❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s
❊s❝♦❧❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❊❝♦♥♦♠✐❛ ❞❛ ❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s
◆◦ ✸✽✹ ■❙❙◆ ✵✶✵✹✲✽✾✶✵
❚❤❡ ▼❛❝r♦❡❝♦♥♦♠✐❝ ❊✛❡❝ts ♦❢ ■♥❢r❡q✉❡♥t ■♥✲
❢♦r♠❛t✐♦♥ ❲✐t❤ ❆❞❥✉st♠❡♥t ❈♦sts
▼❛r❝♦ ❆♥tô♥✐♦ ❈❡s❛r ❇♦♥♦♠♦✱ ❘❡♥é ●❛r❝✐❛
❖s ❛rt✐❣♦s ♣✉❜❧✐❝❛❞♦s sã♦ ❞❡ ✐♥t❡✐r❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❞❡ s❡✉s ❛✉t♦r❡s✳ ❆s
♦♣✐♥✐õ❡s ♥❡❧❡s ❡♠✐t✐❞❛s ♥ã♦ ❡①♣r✐♠❡♠✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛
❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s✳
❊❙❈❖▲❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❈❖◆❖▼■❆ ❉✐r❡t♦r ●❡r❛❧✿ ❘❡♥❛t♦ ❋r❛❣❡❧❧✐ ❈❛r❞♦s♦
❉✐r❡t♦r ❞❡ ❊♥s✐♥♦✿ ▲✉✐s ❍❡♥r✐q✉❡ ❇❡rt♦❧✐♥♦ ❇r❛✐❞♦ ❉✐r❡t♦r ❞❡ P❡sq✉✐s❛✿ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r
❉✐r❡t♦r ❞❡ P✉❜❧✐❝❛çõ❡s ❈✐❡♥tí✜❝❛s✿ ❘✐❝❛r❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❈❛✈❛❧❝❛♥t✐
❆♥tô♥✐♦ ❈❡s❛r ❇♦♥♦♠♦✱ ▼❛r❝♦
❚❤❡ ▼❛❝r♦❡❝♦♥♦♠✐❝ ❊❢❢❡❝ts ♦❢ ■♥❢r❡q✉❡♥t ■♥❢♦r♠❛t✐♦♥ ❲✐t❤ ❆❞❥✉st♠❡♥t ❈♦sts✴ ▼❛r❝♦ ❆♥tô♥✐♦ ❈❡s❛r ❇♦♥♦♠♦✱ ❘❡♥é ●❛r❝✐❛ ✕ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✿ ❋●❱✱❊P●❊✱ ✷✵✶✵
✭❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s❀ ✸✽✹✮
■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛❢✐❛✳
T he Macroeconomic E®ects of Infrequent
Informat ion wit h Adjust ment Cost s
Marco Bonomo
Graduate School of Economics, Funda»c~ao Get¶ulio Vargas Ren¶e Garcia
D¶epartement de sciences ¶economiques, CI RANO and C.R.D.E. Universit¶e de Montr¶eal
First version: August 1995 T his version: May 2000
K eywor ds: Aggregation, Infrequent Informat ion, Adjust ment Cost Mod-els, T ime and St at e Dependent Rules, St at e Dependent Rules.
JEL Cl assi ¯ cat ion: E0,E1,E2,E3
Address for correspondence: D¶epart ment de Sciences ¶Economiques et C.R.D.E.,
Uni-versit¶e de Mont r¶eal, C.P. 6128, Succ. Cent re-ville, Mont r¶eal, Qu¶ebec, H3C 3J7, Canada.
We would like t o t hank Eduardo Engel, St eve A mbler, two anonymous referees and t he co-edit or, Gregor Smit h, for t heir comment s, as well as Carlos Viana de Carvalho and Benoit Brillon for t heir excellent research assist ance. Financial support from t he PA RADI research program funded by t he Canadian Int ernat ional Development Agency (CIDA) is grat efully acknowledged. T he ¯ rst aut hor would also like t o t hank CNPq for a research fel-lowship and PRONEX/ CNPq for ¯ nancial support . T he second aut hor t hanks t he Fonds
de la Format ion de Chercheurs et l' Aide µa la Recherche du Qu¶ebec (FCAR) t he Social
A bst r act
1 I nt r oduct ion
In t he last decade, t he macroeconomic lit erat ure paid considerable at tent ion t o t he pot ent ial aggregat e e®ect s of int ermit t ent large adjust ment s in mi-croeconomic decision variables1. Examples of decision variables modeled in
t hat way are prices, invest ment , consumpt ion of durables and employment2.
A dist inct ive feat ure of t his lit erat ure is t hat explicit aggregat ion of indi-vidual rules is undert aken, result ing in rich dynamic pat t erns for aggregat e variables which are in sharp cont rast wit h t he inert ial microeconomic behav-ior. In t hese adjust ment cost models, economic agent s always observe t he frict ionless opt imal level of t he cont rol variable and infrequent adjust ment s may be justi¯ ed by opt imal behavior in t he presence of kinked adjust ment cost s (Bert ola and Caballero, 1990)3 4.
We develop a simple model which int roduces imperfect informat ion in a kinked adjust ment cost model by assuming t hat agent s do not observe cont inuously t he frict ionless optimal level of t he cont rol variable. T hey re-ceive infrequent ly a ° ow of informat ion which is considered exogenous t o t he
1We are referring t o models of st at e-dependent rules. In fact , t en years before t he
combinat ion of t ime-dependent pricing rules became an essencial ingredient for t he K ey-nesian react ion t o t he Rat ional Expect at ions revolut ion. However, t he focus was in only one microeconomic decision variable: price. Examples of macroeconomic models built on t ime-dependent pricing rules are Fischer (1977), Taylor (1979), and more recent ly, Ball (1994), Ireland (1997), and Bonomo and Carvalho (1999).
2For prices, see Caplin and Spulber (1987), Caplin and Leahy (1991), Caballero and
Engel (1992 and 1993), T siddon (1993) and Almeida and Bonomo (1999); for invest ment , Caballero and Engel (1999); for invent ories, Caplin (1985); for consumpt ion of durables, Caballero (1993) and for employment , Caballero, Engel and Halt iwanger (1994).
3When t he adjust ment cost funct ion has a kink at t he point of no-adjust ment , it is best
for t he agent not t o adjust for small changes of t he frict ionless opt imal level. Adjust ment is t riggered when t he discrepancy between t he cont rol variable and it s opt imal level becomes large enough.
4Some recent papers ext ended t he st at e-dependent rules framework t o a general
agent 's decision. Examples of such exogenous int ermit t ent ° ows are perva-sive: macroeconomic st at ist ics such as in° at ion, level of employment or GNP are published periodically, dividends of ¯ rms are announced only at cert ain dat es, informat ion arrives in asset market s aft er regular closings on weekdays and holidays. In all t hese cases, agent s do not observe cont inuously t he vari-able of int erest . Such an int ermit t ent informat ion arrival has t he int erest ing implicat ion t hat a large number of agent s receive t he same informat ion at t he same t ime, creat ing t he condit ions for a pot ent ial mass react ion5. Indeed,
increased volat ility of ¯ nancial market s around dividend announcement s and macroeconomic dat a releases have been document ed in numerous art icles6.
Our model has t he dist inct charact erist ic t hat a vast number of agent s t end t o act t oget her, and more so when uncertainty is large7. We show t hat lump-sum adjustment cost s int eract wit h infrequent informat ion t o gener-at e e®ect s of aggreggener-at e shocks on macroeconomic variables t hgener-at di®er sub-st ant ially from t hose obt ained wit h cont inuous informat ion adjusub-st ment cosub-st models. First , the relat ive e®ect of cumulat ive aggregat e shocks decreases sharply wit h t he size of t he shock. Second, t he relat ive average e®ect of t hese shocks decreases when aggregat e uncert ainty increases. Ot her result s are more similar in bot h models. When idiosyncrat ic uncert ainty increases, t he average e®ect decreases, while an increase in t he adjust ment cost raises
5Imperfect informat ion could also result from t he opt imal decision of t he agent t o gat her
informat ion infrequent ly in t he presence of informat ion collect ion cost s. T his endogenous infrequent gat hering would not necessarily coordinat e agent s' react ions and will have dif-ferent macroeconomic implicat ions. We focus our at t ent ion only on t he exogenous arrival of macroeconomic informat ion for all agent s at t he same t ime.
6See Cornell (1978), for dividend announcement s and Harvey and Huang (1992),
Ed-eringt on and Lee (1993), for macroeconomic dat a releases.
7Ot her aut hors have explained t hese mass react ions by di®erent informat ion ext ract ion
t he average e®ect .
In order t o perform t he comparisons above, we ¯ rst solve t he microeco-nomic problem of ¯ nding t he opt imal policy in t he presence of bot h lump-sum adjust ment cost s and infrequent informat ion about t he value of t he frict ion-less opt imal level of t he cont rol variable. To make t he condit ions which det ermine t he opt imal policy as simple as possible while keeping t he main insights of t he model, we assume t hat t he st ochast ic process of t he frict ion-less opt imal value of t he cont rol variable has no drift . We ¯ nd t hat t he opt imal rule is for agent s t o adjust or not depending on t he st at e at t imes of informat ion arrivals.8 T herefore, it is bot h st at e and t ime dependent . Such
a rule was conject ured by Blanchard and Fischer (1989, p. 413) and Ca-ballero (1989) as t he rule t hat could result from a combinat ion of infrequent informat ion about t he st at e variable and adjust ment cost s. T he di®erence wit h our model is t hat t hey just ify t he infrequent ° ow of informat ion by t he exist ence of cost s of collect ing informat ion.
T he opt imal rule is charact erized by a single paramet er s, which det er-mines t he inact ion range (¡ s; s) for t he discrepancy between t he frict ionless opt imal value of the cont rol variable and it s act ual value, at t imes of infor-mat ion arrival. We show t hat t he inaction barriers are much t ight er t han in t he cont inuous informat ion model. When t he adjust ment cost is su± -cient ly low, t he barriers are quit e insensit ive to t he uncert ainty governing t he st ochast ic process assumed for t he opt imal level of t he cont rol variable. On t he ot her hand, an increase in t he adjust ment cost brings about a rel-at ively larger increase in t he barriers when informrel-at ion is infrequent t han when it is cont inuous.
Ball and Mankiw (1994) also explore t he consequences of a price rule t hat is both t ime and stat e dependent . T he agent s adjust wit hout paying a
8T he presence of a large drift will make it opt imal for agent s t o adjust between
menu cost at even periods. Adjust ment s at odd periods will be made only if t he bene¯ t of doing so is great er t han t he menu cost . T he frict ionless opt imal price is always known. T hey focus mainly on t he e®ect of t he drift in t he frict ionless opt imal price process on out put dynamics. In our model, we assume t he drift t o be zero and adjust ment cost s are always present . Our main goal is t o illust rat e t he int eract ion between adjust ment cost s and infrequent informat ion.
T he rest of t he paper is organized as follows. In Sect ion 2, we derive t he opt imal rule in t he presence of bot h lump-sum adjust ment cost s and infrequent observat ion of t he opt imal level of t he cont rol variable. Sect ion 3 evaluat es t he aggregat e e®ect of macroeconomic shocks t hrough t he opt imal adjust ment of microeconomic unit s. Concluding remarks are present ed in Sect ion 4.
2 T he Opt i m al R ul e
In t his sect ion, we set up t he opt imizat ion problem of agent s confront ed wit h infrequent informat ion and adjust ment cost s and derive t he opt imal decision rule. We also invest igat e t he implicat ions of t his rule for various con¯ gu-rat ions of adjust ment cost s and uncert ainty in relat ion wit h t he cont inuous informat ion case.
2.1 A ssum pt ions and for mulat ion of t he opt im i zat ion pr obl em
For simplicity, we will assume a quadrat ic form (x¡ x¤)2for t he lat t er cost9.
T ime is discount ed by a const ant inst ant aneous rat e½. We depart from t he previous lit erat ure by assuming t hat information about t he opt imal level x¤
arrives at discret e t ime int ervals10. Alt hough t he agent does not observex¤
between two successive informat ion arrivals, he can form probabilist ic assess-ment s about t he value ofx¤given his informat ion, which consist s of t he past
observat ions of x¤ at t he discret e informat ion t imes. We assume, again for
simplicity, t hat x¤ is a drift less Brownian mot ion wit h di®usion paramet er¾;
t hat is:
dx¤t = ¾dWt; (1)
whereW is a Wiener process.11
T he dist ribut ion of x¤
t; condit ional on past observat ions of x¤ at t he
dis-cret e informat ion t imes, depends only on t he last observation x¤
u, whereu is
t he t ime of t he last informat ion arrival. T he dist ribut ion of x¤
t condit ioned
on t he knowledge of x¤
u, for u < t , is normal wit h mean zero and variance
¾2(t¡ u).
Given init ial values for t he cont rol variable and t he frict ionless opt imal level, t he agent minimizes t he expect ed present value of bot h the adjust ment cost and t he ° ow cost of deviat ing from t he frict ionless opt imal level of t he cont rol variable. T he expect ed value, at t he t ime of t he last informat ion arrival u, of t he ° ow cost at t ime t is Eu(x¡ x¤t)2 and can be decomposed as
follows:
Eu(x¡ x¤t)
2= (x¡ E ux¤t)
2+ E
u(x¤t ¡ Eux¤t) 2
9Quadrat ic ° ow cost s could be just i¯ ed as a second-order approximat ion t o t he loss in
pro¯ t or ut ility caused by a non-opt imal level of t he cont rol variable.
10An excellent exposit ion of opt imal cont rol problems under adjust ment cost s when t he
frict ionless opt imal value of t he cont rol variable is always known is found in Dixit (1993).
11T he assumpt ion of an exogenous process for x¤ might appear unrealist ic in several
set t ings. However, modeling x¤ wit h an endogenous component remains speci¯ c t o t he
part icular set t ing considered. We believe t hat t he main insight s derived from t his simple
T he second t erm represent s t he irreducible cost of not being informed about t he opt imal value of t he frict ionless opt imal valuex¤
t. If t here were
no adjust ment cost s, t he agent will minimize t he expect ed quadrat ic ° ow cost s by set ting xt equal t o Eux¤t. Sincex¤ is drift less, it is a mart ingale,
and Eux¤t = x¤u, t he value of t he opt imal variable when t he last informat ion
arrived. T herefore, even in t he absence of adjust ment cost s, t here will be no adjust ment between informat ion arrivals12.
From t he st ruct ure of t he problem and from t he Markovian nat ure of t he st ochast ic process for x¤, it is clear t hat , given a discrepancy x
u¡ x¤u at t he
t ime of informat ion arrival u; t he value of t he minimized cost st art ing at u will be ident ical t o t he value at u + n (n being an int eger) if t he discrepancy is t he same at t hat t ime. T he discrepancy x ¡ x¤ is t herefore a su± cient st at e variable for t he value funct ion at t imes of informat ion arrival. Since t here will never be an adjust ment between information arrivals, it su± ces t o consider t he value funct ion just at t imes of informat ion arrivals.
2.2 Solv ing t he Opt im izat i on P r obl em
It is never wort hwhile t o correct small deviat ions from t he opt imal level of t he cont rol variable because adjust ment cost s are lump-sum. Also, as t he adjust ment cost s incurred depend neit her on t he st at e before adjust ing nor on t he size of t he adjust ment , t he agent always adjust s t o t he same level of discrepancy (x¡ x¤), which is zero in t he case of a drift less process.
Given t he quadrat ic nat ure of t he ° ow cost s incurred by depart ing from t he frict ionless opt imal value, t he discrepancies which t rigger an upward
12When t here are adjust ment cost s, if an adjust ment t akes place at t he t ime of an
adjust ment and a downward adjust ment are symmet ric around zero. T hus, t he opt imal control policy is not t o adjust between informat ion arrival t imes, and t o reset discrepancy t o zero just after informat ion arrival if it s absolut e level is great er or equal t o a given value, which we call s. T his rule is clearly t ime-and-st ate dependent . T he value ofs is what remains t o be det ermined. We can det ermine it by solving a discret e t ime st ochast ic dynamic pro-gramming problem, where at each t ime of informat ion arrival, t he agent decides eit her t o pay t he adjust ment cost and to adjust t he discrepancy t o zero, and consequent lyx t o x¤; or t o wait unt il t he next period of informat ion
arrival. Wit hout loss of generality, we set the lengt h of t he int erval between informat ion arrivals to one. Formally,
V (y) = minfB (y) + e¡ ½EtV (y¡ ²); k + B (0) + e¡ ½EtV (¡ ²)g (2)
where t he funct ion B represent s t he expect ed discount ed cost of depart ing from t he frict ionless opt imal level of t he cont rol variable between now and t he t ime of t he next informat ion arrival and ² is t he shock t o t he frict ionless opt imal process between t and t + 1; which is a normal variable wit h mean zero and variance¾2. T he expression for B is given by13:
B (y) = y
2(1¡ e¡ ½)
½ ¡
¾2e¡ ½
½ +
¾2(1¡ e¡ ½)
½2 (3)
Equat ion (2) is valid for every y. T he right -hand side can be viewed as a t ransformat ion T of t he funct ion V . T he right V is found when T(V ) = V . BecauseT is a cont ract ion mapping, Tn(V ) t ends t o V as n becomes large.
T herefore, V can be found by guessing an init ial value for V and it erat ing unt il convergence. After t he value funct ion is found, the opt imal policy can be evaluat ed.
13All mat hemat ical derivat ions are included in t he Appendix of t he working
T he second argument of t he min funct ion does not depend on y. More-over, sinceB (:) is increasing injyj, and t he shock ² has normal dist ribut ion wit h mean zero14, it is int uit ive t hat t he value funct ion it self is increasing
in jyj. T herefore, t here exist s a cut o® discrepancy s such t hat , above it , it is opt imal t o adjust and, below it , inact ion is opt imal. T he level s is t he discrepancy t hat makes t he agent indi®erent between adjust ing or not adjust ing:
B (s) + e¡ ½EtV (s¡ ²) = k + B (0) + e¡ ½EtV (¡ ²)
When t he discrepancy is zero, it is clearly opt imal not t o adjust , and t he right hand side of t he equat ion is equal t o V (0) + k. As s is t he point t hat makes t he agent indi®erent between adjust ing or not , t he left hand-side is equal t o t he value funct ion evaluated at s. T hus, t he condit ion above can be rest at ed as a value mat ching condit ion, which is a familiar one in t he problem of opt imal cont rol wit h full informat ion.
V (s) = V (0) + k (4)
T his condit ion must be sat is¯ ed bot h at s and at ¡ s, since t he value func-t ion is symmefunc-t ric. T hen, once func-t he value funcfunc-t ion is obfunc-t ained, func-t his equafunc-t ion can be used t o det ermines:
Not e t hat with infrequent information, t he value mat ching condit ion plays a di®erent role t han in t he full informat ion reset t ing problems. In t he lat t er, t he value mat ching condit ion is a condit ion of consist ency which is always sat is¯ ed by t he value funct ion at the resett ing and t rigger point s, even if t hese point s are not opt imal. In our problem, it is t ruly an opt imality condit ion in t he sense t hat it is only sat is¯ ed if an opt imal s is chosen. In ot her words, if we chose a non-opt imal reset t ing point and calculat ed t he value funct ion for
14T he relevant feat ure of t he normal dist ribut ion for t his result is t hat it s density is
t his reset t ing policy, t here would be a discont inuity in t he value funct ion at such chosen reset t ing point .
2.3 N umer ical R esul t s
Table 1 report s t he opt imal rule paramet er s found numerically15for di®erent
di®usion paramet ers of t he frict ionless opt imal cont rol process. For purposes of comparison, we also report t he barrier value for t he opt imal rule when t he agent has cont inuous informat ion about t he opt imal value of t he cont rol variable. T he opt imal rule in t his case is also symmet ric and two-sided16, but adjust ment occurs whenever t he absolut e value of t he discrepancy is equal t o t he barrier.
T he ¯ rst pat t ern t o not ice is t hat t he infrequent informat ion bands are much narrower than t he cont inuous informat ion ones. To build one's int u-ition, let us not e t hat changing t he size of t he band ent ails a t rade-o®. Wit h a barrier lower t han t he opt imal level, t he cost s of being away from t he opt i-mal level decrease, but t he adjust ment cost s increase in a larger magnitude. In t he infrequent informat ion case, a reduct ion in the barrier level ent ails a much smaller increase in adjust ment cost s t han in t he cont inuous cont rol case since cont rol is only exert ed at t imes of informat ion arrival. So, relat ively, reducing t he inact ion range does not subst ant ially increase cost s.
T he second pat tern is t hat t he size of t he bands is much less sensit ive t o t he variability of t he frict ionless opt imal process in t he infrequent informat ion case. When informat ion arrives cont inuously, t he size of t he band increases wit h t he di®usion paramet er because maint aining t he size const ant will im-ply a subst ant ial increase in adjust ment cost s, while not changing much t he
15T he value funct ion is comput ed numerically using a piecewise linear approximat ion
along a grid wit h a large number of point s. We st art wit h some init ial value for t his set of point s and it erat e unt il convergence. Once t he value funct ion is obt ained, t he s value is found by using condit ion (3).
16To comput e t he opt imal s for t he cont inuous informat ion case, we use t he formula in
cost s of being away from t he optimal level. T he unavoidable increase in cost s is minimized when t he size of t he band becomes wider and bot h cost s are increased. In t he infrequent informat ion case, more variability increase t he probability of larger discrepancies at t imes of no informat ion. So, t he expect ed cost of being away from t he opt imal level increases subst ant ially, since it is convex, even wit h t he same barriers. On t he ot her hand, t he in-crease in adjust ment cost s is lower relat ive t o t he cont inuous case. T herefore, in t he infrequent informat ion case, t here is less need t o balance t he increase in cost s by enlarging t he region of inact ion.
A t hird pat tern emerges from t he t able: t he infrequent informat ion bar-riers t end t o respond less t o t he change in t he di®usion paramet er when t his parameter is relatively large or when t he adjust ment cost s are small. To underst and t his result , not ice t hat t he di®erence between t he two cases accent uat es when t he adjust ment s in t he cont inuous case t end t o occur at int ervals t hat are small compared t o t he int erval between informat ion ar-rivals. Adjust ment s t end t o occur more oft en in t he cont inuous case when t he adjust ment cost is smaller or when t he di®usion paramet er get s larger. T herefore, it is in t hose cases t hat t he infrequent informat ion barriers are less responsive t o changes in t he di®usion paramet er. As illust rat ed in Table 1, for a high adjust ment cost , t he barriers become less responsive t o changes in t he uncert ainty paramet er when it get s larger. For a small adjust ment cost , t he opt imal barriers st ay pract ically const ant for all values of t he di®usion paramet er.17
Finally, it can be seen in t he t able t hat t he size of t he band seems t o be relat ively more sensit ive t o t he adjust ment cost in t he infrequent informa-t ion case. T he opinforma-t imal band size should equalize informa-the adjusinforma-t meninforma-t cosinforma-t informa-t o informa-t he bene¯ t of adjust ing now rat her t han continuing wit h discrepancy s. In t he infrequent informat ion case, t his bene¯ t is less sensit ive t o s, since adjust
-17We say " pract ically" because when calculat ed wit h a higher level of precision, it can
ment s are only part ially st at e-dependent . A more subst ant ial increase in s is t herefore necessary in order t o make t he bene¯ t of adjust ing from s now (and t hen following t he opt imal policy) equal t o a higher adjust ment cost level.
3 A ggr egat e E®ect s of M acr oeconom ic Shock s
We saw t hat infrequent informat ion changes some feat ures of t he opt imal rules. We will see that infrequent informat ion changes also t he pat t ern of response of macroeconomic variables t o aggregat e shocks.
We are int erest ed in modelling the e®ect of an aggregat e shock on t he level of disequilibrium of a macroeconomic variable, as it is done in Bert ola and Caballero (1990). Disequilibrium is de¯ ned as t hedi®erence between t he level of an aggregat e variable in a frict ionless world (wit hout adjust ment cost s) and it s level when infrequent informat ion and adjust ment cost s are present . We t hen compare t he e®ect s predict ed by our model wit h t he e®ect s obt ained when only one kind of imperfect ion prevails: a full informat ion adjust ment cost model and an infrequent informat ion model wit h no adjust ment cost s.
3.1 A ver age E®ect of Shocks in t he M odel wi t h I nfr equent I nfor -mat i on and A dj ust ment Cost s
In t his subsect ion we ¯ rst develop expressions for t he e®ect s of cumulat ive aggregat e shocks of di®erent sizes and for t he average of t hese e®ect s in t he model wit h adjust ment cost s and infrequent informat ion. We t hen comput e and analyze t he e®ect s for various con¯ gurat ions of t he uncert ainty and ad-just ment cost paramet ers.
3.1.1 A naly t ical ex pr ession for t he aver age e®ect of shocks
opt imal level, where t he average is t aken over all agent s. T he analysis can be applied t o any macroeconomic variable where t he frict ionless opt imal level can be considered as an exogenous and drift less process. For example, in a st able economy wit h no in° at ion, we can t hink of t he individual cont rol variablexi as being the individual pricepi, and t he frict ionless opt imal
in-dividual price as being t he sum of t he money supply and an idiosyncrat ic component (x¤
i = m + ei). In t his case, t he disequilibrium level of aggregat e
price will be t he real money supply (y = x¤¡ x = m¡ p). T hus, in t his
example, t he disequilibrium level of t he macroeconomic variable corresponds t o t he out put level or some increasing funct ion of it .18.
Let y be t he disequilibrium level of t he macroeconomic variable and con-sider t he variation of y in one period. During t his period, aggregat e shocks and idiosyncrat ic shocks t o individual frictionless opt imal levels of t he con-t rol variable accumulacon-t e, bucon-t no adjuscon-t mencon-t is made uncon-t il con-t he informacon-t ion is received. T hen, adjust ment is made or not , depending on t he level of t he individual disequilibrium revealed. We explore t he e®ect of aggregate shocks when bot h t he informat ion about t he aggregat e shock and t he informat ion about the idiosyncrat ic shock are released simult aneously t o all agent s.
T he change in t he average disequilibrium between adjust ment periods ¢y is t he average, over all individual unit s, of t he changes in t he individual frict ionless levels ¢x¤
i less t he average of t he act ual changes ¢xi :
¢y =
Z
(¢x¤ i)di ¡
Z
(¢xi)di : (5)
However, since idiosyncrat ic shocks sum t o zero, t he average change in t he frict ionless levels is just t he aggregate shock w accumulat ed in t he period between two adjust ment s:
18T his charact erizat ion of t he macroeconomic variable is st andard in models for money
Z
(¢x¤i)di =
Z
(w + ei)di = w; (6)
whereei is t he accumulat ed idiosyncrat ic shock of agent i in one period. On
t he ot her hand, t he average of t he act ual changes can be broken down in two
part s: Z
(¢xi)di = puEuj¢xij+ pdEdj¢xij; (7)
wherepu is t he fract ion of upward adjust ment s and pd t he fract ion of
down-ward adjust ments, whileEuj¢xijandEdj¢xij are t he average size of upward
and downward adjust ment s respect ively. T he agent s who adjust upwards are t hose who realize upon receiving t he informat ion t hat t he opt imal level of t heir cont rol variable exceeds the act ual level by more t han s. T hus, each agent adjust s it s cont rol variable by at least s. A similar reasoning applies t o t he fract ion of agent s which adjust s downwards. T he overall change can t herefore be writ t en as:
¢y = w¡ puEuj¢xij+ pdEdj¢xij: (8)
To know the e®ect of a cumulat ive aggregat e shock w it is t herefore necessary t o know both t he fract ion and t he average size of upward and downward adjust ment s. T hese quant it ies depend only on t he dist ribut ion of t he init ial individual disequilibria when there are no idiosyncrat ic shocks.
Consider t hen a world wit h no idiosyncrat ic uncertainty. Whenever there is a posit ive accumulat ed aggregat e shock, w > 0, some unit s will adjust upwards if t heir discrepancy is lower t han ¡ s. T his is illust rat ed in Figure 1. Let Ft be t he cumulat ive dist ribut ion of t he discrepancy level before t he
cumulat ive aggregat e shock w. T hen pu is given by Ft(¡ s + w)19 and pd
is zero, since no unit will increase it s discrepancy. T he size of t he upward
19Let d
i be t he discrepancy of an unit before receiving t he shock. T hus t he discrepancy
aft er receiving a posit ive shock w t o t he frict ionless opt imal level is di ¡ w. T he unit will
adjust ment for a unit wit h disequilibrium z is always equal t o t he absolut e value of t he ¯ nal disequilibrium, t hat isjz¡ wj = ¡ (z¡ w). T he e®ect is t ot ally symmet ric in t he case of a negat ive aggregat e shock.
T he exist ence of idiosyncrat ic shocks increases t he frequency of bot h pos-itive and negat ive adjust ment s at t imes of informat ion arrival. Wit hout furt her assumpt ions, we cannot t ell if it will magnify or dampen t he e®ect of aggregat e shocks. If t he dist ribut ion of price deviat ions is decreasing in t he absolut e size of t he deviat ions, t hen t he idiosyncrat ic uncert ainty t ends t o at t enuat e t he e®ect of aggregat e shocks. T his is because a posit ive aggregat e shock, for example, t ends t o simult aneously make t he left side of t he dist ri-but ion of price deviat ions t hicker while leaving t he right side empty. T hen t he added idiosyncrat ic shocks cause many more upward adjust ments t han dowward adjust ment s. As a result , t he e®ect of a positive shock is dampened. Obviously, a symmet ric version of t he same mechanism works for negat ive shocks t oo.
We ¯ rst derive an analyt ical expression for t he average relat ive e®ect of a cumulat ive aggregat e shock of a given size20:
E
·
¢y w j w
¸
= 1¡ 1 w s Z ¡ s 2 4 1 Z
z¡ w+ s
¡
µ
z¡ w¡ ei
¾i ¶
Á(ei ¾i
)dei (9)
¡
z¡ w¡ sZ
¡ 1
(z¡ w¡ ei ¾i
)Á(ei ¾i
)dei 3
5dF (z)
whereF is t he ergodic (average) dist ribut ion of price deviat ions (see next subsect ion) and Á is t he normal dist ribut ion density. T hen t hese e®ect s are averaged according t o t he likelihood of each shock size t o yield an expression for t he average relat ive e®ect of an unspeci¯ ed aggregat e shock:
20T he derivat ion of t he next two formulas and ot her t echnical det ails are included in
E
·
¢y w
¸
= 1¡
1 Z ¡ 1 1 w:¾A Á( w ¾A ) 8 < : s Z ¡ s 2 4 1 Z
z¡ w + s
¡
µ
z¡ w¡ ei
¾i ¶
Á(ei ¾i
)dei¡
z¡ w¡ sZ
¡ 1
(z¡ w¡ ei ¾i
)Á(ei ¾i
)dei 3
5dF (z) 9 =
; dw (10)
3.1.2 T he er godic di st r i but i on of dev iat i ons
Alt hough t here is no invariant dist ribut ion of deviat ions in t he presence of aggregat e shocks, we can comput e t he average (ergodic) dist ribut ion which coincides wit h t he dist ribut ion t hat would remain invariant if all unit s only had idiosyncrat ic shocks. T he ergodic dist ribut ion for t his class of rules has an at om at zero, since many unit s adjust t heir discrepancy simult aneously t o zero at t he t ime of informat ion arrival21.
Despit e t he fact t hat aggregat e shocks occur cont inuously and are always small in magnit ude, adjust ment s are large, infrequent and have a large degree of simult aneity. Simult aneity result s from t he infrequent release of informa-t ion abouinforma-t aggregainforma-t e shocks, which makes informa-t he magniinforma-t ude of news relainforma-t ively large, even t hough innovat ions are small and occur cont inuously.
21T he condit ions t hat det ermine t he ergodic dist ribut ion are given in t he A ppendix
referred t o in foot not e 13. Basically, t he dist ribut ion is derived by it erat ing on t he following condit ion:
ft + 1(z) =
Z ¡ z+ s
¡ z¡ s
ft(z + v)
1 ¾Á ³ v ¾ ´ dv + (1 ¡ Z s ¡ s
ft(z)dz)Á
µ ¡ z
¾ ¶
;
3.1.3 R esult s
Figure 2 shows t he ergodic dist ribut ion corresponding t o two di®erent values of t he di®usion paramet er. We observe t hat t he higher t he di®usion param-et er, t he ° at t er t he density curve and t he higher t he probability associat ed wit h t he at om at zero. T his is consist ent wit h t he fact t hat a higher vari-ance t riggers more adjust ment s at t imes of informat ion arrival, giving more weight t o t he at om. Addit ionally, as t here is more movement in t he deviat ion between adjust ment t imes, t he density get s ° at t er. T hese result s t ell us t hat t he higher t he aggregat e uncert ainty, t he great er t he simult aneity of act ions. Int uit ively, wit h higher uncertainty, informat ion arrivals bring about more news. A large piece of news, t hat is a large cumulat ive aggregat e shock since t he last informat ion arrival, t riggers simult aneous adjust ment s from a large number of agent s.
Since t he ergodic dist ribut ions are symmet ric and decreasing in t he ab-solut e size of t he price deviat ion, we can rely on t he analysis of t he e®ect of cumulat ive aggregat e shocks developed in sect ion 3.1.1. Figure 3 shows t hat t he relat ive e®ect of a shock is decreasing wit h t he absolut e size of t he shock, as ant icipat ed in t heprevious description about t he e®ect of aggregat e shocks. Table 2 shows t he average e®ect of a shock for di®erent t ot al variances of shocks and di®erent decomposit ions of t his t ot al variance between aggregat e and idiosyncrat ic variances of shocks. T he average e®ect decreases when we keep aggregat e uncert ainty const ant and increase idiosyncrat ic uncert ainty, as ant icipat ed in sect ion 3.1.1. When we keep idiosyncrat ic variance con-st ant and increase aggregat e variance, t he average e®ect is also reduced22.
T his result follows solely from t he higher likelihood of large shocks, which have a lower relat ive e®ect , since t he aggregat e uncert ainty does not a®ect t he impact of any speci¯ c aggregat e shock. When we keep t ot al uncert ainty
22It should be not ed t hat t he e®ect is not reduced as much as one might have expect ed,
const ant but increase t he relat ive weight of idiosyncrat ic shocks, t he e®ect is reduced indicat ing t hat t he in° uence of idiosyncrat ic uncert ainty on each speci¯ c shock t akes pre-eminence over t he way shocks are averaged. Finally, we observe t hat t he size of adjust ment cost s has a very import ant impact on t he result s. T his is because t he size of t he bands is very sensit ive t o ad-just ment cost s. A reduct ion in adad-just ment cost s reduces t he size of t he band subst ant ially, decreasing t he proportion of unit s t hat do not adjust and, as a consequence, t he relat ive e®ect of an aggregat e shock.
3.2 Com par i son wi t h ot her m odel s
In t his sect ion we compare t he feat ures of aggregat e e®ect s in t he model wit h infrequent informat ion and lump-sum adjust ment cost s t o t he ones obt ained in models where only one kind of imperfect ion is present .
3.2.1 A dj ust m ent cost wi t h full i nfor m at ion
A lot of work has been done on t he aggregat e e®ect s of shocks in adjust -ment cost models wit h full informat ion, as referred t o in t he int roduct ion. Caballero and Engel (1992) develop a met hod t o quant ify t he average e®ect of aggregat e shocks and apply it t o assess out put e®ect s. However, t heir met hod measures instant aneous e®ect s, and our object ive is t o compare ag-gregat e e®ect s in bot h infrequent information and full informat ion models during t he same t ime horizon23. T he convenient t ime horizon we choose is
t he t ime between informat ion arrivals.
We want t o evaluat e t he e®ect generat ed by cumulat ive aggregat e shocks of di®erent sizes. First , it should be not iced t hat two cumulat ive aggregat e shocks of t he same size may have di®erent e®ect s even if t he init ial dist
ribu-23T he relat ive inst ant aneous average e®ect of an aggregat e shock in a full informat ion
t ion of price deviat ions is t he same. T his is due t o t he hyst eresis built in t he models: t he pat h of aggregat e shocks mat t er, not just t heir cumulat ive sum. Given t his feature, t o obt ain a unique measure of t he relat ive e®ect of a cu-mulat ive aggregate shock of a given size, we have t o average t he e®ect of each shock of a given size wit h di®erent component s by the relat ive likelihood of its components. We t herefore discret ize t he t ime and st ate space according t o t he met hodology described in Bert ola and Caballero (1990). We perform Mont e Carlo simulat ions of aggregat e shocks, drawing pat hs according t o t heir likelihood, and calculat e t he relat ive aggregat e e®ect for each pat h. We t hen classify t he cumulat ive shock sizes in small int ervals and average t he e®ect s of cumulat ive shocks in each class, in order t o obt ain a represent at ive average e®ect for each cat egory of shock size. We also calculate a global average of all simulat ions.
In Figure 4 we graph our simulat ions according t o each class of size shock24. We not ice t hat , in contrast wit h t he infrequent informat ion case in Figure 3, t he relat ive e®ect of shocks t ends t o st ay const ant , except for shocks t hat are very small. Average result s are shown in Table 2. Average e®ect s t end t o be of similar magnit ude but do not always go in t he same direct ion. A higher aggregat e uncert ainty, all ot her paramet ers being kept const ant , t ends t o increase t he average e®ect , cont rary t o what we found in t he infrequent informat ion model.
If we int erpret our model in t erms of t he pricing applicat ion described earlier, we could t hink of t he aggregat e shock as being an aggregat e demand shock. T hen, our model implies t hat a small aggregat e shock has a relat ively large e®ect on out put , while large shocks t end t o have a relat ively lower e®ect . Anot her implicat ion is t hat in more unst able periods (subject t o more aggregat e shocks), t he e®ect of a given aggregate shock t ends t o be reduced.
24T he Mont e Carlo simulat ion is based on 10000 replicat ions. T he st at e space is 2N+ 1,
where N= round( s
si g ¤ sqr t(1000)): Test s done for a ¯ ner grid and a higher number of
T hese implicat ions const rast wit h t hose of the full informat ion model, where t he average e®ect of a shock does not depend on it size and shocks t end t o have more e®ect in unst able macroeconomic environment s. Bot h models predict t hat in an unst able microeconomic environment aggregat e demand shocks have smaller e®ect s.
3.2.2 I nfr equent i nfor m at ion wi t h no adj ust m ent cost s
Caballero (1989) has worked out a model wit h no adjust ment cost s where part of t he informat ion relevant t o each ¯ rm arrives at infrequent intervals, eit her t hrough t he payment of an informat ion gat hering cost or by observing t he act ion of anot her ¯ rm which just paid it s informat ion gat hering cost . T here are some aggregat e e®ect s t hat come from sluggish adjust ment t o innovat ions in t he frictionless opt imal level of t he control variable due t o infrequent informat ion. However, t hese e®ect s do not last more t han t he t ime int erval between informat ion collect ions.
A nest ed simple version of our model wit h no adjust ment cost would ent ail a full adjust ment at every period of informat ion arrival. Any shock will have a full e®ect unt il t he t ime of informat ion arrival, when t he e®ect will be eliminat ed by t he full adjust ment of all unit s. T herefore, if we use t he same t ime int erval we used above t o measure t he e®ect of a cumulat ive aggregat e shock in t he models wit h adjust ment cost s, we ¯ nd no aggregat e e®ect .
4 Fi nal C om ment s and Ex t ensi ons
-ing opt imal rules t hat are bot h t ime and st at e dependent . From t his point it does not seem di± cult t o endogenize informat ion arrival by int roducing cost s of information collect ion. However, as argued in t he int roduct ion, infrequent exogenous informat ion arrival is realist ic per se in various cont ext s. A more di± cult t ask is t o generalize t he current model t o st ochast ic processes t hat are not mart ingales: t hen t here may be adjust ment s between int ervals of informat ion collect ion, and inert ia bands in t his int erval should depend on t he remaining t ime before t he next informat ion arrival. Also, t he same di± -cult ies would appear if we ext ended t he model t o allow part of t he st ochast ic component t o be cont inuously observed.
R efer ences
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Blanchard, O. and S. Fischer (1980), " Lect ures on Macroeconomics," MIT Press.
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Table 1
Optimal Barriers for Various Levels of Uncertainty (ρ= 0. 025)
σ Infrequent Information Continuous Information
k= 0. 01 0. 05 0. 081 0. 1113
0. 10 0. 094 0. 1570
0. 15 0. 098 0. 1921
0. 20 0. 100 0. 2217
k= 0. 001 0. 05 0. 032 0. 0624
0. 10 0. 032 0. 0881
0. 15 0. 032 0. 1079
0. 20 0. 032 0. 1245
Table 2
Average Effect of Shocks
for Various Configurations of Aggregate and Idiosyncratic Uncertainty
σ σA σI k
Infrequent Information Continuous Information
s Average
Effect
s Average
A pp endi x
A : Cal cul at ing t he funct i on B
The funct ion B represent s t he expect ed discount ed cost of departing from t he frict ionless opt imal level of t he cont rol variable between now and t he t ime of t he next informat ion arrival. It is derived as follows
B (y) = Et 0
@ 1 Z
0
e¡ ½z(x
t ¡ x¤t + z)2dz 1
A
= Et 0
@ 1 Z
0
e¡ ½z(y + ¾(w
t+ z ¡ wt))2dz 1 A = 1 Z 0
e¡ ½zE
t (y + ¾(wt + z¡ wt))2dz
=
1 Z
0
e¡ ½z(y2+ ¾2Et(wt + z ¡ wt)2)dz
=
1 Z
0
e¡ ½zy2dz + 1 Z
0
e¡ ½z¾2zdz
= y2(1¡ e½¡ ½) ¡ ¾2½e¡ ½+ ¾2(1¡ e½2¡ ½)
In t he second equality, we decompose t he discrepancy int + zint o t he sum of t he discrepancy in t, which isy; and t he change in x¤ between t andt + z. The
t hird equality result s from applying Fubini’s t heorem, while t he two next ones use respect ively t he condit ional independence of increment s of t he Wiener process and t he formula for t heir variance. T he last equality is obt ained by calculat ing t he int egrals, t he second one wit h an int egration by part s.
B : D er ivat ion for t he ex pr ession of t he e¤ect of a cumul at ive aggr e-gat e shock when t her e i s idi osy ncr at i c uncer t ai nt y
a¤ect ed t he opt imal level of t he cont rol variable for each agent . Since realizat ions of t he idiosyncrat ic shocks across t he economy are generally unknown, we evaluat e t he average e¤ect of a known aggregat e shock w, by averaging over all possible realizat ions of t heei shocks weight ed by t heir likelihood. As a …rst st ep, suppose
t hat all agent s have t he same init ial discrepancyz. Then t he e¤ect of an aggregat e shock w will be:
E [¢ yjw; z] = w¡
³
1¡ ©
³ z¡ w+ s
¾i
´ ´ Z1
z¡ w + s
¡ (z¡ w¡ ei)
Á(ei ¾i) ¾i ³ 1¡ © ³
z ¡ w + s ¾i
´ ´dei
+ ©
³ z¡ w¡ s
¾i
´ z¡ w¡ sZ
¡ 1
(z¡ w¡ ei)
Á(¾iei)
¾i© ³
z ¡ w ¡ s ¾i
´dei
The t erm between parent heses mult iplying t he …rst int egral is t he probability t hat a discrepancy of level z, aft er account ing for t he known aggregat e shock w and t he normally dist ribut ed idiosyncrat ic shocks, becomes smaller than ¡ s, t riggering an upward adjust ment . Thus, t he …rst int egral is t he expected size of t he upward adjust ment , condit ioned on t he occurence of such an adjust ment . The second int egral and t he t erm t hat mult iplies it apply t o downward adjust ment s and have similar int erpret at ions.
The init ial discrepancies of t he unit s at t, rat her t han being concent rat ed on a speci…c value of z, are dist ribut ed according t o some dist ribut ion Ft.
Assum-ing t hat t here are many unit s at each posit ion z, such t hat t he frequency of idiosyncrat ic shocks for all unit s at a given posit ion can be well approximat ed by it s probability dist ribut ion, we can average t he e¤ect of an aggregat e shock, as calculat ed above for a given z, according t o t he dist ribut ion Ft of t hez0s. T hen,
E [¢ y j w; Ft] = w¡ s Z ¡ s 2 4 1 Z
z¡ w+ s
¡
µ
z¡ w¡ ei
¾i ¶
Á(ei ¾i
)dei ¡
z¡ w¡ sZ
¡ 1
(z¡ w¡ ei ¾i
)Á(ei ¾i
)dei 3
5dFt(z)
Next , we express t he e¤ect as a rat io, dividing t he above expression by w, result ing in:
E
h ¢ y
w jw; Ft i
= 1¡ w1
s Z ¡ s 2 4 1 Z
z¡ w+ s
¡
³ z¡ w¡ ei
¾i ´
Á(ei ¾i)dei
¡
z¡ w¡ sZ
¡ 1
(z¡ w¡ ei ¾i )Á(
ei ¾i)dei
3
Now we t ake t he expect at ion wit h respect t ow, yielding:
E
h ¢ y
w j Ft i
= 1¡
1 Z
¡ 1 1 w:¾AÁ(
w ¾A)
8 < : s Z ¡ s 2 4 1 Z
z¡ w+ s
¡
³ z¡ w¡ ei
¾i ´
Á(ei ¾i)dei¡ z¡ w¡ sZ
¡ 1
(z¡ w¡ ei ¾i )Á(
ei ¾i)dei
3
5dFt(z)
9 =
; dw
The expression above evaluat es t he average e¤ect of a shock for a given ini-t ial disini-t ribuini-t ion of deviaini-t ions. Finally, ini-t aking expecini-t aini-t ions wiini-t h respecini-t ini-t o ini-t he dist ribut ion of deviat ions (and using Fubini’s t heorem), we arrive at :
E
h ¢ y
w i
= 1¡
1 Z
¡ 1 1 w:¾AÁ(
w ¾A)
8 < : s Z ¡ s 2 4 1 Z
z¡ w+ s
¡
³ z¡ w¡ ei
¾i ´
Á(ei ¾i)dei¡ z¡ w¡ sZ
¡ 1
(z¡ w¡ ei ¾i )Á(
ei ¾i)dei
3
5dF (z) 9 =
; dw
whereF is t he ergodic (average) dist ribut ion of deviat ions.
C : D er i vat i on of t he condi t ions det er m ining t he er godi c di st r i but i on
In t his appendix we derive t he equat ions which det ermine t he ergodic dist ri-but ion.
Let ft(:)be t he density funct ion for price deviat ions di¤erent from zero at t ime
t , immediat ely aft er informat ion arrival and adjust ment s are made. Let Pt(0) be
t he fract ion of unit s wit h price deviat ion zero at t ime t , aft er t he adjust ment s are made. Let vi be t he t ot al cumulat ive shock t o t he frict ionless opt imal level
of t he cont rol variable of unit i during t he period of t ime wit hout informat ion. So, vi = w + ei and vi is normally dist ribut ed wit h zero mean and variance
¾2= ¾2 A + ¾2I.
It is clear t hat ft + 1(z) = 0for z < ¡ s or z > s. For ¡ s < z < s, and z6= 0,
ft + 1 is given by:
ft + 1(z) =
Z ¡ z+ s
¡ z¡ s ft(z + v)
1 ¾Á µ v ¾ ¶
dv + Pt(0)Á µ
¡ z ¾
The fract ion of units at zero in t + 1 relat es t o t he dist ribut ion in t in t he following way:
Pt + 1(0) =
Zs
¡ sft(z) ·
©
µ
z¡ s ¾
¶
+ 1¡ ©
µ
z + s ¾
¶ ¸
dz + Pt(0)Á(0)
For t he dist ribut ion t o be well de…ned it has t o sat isfy for all t:
Pt(0) +
Z s
¡ sft(z)dz = 1
MakingPt+ 1= Pt andft + 1 = ft in t he condit ions above det ermines t he ergodic
