cid:41)(cid:33) . Last, we multiply the equation by ( − γ ) ; and we eliminate H j ( t + ) using (A33) and K j ( t ) and K j ( t + ) using (A35). We obtain firm j ’s pricing equation, which links its markup to its perceivedmarkup: M j ( t ) − M j ( t ) E j ( M pj ( t )) = (A36) 1 + (cid:69) t (cid:18) Γ ( t + ) Y j ( t + ) P j ( t + ) Γ ( t ) Y j ( t ) P j ( t ) (cid:26) M j ( t + ) − M j ( t + ) [ E j ( M pj ( t + )) − ( − γ ) ϵ ] − γ (cid:27)(cid:19) . B.2. Equilibrium
We present the equilibrium of the model. Because all households and firms face the same condi-tions, they all behave the same in equilibrium, so we drop the subscripts j and k on all variables.The equilibrium can be described by seven variables: output Y ( t ) , employment N ( t ) , theprice level P ( t ) , the wage W ( t ) , the bond price Q ( t ) , the price markup M ( t ) , and the perceivedprice markup M p ( t ) . These seven variables are determined by seven equations.The first equation is the monetary-policy rule, given by (9). The second equation is theproduction function, which is directly obtained from (8):(A37) Y ( t ) = A ( t ) N ( t ) α . The third equation is the usual consumption Euler equation, which is obtained by simplifying(A19). By symmetry X ( t ) = P ( t )/ F ( t ) and Z k ( t ) = F ( t ) Y ( t ) , so (A19) simplifies to(A38) Q ( t ) = δ (cid:69) t (cid:18) P ( t ) Y ( t ) P ( t + ) Y ( t + ) (cid:19) . The fourth equation is the usual expression for the real wage, which is obtained by simplifying62A29). Once again, by symmetry X ( t ) = P ( t )/ F ( t ) and Z k ( t ) = F ( t ) Y ( t ) , so (A29) yields W ( t ) P ( t ) = νν − N ( t ) η Y ( t ) . Combining this equation with (A37), we express the real wage as a function of employment:(A39) W ( t ) P ( t ) = νν − A ( t ) N ( t ) η + α . The fifth equation is the standard link between price markup and employment, which isobtained from the definition of the price markup. In a symmetric economy the price markupis just the inverse of the real marginal cost: M ( t ) = P ( t )/ C ( t ) . Combining the expression of thenominal marginal cost given by (A31) with the value of the real wage given by (A39), we infer thereal marginal cost: C ( t ) P ( t ) = ν ( ν − ) α N ( t ) + η . Since the price markup is the inverse of the real marginal cost, we find(A40) N ( t ) = (cid:20) ( ν − ) αν · M ( t ) (cid:21) /( + η ) . The sixth equation is a pricing equation, which is obtained by simplifying (A36). Recall that Γ ( t ) = δ t X ( ) Z ( )/[ X ( t ) Z ( t )] . Since by symmetry Z ( t ) = F ( t ) Y ( t ) and X ( t ) = P ( t )/ F ( t ) , wehave Γ ( t + ) Γ ( t ) = δ · X ( t ) X ( t + ) · Z ( t ) Z ( t + ) = δ · P ( t ) P ( t + ) · Y ( t ) Y ( t + ) . Hence, (A36) simplifies to(A41) M ( t ) − M ( t ) E ( M p ( t )) = − δγ + δ (cid:69) t (cid:18) M ( t + ) − M ( t + ) (cid:2) E ( M p ( t + )) − ( − γ ) ϵ (cid:3) (cid:19) . This pricing equation shows the dynamic relationship between actual and perceived pricemarkups. Unlike the other equilibrium conditions—which are the same as in the textbook model—the pricing equation is unique to the model with fairness.The seventh and last equation is the law of motion of the perceived price markup. It derivesfrom the law of motion of the perceived marginal cost, given by (7). Since M p ( t ) = P ( t )/ C p ( t ) , (7)implies M p ( t ) = (cid:20) P ( t )( ϵ − ) P ( t )/ ϵ (cid:21) − γ (cid:20) P ( t ) C p ( t − ) (cid:21) γ = (cid:16) ϵϵ − (cid:17) − γ (cid:20) P ( t ) P ( t − ) (cid:21) γ (cid:20) P ( t − ) C p ( t − ) (cid:21) γ . M p ( t ) = (cid:16) ϵϵ − (cid:17) − γ (cid:20) P ( t ) P ( t − ) (cid:21) γ (cid:2) M p ( t − ) (cid:3) γ . B.3. Steady-state equilibrium
We now apply the equilibrium conditions to a steady-state environment, in which all real variablesare constant and all nominal variables grow at the inflation rate, π . We then use these steady-stateconditions to prove lemma 7 and proposition 3.We describe the steady-state equilibrium by six variables: output Y , employment N , inflation π , real interest rate r , price markup M , and perceived price markup M p . These six variables aregoverned by six equations. Steady-state equilibrium conditions.
First, in steady state the consumption Euler equation (A38)gives Q = δ · P ( t ) P ( t + ) . Taking the logarithm of this equation yields − i = − ρ − π , where ρ ≡ − ln ( δ ) is the time discountrate. Hence the steady-state real interest rate r = i − π satisfies r = ρ . Second, in steady state the monetary-policy rule (9) implies that r = i + ( ψ − ) π . Since r = ρ ,the steady-state inflation rate is π = ρ − i ψ − . Third, in steady state the law of motion of the perceived price markup (A42) implies that ( M p ) − γ = (cid:16) ϵϵ − (cid:17) − γ (cid:20) P ( t ) P ( t − ) (cid:21) γ . Taking this expression to the power of 1 /( − γ ) , and noting that in steady state P ( t )/ P ( t − ) = exp ( π ) , we find that the steady-state perceived price markup is(A43) M p = ϵϵ − (cid:18) γ − γ π (cid:19) . = − δγ − M − M E ( M p ) + δ M − M (cid:104) E ( M p ) − ( − γ ) ϵ (cid:105) . Reshuffling this expression, we obtain the following:0 = ( − δγ ) M − ( M − ) E ( M p ) + δ ( M − ) (cid:104) E ( M p ) − ( − γ ) ϵ (cid:105) = (cid:104) − δγ − ( − δ ) E ( M p ) − δ ( − γ ) ϵ (cid:105) M + ( − δ ) E ( M p ) + δ ( − γ ) ϵM = ( − δ ) E ( M p ) + δ ( − γ ) ϵ ( − δ ) E ( M p ) + δ ( − γ ) ϵ − ( − δγ ) M = + ( − δγ )( − δ ) E ( M p ) + ( δ − δγ ) ϵ − ( − δγ ) . (A45)In addition, (A17) shows that in steady state the price elasticity of demand is E ( M p ) = ϵ + ( ϵ − ) γ ϕ ( M p ) . Using this expression, we rewrite the denominator of the fraction in (A45) as ( − δ ) ϵ + ( − δ )( ϵ − ) γ ϕ ( M p ) + ( δ − δγ ) ϵ − ( − δγ ) = ( ϵ − ) (cid:104) ( − δγ ) + ( − δ ) γ ϕ ( M p ) (cid:105) . Plugging this result back into (A45), we obtain the steady-state price markup:(A46) M = + ϵ − · + ( − δ ) γ − δγ ϕ ( M p ) . Fifth, we apply the markup-employment relation (A40) to the steady state. We obtain steady-state employment:(A47) N = (cid:20) ( ν − ) αν · M (cid:21) /( + η ) . Sixth, we apply the production function (A37) to the steady state. We obtain steady-stateoutput: Y = A · N α . Proof of lemma 7.
The expression for the steady-state perceived price markup M p in lemma 7comes from (A43). The expression for the steady-state fairness factor F = F ( M p ) is obtainedby combining (14) with (15). Last, the expression for the steady-state elasticity of the fairnessfunction ϕ = ϕ ( M p ) = − F (cid:48) ( M p ) · M p / F ( M p ) is obtained by noting that with the fairness function6514), F (cid:48) ( M p ) = θ . The properties that M p and ϕ are strictly increasing in π , and that F is weaklydecreasing in π , follow from the assumptions that ϵ > γ ∈ ( , ) , θ >
0, and 1 − χ ≥ Proof of proposition 3.
The expression for the steady-state price markup M in proposition 3comes from (A46). The expression for steady-state employment N comes from (A47). Since δ < γ ∈ ( , ) , and ϕ > π (lemma 7), it follows that M is strictly decreasingin π . And since α > ν > η >
0, and M > π , it follows that N isstrictly increasing in π . B.4. Log-linearized equilibrium
We log-linearize the equilibrium conditions around a steady state. We then use these log-linearizedconditions to prove lemma 6 and proposition 2. We also use these conditions to compute theimpulse responses to monetary and technology shocks that are presented in figures 1 and 2.We describe the log-linearized equilibrium by six variables. The first four variables are thelog-deviations from steady state of output, employment, price markup, and perceived pricemarkup: (cid:98) y ( t ) , (cid:98) n ( t ) , (cid:98) m ( t ) , and (cid:99) m p ( t ) . The last two variables are the deviations from steady state ofthe real-interest and inflation rates: (cid:98) r ( t ) and (cid:98) π ( t ) . These six variables are governed by six linearequations. Log-linear equilibrium conditions.
Several of the original equilibrium conditions take a log-linear form, so they can immediately be log-linearized. The first is the monetary-policy rule (9),which implies(A48) (cid:98) r ( t ) = (cid:98) i ( t ) + ( ψ − ) (cid:98) π ( t ) . The second is the production function (A37), which gives(A49) (cid:98) y ( t ) = (cid:98) a ( t ) + α (cid:98) n ( t ) . The third is markup-employment relation (A40), which yields(A50) (cid:98) m ( t ) = −( + η ) (cid:98) n ( t ) . The fourth is the law of motion for the perceived price markup (A42), which gives(A51) (cid:99) m p ( t ) = γ (cid:104)(cid:98) π ( t ) + (cid:99) m p ( t − ) (cid:105) . S equation.
The fifth equation is the IS equation, which is based on the consumption Eulerequation (A38). We start by computing a log-linear approximation of (A38), as in Gali (2008,pp. 35–36): ln ( Y ( t )) = (cid:69) t ( ln ( Y ( t + ))) + (cid:69) t ( π ( t + )) + ρ − i ( t ) , where ρ = − ln ( δ ) is the time discount rate. Subtracting the steady-state values of both sidesyields (cid:98) y ( t ) = (cid:69) t ( (cid:98) y ( t + )) + (cid:69) t ( (cid:98) π ( t + )) − (cid:98) i ( t ) . Finally, we introduce the values of (cid:98) y ( t ) and (cid:98) y ( t + ) given by (A49), an the value of (cid:98) i ( t ) given bythe monetary-policy rule (9). We obtain the IS equation:(A52) α (cid:98) n ( t ) + ψ (cid:98) π ( t ) = α (cid:69) t ( (cid:98) n ( t + )) + (cid:69) t ( (cid:98) π ( t + )) − (cid:98) i ( t ) − (cid:98) a ( t ) + (cid:69) t ( (cid:98) a ( t + )) . Short-run Phillips curve.
The sixth and final equation is the short-run Phillips curve. It is basedon the pricing equation (A41).As a first step toward computing the Phillips curve, we compute the elasticity of the priceelasticity of demand, E ( M p ) = ϵ + ( ϵ − ) γ ϕ ( M p ) . Given that the elasticity of ϕ ( M p ) is σ (lemma 2),the elasticity of E ( M p ) at the steady state is(A53) d ln ( E ) d ln ( M p ) = ( ϵ − ) γ ϕϵ + ( ϵ − ) γ ϕ · σ ≡ Ω . Second, we introduce the auxiliary function Λ ( M ) = M − M . Its elasticity at the steady state is d ln ( Λ ) d ln ( M ) = MM − − = M − ≡ Ω . Using the value of M in (A46), we find that Ω satisfies(A54) Ω = ( ϵ − ) (cid:20) + ( − δ ) γ − δγ ϕ (cid:21) . The left-hand side of (A41) can be written
LHS = Λ ( M ( t )) · E ( M p ( t )) . Accordingly, around the67teady state the log-linear approximation of LHS is(A55) ln ( LHS ) − ln ( LHS ) = Ω (cid:98) m ( t ) + Ω (cid:99) m p ( t ) . Next, we introduce another auxiliary function: Λ ( M p ) = E ( M p ) − ( − γ ) ϵ = γ (cid:2) ϵ + ( ϵ − ) ϕ ( M p ) (cid:3) . Its elasticity at the steady state is(A56) d ln ( Λ ) d ln ( M p ) = ( ϵ − ) ϕϵ + ( ϵ − ) ϕ · σ ≡ Ω . We also introduce the auxiliary function Λ ( x ) = − δγ + δx . Its elasticity is d ln ( Λ ) d ln ( x ) = δxΛ ≡ Ω . The right-hand side of (A41) (abstracting from the expectation operator) can be written
RHS = Λ ( Λ ( M ( t + )) · Λ ( M p ( t + )) . Hence, around the steady state the log-linear approximation of RHS is(A57) ln ( RHS ) − ln ( RHS ) = Ω · (cid:104) Ω (cid:98) m ( t + ) + Ω (cid:99) m p ( t + ) (cid:105) , where the elasticity Ω is evaluated at Λ = RHS = LHS = E · Λ and x = Λ · Λ . This impliesthat in (A57) we have(A58) Ω = δ Λ · Λ E · Λ = δγ ϵ + ( ϵ − ) ϕϵ + ( ϵ − ) γ ϕ . We now bring these results together. Equation (A41) can be written
LHS = (cid:69) t ( RHS ) . Thisequation also holds in steady state so LHS = RHS . Combining these two equations, we inferexp (cid:16) ln ( LHS ) − ln ( LHS ) (cid:17) = (cid:69) t (cid:16) exp (cid:16) ln ( RHS ) − ln ( RHS ) (cid:17) (cid:17) . Around x =
0, we have exp ( x ) = + x . Applying this approximation to both sides of the previous68quation, we find 1 + ln ( LHS ) − ln ( LHS ) = + (cid:69) t (cid:16) ln ( RHS ) − ln ( RHS ) (cid:17) . We then use the results in (A55) and (A57): Ω (cid:98) m ( t ) + Ω (cid:99) m p ( t ) = Ω · (cid:104) Ω (cid:69) t ( (cid:98) m ( t + )) + Ω (cid:69) t (cid:16) (cid:99) m p ( t + ) (cid:17) (cid:105) . We now divide this equation by Ω ; insert the values of (cid:98) m ( t ) and (cid:98) m ( t + ) given by (A50); andinsert the value of (cid:99) m p ( t + ) given by (A51). We obtain(A59) − ( + η ) Ω Ω (cid:98) n ( t ) + (cid:99) m p ( t ) = − ( + η ) Ω Ω Ω (cid:69) t ( (cid:98) n ( t + )) + γ Ω Ω Ω (cid:69) t (cid:16)(cid:98) π ( t + ) + (cid:99) m p ( t ) (cid:17) . Using (A53), (A54), (A56), and (A58), we find that ( + η ) Ω Ω = ( + η ) ϵ + ( ϵ − ) γ ϕγ ϕσ (cid:20) + ( − δ ) γ − δγ ϕ (cid:21) ≡ λ ( + η ) Ω Ω Ω = ( + η ) δ ϵ + ( ϵ − ) ϕϕσ (cid:20) + ( − δ ) γ − δγ ϕ (cid:21) ≡ λ γ Ω Ω Ω = δγ ϵ + ( ϵ − ) ϕϵ + ( ϵ − ) γ ϕ · ( ϵ − ) ϕσϵ + ( ϵ − ) ϕ · ϵ + ( ϵ − ) γ ϕ ( ϵ − ) γ ϕσ = δγ . Bringing these results into (A59), we obtain the short-run Phillips curve:(A60) ( − δγ ) (cid:99) m p ( t ) − λ (cid:98) n ( t ) = δγ (cid:69) t ( (cid:98) π ( t + )) − λ (cid:69) t ( (cid:98) n ( t + )) . Proof of lemma 6.
The law of motion (12) for the perceived price markup comes from (A51). Theexpression of the perceived price markup as a discounted sum of past inflation rates is obtainedby iterating (A51) backward; and by noting that lim T →∞ γ T · (cid:99) m p ( t − T ) = γ ∈ ( , ) and (cid:99) m p isbounded. Proof of proposition 2.
The short-run Phillips curve (13) comes from (A60). The hybrid expressionof the short-run Phillips curve is obtained by combining (13) with (12).
Blanchard-Kahn representation.
To complete the description of the log-linearized equilibrium,we combine the equilibrium conditions (A51), (A52), and (A60) into a dynamical system of theform proposed by Blanchard and Kahn (1980). Such system is useful to determine the existence69nd uniqueness of an equilibrium, and to solve numerically for the unique equilibrium when itexists.We first combine (A51), (A52), and (A60) into a linear dynamical system: γ γ ψ α λ (cid:99) m p ( t − ) (cid:98) π ( t ) (cid:98) n ( t ) = α − δγ − δγ λ (cid:99) m p ( t ) (cid:69) t ( (cid:98) π ( t + )) (cid:69) t ( (cid:98) n ( t + )) − ζ ( t ) , where ζ ( t ) = (cid:98) i ( t ) + (cid:98) a ( t ) − (cid:69) t ( (cid:98) a ( t + )) is an exogenous shock realized at time t . The inverse of the matrix on the right-hand side is α − δγ − δγ λ − = ( − δγ ) αλ + αδγ λ λ + αδγ − αλ + αδγδγ − λ + αδγ δγλ + αδγ λ + αδγ . Premultiplying the dynamical system by the inverse matrix, we obtain the Blanchard-Kahn formof the system: (cid:99) m p ( t ) (cid:69) t ( (cid:98) π ( t + )) (cid:69) t ( (cid:98) n ( t + )) = γ γ ( − δγ ) αγλ + αδγ λ ψ + αγ ( − δγ ) λ + αδγ ( λ − λ ) αλ + αδγ −( − δγ ) γλ + αδγ [ δψ + δγ − ] γλ + αδγ λ + αδγλ + αδγ (cid:99) m p ( t − ) (cid:98) π ( t ) (cid:98) n ( t ) + λ λ + αδγδγλ + αδγ ζ ( t ) . This dynamical system determines perceived price markup (cid:99) m p ( t ) , inflation (cid:98) π ( t ) , and employment (cid:98) n ( t ) . All the other variables directly follow.Under the calibration in table 4, the Blanchard-Kahn conditions are satisfied, so the equilib-rium exists and is determinate. Indeed, under such calibration, the eigenvalues of the matrix inthe Blanchard-Kahn system are 0 .
30, 1 . + . i , and 1 . − . i : one eigenvalue is within the unitcircle, and two are outside the unit circle. Further, the dynamical system has one predeterminedvariable at time t ( (cid:99) m p ( t − ) ) and two nonpredetermined variables ( (cid:98) n ( t ) and (cid:98) π ( t ) ). As the numberof eigenvalues outside the unit circle matches the number of nonpredetermined variables, thereexists a unique solution to the dynamical system (Blanchard and Kahn 1980, proposition 1).70 .5. Calibration We now calibrate the fairness-related parameters of the New Keynesian model. We do so bymatching the cost passthroughs estimated in microdata and those obtained by simulating thebehavior of a single firm facing a stochastic marginal cost.
Firm problem.
This is a simplified version of the New Keynesian firm problem, which abstractsfrom hiring decisions. The firm chooses price P ( t ) and output Y ( t ) to maximize the expectedpresent-discounted value of profits (cid:69) ∞ (cid:213) t = δ t [ P ( t ) − C ( t )] Y ( t ) , subject to the demand(A61) Y d ( P ( t ) , C p ( t − )) = P ( t ) − ϵ F (cid:18) (cid:16) ϵϵ − (cid:17) − γ (cid:20) P ( t ) C p ( t − ) (cid:21) γ (cid:19) ϵ − and to the law of motion (7) for the perceived marginal cost C p ( t ) . The nominal marginal cost C ( t ) is exogenous and stochastic.To solve the firm’s problem, we set up the Lagrangian: L = (cid:69) ∞ (cid:213) t = δ t (cid:26) [ P ( t ) − C ( t )] Y ( t ) + H ( t ) (cid:2) Y d ( P ( t ) , C p ( t − )) − Y ( t ) (cid:3) + K( t ) (cid:34) (cid:2) C p ( t − ) (cid:3) γ (cid:20) ϵ − ϵ P ( t ) (cid:21) − γ − C p ( t ) (cid:35) (cid:27) , where H ( t ) is the Lagrange multiplier on the demand constraint in period t , and K( t ) is theLagrange multiplier on the perceived marginal cost’s law of motion in period t . First-order condition with respect to output.
The first-order condition with respect to Y ( t ) is ∂ L/ ∂ Y ( t ) =
0. It yields
H ( t ) = P ( t ) (cid:20) − C ( t ) P ( t ) (cid:21) , which is the same equation as (A32) and thus can be rewritten as (A33).71 irst-order condition with respect to price. The first-order condition with respect to P ( t ) is ∂ L/ ∂ P ( t ) =
0, which gives0 = Y ( t ) + H ( t ) ∂ Y d ∂ P + ( − γ )K( t ) C p ( t ) P ( t ) . This equation is the same as (A34); therefore, it can be re-expressed as (A35).
First-order condition with respect to perceived marginal cost.
Finally, the first-order conditionwith respect to C p ( t ) is ∂ L/ ∂ C p ( t ) =
0, which yields0 = δ (cid:69) t (cid:18) H ( t + ) ∂ Y d ∂ C p + γ K( t + ) C p ( t + ) C p ( t ) (cid:19) − K( t ) . Using the elasticity given by (A18), we get K( t ) = δ (cid:69) t (cid:18) H ( t + ) Y ( t + ) C p ( t ) (cid:2) E ( M p ( t + )) − ϵ (cid:3) + γ K( t + ) C p ( t + ) C p ( t ) (cid:19) . Next we multiply the equation by C p ( t )/[ Y ( t ) P ( t )] , and we insert the perceived price markups M p ( t ) = P ( t )/ C p ( t ) and M p ( t + ) = P ( t + )/ C p ( t + ) . We get K( t ) Y ( t ) M p ( t ) = δ (cid:69) t (cid:18) Y ( t + ) P ( t + ) Y ( t ) P ( t ) (cid:26) H ( t + ) P ( t + ) (cid:2) E ( M p ( t + )) − ϵ (cid:3) + γ K( t + ) Y ( t + ) M p ( t + ) (cid:27)(cid:19) . To conclude, we multiply the equation by ( − γ ) ; and we eliminate H ( t + ) using (A33) and K( t ) and K( t + ) using (A35). We obtain the following pricing equation:(A62) M ( t )− M ( t ) E ( M p ( t )) = + δ (cid:69) t (cid:18) Y ( t + ) P ( t + ) Y ( t ) P ( t ) (cid:26) M ( t + )− M ( t + ) (cid:2) E ( M p ( t + ))−( − γ ) ϵ (cid:3) − γ (cid:27)(cid:19) . In steady state, this equation becomes (A44) and can therefore be written as (A46).
Firm pricing.
The firm’s pricing behavior is described by four variables: the price P ( t ) , markup M ( t ) , output Y ( t ) , and perceived markup M p ( t ) . These four variables are determined by fourconditions: the pricing equation (A62), M ( t ) = P ( t )/ C ( t ) , the demand curve (A61), and theperceived markup’s law of motion (A42). Simulations.
We start from a steady-state situation. To be consistent with the simulations offigures 1 and 2, we assume that steady-state inflation is zero, so the marginal cost C is constant72 Years C o s t pa ss t h r ough Instantaneous passthrough = 0.4Two-year passthrough = 0.7
Figure A1.
Simulated dynamics of cost passthrough
The cost passthrough represents the percentage increase in price when the marginal cost increases by 1%. Theempirical estimates of the cost passthrough (0.4 and 0.7) are obtained in section 5.3. The simulations are obtainedfrom the pricing model in appendix B.5 under the calibration in table 4. in steady state. Then we impose an unexpected permanent 1% increase in C . We compute theprice response to this shock by solving the nonlinear dynamical system of four equations thatdescribes firm’s pricing. We then obtain the dynamics of the cost passthrough by calculating thepercentage change in price over time: β ( t ) = P ( t ) − PP × . Calibration procedure.
As explained in section 5.3, we set the shape of the fairness function to(6). We also set the discount factor to δ = .
99. Then, using the simulations, we calibrate thethree main parameters of the model: the concern for fairness, θ , the degree of underinference, γ ,and the elasticity of substitution between goods, ϵ . Our goal is to produce an instantaneous costpassthrough of β = . β = .
7, together with a steady-stateprice markup of M = . θ and γ to some values. Using these valuesand the target M = .
5, we compute ϵ from (A46). In (A46) we use (A8), which holds becausethe fairness function is (6), and because customers are acclimated in steady state as there is noinflation. 73sing the values of θ , γ , and ϵ , we simulate the dynamics of the cost passthrough. We repeatthe simulation for different values of θ and γ until we obtain a passthrough of 0.4 on impact and0.7 after two years. We reach these targets with θ = γ = .
8; the corresponding value of ϵ is2.2. The dynamics of the cost passthrough under this calibration are displayed in figure A1. Appendix C. Textbook New Keynesian model
We describe the textbook New Keynesian model used as benchmark in the simulations of figures 1and 2. The model is borrowed from Gali (2008). The pricing friction in the model is the staggeredpricing of Calvo (1983). The literature alternatively use the price-adjustment cost of Rotemberg(1982). However, both pricing frictions yield the same linearized Phillips curve around the zero-inflation steady state, so the simulations are the same in both cases (Roberts 1995, pp. 976–979).The model’s dynamics around the zero-inflation steady state are governed by an IS equationand a short-run Phillips curve. The IS equation is given by (A52), as in the model with fairness.This IS equation is obtained from equation (12) in Gali (2008, chap. 3), by using logarithmicconsumption utility, and by incorporating the production function (A49) and the monetary-policyrule (9).The short-run Phillips curve is given by (cid:98) π ( t ) = δ (cid:69) t ( (cid:98) π ( t + )) + κ (cid:98) n ( t ) , where κ ≡ ( + η ) · ( − ξ )( − δ ξ ) ξ · αα + ( − α ) ϵ , and ξ is the fraction of firms keeping their prices unchanged each period. This Phillips curve isobtained from equation (21) in Gali (2008, chap. 3), by using logarithmic consumption utility, andby replacing the output gap by α (cid:98) n ( t ) . The IS equation and short-run Phillips curve jointly determine employment (cid:98) n ( t ) and inflation (cid:98) π ( t ) . The other variables directly follow from (cid:98) n ( t ) and (cid:98) π ( t ) . Output (cid:98) y ( t ) is given by (A49). The realinterest rate (cid:98) r ( t ) is given by (A48). The price markup (cid:98) m ( t ) is given by (A50). And since householdsobserve both prices and costs, perceived and actual price markups are equal: (cid:99) m p ( t ) = (cid:98) m ( t ) . The output gap is the logarithmic difference between the actual and natural levels of output. The natural levels ofoutput and employment are reached when prices are flexible, so when the price markup is ϵ /( ϵ − ) . Since ϵ /( ϵ − ) is also the steady-state price markup, we infer from (A40) that the natural level of employment equals steady-stateemployment. Hence, we infer from (8) that the natural level of output is Y n ( t ) = A ( t )( N ) α . Consequently the outputgap is ln ( Y ( t )) − ln ( Y n ( t )) = α [ ln ( N ( t )) − ln ( N )] = α (cid:98) n ( t ) ..