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Price dynamics under imperfect information
Journal of Economic Dynamics and Control 9 (1985) 339-361. North-Holland
PRICE DYNAMICS
UNDER IMPERFECT
INFORMATION
Torben M. ANDERSEN* University
ofrlarhus,
DK-8000
Aarhus
C. Denmark
Received August 1984, final version received November 1985 The paper considers the pricing decision of a monopolist firm having demand and costs exposed to nominal and real shocks which include both permanent and transitory changes. The firm obtains information through both price and quantity signals and the price equation is found by use of a filtering technique. It is shown that imperfect information implies nominal price smoothing where the price adjusts only partially relative to the past price by incorporating new information observed through price and quantity signals.
1. Introduction Recent work on the problem of price flexibility or price smoothing has analysed the role of inventories and adjustment costs of price changes in relation to a monopolist firm. The possibility of inventory holding (and backlogs) makes the firm to operate on two margins, namely on how much output to produce for inventory and on how much inventory to sell. As a consequence the firm engages in a sort of intertemporal substitution, and changes in current market conditions do not in general become reflected in prices instantaneously, that is, the adjustment process tends to be smoothed out over time.’ The same mechanism is at work if the effects of the adjustment of the capital stock is taken into account.2 On the other hand, if the firm holds no inventory but faces adjustment costs of price changes, it is similarly found that it is not necessarily profitable to adjust prices fully and instantaneously to current changes in market conditions, and even perfectly foreseen nominal changes do not necessarily lead to price changes.3 *I have benefited from comments on previous versions by Jacques D&e, an anonymous referee, and participants at the Sixth Conference of the Society of Economics Dynamics and Control, Nice, 1984. The usual disclaimer applies. Financial support from the Danish Social Science Research Council is gratefully acknowledged. ‘See, e.g., Phlips (1980), Blinder (19X2), Reagan (1982) and Amihud and Mendelson (1983). ‘See, e.g., Maccini (1984). ‘See. e.g., Barre (1972), Sheshinski and Weiss (1977), Rotemberg (1982), Kuran (1983) and Danziger (1983). 0165-1889/85/$3.3001985,
Elsevier Science Publishers B.V. (North-Holland)
340
T. M. Andersen,
Price
~namics
under
imperfect
information
The purpose of the present paper is to analyse another possible source of price smoothing by a monopolist firm, namely imperfect information on current market conditions. The importance of imperfect information for the adjustment of prices has often been stressed [see, e.g., Nordhaus (1972) Gordon (1981), Shaw (1984)], but has not hitherto been rigourously analysed.4*s To point out how far imperfect information can bring us in terms of explaining price smoothing by a monopolist firm we shall disregard both inventories and costs of price changes. This should not be taken to imply that a monocausal explanation of price inflexibility is sought for, but serves the analytical purpose of isolating the effects of imperfect information on the dynamics of prices. The different sources of price inflexibility should be seen as complementary rather than competing theories of price adjustment. Price sluggishness is often modelled by use of the so-called partial adjustment model which says that the current price (p,) is given as a convex combination of the past price (p,-r) and the optimal price (p:) defined in some specific sense, i.e.,
p,=XP,-1 + (1- a+?
Ojhjl.
Price sluggishness or price smoothing is accordingly identified with the fact that the current price depends on the past price (X # 0). Although widely used, the adjustment model (1) is an ad hoc model, but we shall in the present paper show how imperfect information can rationalize a price equation which comes very close to (1). To be specific we impose no restrictions on the adjustment of prices except that it is accepted as an institutional fact that prices for many products are set prior to trading6 [Alchian (1970), Gordon (1981), Okun (1981)], which is a reasonable assumption to make in the case of a monopolist firm. Given this observation the interesting question becomes the adjustment of prices between periods, that is, the dynamics of prices over time. The present analysis considers the price decision made by a single monopolist firm’ under less than perfect information on the demand and cost conditions, and the specific environment which is considered is quite general since both demand and costs are subject to permanent and transitory real and nominal shocks. Attention is paid to how the firm obtains information on market conditions through signals readily observable in a decentralized market economy. The main sources of information to the firm in such a setting is 4See, however, D&e (1979) who shows that imperfect knowledge about the elasticity of demand on the part of a monopolist firm has implications comparable to a kink in the demand curve, and thus explains some price rigidity. ‘This paper can be seen as a generalization of Andersen (1983). 6 With pre-set prices it becomes furthermore necessary to specify a trading rule; see section 2. ‘The case of many interdependent firms and hence the problem of coordinating price decisions is considered in Andersen (1985).
T.M.
Andersen,
Price
dynamics
under
imperfect
information
341
observation of costs (price signal) and sales (quantity signal), and it is shown how both the price and the quantity signal provide valuable information to the firm! Hence, not only price signals, but also quantity signals are proven to be important to the adjustment process in a decentralized market economy [compare to Hayek (1945)]. The main conclusion to come out of this analysis is that imperfect information implies nominal price smoothing, since the price quoted by the firm adjusts only partially relative to the past price by taking into account the new information received through the price and quantity signals. Consequently the resulting price equation gets a form which comes very close to that of the partial adjustment model in (1). This shows how imperfect information can explain nominal price smoothing without taking resort to inventories or costs of price changes. That is, imperfect information can qualitatively explain the same sort of sluggishness in the adjustment process as has been previously explained by inventories or cost of price changes. Blinder (1982) has criticized most models of price inflexibility for explaining only relative and not nominal price rigidity,9 since rigidity of prices is explained relative to assumed fixed costs. Hence, unless costs respond less than proportionally to nominal changes, prices will respond proportionally to nominal changes. This criticism does not apply to the present study, since the price smoothing explained in this paper is smoothing of the nominal price quoted by the firm. It turns out that even if the price adjusts proportionally to the expected nominal change, the price equation does not satisfy the traditional homogeneity properties. Brunner, Cukierman and Meltzer (1980) used the idea of confusion between transitory and permanent shocks to explain persistence in the effects of real and nominal shocks on the unemployment rate within an equilibrium model sustaining the neutrality property of systematic aggregate demand management policies. More related to the issue of price adjustment analysed here is the work of Iwai (1981) and Gertler (1982). Iwai (1981) considers the adjustment problems of a firm which acts as a monopolist in the output market and as a monopsonist in the input market. Price dynamics arising from learning over time of market conditions is analysed under specific assumptions conceming the sequential structure of the decisions taken by the firm. The present study differs from Iwai (1981) on several aspects of which the most important ones are that we (i) distinguish explicitly between real and nominal shocks, (ii) consider a more elaborate learning scheme where information acquisition by signals readily observable to the firm is stressed, and (iii) that price and output ‘The same would apply if inventories/backlogs existed. The informational role of sales or inventory signals has not, however, been analysed in the literature dealing with the role of inventories for price adjustment. ‘This criticism is not applicable to models incorporating adjustment costs of price changes. The importance of such costs, especially in relation to macroeconomic phenomena, remains still an open question.
342
T. M. Andersen,
Price
dynamics
under
imperfect
information
decisions are made simultaneously. More closely related is the study by Gertler (1982) where the dynamics of nominal wages is considered under the assumption that nominal wages each period are predetermined at a level equal to the expected Walrasian equilibrium price in an environment exposed to nominal shocks and productivity shocks. The paper is organized as follows. The general model of the monopolist firm is developed in section 2. In the following section it is considered how the information available to the firm is used to make inferences about market conditions, and the dynamic price equation is derived. Since the general version of the model is difficult to interpret, we consider in section 4 the special case where the firm is exposed only to nominal shocks, and prove some detailed results for this case. Section 5 concludes the paper. 2. The model of the monopolist firm
To analyse the dynamics of prices under imperfect information we shall set-up a relatively simple model of a monopolist firm, which for analytical reasons builds on a specific structural form. The main consideration has been (as in much of the literature referred to above) to adopt a formulation which allows an explicit solution to the pricing problem of a monopolist firm, viz. a log-linear model. Needless to say, by adopting this specific structure we are considering only an example of how imperfect information can induce price smoothing. Consider a monopolist facing the following demand function:
D,= Do(W’hdUm
d>
(2)
1.
N, is nominal income defined as RU,,, where R is real income” and Vi, is the nominal shock variable at time t. P, is the price quoted by the firm, and U,, is .le real shock variable to demand at time 1. The monopolist produces the commodity using a single input, say labour, and the production technology is given by a Cobb-Douglas production function
Q, = LP,
O
(3)
Hence, the costs of producing Q, units are C(Q,) = W,(Q,)”
where
(4)
a = cx-‘.
The main concern of this analysis is the dynamics of output prices and for this reason we assume wages to be determined exogenously. To avoid that the ‘OR is assumed to be constant, but a non-constant implications as the real shock U,, to demand.
R
will qualitatively
have the same
T. hi. Andersen,
Price
dynamics
under
imperfect
343
information
argument made for price sluggishness depends on sluggish wage adjustment we shall impose the strong assumption” that wages adjust proportionally to nominal changes, i.e., let W, denote the nominal cost of input at time t, we assume
w,=WUl,v,,,
(5)
where W is the real-wage rate, I2 Vi, is the nominal shock variable at time 1, and U,, is a real-shock to costs at time t. Notice, that the model fulfills the homogeneity requirements, that is, demand is homogeneous of degree zero in the price and the nominal income, and the cost function is homogeneous of degree one in the wage rate. Moreover, the nominal income and the nominal wage rate change proportionally to nominal changes. It should be remarked that a nominal change (shock) is defined as any change which, if perfectly foreseen, implies only a price change, and a real change (shock) is defined as any change which even if perfectly foreseen leads to output changes. The firm announces prior to period t a price conditional on alI available information at the end of period t - 1, 1,-i. We shall adopt the assumption employed in many similar models, that the firm is compelled to meet whatever demand is forthcoming at the announced price, and that the firm can instantaneously adjust production within the period, i.e.,
Q,= 4.
(6)
The commodity is not storable, and the objective of the firm is to set a price for period t, P,, so as to maximize expected profits K=
W,Q,- c(Q,)IL~).
(7)
Inserting we find that (7) can be written
Maximizing
V, with respect to P, we find the following first-order condition:
pl+d~-d_ WD,f-ldaR’-d’“-” I d-l
E[U3,Uj,‘daU,411,-l] ~[4%,l4-11
’
(9)
“This is a strong assumption since the argument advanced here concerning nominal price stickiness is applicable to nominal wages as well. “The remark made on R applies to W mutatis mutandis.
T. M. Andersen,
344
Price
dynamics
under
imperfect
information
The second-order condition reads E(D,,N;(d-
~)LP,-‘-~U,,II,-~)
- E(WUl,U3,Np”du(l + d~)~,-~-~“~,4l~,-,)< 0. To proceed we have to properties of the nominal real shock to costs (us,). log-normally distributed.
make some assumptions concerning the stochastic shock (Q,), the real shock to demand (KY,,) and the We shall assume these shocks to be independently That is, for i = 1,2,3, we assume l3
U,, = In Q, = Ui, + e,,, Ui, =
Pi”ir-1
+ nj,V
01) 02)
lP;l s 19
e,, - N(0, ~2).
“it - N(O, ui),
Furthermore,
(10)
it is assumed
En;,ej, = 0,
Vi, j= 1,2,3,
Eni,nj, = 0,
Vi, j=1,2,3,
i+j,
Ee,,ej, = 0,
Vi, j=1,2,3,
i#j.
Eqs. (11) and (12) say that any shock is composed of two parts, a completely transitory part (e,,) and a permanent part ( ui,) which is made up of a first-order autoregressive process in ni,. Using that for any log-normally distributed variable x, we have that [see Aitchison and Brown (1957)] In E[ x:11,-,]
= aE[lnx,ll,-,]
+ $2var[lnx,lI,-,].
We find that (9) can be written in logs as P,= co+ Eb,,l4-11 13For
any variable
+@b,,lh-11
X, we shall define
x, as x, = In A’,.
+
C2Eb3tl~~-ll~
03)
T. M. Andersen,
Price
dynamics
under
imperfect
informurion
345
where co = (1 + da - d)-‘(
+f((l
ln( D,“-‘WR’-J(“-l)/(d
+da)2-d2)var[u,,lZ,-,]
- 1)) + ivar[ u,,lZ,-,]
+:(a’-l)var[u,,ll,-,I),
c,=(a-l)(l+da-d)-‘>O,
It should be remarked that it follows from (13) that, even though the firm is imperfectly informed about nominal shocks, the price quoted by the firm is adjusted proportionally to the expected nominal shock, i.e., the nominal price is fully flexible with respect to the expected nominal shock. However, the expected real shocks to demand and costs do not lead to proportional changes in the nominal price since these shocks, even under full current information, should not be completely passed on to the nominal price. Notice, that the Gaussian assumption made on the random variables implies that the conditional variances become independent of the actual realizations of the random variables, and hence we can treat them as constant for the purpose of the present analysis. The actual quantity traded is found from (2) and (6) to be
4, = d, = lnD, + d(lnR + I(~, - p,) + Us, = InDO + d( ul, - E( r~,lZ,-~)) + d 1nR + (~2, - dc,E(u,,lL,))
- dc,%,lL)~
(14)
Eq. (14) shows that output responds only to the unanticipated part of the nominal change, whereas it responds with different coefficients to the actual and expected real change to demand and to the expected real change to costs. Finally, we need to determine the information set of the firm at the end of period r - 1, 1,-i. In a decentralized market economy the firm obtains information on market conditions from two important private sources,14 14The firm might also have access to some public (aggregate) information which may be useful to the firm in identifying the,market conditions. We leave out this complication by noting that nothing qualitatively would be changed by introducing this aspect into the model. For an analysis which also includes this type of information in a model with many price-setting firms, see Andersen (1985).
T. M. Andersetr,
346
Price ~vnamics
uder
iniperfect
informalioti
namely, the sales it experiences (quantity signal) and the costs (price signal) of the inputs the firm uses. We shall analyse what kind of information the firm obtains through these two signals and how this knowledge influences the price decision. Given the initially announced price ( p,- t) for period t - 1, actual sales (q,r ) is observed at the end of the period. It is easily seen that the sales figure q,- i is informationally equivalent to q;- Ir where q;-1
= q,-*
Da- dp,-i - dln R
-1n
= dUl,-1+
UZ,-l.
(15)
That is, the quantity signal qiml reveals information on the nominal shock ~i,-i and the real shock to demand u2,-i in period t - 1. Moreover, the firm can infer information from the price signal it observes on nominal costs, since the wage signal is informationally equivalent to w/-i, where w/-1=w,-1-w =
Ul,-1+
U3,-1.
06)
Hence, the price signal reveals information on the nominal shock ul,- t and the real shock to costs u3,-i in period t - 1. Let us denote by s,-i the 2 X 1 observation vector of the firm in period t - 1 given as s,-1
=
[ 1 4:- 1 )$ql
.
(17)
The information set of the firm at the end of period t - 1 is made up of s,-~ and all past observation vectors, i.e., I,-,=
{S,49~,-2r...}.
08)
The information problem faced by the firm is thus two-sided. Firstly, given the signals q;- 1 and w;-~, it must evaluate how much can be regarded as a nominal change and how much as real changes to demand and to costs. Secondly, it must sort out how much of each change constitutes a transitory change and how much constitutes a permanent change. We shall in the next section analyse how this information-expectations problem is solved, and derive the resulting price equation. 3. Filtering and price adjustment We can now turn to the interesting problem of deriving the equation describing price adjustments over time, and to this end we have to start by
T. M. Andersen,
Price dynamics
under imperfect
itlformariott
341
finding the conditional expectations of the state variables given the information available to the firm, cf. (13). Given that ei, is specified as a white-noise random variable, we have that
E[uitllt-,I= E[uitl~,-lI~
Vi = 1,2,3.
(19)
Eq. (19) says that the tim wants to find out the permanent part of each shock since this is what matters for future market conditions. Using (19) we find that (13) can be written P, = co + W[m,lL,l~
where
4 = h, I;=
[l
ClU2,
C93,l~
1 11.
We can now prove the following result on the conditional expectation of m, given 1,-r. Lemma I. By use of the Kalman-filtering technique the conditional expectation of m I is found by the following updating equation:
E[m,l~,-ll =AE[m,-d~,-21++($,-I - E[s,-#,-21).
(20)
Alternatiuely, the conditional expectation can be written as a distributed lag in the observed signals:
E[m,ll,-r]
=Il/e(L)s,-~~
(21)
where B(L)=I+(A-#B)L+(A-#B)2L2+
-0. .
A, B and +!I are matrices defined in the appendix. Proof.
See appendix.
Eq. (20) shows that the conditional expectation of m, given 1,-r is found by updating the past expectation of the state variables (i.e., of m,-, given 1,-2) according to the difference between the actually observed signals on the current market conditions (s,-r) and the expectations of these market condi-
348
T. M. Andersen,
Price
&vtamics
under
imperfecr
information
tions given the hitherto accumulated information (~s,-,]Z,,~]).‘~ Eq. (21) says that the conditional expectation of m, given I,-t alternatively can be written as a distributed lag in all past observed signals on market conditions. Having found the conditional expectations of the state variables, it is straightforward to proceed to determine the price equation for the firm. Proposition 1. Under imperfect information about the current market conditions, the nominal price quoted by the monopolist firm for period t ( p,) is a function of the previously quoted price ( p,- 1) and the most recent observations on market conditions (4;- 1, w,‘- 1), p, = 6, + 6,p,-,
+ Qy:-,
+ &W:_l,
(22)
where
i&=c,(l
-z;(A
-l/B)Z,),
6, = z&4 - JIB)Z,,
(S,-s,>=z;+. Proof.
Using Lemma 1 we find p,=c,+Z;(A-#B)E[m,-,lZ,-,]+I#-,.
Lagging (23) and premultiplying (CIA
- W)Z,)P,-,
(23)
by Z;(A - #B)Z,, we find
= (Z&4
- WZAc,
+ (Z;(A - ~B)Z,Z;)E[m,-,lZ,-,l.
Using that Z,Z-,’= I, where Z is a 3 x 3 matrix with all entries equal to one, eq. (22) follows straightforward. Proposition 1 shows that the imperfect information implies a sluggish adjustment of the nominal price set by the monopolist firm. The price equation (22) shows that the price quoted by the firm for period t ( p,) adjusts partially relative to the past price ( P,-~) by taking into account the new information contained in the quantity signal (q;- r) and the price signal (w;- J. The price smoothing arises from imperfect information due to the fact that the firm does not want to make the current price to depend only on the most recent ISLemma derived from
1 illustrates the well-known fact that adaptive-like expectations statistical principles; cf., e.g., Mutb (1960) and Lawson (1980).
formula
can
be
T. M. Andersen.
Price dynamics
under imperfect
information
349
information since it does not know whether this information reflects transitory or permanent influences. The lagged price term in (22) captures in this way the past information of relevance to the current price decision, and we End that the resulting price equation comes very close to that of the partial adjustment model; cf. (1). Finally, it should be noted that the gradual price adjustment, of course, implies a sluggish adjustment of output; cf. (14). Eq. (22) states the price equation as a first-order difference equation, and the question arises whether this equation is stable. To answer this question we prove: Corollary 1.
The price equation (22) is stable.
Proof. Stability of (22) is ensured if 8; goes to zero for n going to infinity. We find that 8; = I;D”13, where D = A - $B, and it follows from the fact that @ is the unique solution to (A.5) within the class of positive semi-definite symmetric matrices that D is stable, i.e., D” + 0 for n --) co; cf. Bertsekas (1976, sect. 3.1, ch. 6, app. A).
To interpret the coefficients in the price equation (22), we start by observing that the coefficients 6, and S, are given as
where #ij (i = 1,2,3, j = q’, w’) is the weight given to the innovation in the jth signal in predicting the ith element of the m vector; cf. (20). It follows that the higher weight a given signal gets in the prediction formula, the higher becomes the signals weight in the price equation. The relation between the weight put on the past price (6,) and the weight given to the new information (a,, 6,) in (22) is easily seen since we have that
This shows that the more weight the new information is given, the less weight is given to the past price in the price equation. Hence, to the extent that the recent information is a reliable guide to future market conditions, less weight is put on the past price when setting the price for any given period and vice versa. The extent of price smoothing depends thus on the quality of the information signals q;- i and w,‘- i as indicators of future market conditions. The inverse relation between the weights to current and past information is not found in the case where all shocks are completely transitory, i.e., p, = pz = pJ = 0. In this case we find that 6, = 6, = 0, since the current information
350
T. M. Andersen,
Price dynamics
under imperfect
information
signals contain no valuable information on future market conditions when all shocks are completely transitory. Consequently, the past price is also void of any relevant information for current market conditions, and 6, becomes equal to zero, and we find that Pt = co*
That is, the nominal price is completely fixed when all shocks are completely transitory. Unfortunately, it is very difficult to obtain analytical results on how 6,, 6, and 6, depend on the underlying stochastic structure of the model, and although the result of nominal price smoothing comes out clearly from the general model (cf. Proposition l), we shall consider a special version of the model to obtain more insight in the specific characteristics of the price equation. Hence, to gain more specific knowledge of the nominal price smoothing implied by imperfect information, we shall proceed by considering nominal shocks in isolation from any real shocks in section 4. It should be obvious from the exposition that the same qualitative results can be obtained by considering any of the shocks in isolation. 4. Nominal shocks To gain further insight into the effects of imperfect information for the adjustment of prices, it is useful to consider the case where the firm is exposed only to nominal shocks (i.e., ui’, = ei’, = 0, i = 2,3). In this case the firm faces the information problem of disentangling the permanent part of the nominal shock from the transitory part. The reason for considering nominal shocks is twofold: firstly this allows us to address a number of questions raised in macroeconomics and monetary theory, and secondly nominal changes provide a good point of reference since such shocks have only nominal price effects and no real effects under full information. Before analysing this information problem in detail, we shall note that, if the firm were setting the price for period t given full information about the period f nominal shock, we would find the price for period t to be P: = co + q,,
(24)
i.e., under full information the price adjust proportionally to any nominal shock. Turning to the imperfect information case, we find that the quantity signal q:-1 and the price signal w:-i are informationally equivalent, since they both reveal the period t - 1 nominal shock, I(~,-~. Without loss of generality we can thus consider only one of these signals, and we choose the quantity signal, i.e., s,-1=
q;-1.
T. M. Andersen,
Price
dynamics
under
imperfect
information
351
Proposition 2. In an environment subject to nominal shocks about khich the firm has imperfect information, the price equation for the monopolist firm is given as
&I,-,,
(29
For ai’,=a$=O, i= and g, = n,,; cf. proof We shall start by solving variance (see proof of Lemma
2,3, we have that B= d, L=de,,, m,=vl,, of Lemma 1. _ _.. . for the stationary solution to the contihonal l), and we find that
=
6, +
&P,-,
+
4
=
(1 -
UP,/4
A,
= a:,/(+
where
+ ale),
S;=d&. Proof. A=P,
@=p;@+a:,-p,@d(d2@+d20,2,)-‘p,@d
= a;, + p;!S - pi@p’( @ + a$! This is a second-order equation in @, and solving for @ we find ~-
-(P:-l)a:,-o:,fd((P:-l)a:,+a:,)2+40~~a,Z, -2
The unique positive stationary solution for @ is found to be 4P=+
I(
p; - 1 UI’C+ a:, +
)
((p:-l)~;a+cf~“)‘+4o:p:,}.
Hence, the # coefficient becomes +=wVd(@+a:e)The price equation (25) follows straightforward.
352
T. M. Andersen,
Price
dynamics
under
intperject
information
Eq. (25) shows that, even when nominal shocks are considered in isolation, imperfect information about nominal shocks implies that the adjustment of the nominal price to nominal shocks becomes smoothedf6 whereas the nominal price under full information would adjust instantaneously and proportionally to nominal changes. The reason for the nominal price smoothing is that the firm cannot know whether a nominal shock is a permanent or a transitory shock, and hence by how much to adjust the price for the next period given the nominal shock of the current period. To see clearly how imperfect information implies price inflexibility or price smoothing it is natural to compare with the case of full information, and we find the following result: Proposition 3. Let p: and pI denote the price quoted by the monopolist firm in an environment exposed to nominal shocks under full and imperfect information, respectively. We find that
4 Proof.
PI1-4
P:l-
(26)
From (24) it follows that
4
p;l = v=[u,,l = u1’, + a,2,/( 1 - p: ) .
The steady-state variance of p, is found from (25) to be 8;
4
P,l = y-pd%,l. -:
Using that -=
6:
1-s;
GPi
1 - (1 - x,)‘p;
’
we find that var[ p,] < var[ p,?] since x:p; c 1 - (1 - A,)2p:, l6 With correlated state variables it is, of course, generally possible to find a relationship between and p,- 1 even if no price smoothing is taking place. In the present model p,-, appears only for its price smoothing role, which is seen by noting that P,-~ drops out of (25) if there is no confusion between transistory and permanent shocks. p,
T. M. Andersen,
Price +lumics
under
imperfecr
informution
353
or 0 < (1 - p:> + 2A,p31 -A,). This shows that to an outside observer the price variability is lesser in the world with an imperfect, information structure than in a world with full information, hence the conclusion that imperfect information implies price inflexibility or price smoothing. To see clearly how the confusion between permanent and transitory changes induces sluggish price adjustment, we find that the price smoothing disappears if the nominal shock does not include any transitory part, i.e., ~12,= 0, since (25) reduces to PI =
60 +
PlUl,-1.
(27)
Eq. (27) shows that the price depends no longer on the lagged price, and p, is adjusted to the past nominal shock ~i,-i, multiplied by the fraction (pi) of this change which carries over to period f. In this case the quantity signal #-i reveals the permanent nominal change with full certainty and no price smoothing evolves. However, the firm is still imperfectly informed about the innovation in the nominal shock, and the price path diverges from the full current information price path. Another special case should be mentioned where all nominal changes are completely transitory, pi = 0. In this case we find [cf. (25)]
This case implies a completely fixed price since with completely transitory shocks there is nothing to be learnt about the future and hence no reason to adjust the price to take account of current and past information. Consider the case where pi = 1. In this case we find from (25) that the price equation becomes” P, = hP,-,
+ (1 - hh-1.
The Xi coefficient is determined by the relation between the variance of the transitory component and the sum of the variance of the transitory and the permanent component of the nominal change.” As shown above, the full current information price for period t - 1 is p,Y1 = ~i,-i, and we find that (29) can be written as Pr=hP,-,
+ (1 -UPr%
“Generally, (25) can be written p, = 80 f PI(AI Pr- L +o -h)ul,-,). “Recall that u,, = D,, + e,,. Hence, var(u,,ll,- 1) = var(ol,ll,- 1) + d = 0 + ‘6
T. M. A tdersen.
354
PI-P,-1
=
(1 -MPL
Price +rutvics
under imperfect
irifonnutiorr
-PA.
(30)
Eq. (30) is identical to the partial adjustment equation (l), except for the fact that the price adjustment is a linear function of the difference between the optimal and the actual period t - 1 price, rather than the difference between the optimal period t price and the actual period r - 1 price. This change is caused by the imperfect information set-up of the present analysis, and (30) says that the firm ex post in period f - 1 knows the nominal shock Us,- 1 and hence the optimal full information period 1- 1 price, p,? 1. The extent of price adjustment is thus determined by the divergence between the optimal and the actual price in period t - 1. In the present context the partial adjustment is not caused by adjustment costs, etc., but by the fact that the firm does not know until the end of the period what the optimal price would have been for the period. Interestingly we find that the price equation has the following property: Proposition 4. The nominal price quoted by the monopolist firm ( p,) in an environmenl subject to nominal shocks is homogeneous of degree 6, + 8; in the nominal variables ptel and ulr-,. In general 6, +- Si is different from one since
Proof.
Follows immediately from (25) and the definition of 6, and Si.
This result is remarkable because it says that even though the firm adjusts its price proportionally to the expected nominal change [cf. (13)], we do not find that the price equation (25) displays homogeneity of degree one in the two nominal variables p,- 1 and ulrel which determine p,. Such homogeneity of degree one prevails only in the special case where the permanent part of the nominal shock is given by a random walk process ( p1 = 1). The reason for this is obviously that the firm does not want in a proportional way to pass on nominal changes to future nominal prices unless the permanent part of the nominal shock is of a complete permanent character (pl = 1). Notice, that this result holds even if the nominal variable does not include any transitory part, i.e., u:, k 0. Although this point may seem clear enough in the present context, it has important implications for empirical work on macro price equations, where long-run homogeneity of the price equation is either imposed or tested19 for models based on a price adjustment model close to that of (22) or (25). The “See.
e.g., Gordon
(1981,
1982).
T.M.
Andersen,
Price dynamics
under imperfect
information
355
present model can thus rationalize such price equations which have previously been stated as ad hoc equations, and moreover we are able to explain (Proposition 4) that long-run homogeneity is the exception rather than the rule, even though prices may adjust proportionally to expected nominal changes.20 Moreover, we find that changes in the structure of the price equation between different subperiods can be explained by changes in the underlying stochastic properties of the random variables; cf. Proposition 5. Proposition 5. In an environment subject to nominal shocks, we find under imperfect information that (i)
the coefficient to the lagged price term in the price equation (6,) increasing in ofe, decreasing in a:,, and increasing in p,, and
(ii)
the coeBcient to the current observation on market conditions in the price equation (6,) is decreasing in a:=, increasing in a:,, and increasing in pl.
Proof.
is
See appendix.
Proposition 5 shows the sign of the changes in the coefficients in the price equation to changes in the stochastic properties of the nominal shock variable. An mcrease m u1’, implies a larger variance of the transitory part of the nominal change, and more weight is put on the past price and a lower weight is given to the recent information, since changes in the latter to a larger extent reflect transitory changes when u1’, increases. On the other hand, if a;,,, increases, it means that the variance of the permanent part of the nominal variable increases, and lesser weight is given to the past price and more weight is given to the recent information, since variation in the latter now to a larger extent reflects permanent changes. A change in p1 has two effects on 6, and 13,: one is a direct effect which tends to increase both, and the other is an indirect effect which operates through the effect of p1 on the conditional variance @. A higher p1 (pl > 0)21 means more persistence in the permanent part of the nominal change, and this tends to increase @ in the same way as for an increase in u&. This makes it clear that S, increases for p1 increasing, and it turns out that the direct effect on St dominates over the indirect effect and 6, increases also due to an increase in pl. “This non-homogeneity of degree one does not leave any clear-cut reason for non-neutrality of money and hence for an active monetary policy [see Andersen (1985) for a further discussion of this issue]. *‘Notice, that for p1 < 0 we have that 6, < 0, hence higher p1 means less persistence and the numerical value of 6, falls. The same holds for 82.
356
T. M. Andersett,
Price
&un~ics
wder
intper/ect
ittforttturior~
6. Conclusion The adjustment of prices is a controversial issue in both micro- and macro-economics, and the present model has been formulated such that it is of relevance to both branches of literature. The conclusion that price smoothing evolves under imperfect information about the permanent and transitory components of real and nominal shocks provides thus a rigourous dynamic formulation of some theoretical and empirical microeconomic models of price behaviour,** while at the same time explaining price sluggishness and especially nominal price rigidity which is of great importance to a number of macro-economic problems. Since a summary of the findings have been given in the introduction we shall conclude by a few remarks on some aspects of the model. We have in the present context allowed only for privately observed information, viz. price and quantity signals, but to place the present model in a more specific macroeconomic context it would be relevant to include public information on aggregate variables as information variables to the firm. The model could easily be modified to allow for such information, and nothing qualitatively is changed as long as this public information combined with the private information is insufficient to allow the firm full information on the current state of the market. Moreover, a single firm has been analysed in isolation and the interaction between many firms is essential to the analysis of macroeconomic problems and especially the problem of monetary policy [see Andersen (1985)). It has been assumed that the firm sets the price before it is informed about the current costs. Nothing qualitatively would, however, be changed if costs were known before the price is to be set as long as costs are subject to both nominal and real shocks [cf. Andersen (1983)]. For analytical convenience use has been made of specific functional forms of the demand and the cost functions. Though, the qualitative result of the paper relies more on the imperfect information structure than the specific functional form of the model. More troublesome is the assumption that demand determines quantities traded given the prices set by firms. This is in a sense a very Keynesian assumption since it implies that effective demand determines output, but the empirical importance of such arrangements remains an open question. This is so because it implies that ftrms continue to supply at the announced price even if marginal costs exceed the price. It could, of course, be argued that firms have an incentive to act in this way to make their price announcements credible, i.e., to gain reputation that prices announced are reliable signals for the decisions of consumers by ensuring that consumers can ‘*The present model can thus be seen as a rigourous dynamic formulation of the mod& by Nordhaus (1972) and de Menil (1974), with specific attention to learning over time ol cost and demand conditions.
T. M. Andersen,
Price
dynamics
under
imperfect
information
351
actually buy at the quoted prices. However, the assumption implies a considerable analytical simplification, and it seems difficult to do without it due to the problems of handling the non-linearities which arise under other assumptions; cf. Taylor (1985). Finally, it should be noted that the present model yields results which could fairly easily be exposed to empirical tests, an interesting topic which I hope to be able to pursue in the future. However, given that the theoretical analysis justifies the ad hoc models used in many empirical studies partial support for this model has already been made.23 Appendix Proof of Lemma 1
The vector m, can be written as
m, = Am,-l f g,,
(A.1)
where
A=
[“If 1 Pl
0
0
00
0 p2
P3 0,
g,= [c2n3r 1. w2,
The observation vector s,-i can be written s,-~ = Bm,-l+.ft-lv
64.2)
where
23To refer to just a small and recent sample of such work (which fits into this framework), see, e.g., Gordon (1981, 1982) and Paldam and Hylleberg (1984).
358
T. M. Andersen,
Price
dynamics
under
imperfect
information
The firm is faced with the problem of making an estimate of m, given the information 1,-i. This problem can be solved by use of the Kalman Filtering technique; cf. Bertsekas (1976). Using this method we can write the conditional expectation of 112, given I,-i as E(m,JZ,-,)
=AE(m,-ill,-,)
+ *,&,-i
- BE(m,-ilZ,-i)),
(A-3)
where
The conditional variance of m, given I,-, Bicatti equation as var(m,IZ,-,)
is determined from the so-called
=Avar(m,-,lZ,-,)A’+
G
- (Avar(m,-,lZ,-,)B’)(Bvar(m,-#,-,)B’+F)-’
X (A v4m,-114-Z)B’)‘,
(~4.4)
where
io2 *’ le
and
G= E(g:g,)=
[ 01” 002
c2a2 10 2n
1
da,2,,
d*a* + a*
4,
+ 4e
c*(J3 **0 ”
I .
Notice, that the sequences of {et-,} and {var(m,lZ,-J} are independent of the actual realizations of the random variables. It follows that they can be calculated independently of knowledge of the actual realizations of the random variables, and we shall solve for their steady-state values. To solve for the steady-state value of var(m,lZ,-J we set var(m,lZ,-i) = var(m,-,)Z,-2) = @ for all I, and we find from (A-4) the steady-state value 4 for any arbitrary positive semidefinite initial matrix to be @=AQiA’+
G- (A@B’)(B@B’+F)-‘(AW)‘.
(A.5)
@ is the unique solution to (AS) within the class of positive semidefinite
T. IU. Andersen,
Price
dynamics
under imperfect
injormation
359
matrices.24 By using this steady- state solution the estimation can be simplified considerably since ‘k;_,=\k=A~B’(B~~‘+F)-’
forallt.
(‘4.6)
It follows from (A.3) that we can write the conditional given 1,-I as a distributed lag in all past observations: E(m,l&l)
expectations of m,
= *d(L)s,-l,
(A-7)
where
B(L)=I+(A-‘kB)L+(A-wq2L2+
*** .
Proof of Proposition 5
We have that
where
Change in a-&
We have that
= Pl
Q- &,(a@/a0:J (@+crfJ2
’
and from (33) that
24Exi~tence and uniqueness depends on the requirements of controllability Bertsekas (1976, sect. 3.1)] which are fulfilled for the present model.
and observability [cf.
360
T. M. Andersen,
Price
dynamics
under imperfect
information
Hence,
=$[o:.+~-((u~e)2(P:-l)2+u:,o~,,(p:+l)(JT))-1] =f[~:.+(((~:-~)u:,+u:,,)*+4u~~u~ - b&)'(d
- q*-
u:,a:,(P:
+ l))(W]
where T = (( p: - l)*u;, + u;“)’ + ~u&J;,.
It follows that 861
->O au:,
and
882
-= ad
i as,
---
d au:,
Similarly, the other results of Proposition 5 can be found, see Andersen (1984) (available upon request) for details. References Aitchison, J. and J.A.C. Brown, 1957, The lognormal distribution (Cambridge University Press, Cambridge). Al&ion, A.A., 1970, Information costs, pricing and resource unemployment, in: ES. Phelps, cd., Microeconomic foundations of employment and inflation theory (Norton, New York) Amihud. Y. and H. Mendelson, 1983. Price smoothing and inventory, Review of Economic Studies 50. 87-98.
T. M. Anderseu.
Price
~ynotnics
under
itnperfecr
infornuuion
361
Anderscn, T.M.. 1983, Relative vs. absolute price rigidity, Economic Letters 13, 123-128. Andersen, T.M., 1984, Price dynamics under imperfect information, Memo. 84-5 (Institute of Economics, Aarhus). Andersen. T.M., 1985, Price and output responsiveness to nominal changes under difl’erential information, European Economic Review 29.63-87. Barre. R.J.. 1972, h theory of monopolistic price adjustment, Review of Economic Studies 39, 17-26. ’ Bertsekas. D.P., 14 76. Dynamic programming and stochastic control (Academic Press, New York). Blinder, A.. 1982, Inventories and sticky prices: More on the microfoundations of macroeconomits. American Economic Review 72,-334-348. Bnmner. K.. A. Cukierman and A.H. Meltzer. 1980. Staatlation, persistent unemolovment and the permanence of economic shocks, Journal of Monetary Economics 6,467-492. Danxiger. L.. 1983, Price adjustments with stochastic inflation, International Economic Review 24, 699-707. dc Menil, G.. 1974. Aggregate price dynamics, Review of Economics and Statistics 56, 129-146. D&c. J.H.. 1979, Demand estimation, risk aversion and sticky prices, Economic Letters 4, 1-6. Gcrtlcr. M., 1982, Imperfect information and wage inertia in the business cycle, Journal of Political Economy 90, 967-987. Gordon, R.J.. 1981, Output Ructuations and gradual price adjustment, Journal of Economic Literature 45, 493-530. Gordon, R.J., 1982, Price inertia and policy ineffectiveness in the United States, 1890-1980, Journal of Political Economy 90,1087-1117. Hayek. F.H.. 1945. The use of knowledge in society, American Economic Review 35, 519-530. Iwai, K., 1982, Disequilibrium dynamics: A theoretical analysis of inflation and unemployment, Cowles Foundation Monograph 27 (Yale University Press, New Haven, CT). Kuran. T.. 1983a. Asymmetric price rigidity and inflationary bias, American Economic Review 73, 373-382. Lawson. T., 1980, Adaptive expectations and uncertainty, Review of Economic Studies 47, 305-320. Maccini. L., 1984. The interrelationship between price and output decisions: Microfoundations and aggregate implications, Journal of Monetary Economics 13,41-65. Muth. J.F., 1960, Optimal properties of exponentially weighted forecasts, Journal of American Statistical Association 55. 299-.306. Nordhaus. W.D., 1972, Recent developments in price dynamics, Ch. 2 in: 0. Eckstein, ed., The economics of price determination (Boards of Governors, Federal Reserve System, Washington, DC). Okun. A.. 1981, Prices vs. quantities: A macroeconomic analysis (The Brookings Institution, Washington, DC) Paldam. M. and S. Hylleberg, 1984, Prices and wages in the OECD area, 1913-80: A study of the time series evidence, Memo. 84-7 (Institute of Economics, Aarhus). Phlips. L., 1980, Intertemporal price discrimination and sticky prices, Quarterly Journal of Economics 94, 525-542. Reagan, P.B., 1982. Inventory and price behavior, Review of Economic Studies 49,137-142. Rotemberg. J.J., 1982, Monopolistic price adjustment and aggregate output, Review of Economic Studies 49, 517-531. Shaw, G.K., 1984, Rational expectations: An elementary exposition (Wheatsheaf Books, Brighton). Sheshinski. E. and Y. Weiss. 1977. Inflation and cost of mice adiustmenf Review of Economic Studies’44, 287-304. Taylor, J.B.. 1985, Rational expectations in macroeconomic models, in: K.J. Arrow and S. Honkapohja, eds., Frontiers of economics (Basil Blackwell, Oxford).
PRICE DYNAMICS
UNDER IMPERFECT
INFORMATION
Torben M. ANDERSEN* University
ofrlarhus,
DK-8000
Aarhus
C. Denmark
Received August 1984, final version received November 1985 The paper considers the pricing decision of a monopolist firm having demand and costs exposed to nominal and real shocks which include both permanent and transitory changes. The firm obtains information through both price and quantity signals and the price equation is found by use of a filtering technique. It is shown that imperfect information implies nominal price smoothing where the price adjusts only partially relative to the past price by incorporating new information observed through price and quantity signals.
1. Introduction Recent work on the problem of price flexibility or price smoothing has analysed the role of inventories and adjustment costs of price changes in relation to a monopolist firm. The possibility of inventory holding (and backlogs) makes the firm to operate on two margins, namely on how much output to produce for inventory and on how much inventory to sell. As a consequence the firm engages in a sort of intertemporal substitution, and changes in current market conditions do not in general become reflected in prices instantaneously, that is, the adjustment process tends to be smoothed out over time.’ The same mechanism is at work if the effects of the adjustment of the capital stock is taken into account.2 On the other hand, if the firm holds no inventory but faces adjustment costs of price changes, it is similarly found that it is not necessarily profitable to adjust prices fully and instantaneously to current changes in market conditions, and even perfectly foreseen nominal changes do not necessarily lead to price changes.3 *I have benefited from comments on previous versions by Jacques D&e, an anonymous referee, and participants at the Sixth Conference of the Society of Economics Dynamics and Control, Nice, 1984. The usual disclaimer applies. Financial support from the Danish Social Science Research Council is gratefully acknowledged. ‘See, e.g., Phlips (1980), Blinder (19X2), Reagan (1982) and Amihud and Mendelson (1983). ‘See, e.g., Maccini (1984). ‘See. e.g., Barre (1972), Sheshinski and Weiss (1977), Rotemberg (1982), Kuran (1983) and Danziger (1983). 0165-1889/85/$3.3001985,
Elsevier Science Publishers B.V. (North-Holland)
340
T. M. Andersen,
Price
~namics
under
imperfect
information
The purpose of the present paper is to analyse another possible source of price smoothing by a monopolist firm, namely imperfect information on current market conditions. The importance of imperfect information for the adjustment of prices has often been stressed [see, e.g., Nordhaus (1972) Gordon (1981), Shaw (1984)], but has not hitherto been rigourously analysed.4*s To point out how far imperfect information can bring us in terms of explaining price smoothing by a monopolist firm we shall disregard both inventories and costs of price changes. This should not be taken to imply that a monocausal explanation of price inflexibility is sought for, but serves the analytical purpose of isolating the effects of imperfect information on the dynamics of prices. The different sources of price inflexibility should be seen as complementary rather than competing theories of price adjustment. Price sluggishness is often modelled by use of the so-called partial adjustment model which says that the current price (p,) is given as a convex combination of the past price (p,-r) and the optimal price (p:) defined in some specific sense, i.e.,
p,=XP,-1 + (1- a+?
Ojhjl.
Price sluggishness or price smoothing is accordingly identified with the fact that the current price depends on the past price (X # 0). Although widely used, the adjustment model (1) is an ad hoc model, but we shall in the present paper show how imperfect information can rationalize a price equation which comes very close to (1). To be specific we impose no restrictions on the adjustment of prices except that it is accepted as an institutional fact that prices for many products are set prior to trading6 [Alchian (1970), Gordon (1981), Okun (1981)], which is a reasonable assumption to make in the case of a monopolist firm. Given this observation the interesting question becomes the adjustment of prices between periods, that is, the dynamics of prices over time. The present analysis considers the price decision made by a single monopolist firm’ under less than perfect information on the demand and cost conditions, and the specific environment which is considered is quite general since both demand and costs are subject to permanent and transitory real and nominal shocks. Attention is paid to how the firm obtains information on market conditions through signals readily observable in a decentralized market economy. The main sources of information to the firm in such a setting is 4See, however, D&e (1979) who shows that imperfect knowledge about the elasticity of demand on the part of a monopolist firm has implications comparable to a kink in the demand curve, and thus explains some price rigidity. ‘This paper can be seen as a generalization of Andersen (1983). 6 With pre-set prices it becomes furthermore necessary to specify a trading rule; see section 2. ‘The case of many interdependent firms and hence the problem of coordinating price decisions is considered in Andersen (1985).
T.M.
Andersen,
Price
dynamics
under
imperfect
information
341
observation of costs (price signal) and sales (quantity signal), and it is shown how both the price and the quantity signal provide valuable information to the firm! Hence, not only price signals, but also quantity signals are proven to be important to the adjustment process in a decentralized market economy [compare to Hayek (1945)]. The main conclusion to come out of this analysis is that imperfect information implies nominal price smoothing, since the price quoted by the firm adjusts only partially relative to the past price by taking into account the new information received through the price and quantity signals. Consequently the resulting price equation gets a form which comes very close to that of the partial adjustment model in (1). This shows how imperfect information can explain nominal price smoothing without taking resort to inventories or costs of price changes. That is, imperfect information can qualitatively explain the same sort of sluggishness in the adjustment process as has been previously explained by inventories or cost of price changes. Blinder (1982) has criticized most models of price inflexibility for explaining only relative and not nominal price rigidity,9 since rigidity of prices is explained relative to assumed fixed costs. Hence, unless costs respond less than proportionally to nominal changes, prices will respond proportionally to nominal changes. This criticism does not apply to the present study, since the price smoothing explained in this paper is smoothing of the nominal price quoted by the firm. It turns out that even if the price adjusts proportionally to the expected nominal change, the price equation does not satisfy the traditional homogeneity properties. Brunner, Cukierman and Meltzer (1980) used the idea of confusion between transitory and permanent shocks to explain persistence in the effects of real and nominal shocks on the unemployment rate within an equilibrium model sustaining the neutrality property of systematic aggregate demand management policies. More related to the issue of price adjustment analysed here is the work of Iwai (1981) and Gertler (1982). Iwai (1981) considers the adjustment problems of a firm which acts as a monopolist in the output market and as a monopsonist in the input market. Price dynamics arising from learning over time of market conditions is analysed under specific assumptions conceming the sequential structure of the decisions taken by the firm. The present study differs from Iwai (1981) on several aspects of which the most important ones are that we (i) distinguish explicitly between real and nominal shocks, (ii) consider a more elaborate learning scheme where information acquisition by signals readily observable to the firm is stressed, and (iii) that price and output ‘The same would apply if inventories/backlogs existed. The informational role of sales or inventory signals has not, however, been analysed in the literature dealing with the role of inventories for price adjustment. ‘This criticism is not applicable to models incorporating adjustment costs of price changes. The importance of such costs, especially in relation to macroeconomic phenomena, remains still an open question.
342
T. M. Andersen,
Price
dynamics
under
imperfect
information
decisions are made simultaneously. More closely related is the study by Gertler (1982) where the dynamics of nominal wages is considered under the assumption that nominal wages each period are predetermined at a level equal to the expected Walrasian equilibrium price in an environment exposed to nominal shocks and productivity shocks. The paper is organized as follows. The general model of the monopolist firm is developed in section 2. In the following section it is considered how the information available to the firm is used to make inferences about market conditions, and the dynamic price equation is derived. Since the general version of the model is difficult to interpret, we consider in section 4 the special case where the firm is exposed only to nominal shocks, and prove some detailed results for this case. Section 5 concludes the paper. 2. The model of the monopolist firm
To analyse the dynamics of prices under imperfect information we shall set-up a relatively simple model of a monopolist firm, which for analytical reasons builds on a specific structural form. The main consideration has been (as in much of the literature referred to above) to adopt a formulation which allows an explicit solution to the pricing problem of a monopolist firm, viz. a log-linear model. Needless to say, by adopting this specific structure we are considering only an example of how imperfect information can induce price smoothing. Consider a monopolist facing the following demand function:
D,= Do(W’hdUm
d>
(2)
1.
N, is nominal income defined as RU,,, where R is real income” and Vi, is the nominal shock variable at time t. P, is the price quoted by the firm, and U,, is .le real shock variable to demand at time 1. The monopolist produces the commodity using a single input, say labour, and the production technology is given by a Cobb-Douglas production function
Q, = LP,
O
(3)
Hence, the costs of producing Q, units are C(Q,) = W,(Q,)”
where
(4)
a = cx-‘.
The main concern of this analysis is the dynamics of output prices and for this reason we assume wages to be determined exogenously. To avoid that the ‘OR is assumed to be constant, but a non-constant implications as the real shock U,, to demand.
R
will qualitatively
have the same
T. hi. Andersen,
Price
dynamics
under
imperfect
343
information
argument made for price sluggishness depends on sluggish wage adjustment we shall impose the strong assumption” that wages adjust proportionally to nominal changes, i.e., let W, denote the nominal cost of input at time t, we assume
w,=WUl,v,,,
(5)
where W is the real-wage rate, I2 Vi, is the nominal shock variable at time 1, and U,, is a real-shock to costs at time t. Notice, that the model fulfills the homogeneity requirements, that is, demand is homogeneous of degree zero in the price and the nominal income, and the cost function is homogeneous of degree one in the wage rate. Moreover, the nominal income and the nominal wage rate change proportionally to nominal changes. It should be remarked that a nominal change (shock) is defined as any change which, if perfectly foreseen, implies only a price change, and a real change (shock) is defined as any change which even if perfectly foreseen leads to output changes. The firm announces prior to period t a price conditional on alI available information at the end of period t - 1, 1,-i. We shall adopt the assumption employed in many similar models, that the firm is compelled to meet whatever demand is forthcoming at the announced price, and that the firm can instantaneously adjust production within the period, i.e.,
Q,= 4.
(6)
The commodity is not storable, and the objective of the firm is to set a price for period t, P,, so as to maximize expected profits K=
W,Q,- c(Q,)IL~).
(7)
Inserting we find that (7) can be written
Maximizing
V, with respect to P, we find the following first-order condition:
pl+d~-d_ WD,f-ldaR’-d’“-” I d-l
E[U3,Uj,‘daU,411,-l] ~[4%,l4-11
’
(9)
“This is a strong assumption since the argument advanced here concerning nominal price stickiness is applicable to nominal wages as well. “The remark made on R applies to W mutatis mutandis.
T. M. Andersen,
344
Price
dynamics
under
imperfect
information
The second-order condition reads E(D,,N;(d-
~)LP,-‘-~U,,II,-~)
- E(WUl,U3,Np”du(l + d~)~,-~-~“~,4l~,-,)< 0. To proceed we have to properties of the nominal real shock to costs (us,). log-normally distributed.
make some assumptions concerning the stochastic shock (Q,), the real shock to demand (KY,,) and the We shall assume these shocks to be independently That is, for i = 1,2,3, we assume l3
U,, = In Q, = Ui, + e,,, Ui, =
Pi”ir-1
+ nj,V
01) 02)
lP;l s 19
e,, - N(0, ~2).
“it - N(O, ui),
Furthermore,
(10)
it is assumed
En;,ej, = 0,
Vi, j= 1,2,3,
Eni,nj, = 0,
Vi, j=1,2,3,
i+j,
Ee,,ej, = 0,
Vi, j=1,2,3,
i#j.
Eqs. (11) and (12) say that any shock is composed of two parts, a completely transitory part (e,,) and a permanent part ( ui,) which is made up of a first-order autoregressive process in ni,. Using that for any log-normally distributed variable x, we have that [see Aitchison and Brown (1957)] In E[ x:11,-,]
= aE[lnx,ll,-,]
+ $2var[lnx,lI,-,].
We find that (9) can be written in logs as P,= co+ Eb,,l4-11 13For
any variable
+@b,,lh-11
X, we shall define
x, as x, = In A’,.
+
C2Eb3tl~~-ll~
03)
T. M. Andersen,
Price
dynamics
under
imperfect
informurion
345
where co = (1 + da - d)-‘(
+f((l
ln( D,“-‘WR’-J(“-l)/(d
+da)2-d2)var[u,,lZ,-,]
- 1)) + ivar[ u,,lZ,-,]
+:(a’-l)var[u,,ll,-,I),
c,=(a-l)(l+da-d)-‘>O,
It should be remarked that it follows from (13) that, even though the firm is imperfectly informed about nominal shocks, the price quoted by the firm is adjusted proportionally to the expected nominal shock, i.e., the nominal price is fully flexible with respect to the expected nominal shock. However, the expected real shocks to demand and costs do not lead to proportional changes in the nominal price since these shocks, even under full current information, should not be completely passed on to the nominal price. Notice, that the Gaussian assumption made on the random variables implies that the conditional variances become independent of the actual realizations of the random variables, and hence we can treat them as constant for the purpose of the present analysis. The actual quantity traded is found from (2) and (6) to be
4, = d, = lnD, + d(lnR + I(~, - p,) + Us, = InDO + d( ul, - E( r~,lZ,-~)) + d 1nR + (~2, - dc,E(u,,lL,))
- dc,%,lL)~
(14)
Eq. (14) shows that output responds only to the unanticipated part of the nominal change, whereas it responds with different coefficients to the actual and expected real change to demand and to the expected real change to costs. Finally, we need to determine the information set of the firm at the end of period r - 1, 1,-i. In a decentralized market economy the firm obtains information on market conditions from two important private sources,14 14The firm might also have access to some public (aggregate) information which may be useful to the firm in identifying the,market conditions. We leave out this complication by noting that nothing qualitatively would be changed by introducing this aspect into the model. For an analysis which also includes this type of information in a model with many price-setting firms, see Andersen (1985).
T. M. Andersetr,
346
Price ~vnamics
uder
iniperfect
informalioti
namely, the sales it experiences (quantity signal) and the costs (price signal) of the inputs the firm uses. We shall analyse what kind of information the firm obtains through these two signals and how this knowledge influences the price decision. Given the initially announced price ( p,- t) for period t - 1, actual sales (q,r ) is observed at the end of the period. It is easily seen that the sales figure q,- i is informationally equivalent to q;- Ir where q;-1
= q,-*
Da- dp,-i - dln R
-1n
= dUl,-1+
UZ,-l.
(15)
That is, the quantity signal qiml reveals information on the nominal shock ~i,-i and the real shock to demand u2,-i in period t - 1. Moreover, the firm can infer information from the price signal it observes on nominal costs, since the wage signal is informationally equivalent to w/-i, where w/-1=w,-1-w =
Ul,-1+
U3,-1.
06)
Hence, the price signal reveals information on the nominal shock ul,- t and the real shock to costs u3,-i in period t - 1. Let us denote by s,-i the 2 X 1 observation vector of the firm in period t - 1 given as s,-1
=
[ 1 4:- 1 )$ql
.
(17)
The information set of the firm at the end of period t - 1 is made up of s,-~ and all past observation vectors, i.e., I,-,=
{S,49~,-2r...}.
08)
The information problem faced by the firm is thus two-sided. Firstly, given the signals q;- 1 and w;-~, it must evaluate how much can be regarded as a nominal change and how much as real changes to demand and to costs. Secondly, it must sort out how much of each change constitutes a transitory change and how much constitutes a permanent change. We shall in the next section analyse how this information-expectations problem is solved, and derive the resulting price equation. 3. Filtering and price adjustment We can now turn to the interesting problem of deriving the equation describing price adjustments over time, and to this end we have to start by
T. M. Andersen,
Price dynamics
under imperfect
itlformariott
341
finding the conditional expectations of the state variables given the information available to the firm, cf. (13). Given that ei, is specified as a white-noise random variable, we have that
E[uitllt-,I= E[uitl~,-lI~
Vi = 1,2,3.
(19)
Eq. (19) says that the tim wants to find out the permanent part of each shock since this is what matters for future market conditions. Using (19) we find that (13) can be written P, = co + W[m,lL,l~
where
4 = h, I;=
[l
ClU2,
C93,l~
1 11.
We can now prove the following result on the conditional expectation of m, given 1,-r. Lemma I. By use of the Kalman-filtering technique the conditional expectation of m I is found by the following updating equation:
E[m,l~,-ll =AE[m,-d~,-21++($,-I - E[s,-#,-21).
(20)
Alternatiuely, the conditional expectation can be written as a distributed lag in the observed signals:
E[m,ll,-r]
=Il/e(L)s,-~~
(21)
where B(L)=I+(A-#B)L+(A-#B)2L2+
-0. .
A, B and +!I are matrices defined in the appendix. Proof.
See appendix.
Eq. (20) shows that the conditional expectation of m, given 1,-r is found by updating the past expectation of the state variables (i.e., of m,-, given 1,-2) according to the difference between the actually observed signals on the current market conditions (s,-r) and the expectations of these market condi-
348
T. M. Andersen,
Price
&vtamics
under
imperfecr
information
tions given the hitherto accumulated information (~s,-,]Z,,~]).‘~ Eq. (21) says that the conditional expectation of m, given I,-t alternatively can be written as a distributed lag in all past observed signals on market conditions. Having found the conditional expectations of the state variables, it is straightforward to proceed to determine the price equation for the firm. Proposition 1. Under imperfect information about the current market conditions, the nominal price quoted by the monopolist firm for period t ( p,) is a function of the previously quoted price ( p,- 1) and the most recent observations on market conditions (4;- 1, w,‘- 1), p, = 6, + 6,p,-,
+ Qy:-,
+ &W:_l,
(22)
where
i&=c,(l
-z;(A
-l/B)Z,),
6, = z&4 - JIB)Z,,
(S,-s,>=z;+. Proof.
Using Lemma 1 we find p,=c,+Z;(A-#B)E[m,-,lZ,-,]+I#-,.
Lagging (23) and premultiplying (CIA
- W)Z,)P,-,
(23)
by Z;(A - #B)Z,, we find
= (Z&4
- WZAc,
+ (Z;(A - ~B)Z,Z;)E[m,-,lZ,-,l.
Using that Z,Z-,’= I, where Z is a 3 x 3 matrix with all entries equal to one, eq. (22) follows straightforward. Proposition 1 shows that the imperfect information implies a sluggish adjustment of the nominal price set by the monopolist firm. The price equation (22) shows that the price quoted by the firm for period t ( p,) adjusts partially relative to the past price ( P,-~) by taking into account the new information contained in the quantity signal (q;- r) and the price signal (w;- J. The price smoothing arises from imperfect information due to the fact that the firm does not want to make the current price to depend only on the most recent ISLemma derived from
1 illustrates the well-known fact that adaptive-like expectations statistical principles; cf., e.g., Mutb (1960) and Lawson (1980).
formula
can
be
T. M. Andersen.
Price dynamics
under imperfect
information
349
information since it does not know whether this information reflects transitory or permanent influences. The lagged price term in (22) captures in this way the past information of relevance to the current price decision, and we End that the resulting price equation comes very close to that of the partial adjustment model; cf. (1). Finally, it should be noted that the gradual price adjustment, of course, implies a sluggish adjustment of output; cf. (14). Eq. (22) states the price equation as a first-order difference equation, and the question arises whether this equation is stable. To answer this question we prove: Corollary 1.
The price equation (22) is stable.
Proof. Stability of (22) is ensured if 8; goes to zero for n going to infinity. We find that 8; = I;D”13, where D = A - $B, and it follows from the fact that @ is the unique solution to (A.5) within the class of positive semi-definite symmetric matrices that D is stable, i.e., D” + 0 for n --) co; cf. Bertsekas (1976, sect. 3.1, ch. 6, app. A).
To interpret the coefficients in the price equation (22), we start by observing that the coefficients 6, and S, are given as
where #ij (i = 1,2,3, j = q’, w’) is the weight given to the innovation in the jth signal in predicting the ith element of the m vector; cf. (20). It follows that the higher weight a given signal gets in the prediction formula, the higher becomes the signals weight in the price equation. The relation between the weight put on the past price (6,) and the weight given to the new information (a,, 6,) in (22) is easily seen since we have that
This shows that the more weight the new information is given, the less weight is given to the past price in the price equation. Hence, to the extent that the recent information is a reliable guide to future market conditions, less weight is put on the past price when setting the price for any given period and vice versa. The extent of price smoothing depends thus on the quality of the information signals q;- i and w,‘- i as indicators of future market conditions. The inverse relation between the weights to current and past information is not found in the case where all shocks are completely transitory, i.e., p, = pz = pJ = 0. In this case we find that 6, = 6, = 0, since the current information
350
T. M. Andersen,
Price dynamics
under imperfect
information
signals contain no valuable information on future market conditions when all shocks are completely transitory. Consequently, the past price is also void of any relevant information for current market conditions, and 6, becomes equal to zero, and we find that Pt = co*
That is, the nominal price is completely fixed when all shocks are completely transitory. Unfortunately, it is very difficult to obtain analytical results on how 6,, 6, and 6, depend on the underlying stochastic structure of the model, and although the result of nominal price smoothing comes out clearly from the general model (cf. Proposition l), we shall consider a special version of the model to obtain more insight in the specific characteristics of the price equation. Hence, to gain more specific knowledge of the nominal price smoothing implied by imperfect information, we shall proceed by considering nominal shocks in isolation from any real shocks in section 4. It should be obvious from the exposition that the same qualitative results can be obtained by considering any of the shocks in isolation. 4. Nominal shocks To gain further insight into the effects of imperfect information for the adjustment of prices, it is useful to consider the case where the firm is exposed only to nominal shocks (i.e., ui’, = ei’, = 0, i = 2,3). In this case the firm faces the information problem of disentangling the permanent part of the nominal shock from the transitory part. The reason for considering nominal shocks is twofold: firstly this allows us to address a number of questions raised in macroeconomics and monetary theory, and secondly nominal changes provide a good point of reference since such shocks have only nominal price effects and no real effects under full information. Before analysing this information problem in detail, we shall note that, if the firm were setting the price for period t given full information about the period f nominal shock, we would find the price for period t to be P: = co + q,,
(24)
i.e., under full information the price adjust proportionally to any nominal shock. Turning to the imperfect information case, we find that the quantity signal q:-1 and the price signal w:-i are informationally equivalent, since they both reveal the period t - 1 nominal shock, I(~,-~. Without loss of generality we can thus consider only one of these signals, and we choose the quantity signal, i.e., s,-1=
q;-1.
T. M. Andersen,
Price
dynamics
under
imperfect
information
351
Proposition 2. In an environment subject to nominal shocks about khich the firm has imperfect information, the price equation for the monopolist firm is given as
&I,-,,
(29
For ai’,=a$=O, i= and g, = n,,; cf. proof We shall start by solving variance (see proof of Lemma
2,3, we have that B= d, L=de,,, m,=vl,, of Lemma 1. _ _.. . for the stationary solution to the contihonal l), and we find that
=
6, +
&P,-,
+
4
=
(1 -
UP,/4
A,
= a:,/(+
where
+ ale),
S;=d&. Proof. A=P,
@=p;@+a:,-p,@d(d2@+d20,2,)-‘p,@d
= a;, + p;!S - pi@p’( @ + a$! This is a second-order equation in @, and solving for @ we find ~-
-(P:-l)a:,-o:,fd((P:-l)a:,+a:,)2+40~~a,Z, -2
The unique positive stationary solution for @ is found to be 4P=+
I(
p; - 1 UI’C+ a:, +
)
((p:-l)~;a+cf~“)‘+4o:p:,}.
Hence, the # coefficient becomes +=wVd(@+a:e)The price equation (25) follows straightforward.
352
T. M. Andersen,
Price
dynamics
under
intperject
information
Eq. (25) shows that, even when nominal shocks are considered in isolation, imperfect information about nominal shocks implies that the adjustment of the nominal price to nominal shocks becomes smoothedf6 whereas the nominal price under full information would adjust instantaneously and proportionally to nominal changes. The reason for the nominal price smoothing is that the firm cannot know whether a nominal shock is a permanent or a transitory shock, and hence by how much to adjust the price for the next period given the nominal shock of the current period. To see clearly how imperfect information implies price inflexibility or price smoothing it is natural to compare with the case of full information, and we find the following result: Proposition 3. Let p: and pI denote the price quoted by the monopolist firm in an environment exposed to nominal shocks under full and imperfect information, respectively. We find that
4 Proof.
PI1-4
P:l-
(26)
From (24) it follows that
4
p;l = v=[u,,l = u1’, + a,2,/( 1 - p: ) .
The steady-state variance of p, is found from (25) to be 8;
4
P,l = y-pd%,l. -:
Using that -=
6:
1-s;
GPi
1 - (1 - x,)‘p;
’
we find that var[ p,] < var[ p,?] since x:p; c 1 - (1 - A,)2p:, l6 With correlated state variables it is, of course, generally possible to find a relationship between and p,- 1 even if no price smoothing is taking place. In the present model p,-, appears only for its price smoothing role, which is seen by noting that P,-~ drops out of (25) if there is no confusion between transistory and permanent shocks. p,
T. M. Andersen,
Price +lumics
under
imperfecr
informution
353
or 0 < (1 - p:> + 2A,p31 -A,). This shows that to an outside observer the price variability is lesser in the world with an imperfect, information structure than in a world with full information, hence the conclusion that imperfect information implies price inflexibility or price smoothing. To see clearly how the confusion between permanent and transitory changes induces sluggish price adjustment, we find that the price smoothing disappears if the nominal shock does not include any transitory part, i.e., ~12,= 0, since (25) reduces to PI =
60 +
PlUl,-1.
(27)
Eq. (27) shows that the price depends no longer on the lagged price, and p, is adjusted to the past nominal shock ~i,-i, multiplied by the fraction (pi) of this change which carries over to period f. In this case the quantity signal #-i reveals the permanent nominal change with full certainty and no price smoothing evolves. However, the firm is still imperfectly informed about the innovation in the nominal shock, and the price path diverges from the full current information price path. Another special case should be mentioned where all nominal changes are completely transitory, pi = 0. In this case we find [cf. (25)]
This case implies a completely fixed price since with completely transitory shocks there is nothing to be learnt about the future and hence no reason to adjust the price to take account of current and past information. Consider the case where pi = 1. In this case we find from (25) that the price equation becomes” P, = hP,-,
+ (1 - hh-1.
The Xi coefficient is determined by the relation between the variance of the transitory component and the sum of the variance of the transitory and the permanent component of the nominal change.” As shown above, the full current information price for period t - 1 is p,Y1 = ~i,-i, and we find that (29) can be written as Pr=hP,-,
+ (1 -UPr%
“Generally, (25) can be written p, = 80 f PI(AI Pr- L +o -h)ul,-,). “Recall that u,, = D,, + e,,. Hence, var(u,,ll,- 1) = var(ol,ll,- 1) + d = 0 + ‘6
T. M. A tdersen.
354
PI-P,-1
=
(1 -MPL
Price +rutvics
under imperfect
irifonnutiorr
-PA.
(30)
Eq. (30) is identical to the partial adjustment equation (l), except for the fact that the price adjustment is a linear function of the difference between the optimal and the actual period t - 1 price, rather than the difference between the optimal period t price and the actual period r - 1 price. This change is caused by the imperfect information set-up of the present analysis, and (30) says that the firm ex post in period f - 1 knows the nominal shock Us,- 1 and hence the optimal full information period 1- 1 price, p,? 1. The extent of price adjustment is thus determined by the divergence between the optimal and the actual price in period t - 1. In the present context the partial adjustment is not caused by adjustment costs, etc., but by the fact that the firm does not know until the end of the period what the optimal price would have been for the period. Interestingly we find that the price equation has the following property: Proposition 4. The nominal price quoted by the monopolist firm ( p,) in an environmenl subject to nominal shocks is homogeneous of degree 6, + 8; in the nominal variables ptel and ulr-,. In general 6, +- Si is different from one since
Proof.
Follows immediately from (25) and the definition of 6, and Si.
This result is remarkable because it says that even though the firm adjusts its price proportionally to the expected nominal change [cf. (13)], we do not find that the price equation (25) displays homogeneity of degree one in the two nominal variables p,- 1 and ulrel which determine p,. Such homogeneity of degree one prevails only in the special case where the permanent part of the nominal shock is given by a random walk process ( p1 = 1). The reason for this is obviously that the firm does not want in a proportional way to pass on nominal changes to future nominal prices unless the permanent part of the nominal shock is of a complete permanent character (pl = 1). Notice, that this result holds even if the nominal variable does not include any transitory part, i.e., u:, k 0. Although this point may seem clear enough in the present context, it has important implications for empirical work on macro price equations, where long-run homogeneity of the price equation is either imposed or tested19 for models based on a price adjustment model close to that of (22) or (25). The “See.
e.g., Gordon
(1981,
1982).
T.M.
Andersen,
Price dynamics
under imperfect
information
355
present model can thus rationalize such price equations which have previously been stated as ad hoc equations, and moreover we are able to explain (Proposition 4) that long-run homogeneity is the exception rather than the rule, even though prices may adjust proportionally to expected nominal changes.20 Moreover, we find that changes in the structure of the price equation between different subperiods can be explained by changes in the underlying stochastic properties of the random variables; cf. Proposition 5. Proposition 5. In an environment subject to nominal shocks, we find under imperfect information that (i)
the coefficient to the lagged price term in the price equation (6,) increasing in ofe, decreasing in a:,, and increasing in p,, and
(ii)
the coeBcient to the current observation on market conditions in the price equation (6,) is decreasing in a:=, increasing in a:,, and increasing in pl.
Proof.
is
See appendix.
Proposition 5 shows the sign of the changes in the coefficients in the price equation to changes in the stochastic properties of the nominal shock variable. An mcrease m u1’, implies a larger variance of the transitory part of the nominal change, and more weight is put on the past price and a lower weight is given to the recent information, since changes in the latter to a larger extent reflect transitory changes when u1’, increases. On the other hand, if a;,,, increases, it means that the variance of the permanent part of the nominal variable increases, and lesser weight is given to the past price and more weight is given to the recent information, since variation in the latter now to a larger extent reflects permanent changes. A change in p1 has two effects on 6, and 13,: one is a direct effect which tends to increase both, and the other is an indirect effect which operates through the effect of p1 on the conditional variance @. A higher p1 (pl > 0)21 means more persistence in the permanent part of the nominal change, and this tends to increase @ in the same way as for an increase in u&. This makes it clear that S, increases for p1 increasing, and it turns out that the direct effect on St dominates over the indirect effect and 6, increases also due to an increase in pl. “This non-homogeneity of degree one does not leave any clear-cut reason for non-neutrality of money and hence for an active monetary policy [see Andersen (1985) for a further discussion of this issue]. *‘Notice, that for p1 < 0 we have that 6, < 0, hence higher p1 means less persistence and the numerical value of 6, falls. The same holds for 82.
356
T. M. Andersett,
Price
&un~ics
wder
intper/ect
ittforttturior~
6. Conclusion The adjustment of prices is a controversial issue in both micro- and macro-economics, and the present model has been formulated such that it is of relevance to both branches of literature. The conclusion that price smoothing evolves under imperfect information about the permanent and transitory components of real and nominal shocks provides thus a rigourous dynamic formulation of some theoretical and empirical microeconomic models of price behaviour,** while at the same time explaining price sluggishness and especially nominal price rigidity which is of great importance to a number of macro-economic problems. Since a summary of the findings have been given in the introduction we shall conclude by a few remarks on some aspects of the model. We have in the present context allowed only for privately observed information, viz. price and quantity signals, but to place the present model in a more specific macroeconomic context it would be relevant to include public information on aggregate variables as information variables to the firm. The model could easily be modified to allow for such information, and nothing qualitatively is changed as long as this public information combined with the private information is insufficient to allow the firm full information on the current state of the market. Moreover, a single firm has been analysed in isolation and the interaction between many firms is essential to the analysis of macroeconomic problems and especially the problem of monetary policy [see Andersen (1985)). It has been assumed that the firm sets the price before it is informed about the current costs. Nothing qualitatively would, however, be changed if costs were known before the price is to be set as long as costs are subject to both nominal and real shocks [cf. Andersen (1983)]. For analytical convenience use has been made of specific functional forms of the demand and the cost functions. Though, the qualitative result of the paper relies more on the imperfect information structure than the specific functional form of the model. More troublesome is the assumption that demand determines quantities traded given the prices set by firms. This is in a sense a very Keynesian assumption since it implies that effective demand determines output, but the empirical importance of such arrangements remains an open question. This is so because it implies that ftrms continue to supply at the announced price even if marginal costs exceed the price. It could, of course, be argued that firms have an incentive to act in this way to make their price announcements credible, i.e., to gain reputation that prices announced are reliable signals for the decisions of consumers by ensuring that consumers can ‘*The present model can thus be seen as a rigourous dynamic formulation of the mod& by Nordhaus (1972) and de Menil (1974), with specific attention to learning over time ol cost and demand conditions.
T. M. Andersen,
Price
dynamics
under
imperfect
information
351
actually buy at the quoted prices. However, the assumption implies a considerable analytical simplification, and it seems difficult to do without it due to the problems of handling the non-linearities which arise under other assumptions; cf. Taylor (1985). Finally, it should be noted that the present model yields results which could fairly easily be exposed to empirical tests, an interesting topic which I hope to be able to pursue in the future. However, given that the theoretical analysis justifies the ad hoc models used in many empirical studies partial support for this model has already been made.23 Appendix Proof of Lemma 1
The vector m, can be written as
m, = Am,-l f g,,
(A.1)
where
A=
[“If 1 Pl
0
0
00
0 p2
P3 0,
g,= [c2n3r 1. w2,
The observation vector s,-i can be written s,-~ = Bm,-l+.ft-lv
64.2)
where
23To refer to just a small and recent sample of such work (which fits into this framework), see, e.g., Gordon (1981, 1982) and Paldam and Hylleberg (1984).
358
T. M. Andersen,
Price
dynamics
under
imperfect
information
The firm is faced with the problem of making an estimate of m, given the information 1,-i. This problem can be solved by use of the Kalman Filtering technique; cf. Bertsekas (1976). Using this method we can write the conditional expectation of 112, given I,-i as E(m,JZ,-,)
=AE(m,-ill,-,)
+ *,&,-i
- BE(m,-ilZ,-i)),
(A-3)
where
The conditional variance of m, given I,-, Bicatti equation as var(m,IZ,-,)
is determined from the so-called
=Avar(m,-,lZ,-,)A’+
G
- (Avar(m,-,lZ,-,)B’)(Bvar(m,-#,-,)B’+F)-’
X (A v4m,-114-Z)B’)‘,
(~4.4)
where
io2 *’ le
and
G= E(g:g,)=
[ 01” 002
c2a2 10 2n
1
da,2,,
d*a* + a*
4,
+ 4e
c*(J3 **0 ”
I .
Notice, that the sequences of {et-,} and {var(m,lZ,-J} are independent of the actual realizations of the random variables. It follows that they can be calculated independently of knowledge of the actual realizations of the random variables, and we shall solve for their steady-state values. To solve for the steady-state value of var(m,lZ,-J we set var(m,lZ,-i) = var(m,-,)Z,-2) = @ for all I, and we find from (A-4) the steady-state value 4 for any arbitrary positive semidefinite initial matrix to be @=AQiA’+
G- (A@B’)(B@B’+F)-‘(AW)‘.
(A.5)
@ is the unique solution to (AS) within the class of positive semidefinite
T. IU. Andersen,
Price
dynamics
under imperfect
injormation
359
matrices.24 By using this steady- state solution the estimation can be simplified considerably since ‘k;_,=\k=A~B’(B~~‘+F)-’
forallt.
(‘4.6)
It follows from (A.3) that we can write the conditional given 1,-I as a distributed lag in all past observations: E(m,l&l)
expectations of m,
= *d(L)s,-l,
(A-7)
where
B(L)=I+(A-‘kB)L+(A-wq2L2+
*** .
Proof of Proposition 5
We have that
where
Change in a-&
We have that
= Pl
Q- &,(a@/a0:J (@+crfJ2
’
and from (33) that
24Exi~tence and uniqueness depends on the requirements of controllability Bertsekas (1976, sect. 3.1)] which are fulfilled for the present model.
and observability [cf.
360
T. M. Andersen,
Price
dynamics
under imperfect
information
Hence,
=$[o:.+~-((u~e)2(P:-l)2+u:,o~,,(p:+l)(JT))-1] =f[~:.+(((~:-~)u:,+u:,,)*+4u~~u~ - b&)'(d
- q*-
u:,a:,(P:
+ l))(W]
where T = (( p: - l)*u;, + u;“)’ + ~u&J;,.
It follows that 861
->O au:,
and
882
-= ad
i as,
---
d au:,
Similarly, the other results of Proposition 5 can be found, see Andersen (1984) (available upon request) for details. References Aitchison, J. and J.A.C. Brown, 1957, The lognormal distribution (Cambridge University Press, Cambridge). Al&ion, A.A., 1970, Information costs, pricing and resource unemployment, in: ES. Phelps, cd., Microeconomic foundations of employment and inflation theory (Norton, New York) Amihud. Y. and H. Mendelson, 1983. Price smoothing and inventory, Review of Economic Studies 50. 87-98.
T. M. Anderseu.
Price
~ynotnics
under
itnperfecr
infornuuion
361
Anderscn, T.M.. 1983, Relative vs. absolute price rigidity, Economic Letters 13, 123-128. Andersen, T.M., 1984, Price dynamics under imperfect information, Memo. 84-5 (Institute of Economics, Aarhus). Andersen. T.M., 1985, Price and output responsiveness to nominal changes under difl’erential information, European Economic Review 29.63-87. Barre. R.J.. 1972, h theory of monopolistic price adjustment, Review of Economic Studies 39, 17-26. ’ Bertsekas. D.P., 14 76. Dynamic programming and stochastic control (Academic Press, New York). Blinder, A.. 1982, Inventories and sticky prices: More on the microfoundations of macroeconomits. American Economic Review 72,-334-348. Bnmner. K.. A. Cukierman and A.H. Meltzer. 1980. Staatlation, persistent unemolovment and the permanence of economic shocks, Journal of Monetary Economics 6,467-492. Danxiger. L.. 1983, Price adjustments with stochastic inflation, International Economic Review 24, 699-707. dc Menil, G.. 1974. Aggregate price dynamics, Review of Economics and Statistics 56, 129-146. D&c. J.H.. 1979, Demand estimation, risk aversion and sticky prices, Economic Letters 4, 1-6. Gcrtlcr. M., 1982, Imperfect information and wage inertia in the business cycle, Journal of Political Economy 90, 967-987. Gordon, R.J.. 1981, Output Ructuations and gradual price adjustment, Journal of Economic Literature 45, 493-530. Gordon, R.J., 1982, Price inertia and policy ineffectiveness in the United States, 1890-1980, Journal of Political Economy 90,1087-1117. Hayek. F.H.. 1945. The use of knowledge in society, American Economic Review 35, 519-530. Iwai, K., 1982, Disequilibrium dynamics: A theoretical analysis of inflation and unemployment, Cowles Foundation Monograph 27 (Yale University Press, New Haven, CT). Kuran. T.. 1983a. Asymmetric price rigidity and inflationary bias, American Economic Review 73, 373-382. Lawson. T., 1980, Adaptive expectations and uncertainty, Review of Economic Studies 47, 305-320. Maccini. L., 1984. The interrelationship between price and output decisions: Microfoundations and aggregate implications, Journal of Monetary Economics 13,41-65. Muth. J.F., 1960, Optimal properties of exponentially weighted forecasts, Journal of American Statistical Association 55. 299-.306. Nordhaus. W.D., 1972, Recent developments in price dynamics, Ch. 2 in: 0. Eckstein, ed., The economics of price determination (Boards of Governors, Federal Reserve System, Washington, DC). Okun. A.. 1981, Prices vs. quantities: A macroeconomic analysis (The Brookings Institution, Washington, DC) Paldam. M. and S. Hylleberg, 1984, Prices and wages in the OECD area, 1913-80: A study of the time series evidence, Memo. 84-7 (Institute of Economics, Aarhus). Phlips. L., 1980, Intertemporal price discrimination and sticky prices, Quarterly Journal of Economics 94, 525-542. Reagan, P.B., 1982. Inventory and price behavior, Review of Economic Studies 49,137-142. Rotemberg. J.J., 1982, Monopolistic price adjustment and aggregate output, Review of Economic Studies 49, 517-531. Shaw, G.K., 1984, Rational expectations: An elementary exposition (Wheatsheaf Books, Brighton). Sheshinski. E. and Y. Weiss. 1977. Inflation and cost of mice adiustmenf Review of Economic Studies’44, 287-304. Taylor, J.B.. 1985, Rational expectations in macroeconomic models, in: K.J. Arrow and S. Honkapohja, eds., Frontiers of economics (Basil Blackwell, Oxford).