price setting, imperfect information, and the law of one price

BETTY C. DANIEL State University of New York at Albany

Albany, New York

Price Setting, Imperfect Information, and the Law of One Price*

This paper uses a two-country model to investigate the effects of price-setting behavior, with menu costs, imperfect information, and flexible exchange rates, on deviations from the law of one price. When last minute price revision is costly, and when the variances of real and nominal disturbances are similar such that the information content of financial variables is low, then firms might choose not to adjust domestic currency and foreign currency prices to signals from observable financial variables. When a subset of prices adjusts, the most likely candidates are export prices. However, the adjustment will fall short of that needed for the law of one price.

1. Introduction This paper examines the role of a flexible exchange rate sys-

tem in creating deviations from the law of one price. Empirically, prices in asset markets behave differently from prices in goods markets. Asset market prices exhibit near random walk behavior and high variance. Goods market price changes, in contrast, show sub- stantial serial correlation and much lower variance. These stylized facts imply that the short-run dynamic behavior of deviations from the law of one price should differ when exchange rates are determined in asset markets compared with regimes in which they are officially pegged.

Dornbusch (1976) created a model of exchange rate dynamics which explicitly focused on the differential behavior of prices in goods markets and in asset markets. However, this model and the Mun- dell-Fleming model, on which it is based, impose an asymmetry in the pricing of domestic goods and imported goods which might not be justified by optimizing behavior. Specifically, these models assume that the domestic currency price of domestic goods adjusts slowly to a disturbance, but that the domestic currency price of foreign goods adjusts instantaneously to the exchange rate, contin-

*The author would like to thank Lars Svensson, participants at the 1987 NBER workshop on purchasing power parity, and two anonymous referees for helpful com- ments on earlier drafts.

Journal of Macroeconomics, Summer 1992, Vol. 14, No. 3, pp. 383-415 Copyright © 1992 by Louisiana State University Press 0164-0704/92/$1.50

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Betty C. Daniel

uously maintaining the law of one price for imported goods. Flood and Hodrick (1984) have criticized this asymmetry.

This paper provides an analysis of the pass-through problem, whereby prices might fail to adjust completely to an exchange rate change, in a model in which the exchange rate is endogenously determined. Models which treat the exchange rate as exogenous fail to recognize that the primary disturbances, which are responsible for the exchange rate change, can also exert an independent influ- ence on the firm's desired price (Dornbusch 1987; Krugman 1986).

The paper focuses on a firm's optimal pricing decision subject to imperfect information and menu costs. In a closed economy model, Parkin (1986) showed that menu costs generate different optimal policy behavior depending on the money supply growth process. The setup of the model is as follows. Many firms are assumed to produce differentiated products, implying some degree of market power but no strategic interaction. Following Aizenman (1984), firms are assumed to set prices before the period. Domestic (foreign) residents are assumed to make all their purchases with domestic (foreign) currency, so that traded-goods producers must set both a domestic currency and a foreign currency price. At the beginning of the period they receive financial news which yields new information on the profit maximizing price. Last minute price revision is costly, and price adjustment occurs only if the firm believes the benefits exceed the costs.

The financial variables, which the firm observes, are endogenously determined in response to primary shocks. To maintain tractability, it is necessary to rely on a conventional ad hoc macroeconomic model for these solutions. The primary shocks are disturbances to IS and LM curves. Potential output is assumed to be fixed. The model is therefore a general equilibrium model, but it fails to be fully optimizing.

The primary results of the paper are based on the interaction of menu costs and imperfect information. News on financial variables, including the exchange rate, is used by the firm to extract a signal regarding the change in the profit-maximizing price. Menu costs imply that prices will not adjust whenever the expected gains fall short of the menu costs. It is shown that imperfect information reduces the expected gains, thereby reducing the incidence of adjustment over a given distribution of disturbances.

Furthermore, the interaction of menu costs and imperfect information imply that the gains to adjusting some prices exceed the

384

Price Setting

gains to adjusting others, thereby yielding greater adjustment fre- quency for some types of prices. Specifically, if financial signals are weak, such that exchange rate appreciation yields little information on whether aggregate demand expanded or the money supply con- tracted, then firms will react explicitly to the exchange rate change. Although the exchange rate change is due to aggregate shocks, it behaves as an idiosyncratic shock to desired prices of commodities in different markets (Ball and Romer 1987a). It is shown that foreign currency prices are most likely to adjust, but not by enough to maintain the law of one price. It is also shown that country size plays a role in the decision to adjust foreign currency prices, with firms in the small country more likely to adjust than firms in the large country. The result that foreign currency prices are most likely to adjust to exchange rate changes is consistent with the assumption in the Dornbusch and Mundell-Fleming models, but the result, that the price will not adjust enough to maintain the law of one price, is not.

It is important to emphasize that menu costs and imperfect information together are necessary to the results. Menu costs and perfect information would imply that the prices most likely to adjust would depend on the origin of each shock and not directly on the exchange rate. Imperfect information without menu costs would imply that all prices always adjust, implying continuous maintenance of the law of one price since there is no lack of information on the exchange rate.

It should also be noted that there are other sources of deviations from the law of one price which are not considered in this paper. These include different demand elasticities at home and abroad or different market structures. Law of one price deviations due to these causes would not be substantially different under fixed and flexible exchange rates. Other causes, like strategic interaction among oligopolists and intertemporal linkages giving rise to "hysteresis," are interesting but diflqcult to incorporate into a fully specified macroeconomic model with an endogenous exchange rate (Dornbusch 1987; Krugman 1986).

The paper is organized as follows. Section 2 presents the model of firms. Section 3 contains the macroeconomic model and its solution together with the solution for the full information profit-maximizing prices of the firms. Section 4 presents the issues involved in the firm's price-setting decisions. Section 5 examines the implications for the law of one price, and Section 6 contains conclusions.

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2. Model of Firms

Demand and Cost There are two countries in the world economy, and each

country has traded and non-traded goods producers. There are many producers of each type of good, and goods p roduced by any one firm are imperfect substitutes for goods produced by any other firm. The imperfect substitute assumption gives firms some market power, but the assumption of many firms rules out strategic behavior. Pro- ducers and retailers are assumed to be vertically integrated.

Additional assumptions, which require countr ies to be similar, are imposed. The purpose is to create a simple model with the focus on imperfect information and menu costs as short-run deter- minants of law of one price deviations. D e m a n d elasticities are assumed to be identical across countries, and across firms and goods as well. Identical elasticities in both markets and the assumption that imperfect information is a short-run p h e n o m e n o n el iminate the incent ive for a t raded goods p roduce r to a t t empt to segment markets and charge different prices, t he reby eliminating market seg- menta t ion as a possible cause for the failure of the law of one price. Fu r the rmore , tastes in the two countries are assumed to be identical, so that the foreign and domest ic price indices have identical weights.

D e m a n d for an individual firm's good is specified very simply. It is assumed to depend on the firm's price for the good relative to the aggregate price level and on the level of aggregate demand.1 More specifically, when the firm's price level (Pkt) equals the aggregate price level (Pt), demand facing the non- t raded goods producer is a constant (r), mult ipl ied by aggregate d em an d (Dr ) , di- vided by the n u m b e r of non- t raded goods firms (nk). The constant ('r) represents the fraction of total demand allocated to non- t raded goods when their relative price is unity. 2 A representa t ive domest ic

LAt this stage, the specification looks like that obtained by Blanchard and Fischer (1989) and Ball and Romer (1987a, 1987b) in their static optimizing models. Un- fortunately, it is not possible to use their models here. The results in this paper rely on confusion between IS and LM curve shocks. In a static model, the interest rate disappears, and all aggregate demand shocks become LM shocks. The intertemporal optimization model is a much more ambitious project. Svensson (1986) provides a model which could be used for a benchmark but at the cost of consid- erable complication of the paper.

~This specification assures that aggregation of demand for each commodity by residents of a single country equals total expenditures (Dr) by those residents. A macroeconomic model is used later to specify total expenditure.

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Price Setting

non-traded goods producer faces a demand curve (Dkt) of the form

Dkt = (Pkt/Pt Dr, (ld)

where b is the elasticity of demand. A foreign non-traded goods producer faces an analogous demand function given by

D~=(P~/P*)-b(-~)D*. (it')

An asterisk (*) is used to denote a foreign variable, and r = "r*. Traded-goods producers sell in both domestic and foreign

markets. Consumers are assumed to hold and use only their own country's currency, and prices to consumers in each country are assumed to be quoted in terms of a single num6raire. This requires that producer-sellers quote a domestic currency price for domestic sales, and a foreign currency price for foreign sales. Demand facing a representative domestic traded goods producer is the sum of domestic demand (Dit) and foreign demand (D*), where each component depends on the level of aggregate demand in the country, the fraction of aggregate demand allocated to traded goods when all prices are the same (1 - ~), the number of traded-goods producers, and the relative price quoted by the firm. Letting ni and nj represent respectively the number of domestic and foreign traded- goods producers, it is expressed as

D,, + D * = [(e,Jet) -b D, + ( e ~ / e , * ) - ~ O,*] 1 - (gd)

where the first (second) term represents demand by domestic (foreign) residents. Demand facing a representative foreign traded-goods producer, indexed by j, is analogous:

o~, + o ~ = [(e~,/e,) -~ D, + (e~,/et*) -~ o , * ] 1 - ( ~

These specifications implicitly assume that arbitrage activity does not force the law of one price. This is reasonable empirically (Isard 1977). Theoretically, it requires that arbitrage activity be too

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costly, given the deviations from the law of one price permitted by firms.

The aggregate price level in each country is assumed to be an index with constant identical weights. In the absence of Cobb-Douglas utility, this should be viewed as an approximation to the true index, in which the weights depend on relative prices. Letting the weights be superscripted 0s, the price indices take the form

P, = (e,,)e(pit)°*(pkt)x-e-°* ; (3)

( e , ~0~o, ~0"/o, ~1-0-0. p* ] \ x j t ) \L kt] (4)

Real costs are assumed to be proportional to squared sales by a factor of C/2. The assumption of increasing marginal costs, im- plied by this specification, is important to the analysis. If marginal costs were constant, an aggregate demand disturbance would not generate an increase in a firm's desired price, unless at least some other firms had changed price. Also implicit in the specification is symmetry in wage and price adjustment. Nominal costs are proportional to the nominal domestic price level; thus, costs are sticky whenever the domestic price level is sticky. 3

Optimization Problem Firms are assumed to be price setters who operate in a world

in which financial information is more readily available than other information. Given the firm's choices on prices, production is just sufficient to satisfy demand. Following Aizenman (1984), firms must announce prices for the current period at the end of the immedi- ately preceeding period, before any current period information becomes available. When prices are set, firms are assumed to have full information on all disturbances to date but no information on any disturbances which will occur in the coming period. When the period begins, information on financial variables, which includes interest rates and the exchange rate, becomes available. These variables provide imperfect information on the disturbances, with full information arriving only at the end of the period. Beginning-of- the-period price revision is assumed to be costly. However, firms can choose to adjust their prices based on the imperfect information

3Implicitly, wage and price decisions could be made simultaneously with identical information. Technological diminishing marginal productivity could account for rising marginal costs. The wage sector is not explicitly modeled, however.

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Price Setting

provided by interest rates and the exchange rate and pay a fixed cost for each price they change. Alternatively, they can maintain their quoted price. In either case, they must satisfy the demand resulting from their pricing decision.

The firm therefore faces two kinds of choices. First, before the period begins, it must set prices. Second, when the period begins, it must decide whether to revise prices based on new financial information and pay the associated fixed cost for each price revision. Given the resulting prices, production satisfies demand. It is assumed that output is not storable so these decisions are made to maximize period by period profits.

To solve these two problems it is necessary to have expressions for the values of full information prices. They are determined to maximize real full information profits. For a domestic traded- goods producer, they can be expressed as

~r(e,,,e* ) = [(e,, /e,) ~-b Dt + (s ,e* / e , ) ( e* / e * ) -b D* ] 1 - "r

- (C/2)(D,t + D,*)2, (5)

where St is the domestic currency price of foreign currency. First-order conditions with respect to Pit and P* are, respec-

tively,

/ ] D* = 0 ' (6) P,t \ Pt, I \ e* ] '

b C P t ( X - ' r ~ [ ( P , t ~ -~ (P*~-~ ] - - - - - - D r + D* = 0 . (7) ( 1 - b ) S t + P * \ n , + n / L \ P , ] \ e * /

The first-order conditions imply that optimal full information prices, denoted by (-), are related by the law of one price according to

P,, = St i~, (8)

where

+ ( p, ~-b -])1/~+~ = 0 ,

b > l . (9)

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Note that PPP deviaions (Pt /S tP* ~ 1) affect the desired relative price (Pit /Pt) of traded goods. To understand why, consider a domestic traded-goods firm facing an increase in the exchange rate when all prices are unchanged. An increase in the exchange rate is an upward shift in the firm's marginal revenue curve. The firm maximizes profits by raising sales and prices. Therefore, the firm raises its domestic currency price and reduces its foreign currency price. The reduction is less than proportionate to the change in the exchange rate. The resulting increase in e i t / P t and fall in P * / P * r e -

distribute sales between domestic and foreign markets. This effect is more important the larger the size of the foreign market, since the size of the foreign market determines the magnitude of the shift in the marginal revenue curve.

Logarithmically linear approximations of the full information optimal prices for all firms will be useful at a later stage of the analysis. They are expressed as

1 Pit = Pt + - -

b + l [c + t~dt + (1 - ~)d*

- b(1 - @ ) ( p t - e t - p * ) ] , (iod)

P* = P i , - et = pt - e, + - - 1

b + l [c + d~dt + (I - @)d*

- b ( 1 - @ ) ( p , - e t - p * ) ] , (lid)

P j t = p ~ + e , = p * + e ~ + - - 1

b + l [c + @d, + (I - qJ)d*

+ bqJ(pt - e , - p*)] , (iif)

p ? = p * + ~ 1

b + l [c + d~dt + (1 - t~)d* + b@(p, - et - p*)] , (10f)

Pkt = Pt + - - 1

(c~ + , i t ) , (12d) b + l

p~, = p , , + - - 1

(c~ + d~) ; (120 b + l

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Price Setting

where

et = In St,

c , n

[( ck = In

[( c~' = In

and o ther small let ters represen t the logarithms of capital letters. Given the definitions of ~" and 0 above, note that • = 1 - 0 - 0". Assuming that t rade is balanced with ou tpu t and demand at thei r initial long-run equi l ibr ium levels, that is, imports (0*Y) = exports (0Y*), it is straightforward to show that 0 = ~(1 - x), and 0* = (1 - t~)(1 - "r), where ~ is the fraction of long-run output accounted for by the domest ic country. Let t ing Y denote long-run domestic output , where all prices are identical, ~ = Y/(:Y + :Y*). Note that the optimal price depends on the level of aggregate demand, on the exchange rate, and on the level of o ther prices.

Imper fec t information prevents the firm from maintaining the full information optimizing price. It is useful to wri te an expression for the firm's loss due to imperfect information by taking the negative of a second-order Taylor 's series approximation of real profits about thei r full information optimal level and dropping terms which the firm cannot affect with its pricing decision. 4 Taking second derivatives with respect to Pit and Pi*, s and evaluating the constants at thei r full equi l ibr ium levels, the loss for a domest ic t raded-goods p roducer can be expressed as

Cb = p,,)] L, --:-(Z)~(b[~(p,, - #,,) + (1 - ~J)(p* - -* 2

Z

+ [dg(p,t- #,t) 2 + (1 - ~)(p* - # . )2]}. (13)

4Roternberg (1982) follows this procedure. 5See Appendix A for the derivation.

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B e t t y C. Dan ie l

The p's represent logarithms of P's. Z is equilibrium firm size, which is assumed identical across firms, and is given by

( l - r / ( (n-kk) ( ~ - ) Z = \ n i + n / ~ + ~*) = "r ~ = "r ~ , .

The finn's problem can now be simply stated. It must choose prices conditional on imperfect information and satisfy the resulting demand. The price-setting decision contains two stages. Before the period begins, changing prices for the coining period is costless. However, at this stage, there is no information on any disturbances which might occur during the period. Therefore, at the first stage, the firm chooses preset prices to minimize the expected value of the loss function, conditional on information available at the end of the preceeding period. Let E~-I denote the expectation, conditional on information available at the end of the period, where the + superscript is used to denote the fact that more information is available at the end of the period than at the beginning of the period. Preset prices are:

and

+ -

Pit = E t - l P~,

p* = E ; - _ ¢ * .

The second stage in the firm's price-setting decision comes at the beginning of the next period, when it receives new information provided by interest rates and exchange rates. Conditional on this new information, the firm chooses whether to adjust its preset prices. The expected gain to adjusting, conditional on information provided by financial variables, is compared with the fixed cost of adjusting. Using Equations (10)-(12), for desired prices, in the loss function given by Equation (13) reveals that it is necessary to solve a macroeconomic model for endogenous values of domestic and foreign expenditure and for exchange rates. Aggregate price levels will then be determined by the collective decisions of firms. Also, since disturbances are temporary, it is necessary to solve only for surprises to the endogenous variables. This second-stage decision is the focus of the remainder of the paper.

Foreign traded-goods producers face an analogous problem. Non-traded-goods producers face a more conventional profit max-

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P r i c e S e t t i n g

imization problem, as they must choose only an own-currency price, and they are affected by only their own country's aggregate demand. Their problem is otherwise analogous and is not explicitly presented in this section.

3. Macroeconomic Model

E q u a t i o n s

The next step in the solution is to specify a macro model which permits simultaneous determination of the aggregate demand variables and of the exchange rate, enabling calculation of the expected value of the loss function. The macro model is a reasonably conventional two-country, ad hoc specification, aside from the special role played by information and price setting. Unfortunately, use of an intertemporal-optimizing model to provide the simultaneous solution, while desirable, would vastly complicate the analysis. 8

The equations are as follows, where all variables except the interest rate are expressed as natural logarithms.

m , - p , = - h i t + d, + n,, m* - p* = -h i* + d* + n* ; (14)

d , = 0 + ~ / ( r - r , ) + v , , d * = 0 * + ~ / ( P - r * ) + v * ; (15)

it = i * + E t ( e t+1 - e,) ; (16)

r , = i t - E t ( P t + z - P , ) , r * = i * - E t ( p * + I - p * ) . (17)

Equations (14) specify real demand for each country 's money. Transactions demand for the domestic currency depends positively on expenditure by domestic residents (dr), negatively on the domestic interest rate (it), and on a disturbance term (nt). Both the money supply and the disturbance to money demand are allowed to be stochastic with their difference being white noise according to

m , - n t = rh + m , - i - n , - z + aq,, with ~qt - N(0, z ) .

Foreign variables are assumed to have analogous behavior with an independent variance denoted by cr *~.

6See the first footnote.

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Betty C. Daniel

Equations (15) specify domest ic and foreign expend i tu re (demand) as equal to long-run real ou tput adjusted for the deviation of the country 's real interest rate (rt) from its long-run equi l ibr ium level (P) and for a disturbance te rm (vt). The dis turbance te rms are white noise stochastic processes with zero mean and variances given

2 and .2 by cr, o', , respectively. All dis turbance terms are assumed to be independent . This specification of demand ignores the effects of wealth transfers due to current account imbalance as in other models. 7 It is consistent with the standard specifications in the Dornbusch (1976) and Mundel l -F leming models in which foreign and domest ic demand for domestic goods depends on income, interest rates, and relative prices, s

Equat ion (16) is the interest pari ty condition. Et is the expectations operator , conditional on information available at the beginning of per iod t. Equat ions (17) specify the domest ic and foreign real interest rates as equal to the nominal rate minus expec ted inflation.

Solution The purpose of this section is to provide the solutions from

the macroeconomic model for surprises to both domest ic and foreign expendi ture and for surprises to purchasing power pari ty deviations. These variables en te r the firm's loss function. To solve the model, it is convenient to first solve for the expec ted one-per iod- ahead equil ibrium. Since all disturbances are temporary , this is the steady-state equi l ibr ium which would obtain, were all disturbances to take on their expected values of zero. These values can then be used to substi tute for expectations in solving the model. 9

The expected one-per iod-ahead equi l ibr ium is character ized by constancy of prices and exchange rates, so that, by Equat ions (16) and (17), real and nominal interest rates are equal and thei r domest ic and foreign counterpar ts are equal. In Equat ions (15), r = ~, requir ing cur ren t account balance, that is, in each count ry

rThis feature of the model simplifies the solution procedure, but it is undesirable from a theoretical point of view. Since the purpose of the model is to examine short-run deviations from the law of one price, this should not be a major concern.

8To see this, aggregate across i domestic producers using Equation (2d). This yields an expression for demand by domestic and foreign residents for domestic goods. The logarithms of D and of D* are given by Equations (15).

9In a previous version of the paper, errors were allowed to be permanent with no significant differences to the results on pricing decisions. Requiring errors to be purely temporary simplifies the calculations without sacrificing significant content.

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income (0) must equal expenditure (d). Normalizations sufficient to assure purchasing power parity in the expected steady state are de- rived in Appendix B. Imposing these, purchasing power parity (p = e + p*) and money market equilibrium Equations (I4) can be used to solve for steady-state aggregate price levels as

/ 5 = r h + h ~ - 0 , ~ 5 * = r h * + k ~ - 0 * . (18)

The next step in the solution procedure is determination of the information provided by the observation of financial variables. Using interest rate parity (Equation 16), the exchange rate can be expressed as

e , = E , e t + l + i * - i t . (19)

Using Equations (14) and (15), domestic and foreign interest rates are expressed as

it = - - 1

k + 7 {Y + Pt - rh + v t - 'tit + 'y[f" + E t ( P t + ~ - pt)]} ; (20d)

1 i * = - - . k+- , /

{0" + p,* - rh* + v,* - n,* + ~ [ e + E,(p,*÷, - p ,*) ]} . (9.0t)

Equations (20) demonstrate that observation of each interest rate is equivalent to observation of the difference between the aggregate demand disturbance and the excess money demand disturbance in each country; that is, v t - "qt and v * - ~i* This is because the other variables determining interest rates are either known or can be calculated given knowledge of the model. By Equation (19), observation of the exchange rate provides no additional information. Therefore, observation of financial variables, which include interest rates and the exchange rate, is equivalent to observation of vt - tit and v * - "q*

Letting 2t = x t - . E t + - s X t , it is possible to solve for dr, dr, and d* as functions of ~3t, ~3" and disturbance terms. Aggregate price surprises are determined in the next section as the aggregate of the behavior of individual firms. The solutions for the one-period-ahead expected exchange rate, given by purchasing power parity and Equations (18), together with the result that the expected one-period-ahead level of interest rates is ~, and Equations (14) through (17) allow expression of exchange rate surprises as

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Betty C. Daniel

~, = [ ( 1 - ~)(p,*- p,) + ( , ] ,

where

~, = "q, - v, - "q* + v * . (21)

Note that the price adjustment, such that domestic prices move with e, and foreign prices move inversely, reduces exchange rate surprises for -/ < 1.

Semi-reduced forms for domestic and foreign aggregate demand surprises can be calculated using Equations (14)-(17):

(~, = _ _ 1 [~/~q, + b y , - ~/(1 + h ) ~ , ] . (22d)

~ . = 1 [ T q * + b y * - - /(1 + h ) ~ * ] . (22f) h + ' y

Using Equations (10)-(12), (21), and (22), the semi-reduced forms for unexpected changes in desired prices for domestic and foreign traded- and non-traded-goods producers become

- = to, + b ( 1 - ~)~,

+ {(b + 1)(~ - 1) + (1 + h)[1 + 0(b - ./)1}~,

+ (1 - th)(1 + h)(b - "y)l 3. ) ;

pff -- E+-l ~)~t = ~ t - - bt~-t + {(b + 1)(-/- 1)

(23d)

+ (1 + h)[1 + (1 - O)(b - "y)]}~*

+ th(1 + k)(b - ~)/~,] ; /

(23f)

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Price Setting

~* - / ~ - 1 ~ * = g - $ - S , - (1 + b~)~ ,

;6j, - E+_l/)jr =

+ (1 + h) [1 + ~ (b - 3 ') ]~, + [ (b + 1)(3' - 1)

+ (1 - t~)(1 + ~,)(b - y)]/~* } ;

1 1

(24d)

+ (1 + h) [1 + (1 - qJ)(b - 3')] ~ *

/~k, - E t +_,/~k, =

/5~ - E ~ - I / ~ =

where

+ [(b + 1)(3' - 1)

+ ~(1 + h)(b - 3')]/~, } ; (24f)

+ [ ( b + 1)(3 ' - 1 ) + ( 1 + h ) ( l + b - 3 ' ) ] O , } ;

+ [(b + 1)(3' - 1) + (1 + h)(1 + b - 3')1/~*~ ; (25f) J

to, = q,(x~, + yn , ) + (1 - q , ) (x~* + yn,*);

and where ~, is given following Equations (21). It is useful to point out some features of the solutions. Recall

that observabil i ty of interest rates implies observabil i ty of "qt - vt and its foreign counterpart . Therefore, et is observable due to the observabili ty of financial variables, that is, "qt - vt and "q*- v* The term tot, which represents the demand effect of disturbances, is not observable. In deciding whether to adjust prices, the firm must form an expectat ion of it, conditional on observation of financial variables.

Also, note that changes in domest ic or foreign aggregate price levels generally have an ambiguous effect on a particular desired

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Betty C. Daniel

price for the firm. This is because an increase in the aggregate price allows a firm to raise its own price and maintain sales, but the increase in the aggregate price also lowers total aggregate demand inducing the firm to lower its price.

4. Price-Setting Behavior This section focuses on the firm's beginning-of-the-period de-

cision regarding whether it wants to revise its preset prices, based on the arrival of financial news. This decision is made by comparing the expected gain from adjustment, calculated using the loss function, with the costs of adjustment. Firms can choose to forego adjustment entirely, to adjust a subset of prices, or to adjust all prices. In the following section, each of these decisions is considered in turn.

The Decision Not to Adjus t An equilibrium, in which no firm adjusts prices, will exist if

the expected gain to adjusting, conditional on all firms maintaining price, falls short of the fixed adjustment fee. The expected gain, conditional on observation of financial variables, equals the expected value of the loss function, given that the firm does not adjust price, minus its expected value, given price adjustment.

The expected value of the loss function for a domestic traded- goods producer, conditional on observation of financial variables, when all firms maintain prices, is calculated in Appendix C. Following the arrival of financial information at the beginning of the period, a firm can choose to adjust, thereby reducing the expected loss. This reduction to expected loss is the gain to adjustment. However, due to the absence of full information, an unavoidable expected loss remains. Therefore, total loss is comprised of two components, the expected gain to adjustment conditional on imperfect beginning-of- the-period information, EtGt, and a remaining irreducible component, L. For the domestic traded-goods producer, expected loss can be expressed according to

EtLt = EtGt + L ,

EtGt = K(b + i)[(Eteot) 2 + (1 - d))(1 + bt~)e~ - 2(1 - t~)etEtto,],

[_, = K(b + 1)[qJ2~b + (1 - ,)200"1,

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Price Sett ing

where

K = b C Z ~

2(k + -y)2(b + 1) z '

2 2 ~2 ~2 O'rlO" u O'~ O" v

, = ~ + ~, ** • (26) % cr~ ~ . 2 + ¢ r . 2 ,

Et¢o, = ~(~t- v,) + ~*(xl* - v*) ,

= , [ v a - x ( 1 - a ) ] ; ~* = ( t - , ) [ ~ a * - x ( 1 - a * ) ] ,

z ~r,2 fl = % - ~* -

2 + z , ~r .2 + ~ , z " O'~q O" v

Loss for the foreign traded-goods producer is analogous. Note, first, that the expected gain is proportional to the square

of small disturbances, as in Ackerlof and Yellen (1985) and Blan- chard and Kiyotaki (1985), and is therefore second-order small. Small menu costs are sufficient to justify a failure to adjust.

Next, consider the role of imperfect information. Imperfect information implies that the full loss created by a deviation of price from its optimal full information level cannot be eliminated. There- fore, the absence of full information reduces the profitability of adjustment. The irreducible component of expected loss drives a wedge between the total expected loss and the expected gain to adjustment. To further examine the role of information, it is useful to calculate the average expected gain. In doing so, it is convenient to assume that size adjusted variances are the same in each country, l° that is, ~(r 2 = (1 - t~)cr *z. Under this assumption, the average expected gain to a traded-goods producer, conditional on other firms maintaining prices, is given by

e [ e , c , ] = K(b + 1){,[~a - x(1 - a)] ~ + (1 + b , ) ] ) ( ~ + ~ ) . (27)

When information is imperfect, financial variables serve two roles, a direct role and an information role. The direct role occurs

2 through the exchange rate. It is reflected by the coefficients on e, in Equation (26) and by the 1 + bd~ term in Equation (27). Recall that an increase in the exchange rate raises the desired relative price of sales in the domestic market and lowers the desired relative price

~°This eliminates ambiguous cross product terms, simplifying the exposition. The more general expression is contained in Equation (C10) in Appendix C.

399

Betty C. Daniel

for foreign sales as the domestic producer reacts to the shift in his marginal revenue curve. This is more important the larger the size of the foreign market.

Financial variables also affect desired prices by providing information on the disturbances which directly affect aggregate demand and the reby des i red prices. Recall that information on ~lt - vt and its foreign counterpart are provided by observation of domestic and foreign interest rates. The E t t o t term in Equation (26) and the ~/~ - h(1 - ll) term in Equation (27) each reflect this information role. For example, an increase in the exchange rate accompanied by a reduction in the domestic interest rate could signal an increase in excess money demand or a reduction in domestic aggregate demand. For a disturbance of the first type, prices of domestic goods should rise, while for a disturbance of the second type, prices of domestic goods should fall.

The information role of financial variables makes a smaller contribution to expected gain the poorer the quality of information, that is, the weaker the signals. Recall that tot is unobservable at the beginning of the period. Financial signals are weak when they fail to provide much information on the value of this unobservable term, that is, when E t t o t ~ O. In this model signals are weak when ~/ll

h(1 - Ft). This is the case in which it is not possible to distin- guish an increase in money demand from a reduction in domestic aggregate demand, that is, an IS shock from an LM shock. From Equation (27), the weaker the signal, the smaller the average expected gain. Equation (26) can be used to .show that a weak signal makes the irreducible loss a larger fraction of the total loss. To see this, note that for a given sum of variances, ~b is larger the more similar the variances. This is because a weak signal implies uncer- tainty about how to adjust price.

In summary, the gain to adjusting is proportional to the square of small disturbances, implying that small menu costs are sufficient to impede price adjustment. Furthermore, imperfect information introduces a wedge between the total expected loss and the expected gain to adjustment. The poorer the quality of information, that is, the weaker the signal from financial variables, the smaller is the expected gain to adjusting. Equilibria in which no prices change are likely implying failure of the law of one price.

Prices Most Likely to Adjust with Weak Financial Signals Assume, in this section, that the signal from financial variables

is so weak that these terms are negligible in magnitude. Ettot = O.

400

Price S e t t i n g

This focuses attention on the direct role of the exchange rate, given by et. From Equation (26), the expected gain to adjusting, conditional on the observation of financial variables when all firms maintain prices, becomes

[EtGtlEt~ot = O] = K ( b + 1)(1 - 0)(1 + bO)E,e2t . (26')

Since an exchange rate change affects the desired price of each type of good differently (Equations [23]-[25]), adjustment of some prices should yield higher expected gains than adjustment of others.

Consider the question of which prices are most likely to adjust under the assumption that firms face identical adjustment costs. The possibility of the existence of an equilibrium, in which a subset of prices adjusts, is addressed subsequently.

First, when financial signals are weak, so that firms are re- sponding only to the direct role of the exchange rate, export prices are most likely to adjust. To demonstrate this, it is necessary to determine the desired price change for a traded-goods producer when the firm chooses to adjust price in only one currency. The firm adjusts the single price to minimize the expected value of the loss function in Equation (13), yielding

b 0 p * = Et~i*tt + E t ( ~ , t - E +_x/Sit), (28)

b(1 - 0) + 1

if it adjusts foreign-currency prices only, or

b(1 - , ) = - - - E t - l p . ) , (29) p . EtlS,t + Et(15* + -*

b 0 + l

ff it adjusts domestic currency prices only. Note, first, that the price does not adjust to its expected full information level. This can be explained as follows. If the exchange rate rises, /5. rises and/5" falls. If the firm were to reduce p* (raise p . ) to its expected full information level, it would generate too many (few) total sales since p . (p*) is not being adjusted upwards (downwards).

Next, substitution of Equations (23d) and (24d) into Equation (29), reveals that p . in Equation (29) equals Et+_l~.. The firm ob- tains no gain from adjusting the domestic currency price alone. This can be explained as follows. An increase in the exchange rate raises the firm's marginal revenue, The optimal response is to raise total

401

B e t t y C. Dan ie l

sales and to redistribute them towards the foreign country. If foreign currency prices are constant, the goal of raising total sales tends to decrease desired p., but the goal of redistribution increases desired p.. This implies that when a traded-goods producer chooses to adjust only one price, it will choose the foreign currency price. A reduction in the foreign currency price raises total sales and re- distributes them toward the foreign country.

It is also possible to show that when financial signals are weak (Etto t = 0), such that firms are reacting only to the direct role of the exchange rate, small countries receive greater benefits from adjustment than their counterparts in large countries. By letting the export price be determined to maximize the expected gain, according to Equation (28), Equation (26') can be broken into two components. The average expected gain from adjusting the export price alone is given by n

K ( b + 1)~(1 - ,)] (30)

The remaining loss, which could be eliminated were the own currency price also adjusted, is

K ( b + 1)b2~(1 - ~)2] E'" z (31)

"From Equations (13) and (28). the expected gain to adjusting the export price for the domestic traded-goods producer, before substitution of expressions for prices, is expressed as

b ~ - - ,-) + ~ E,{b,(v,, - f,,,) + [b(1 - , ) + l](v,,* - ~*)}' .

The remaining expected reducible loss is

(b) K*(l+b) E , [ p . - P . ] 2. b'~--~ ~ + 1

To obtain the expression in Equation (30), calculate the remaining expected re- _ + - ducible loss by substituting p . - Et-tp.. and by using Equation (28) to substitute

tbr p~. Recognize that E,(E,#* - -* 2 p.) forms the irreducible component of the loss. The expected gain is the difference between the total expected reducible loss and the remaining expected reducible loss.

402

Price Setting

The gain, given by Equation (30), falls as th rises, holding constant the sum of all variances.l~ Although the full equilibrium change in the export price is large for a large country, the foreign market is small implying a small loss in the absence of adjustment.

Finally, using Equations (25) and the non-traded counterpart to Equation (13), it is possible to show that the average gain to adjusting for non-traded-goods producers, when E t ( o t = 0 and when other firms do not adjust, is zero. This is because the exchange rate plays no direct role in the full information profit-maximizing price for non-traded goods. 13

In summary, since exchange rate changes affect desired prices differently, some prices are more likely to adjust than others. In particular, export prices are more likely to adjust than domestic currency prices of traded or non-traded goods. Also, a small country is more likely to adjust its export prices than a large country.

Equilibria When a Subset of Prices Adjusts These considerations suggest that, when financial signals are

weak such that E t o ) t - - 0 , there are two alternative likely types of equilibria in which only some prices adjust. The first occurs when the only prices to adjust are the export prices of a small country, and the second exists when the only prices to adjust are the export prices of more similarly sized countries. To show that these equilibria are possible, it is necessary to show that the accompanying aggregate price adjustment does not induce adjustment in other prices.

Equilibrium: Small-Country Producers Adjust. Consider first the case in which the only prices to adjust are the export prices of a small country. Let the small country be the domestic country. When its export prices adjust, the change in the foreign price level can be calculated, using Equations (4), (23d), (24d), and (28), as

--O~t

/3" = b ( h + ~/)(1 - 0 ) (1 - th) + • + ~/(1 - 0q0 + 0 + 0h~/(1 - 4 ) " (32 )

~2Equation (31) demonstrates that if b were large enough, the firm would choose to adjust both prices.

13The loss function for non-traded goods producers will depend upon the beginning-of-the-period expectation of the deviation of the preset price from the full- information, profit-maximizing price. Equations (25) reveal that the reducible por- tion of this loss is zero when Et~0t = 0.

403

Betty C. Daniel

Recall that 0 = ~(1 - a'), where a" is the proportion of tradeable goods, and ~ is the fraction of world output accounted for by the small country. It is straightforward to show that the coefficient on et is negative, so that the foreign currency price moves inversely with the exchange rate.

For this equilibrium to be possible, the expected gain for export price adjustment to the small country traded-goods producer, conditional on aggregate price adjustment, has to exceed the adjustment cost. Second, the accompanying aggregate price adjustment must not induce adjustment of other prices. This requires three inequalities. The remaining expected loss for the small country traded- goods producer (due to non-adjustment of the domestic currency price), the expected gain for the large country traded-goods producer from adjusting his export price, and the expected gain to the foreign non-traded goods producer from adjusting his price, must all be smaller than the adjustment cost, and therefore smaller than the small country's expected gain. Using Equations (23), (24), (13), and (28), the small country traded- goods producer's expected gain and the remaining expected reducible loss, conditional on the adjustment of aggregate prices, can be written as 14

K(1 - , ) ( b + 1/2 . { b-(1 ---~-) + 1 L ~ - e , + {31 - 1 + (1 - t~)[b(h + `/)

2 \

-/(1 + , ( 3 o ' s )

K~(1 - ~)2(b + 1) E[b~, + (1 + k)(b - `/)/3*] 2 , (31'S) b(1 - ~) + 1

where these expressions are the counterparts to expressions (30) and (31) when the domestic country is small and the foreign aggregate price level adjusts as a result of adjustment of export prices by domestic firms.

The same equations can be used to calculate the expected gain to the large (foreign) country traded-goods producer from adjusting his export price as

~These calculations differ from those yielding the expressions in Equations (30) and (31) in that the foreign aggregate price level adjusts by 0 multiplied by the change in the export price, from Equation (32). The procedure is identical to that outlined above.

404

K~(b + 1) z E ~e, + [1 b t ~ + l t

Price Setting

2

+ k + ( 1 - d ~ ) ] [ b ( k + ~ / ) - ~ / ( 1 +k)lgt* ) . (30'L)

Using a non-traded goods producer's version of Equation (13), as well as Equations (25~, and assuming Eta0, = 0, the expected gain to a foreign non-traded goods producer becomes

[E,G,NIE, oo, = O, 9,*] = K(b + 1)E{[~/(b - ~/) + k(b + 1)]9-} z , (33)

where the subscript N denotes non-traded goods producers. For an equilibrium in which price adjustment is restricted to foreign currency prices by the small country traded-goods producers, the expressions labeled (31'S), (30'L), and (33) must all be less than the expression labeled (30'S). A small enough @ and b assure these three inequalities. Note, also, that for ~/ < 1, adjustment of aggregate prices reduces the expected gains to adjustment for each of the three remaining prices. ~5 This is in contrast to the results in Blanchard and Fischer (1989) and in Ball and Romer (1987a) in which adjustment of aggregate prices raises the gains to adjusting for other firms. The result occurs here because firms are adjusting to an endogenous variable, the exchange rate. When ~ < 1, price adjustment reduces the exchange rate change, creating a smaller exchange rate surprise. The smaller exchange rate surprise reduces the gains to adjustment.

Therefore, for a set of parameter values, the expected gain to the small country producer is positive and larger than his remaining expected loss, than the large country producer's expected gain, and than the foreign non-traded goods producer's expected gain. An equilibrium, in which foreign currency prices of small country traded- goods producers adjust and other prices remain fixed, is possible.

Equilibrium: Producers in Equal-Sized Countries Adjust. When export prices of equal-sized countries adjust (~ = 1/2), Equations (4), (23), (24), and (28) can be used to evaluate the change in aggregate price levels as

(1 - v ) e , 9, = -/3* = (k + "~)(b + 2) + (1 - v)(2 + k - ~/)" (34)

~'~Sinee P* moves negatively with ~, it is necessary to show that the coefficients on P* in the expressions labeled (31'S), (30'L), and (33) are all positive. Since b > 1, this holds if ~/ < 1.

405

Betty C. Daniel

The expected gain to traded-goods producers from adjusting export prices and the remaining loss can be written as

[E,G, IE,¢o, = 0, ~, = - ~ , * ]

K(b + 1) ~

(b + 2) E l - e , + (X - ~ + 2 ) ~ , ] ~- , (35)

[E,LR, IE, to, = O, ~, = - ~ * ]

K(b + 1)

(b + 2) - - E{(b/2)~, + [k + ~ / - b(1 - ~)]~,}2. (36)

It can be shown that the expected gain exceeds the remaining expected loss, as long as 4(b + 1) > (b + 1 - "r) 2. The larger the proportion of non-traded goods, the less the adjustment in the aggregate price level and the greater the gains to adjusting the export price. The expected gain to non-traded goods producers is given as before in Equation (33). For ~ < 1, the expected gain to non-traded goods producers again falls short of that to traded goods producers.

Therefore, an equilibrium in which equal-sized countries adjust their export prices while other prices remain fixed, is also fea- sible.

Summary and Qualification. The foregoing has shown that there is a set of parameter values which imply equilibria in which a subset of prices adjusts to an exchange rate change. These two types of equilibria will occur with exchange rate shocks of the right magnitude. For very small exchange rate shocks, no prices will adjust. For somewhat larger exchange rate shocks, it is possible that no equilibrium in which identical agents behave identically exists. The expected gain to adjustment to an individual traded-goods producer, conditional on no aggregate price adjustment, could exceed adjustment costs, while the expected gain conditional on adjustment of all foreign currency prices could fall short of adjustment costs. ~8 From Equation (30'S), aggregate price adjustment has an ambiguous effect on the expected gain for the small country firm, and from Equation (35), it reduces the expected gain for the firms in equal-

~A more complicated equilibrium in which a subset of agents adjusts, such that, given adjustment by the subset, each agent is indifferent toward adjustment, might be possible. Investigation of this is beyond the scope of this paper.

406

Price Setting

sized countries. This implies that, for some exchange rate shocks, firms want to adjust export prices if no one else adjusts but that they do not want to adjust if others adjust. For the equilibria in which a subset of prices adjusts to exist, the exchange rate changes must be "large enough." Finally, very large shocks will induce adjustment of all prices.

These results demonstrate that equilibria in which export prices are flexible and other prices are sticky could be the profit-maximizing response of firms to disturbances creating exchange rate changes of a particular magnitude when the signal provided by financial variables is weak. However, other outcomes obtain for larger and smaller exchange rate surprises. Note that the equilibria in which export prices alone adjust represents the pricing asymmetry typically present in sticky price models, like the classics by Mundell (1968), Fleming (1962), and Dornbusch (1976).

5. The Law of One Price Consider the implications of the model for the law of one price.

When financial signals are weak, Et(o t = 0 , then periods in which exchange rate changes are small are likely to be accompanied by fixed prices in both currencies. These periods will obviously be characterized by the failure of the law of one price.

In other periods, exchange rate changes might be large enough to elicit adjustment of export prices only. When the export price of a small country adjusts, Equations (21) and (32) can be used to evaluate exchange rate surprises and export price surprises as

{h + ~ / + (1 - t~)[0(1 + h + k~/) + (1 - 0 )b(h + ~/)]}et

(k + ~/)[b(k + ~/)(1 - 0)(1 - t~) + h + ~/(1 - Okb) + 0 + Ok~/(1 - ~)]'

(37)

and

~*= (X + ")')et

(x + "y)[b(x + "y)(t - o)(t - ~) + x + "v(l - o~,) + o + ex'yO - ~)]

(38)

When export prices of equal-sized countries adjust, Equations (21) and (34) can be used to yield

407

Betty C. Daniel

(b + 3 - a')~, Ot (k + "y)(b + 2) + (1 - r)(2 + k - y) (39)

and

- 2 e t P* (k + y)(b + 2) + (1 - a')(2 + k - "y) (40)

In both cases the export price does not adjust enough to maintain the law of one price. This is contrary to the assumptions made in the Mundel l -Fleming-Dornbusch sticky price models. These models allow purchasing power parity deviations, but the domestic currency price of foreign goods always equals the foreign currency price multiplied by the exchange rate, requiring the law of one price.

For completeness, consider the implications for the law of one price when everyone adjusts bases on incomplete financial information. Whenever both prices respond, the law of one price holds, since the law of one price holds under full information and since exchange rates are readily available information. Therefore, the absence of full information, given financial information, cannot alone be responsible for failure of the law of one price.

6. Conclusion This paper has shown that menu costs and imperfect infor-

mation can be responsible for the short-run failure of the law of one price. When there are fixed costs to adjusting prices, and when the variances of IS and LM disturbances are similar such that the information content of the exchange rate is low, then the gain to the firm from adjusting prices to financial information might not exceed the cost. When prices do not react to financial information, the law of one price does not hold. When exchange rate changes are large enough to induce price adjustment by some firms, the most likely prices to adjust are export prices. However, the export price adjustment will fall short of that needed for the law of one price. This result is due to the assumption of increasing marginal costs so that a producer facing an increase in the exchange rate wants to increase sales and prices. This implies that the adjustment to the export price will fall short of the proportionate change in the exchange rate. The law of one price would hold only if both domestic currency and foreign currency prices were adjusted.

408

Price Setting

The results of the model can be used to partially justify the asymmetry typically assumed in open economy sticky-price models. When some prices respond to the exchange rate, the most likely ones to do so are export prices. However, contrary to the assumptions of these models, the price adjustment should fall short of that necessary for the law of one price.

The model has several empirical implications. First, if prices are sticky, then PPP deviations should be greater under flexible rates than under fixed rates due to the failure of the law of one price. Second, if real and nominal variances are similar such that the information quality of financial variables is low, then the gain to adjusting export prices to exchange rate changes is generally greatest. This implies that export prices should be relatively more volatile, although less volatile than the exchange rate. This is because export prices should move less than the exchange rate, when they do adjust, and because they should not always adjust.

Finally, the paper does not imply that an ex ante law of one price should hold. The assumption that prices are adjusted at the end of every period was imposed as a convenient way to deal with issues posed by the more rapid receipt of financial information compared to complete information. A preferable model would be one in which the decision about how frequently to adjust prices was endogenously determined in the same way as the decision whether or not to react to financial information. In such a model, it is likely that different types of firms will choose different price adjustment frequencies for different prices, implying that current deviations from the law of one price could persist until a firm decides to adjust both prices to the current exchange rate.

Received: November 1990 Final version: August 1991

References Aizenman, Joshua. "Modeling Deviations from Purchasing Power

Parity (PPP)." International Economic Review 25 (February 1984): 175-92.

Akerlof, George A., and Janet L. Yellen. "A Near-Rational Model of the Business Cycle with Wage and Price Inertia.'" Quarterly Journal of Economics C (Supplement 1985): 823-38.

Ball, Lawrence, and David Romer. "The Equilibrium and Optimal Timing of Price Changes.'" NBER Working Paper No. 2412, Feb- ruary, 1987a.

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Betty C. Daniel

- - . "Are Prices Too. Sticky?" NBER Working Paper No. 2171, February, 1987b.

Blanchard, Olivier, and Nobuhiro Kiyotaki. "Monopolistic Compe- tition, Aggregate Demand Externalities, and Real Effects of Nominal Money." NBER Working Paper No. 1770, 1985.

Blanchard, Olivier, and Stanley Fischer. Lectures on Macroeconom- ics. Cambridge: MIT Press, 1989.

Dornbusch, Rudiger. "'Expectations and Exchange Rate Dynamics." Journal of Political Economy 90 (December 1976): 158-65.

"Exchange Rates and Prices.'" American Economic Review 77 (March 1987): 93-106.

Fleming, M. "Domestic Financial Policies under Fixed and Float- ing Exchange Rates." International Monetary Fund Staff Papers 9 (November 1962): 369-79.

Flood, Robert, and Robert Hodrick. "Exchange Rate and Price Dy- namics with Asymmetric Information." International Economic Review 25 (October 1984): 513-26.

Hogg, Robert, and Allen Craig. Introduction to Mathematical Sta- tistics. New York: Macmillan, 1978.

Isard, Peter. "How Far Can We Push the Law of One Price?" American Economic Review 67 (December 1977): 942-48.

Krugman, Paul. "Pricing to Market When the Exchange Rate Changes." NBER Working Paper No. 1926, May 1986.

Mundell, Robert. International Economics. New York: Macmillan, 1968.

Parkin, Michael. "'The Output-Inflation Trade-off When Prices Are Costly to Change." Journal of Political Economy 94 (February 1986): 200-24.

Rotemberg, Julio."Sticky Prices in the United States." Journal of Political Economy 90 (December 1982): 1187-1211.

Svensson, Lars. "Sticky Goods Prices, Flexible Asset Prices, Mo- nopolistic Competition, and Monetary Policy." Review of Eco- nomic Studies 103 (July 1986): 385-405.

Appendix A. Loss Function Derivation Taking second part ial der iva t ives , and le t t ing At

\n~ + n j Dt the terms of the Taylor's Series expansion can be

expressed as

410

C b ( b + 1)

Price S e t t i n g

- -b 2

- A * \ e , /

+ C b A , \ p , ] l \ P * ] \ e* /

To obtain the expression in Equation (13), these derivatives are evaluated at full equil ibrium values. In particular, Dt = 17, Pit = Pt, D* = 17", and /5. = p . and the approximation Pit - 15it

(Pit - /5~t)/ /si t (analogous approximation with foreign prices) is made. Recall that the expressions, with first-order terms on prices, vanish since the expansion is taken about the opt imum.

Appendix B. Purchasing Power Parity Cons ider the normalizat ions necessary for expected future

purchasing power parity. Using Equations (3), (4), and (10)-(12) to form domestic and foreign price indices yields

p, = 0~, + 0*~j, + (1 - 0 - 0*)~kt = p, + O*(p* + e t - p , )

1

b + l {(1 - ~:)[c + 0 d + (1 - 0 ) d * ] + r(ck + dr)} , ( B l d )

p* --- 0p* + 0*pff + (1 - 0 - O*)f~, = p * - O(p* + e, - Pt)

+ - - 1

b + l {(1 - "r)[c + *d + (1 - ~ ) d * ] + a'(c~' + d * ) } . ( B l f )

411

B e t t y C . D a n i e l

Assuming world equil ibrium such that

~d, + (1 - ~)d* = t~0 + (1 - ~ ) 0 " . (B2)

PPP deviations become

1 p* + e, - p, = - - {(1 - r)[c + ~0 + (1 - ¢)0"]

0(b + 1)

+ r(c~" + d*)}

- 1 - {(1 - "r)[c + ¢ 0 + (1 - t ~ )O*]

0*(b + 1)

+ r(ck + d,)). (B3)

This equali ty together with world equil ibrium places a restriction on the cs, such that

- ~ O - (1 - ~ ) 0 " = (1 - ~)c + ~ [~ck + (1 - ~ ) c ~ ] . (B4)

Using the definitions for the cs, given following Equat ions (10)-(12) to subst i tute into Equat ion (B4), the approximation that In{l? + l?*}

t~# + (1 - t~)O* and the assumption that firms are identical in size yields a relationship be tween firm size and cost:

C = (b - 1 ) / b Z .

Using this, the cs become

c = - [ * 0 + (1 - ~ ) 0 " ] , ck = - 0 , c~' = - 0 " ,

and long-run PPP deviations are eliminated. Short-run PPP deviations become

T m T p * + et - p t = - - ( d * - Y * ) = ( d , - Y ) . (B5)

0(b + 1) 0*(b + 1)

High relative domest ic demand raises the relative price of domest ic non-t raded goods.

412


Appendix C. Expectations of the Loss Function Using Equations (13), (23), and (24), expected loss for the do-

mestic traded-goods producer can be expressed as

E , L , = K ( b E t [ t o , - (1 - th)e,] z + thE,[to, + b(1 - th)e,] z

(1 - ~)~,[~,- (i + b~)~,]~}. (Cl)

Recalling that et is observable, this can be simplified to yield

EtL, = K(b + 1)[Et(tot) 2 + (1 - th)(1 + b ~ ) e ~

- 2(1 - th)etEtto,]. ( C 2 )

To obtain the expression in Equation (26) in the text, it is necessary to solve for

Et(o2t) = Et[th('y'q, + hv t ) + (1 - th)(yq* + hv*)] 2 , (C3)

where the information available at t ime t includes "qt - vt and "q* - v* Define

Yt = lPt

and

Xt = ("qt - ])t) "

The conditional variance of y given x is given by Hogg and Craig (1978, 118) as

or~(1 - p2),

where

Using this,

0 - - - (C4) O" x O'y

p _ [ ~ ( ~ - v) ] - ~

9.

413

Betty C. Daniel

This implies that

and

2 O" u

p g ~ - -

2_{_ 2 O'xl or u

2

l - p 2 - o'~ -,0,. (C5) 2 + o r ~ o'.q

Using the expressions (C4) and (C5), the conditional variance of v is therefore

2 2

~_______L~ _ (i - ~)~ = ft~ = (h. (c6) s + o r ~ o'~

By the same argument, the conditional variance of ~q is ~b. To make the necessary calculations, it is necessary to solve for

terms like Et(v~) and Et(v,~,). To obtain the former, note that

Et(v~) = conditional variance + (conditional mean) s

= ( 1 - l ' l ) [ o ' ~ + ( 1 - 12)(11, - vt)~], ( C 7 )

using (C6). The remaining type of term can be calculated in the following

manner:

F.(v,~,) = E , [ v , ( ~ , - v, + ~,)1 = E , [ ~ , ( ~ , - ~ , ) + ~,~1

= - ( 1 - D. ) ( ' r l , - v , ) ~ + (1 - E}.) [cr~ + (1 - D.,)( 'q, - vt) ~]

= (1 - D , ) [ t r~ - ,.Q(vlt - v , ) 2 ] . ( C 8 )

Applying these procedures and defining 1~* and ~b* as the foreign counterparts of f~ and ~b, the expression in (C3) becomes

(h + 7)2[,~d~ + (1 - ~b)z~b] + (E,~o,) ~ . (C9)

The unconditional expectation of the expected loss function can be expressed as

414

Price Setting

E[EtL,] = K(b + 1){(h + 7 )~[ ,26 + (1 - ,)9cb]

+ [ ~ + (1 - t~)(1 + bd~) - 2(1 - ~)~](cr~ + cr2~)

+ [~,2 + (1 - ~)(1 + b~ ) + 2(1 - t~)~*](o ".2 + ,2 )}. (ClO)

N o t e t h a t i d e n t i c a l s i z e - a d j u s t e d v a r i a n c e s e q u a t e

, 2 2 (1 - *)(cr *~ + cr~ ) and d~(cr~ + cry).

415

price setting, imperfect information, and the law of one price

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