- Email: [email protected]
The necessity of delaying economic adjustment
Journal of Economic Behavior md Organisation 10 (1988) 255-286. (Worth-Holland)
onald A. Brigham Young Universi?y,Provo, UT dW602 USA Received June 1987, fmal wersionreceived October 1987 I give an explanation for why agents react with a delay to decision and expectation parameters, by showing that imperfecca agents must adjust very slowly relative to optimally adjusting agents if they begin reacting as soon as the latter agents would react. The rest&s follow from the hypothesis of imperfect choice alone irrespective of whether there are any infsrmation, search, transaction, or other costs of adjusting decisions. The results thereby provide a general reason for observing inertially lagged behavior, and also agree with recent asset market experiments whose dynamics are governed by dizptiue rather than ‘rational’ expectations.
The extent and variety of topics to which distributed-lag analysis has been applied in empirical economics is astounding, but, what is more remarkab!e, is the v;rtual la& sf theoretical justification for the lag structures . . . ark Nerlove 1972, page 223)
This paper offers an explanation for why agents will not respond immediately to shifts in either decision or expectation parameters. It shows that agents who respond imperfectly mzst adjust very slowly relative to agents who always adjust optimally, if they start reacting immediately after a decision parameter begins shifting and the initial rate of parameter change is non-zero. Imperfect response means *,SCiC is a positive probability of mistakenly adjusting either too soon or in the wrong direction at a rate that is not arbitrarily small relative to their adjustments in the right direction (toward a shifting optimal decision or expectation). The paper proceeds in eight steps. (1) The literature on inertial behavior is briefly reviewed, including various distributed lag? partial adjustment, invest *I thank the followine prsons for helpful Robert Glower, Richard bay, Jonathan LeEand and Joseph Stiglitz. Of course, 1 alone am resp OiKP-2681/88,/%3.50@ i988,
eth Arrow, James Buchanan, d, Rulon Pope, Vernon Smith r’s content. ers
256
R.A. Heiner, The necessity of delaying economic adjustment modeis [such as Koyck (1954), Nerlove (1972), Jorgenson (1963,
1970), Cagan (1956), Eisner and Strotz (1 helpful in describing been less fruitful i where the existence w imperfect agents must behave in order to control
justification for The latter problem is overcome by introducing certain concepts about imperfect response to a shifting parameter. the possibility of sometimes respondin tion, or at the wrong rate. equality is developed to determine the relative size of imperfect to optimal adjustment in order for agents’ expected utility to rise conditional on adjusting before a small delay after a decision parameter starts shifting. A parameter change that begins with a non-zero first derivative is called a ‘first-order’ parameter shift. Except at the limit where adjustment becomes fully optimal, the expected rate of (im feet) response to a first-order parameter shift - whether mis&enly soon or in the wrong direction, or correctly toward the new optimum must become arbitrarily small relative to the optimal response rate as adjustment delay goes to zero. e main result of step e is formally summarized and discussed in arison een an imperfectly responding agent and
ctive of whether t
R.A.
iw,
The necessityof delaying economicadjustment
257
explaining why a psy& ological tendency toward ‘habit formation’ e when no outwardly observable adjustment costs normative status of delayed response is also brie 6) argued that constraining adjustment to lag hind an optimal agent’s response is nevertheless fully rational given that an im ted utility will drop if this constraint is violated. costs needed to reduce the incidence of se errors are (7) scus& Perfect agents errors without sacrificing scarce resources in order t On the other hand, imperfect agents are those for whom a decision costs are necessary in order to becoming infinite to reach the direction, or magnitude never occur. Except at t delay relative to responding optimally is still necessary. (8) The analysis of previous steps applies not only to outward behavior, but also to imperfect adjustment of subjective beliefs such as expectations. A simplified example of the latter is constructed. Agents use a forecasting rule that has the same functional form as the optimal Bayesian rule, but the specific coefficients that transform observed information into forecasts may respond imperfectly to shifts in parameters governing the underlying stochastic properties of agents’ environment. Previous results are used to show the necessity of delayed or inertial adjustment of imperfect expectations relative to an optimally adjusting Bayesian agent (i.e., imperfect agents will ‘adaptively’ adjust their expectations to ongoing experience). These results also accord with recent experiments of spot asset markets by Vernon Smith, Gerry Suchanek and Arlington Williams (1987) and Williams (1987); showing that ‘rational expectations’ apply only near the end of a dynamic adjustment process characterized by adaptive expectations. I e1. Earlier models of inertial behavior is now accepted that a general tendency to delay adjustment from &ions (abOUtconsumption, investment, pri employment, and ordingly, a large very widespread empirical phenomenon. econometric and theory literature on dynamic decision maki t
258
R.A. Heiner, The necessity of delaying economic adjustment
adjustment
costs
in information processing costs, transactions costs, and so on. example of this approach are decisions involving investment in capital
decisions about durable capital. extensive demand studies showed
outhakker and Taylor’s
of habits’ (i.e., consumers nding habits to avoid psychological adjustment costs). Doing so used for investment decisions. Yet it leaves unexplained why a particular sychology’ toward habits should arise in the first place. , the idea of habit simply redescribes the behavioral s plained (that is, habits are those behavior patterns that slu inferred from obse
mechanisms deterri
s of slowly adjusting
R.A. Heiner, The necessityof delaying economica&stment
1.2.
losses are
sufficient to
259
inertial behavior
t us first briefly discuss why inertial behavior cannot be just%ed osses from delayed adjustment may be ‘small’. To do so, let V(x, a) represent an agent’s objective function which depends on a decision variable x chosen by the agent and a decision parameter a t as given by the agent. Assume the first and second order maximizing co tions hold, so that an optimal value of x exists for each a, denoted by the function x*(a). Standard analysis then implies &P(a) -a2V(x*(42),a)/axaa -= -da - a2V(x*(a), a)&’
where
’
a2 V(x’(a), a)px2 SO for each
a.
(2)
To insure that the optimal decisitin rule responds to shifts in parameter a (so that i%P/aa#O) also assume a2 V(x+, a)/ax aa #O
for each
a.
!3)
Now suppose parameter a has remained stationary at a=a* long enough for the agent to have previously adjusted x to its optimal value x0 =x*(a*). Then at some point parameter a begins to shift. xt calculate the net rate of loss from not sting away from x0 compared to optimally responding according to x That is, calculate d[ V(x*, a) - V(x*(a), a)]/da.
(4)
Akerlolf and Ye!!en (1985a, b) have recently shown that (4) goes to zero as a -a* goes to zero. Consequently, any ‘first order’ losses from not adjusting . _- 1. i * ahif& _ a__dg&&fisr - rrrvl”.. ~~arr====~== ypLILLIPIoIL&l a-7a.ay from its iinrINxlra~~ny go LU iax c-e ati1 nmnll OAAikbAA Y_..__ Qf initia! value. However, there is a basic problem with the logic of small first order losses; namely, they apply equally to any arbitrary way of responding to holding x temporarily fixed at x0. to a shifting parameter, not sim es from responding too quickly instead of For example, the first order ment rate) also go too slowly (say three times fa resent an to zero as a- a* goes to zero.
R.A. Heiner, The necessityof delaying economicfuijustment
260
ma1 adjustment, including not adjusting as a us, slmalysisb -pm-t@
itself givesno reaso 2 behavior will take the form of delayed reaction to
091negligible first-order losses by
ing that nonopt changes. In order to justify the latter possibility, additional analysis
Accordingly, this derived as a generic feature of response to parameter changes. .
will show that de1 avior by modeling t .
1.3. Imperfect response?to chnge: Preliminary concepts asic motivation is th 8 ‘competence’may not problems so as to ent relative to the always react opt~ally to changing conditions based on their information [called a C-D gap; see r (1983, 198Sa, 1986a)]. th imperfect skills at in re complex and unstable t to changes in those environmental relationships will not
soon or in the response. ods of relative with those used
R.A. Heiner, The necessityof delaving economicadjustment
261
previously adjusted x to its optimal value x0= x t SOme point (denoted to), parameter Q will start shifting toward a new value that remains constant for another uncertain period before shifting again, and so on. the point of time when agents start adjusting to changes in S, y not always equal to. even if they don’t start always in the right direction t may sometimes respond in o when agents react toward the ne-w optimu the optimal rate (so that ax?/&#ax*/&). To simplify notation let the derivative of XI or x denoted with a subscript t (for example, 3x?/&= larly, let the time-de of parameter (I equal u#du@)/dt. c=t-f as the length after agents begin to change X, so time t equals f+ c. Now consider what ha s soon after agents begin to react (so that t =f+c), assuming their s time f is delayed at most negligibly beyond the true parameter shift time to (so that f-to is less than some maximum delay 6 > 0 that shrinks to zero). To ensure that both c and 6 become small together, let S be any continuous, non-negative function of 5, denoted S(e), that goes to zero with {. That is, S(c) 20 for all 5 2 0 and lim,,, S(c) =O. This specification allows the size of t-t to vary arbitrarily relative to f-to so long as both go to zero together. Consequently, the analysis does not depend on the relative convergence rates of these to limiting processes. The maximal delay inequality, f-to 5 S(e), does not determine whether agents’ starting time f occurs before or after the true parameter shift time to (except to limit how far beyond to that f can be). Thus, if d occurs more (less) than c before to, then current time t =f+ c -All a& %bebefore (after) to. Now consider a perfectly optimizing agent who was forced to respond with a maximum delay after to of at most S(e). This constraint would be irrelevant to such a perfect agent who always begins responding according to x0(t) exactly at to, neither early r late; thereby autsmaiically sahfying k to+ wever, the situation is different for imperjkct S(r) so long as S(e)>O. agents. Trying to respond with less delay will shrink the time needed to &digest’the new conditions causing arameter a to shift. Consequently, over a number of prospective trails in w ch parameter a may unexpectedly shift, rfect agents will more frequently respond too soon ot in the i direction rather than in the right direction if they try to react with les after the parameter really begins shifting. This means the probabilities of
262
einer, The necessity ofdelaying econo
6
9
P
3
R.A. Heiw, The nticesaity of delaying economic adg’ustment
264
owever, even if w(e) and W(t) are positive, mistaken responses would eventually make no dserence if they became crabi~~rily SW&~compared. to correct responses [for example, if the ex direction $‘({) bet rate in the right ---*-.-*a* {ei*hpr boa soon or in the wrong d irection) are assumed to be at lG3)pvLI~z, \CIA*tiUW. least some positive fraction A>O of agents’ expected response in the right direction. That is,
of reaction if too soon, so e&l also that 2: is the expected absolut positive. In addition, since that Zy/!$I is positive since I$1 and 2: are %r and $ necessarily refer to reactions in opposite directions (respectively away and toward a shifting optimal decisio x*(t)), then - x’r/$ is always positive. The basic idea of assumptions Al and A2 is that imperfect agents have a ither too soon or in the positive probability of mistaken initial g direction) at rates which are not small compared to their ons are vir1 reactions toward a shiftin tually implied in the very meaning of imperfect dynamic choice. Thus, the low follow without assuming any special properties inherent in the basic concept of imperfict adjustment alone.
‘T3ic s&on
&z&&es the implications of assumptions Al and A2 for sting to ‘first-o r’ parameter shifts. hese refer to shifts whose dynamic ns with a non-ze re precisely, this means that ur5s thg: initiall optimal re/aa is non-zero by (l), (2), (3) agents will also react at a fore an arbitrarily small delay f;,rst= agents may still not benefit from readng to rection e same
R.A. Weiner, The necessity o/delaying economic adjustment
265
order optimal response rate as their response delay goes to zero (i.e., their ‘immediate’ expected response rate must be ‘secon er’ relative to the initial rate of parameter change). For example, this ies that no matter how rapid the initial rate of parameter change might be short of infinity, agents’ initial respons rate must still drop to zero if their maximum reacfion delay also goes to zer 2.1.
en is immediate but imperfhct reaction benefichl?
When will expected utility rise (instead of fall) from a shifting parameter? At the instant when agen absolute change in expected utility is necessarily zero. Th to deteramine when expected utility will begin rising start to react, conditional on starting before a m parameter begins to shift. To fulfill this objective, define the following expected values of at t= r+& They are conditional either on beginning adjustment not 5) beyond to, or on not adjusting up to S(c) beyond to. That is, for any x?(t)@, let
V(c) = E( V(x?(f+<),a(f+ 5))1f--to I W)), (11) Agents’ expected utility will rise from imperfectly responding (instead of not reacting) at c >O beyond a starting point f that is delayed at most S(c) beyond t” sniy if the instantaneous rate of difference betwee r(c) and V’(c) is positive at {. Consequently, the followi g condition must be satisfied in order for the net difference in expected utility between v(e) and P”(Q to rise immediately after agents start to adjust with at most an infinitesimal delay after a decision parameter starts shifting,
The objective is to determine what (12) implies for any decision rule X% that imperfect!y adjusts according to a_ssumpiiuns inequality for answering this question is described proof are nresented in the appendix Thfj proof shows (a (5) above) that condition (12) is equivalent to a ra first-order effects in which both the numerator and
266
R.A. Heiner, The necessity of delaying economic adjustment
and also define the chance of tzt* later than S(c) after to,
given that agents start responding no
can then show (see the a the ratio of x’:(c) to x,S(c), ondition for any first-order parameter shift,
(12) and (Al), (A2) (c!, must satisfy the
where
(Ma) (15b) (1%) Wd! !lSe!
Wf! ow consider what ns if /I#) does not go to zero when < goes to First recall that {, so that f= t at c =Q which in turn implies ~~S)~~t-t”lb-‘3~ci)=0 as &O. ence, 1@)4 as <4. In additL)n, assumptions A and A2 imply tha k,(tJ converges to a positive limit baklded below y a >O, and w(t) also converges to a positive limit, denoted * r 0. Consequently, lSb, c, d, e, f) imply the n
is necessary to satisfy c consider the ratio j&&j)
! 16)
R.A. Heiner, The necessity of delaying economic tijustvnent
Similar reasoning to that used for /$&) also implies /J c) must to zero in order to satisfy (17). applied to the ratio /II,,(&) =Zr(t Therefore, all three response rati must converge to zero as +Q 0 2.2. The necessity of d&ying
267
converge
rfeci 0djusinoepst:A generai resu
The main result shown above is the following.
must all converge to zero as {-+O in order for conditidri (i2) to **@uSing age.st’s e_xpxtd rate of response to a hold. That is, m i_mpe.r$ecr!, first-order parameter chge (whether too soon, in the wrong & t direction) must become arbitmrily maOi relative to t rate in order for expected utility to rise immediately ufter starting to respond, conditional on beginning adjustment immediately after the parameter starts to shift. $~[)/xf({)
Think of Theorem 1 in terms of a comparison between an agent who responds imperfectly to a shifting parameter, and one who responds optimally. The latter ‘perfect’ agent will thereby benefit from immediate response to a shifting parameter. On the other hand, the imperfect agent cannot benefit from immediate response without severely constraining its expected adjustment rate relative to that of the perfect agent. However, this restriction gradually relaxes as the imperfect agent responds with a longer lag after a parameter starts shifting instead of trying to react with little or no delay. Thus, the imperfect agent can try to react soon but at a much slower rate than the perfect agent; or alternatively, he can start adjusting with a noticeable delay compared to the perfect agent. In either case, the cumulative * a while must lag behind that of t adjustment by the imperfect agem4 cro1
268
einer,
necessity of delayi
economic ~j~t~~g~t
. __.
a?.
.
-*
s.
.
272
R.A. Weiner, The necessityof delaying economic adjustment
section, where C now represents t tations (or other subjective beliefs) used to guide be raised to the limit where
plies to imperfect ex
is implication provides a theoretical rationale for distributed lag models ften referred to as ‘adaptive e of expectation adjustme ave been criticized as seemin
Lucas (1981), Sargent (1979)].
owever, once the behavioral effects of
illiams (1987) concerning the
systematic adaptive expectation features3 as indicated in the concluding the general conclusion [is] that rted only as an equilibrium tment process’ [emphasis o rimental study by
al expectations may be a
This conclusion is also relevant to G Sh 0 0
of neoclassical
R.A. Heiner, The necessityof delaying economicadjustment
equilibrium inconsistent
233
to be govern
by an adaptive process that is thian individual rationality.
y
I have constructed a general model to analyze the imperfectly to shifts in decision or ex tation parameters. I ay sometimes means agents tion, or at the wrong rate to a shifting parameter. imperfection exists, agents’ rate of response must become relative to the optimal adjustment rate if the initial rate of par is non-zero and they begin responding immediately after a to shift. This implies that an imperfect agent’s cumulati while must lag behind that of a perfect agent who always reacts optimally, thereby producing a general symptom of inertially delayed response conipared to perfectly adjusting agents. These results are implied irrespective of whether there are positive or zero costs of adjusting decisions or expectations (such as search, information processing, or transactions costs). They also apply without having to assume ‘habit formation’ or related psychological costs of quickly modifying previous decisions or expectations. Instead, such factors can themselves be regarded as havioral mechanisms that arise in order to control the accumulation of dynamic response errors over time. The analytical tools used in the pa r are also compatible with those traditionally used to analyze optimal behavior, and can be applied to a wide variety of other to its where optimal decisions have been assumed. The paper thereby illustrates how we can modify traditional analysis to explicitly model the behavioral effects of both perfect and imperfect decisions.
1
iacr@
&&
Lat the net rate of loss from adjusting to an arbitrary differentiable function x?(a) instead of optimal adjustment x (4 necessarily goes to zero as a approaches a*; assuming both x .(a) and x?(a) start Salrae o
a[V(S(a),a) -
in&ial’ry
opiinla’j
d&rrofi
So
jihaa
is,
do so, we must first compute the following difFeren-
einer, The necessity of
econo
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ecessi~y
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5-
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ote
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ofdelaying
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einer, The necessity of delaying economic adjust
einer,
ecessity
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285
286
einer, The necessity
ofdelaying
economic
nt
onald A. Brigham Young Universi?y,Provo, UT dW602 USA Received June 1987, fmal wersionreceived October 1987 I give an explanation for why agents react with a delay to decision and expectation parameters, by showing that imperfecca agents must adjust very slowly relative to optimally adjusting agents if they begin reacting as soon as the latter agents would react. The rest&s follow from the hypothesis of imperfect choice alone irrespective of whether there are any infsrmation, search, transaction, or other costs of adjusting decisions. The results thereby provide a general reason for observing inertially lagged behavior, and also agree with recent asset market experiments whose dynamics are governed by dizptiue rather than ‘rational’ expectations.
The extent and variety of topics to which distributed-lag analysis has been applied in empirical economics is astounding, but, what is more remarkab!e, is the v;rtual la& sf theoretical justification for the lag structures . . . ark Nerlove 1972, page 223)
This paper offers an explanation for why agents will not respond immediately to shifts in either decision or expectation parameters. It shows that agents who respond imperfectly mzst adjust very slowly relative to agents who always adjust optimally, if they start reacting immediately after a decision parameter begins shifting and the initial rate of parameter change is non-zero. Imperfect response means *,SCiC is a positive probability of mistakenly adjusting either too soon or in the wrong direction at a rate that is not arbitrarily small relative to their adjustments in the right direction (toward a shifting optimal decision or expectation). The paper proceeds in eight steps. (1) The literature on inertial behavior is briefly reviewed, including various distributed lag? partial adjustment, invest *I thank the followine prsons for helpful Robert Glower, Richard bay, Jonathan LeEand and Joseph Stiglitz. Of course, 1 alone am resp OiKP-2681/88,/%3.50@ i988,
eth Arrow, James Buchanan, d, Rulon Pope, Vernon Smith r’s content. ers
256
R.A. Heiner, The necessity of delaying economic adjustment modeis [such as Koyck (1954), Nerlove (1972), Jorgenson (1963,
1970), Cagan (1956), Eisner and Strotz (1 helpful in describing been less fruitful i where the existence w imperfect agents must behave in order to control
justification for The latter problem is overcome by introducing certain concepts about imperfect response to a shifting parameter. the possibility of sometimes respondin tion, or at the wrong rate. equality is developed to determine the relative size of imperfect to optimal adjustment in order for agents’ expected utility to rise conditional on adjusting before a small delay after a decision parameter starts shifting. A parameter change that begins with a non-zero first derivative is called a ‘first-order’ parameter shift. Except at the limit where adjustment becomes fully optimal, the expected rate of (im feet) response to a first-order parameter shift - whether mis&enly soon or in the wrong direction, or correctly toward the new optimum must become arbitrarily small relative to the optimal response rate as adjustment delay goes to zero. e main result of step e is formally summarized and discussed in arison een an imperfectly responding agent and
ctive of whether t
R.A.
iw,
The necessityof delaying economicadjustment
257
explaining why a psy& ological tendency toward ‘habit formation’ e when no outwardly observable adjustment costs normative status of delayed response is also brie 6) argued that constraining adjustment to lag hind an optimal agent’s response is nevertheless fully rational given that an im ted utility will drop if this constraint is violated. costs needed to reduce the incidence of se errors are (7) scus& Perfect agents errors without sacrificing scarce resources in order t On the other hand, imperfect agents are those for whom a decision costs are necessary in order to becoming infinite to reach the direction, or magnitude never occur. Except at t delay relative to responding optimally is still necessary. (8) The analysis of previous steps applies not only to outward behavior, but also to imperfect adjustment of subjective beliefs such as expectations. A simplified example of the latter is constructed. Agents use a forecasting rule that has the same functional form as the optimal Bayesian rule, but the specific coefficients that transform observed information into forecasts may respond imperfectly to shifts in parameters governing the underlying stochastic properties of agents’ environment. Previous results are used to show the necessity of delayed or inertial adjustment of imperfect expectations relative to an optimally adjusting Bayesian agent (i.e., imperfect agents will ‘adaptively’ adjust their expectations to ongoing experience). These results also accord with recent experiments of spot asset markets by Vernon Smith, Gerry Suchanek and Arlington Williams (1987) and Williams (1987); showing that ‘rational expectations’ apply only near the end of a dynamic adjustment process characterized by adaptive expectations. I e1. Earlier models of inertial behavior is now accepted that a general tendency to delay adjustment from &ions (abOUtconsumption, investment, pri employment, and ordingly, a large very widespread empirical phenomenon. econometric and theory literature on dynamic decision maki t
258
R.A. Heiner, The necessity of delaying economic adjustment
adjustment
costs
in information processing costs, transactions costs, and so on. example of this approach are decisions involving investment in capital
decisions about durable capital. extensive demand studies showed
outhakker and Taylor’s
of habits’ (i.e., consumers nding habits to avoid psychological adjustment costs). Doing so used for investment decisions. Yet it leaves unexplained why a particular sychology’ toward habits should arise in the first place. , the idea of habit simply redescribes the behavioral s plained (that is, habits are those behavior patterns that slu inferred from obse
mechanisms deterri
s of slowly adjusting
R.A. Heiner, The necessityof delaying economica&stment
1.2.
losses are
sufficient to
259
inertial behavior
t us first briefly discuss why inertial behavior cannot be just%ed osses from delayed adjustment may be ‘small’. To do so, let V(x, a) represent an agent’s objective function which depends on a decision variable x chosen by the agent and a decision parameter a t as given by the agent. Assume the first and second order maximizing co tions hold, so that an optimal value of x exists for each a, denoted by the function x*(a). Standard analysis then implies &P(a) -a2V(x*(42),a)/axaa -= -da - a2V(x*(a), a)&’
where
’
a2 V(x’(a), a)px2 SO for each
a.
(2)
To insure that the optimal decisitin rule responds to shifts in parameter a (so that i%P/aa#O) also assume a2 V(x+, a)/ax aa #O
for each
a.
!3)
Now suppose parameter a has remained stationary at a=a* long enough for the agent to have previously adjusted x to its optimal value x0 =x*(a*). Then at some point parameter a begins to shift. xt calculate the net rate of loss from not sting away from x0 compared to optimally responding according to x That is, calculate d[ V(x*, a) - V(x*(a), a)]/da.
(4)
Akerlolf and Ye!!en (1985a, b) have recently shown that (4) goes to zero as a -a* goes to zero. Consequently, any ‘first order’ losses from not adjusting . _- 1. i * ahif& _ a__dg&&fisr - rrrvl”.. ~~arr====~== ypLILLIPIoIL&l a-7a.ay from its iinrINxlra~~ny go LU iax c-e ati1 nmnll OAAikbAA Y_..__ Qf initia! value. However, there is a basic problem with the logic of small first order losses; namely, they apply equally to any arbitrary way of responding to holding x temporarily fixed at x0. to a shifting parameter, not sim es from responding too quickly instead of For example, the first order ment rate) also go too slowly (say three times fa resent an to zero as a- a* goes to zero.
R.A. Heiner, The necessityof delaying economicfuijustment
260
ma1 adjustment, including not adjusting as a us, slmalysisb -pm-t@
itself givesno reaso 2 behavior will take the form of delayed reaction to
091negligible first-order losses by
ing that nonopt changes. In order to justify the latter possibility, additional analysis
Accordingly, this derived as a generic feature of response to parameter changes. .
will show that de1 avior by modeling t .
1.3. Imperfect response?to chnge: Preliminary concepts asic motivation is th 8 ‘competence’may not problems so as to ent relative to the always react opt~ally to changing conditions based on their information [called a C-D gap; see r (1983, 198Sa, 1986a)]. th imperfect skills at in re complex and unstable t to changes in those environmental relationships will not
soon or in the response. ods of relative with those used
R.A. Heiner, The necessityof delaving economicadjustment
261
previously adjusted x to its optimal value x0= x t SOme point (denoted to), parameter Q will start shifting toward a new value that remains constant for another uncertain period before shifting again, and so on. the point of time when agents start adjusting to changes in S, y not always equal to. even if they don’t start always in the right direction t may sometimes respond in o when agents react toward the ne-w optimu the optimal rate (so that ax?/&#ax*/&). To simplify notation let the derivative of XI or x denoted with a subscript t (for example, 3x?/&= larly, let the time-de of parameter (I equal u#du@)/dt. c=t-f as the length after agents begin to change X, so time t equals f+ c. Now consider what ha s soon after agents begin to react (so that t =f+c), assuming their s time f is delayed at most negligibly beyond the true parameter shift time to (so that f-to is less than some maximum delay 6 > 0 that shrinks to zero). To ensure that both c and 6 become small together, let S be any continuous, non-negative function of 5, denoted S(e), that goes to zero with {. That is, S(c) 20 for all 5 2 0 and lim,,, S(c) =O. This specification allows the size of t-t to vary arbitrarily relative to f-to so long as both go to zero together. Consequently, the analysis does not depend on the relative convergence rates of these to limiting processes. The maximal delay inequality, f-to 5 S(e), does not determine whether agents’ starting time f occurs before or after the true parameter shift time to (except to limit how far beyond to that f can be). Thus, if d occurs more (less) than c before to, then current time t =f+ c -All a& %bebefore (after) to. Now consider a perfectly optimizing agent who was forced to respond with a maximum delay after to of at most S(e). This constraint would be irrelevant to such a perfect agent who always begins responding according to x0(t) exactly at to, neither early r late; thereby autsmaiically sahfying k to+ wever, the situation is different for imperjkct S(r) so long as S(e)>O. agents. Trying to respond with less delay will shrink the time needed to &digest’the new conditions causing arameter a to shift. Consequently, over a number of prospective trails in w ch parameter a may unexpectedly shift, rfect agents will more frequently respond too soon ot in the i direction rather than in the right direction if they try to react with les after the parameter really begins shifting. This means the probabilities of
262
einer, The necessity ofdelaying econo
6
9
P
3
R.A. Heiw, The nticesaity of delaying economic adg’ustment
264
owever, even if w(e) and W(t) are positive, mistaken responses would eventually make no dserence if they became crabi~~rily SW&~compared. to correct responses [for example, if the ex direction $‘({) bet rate in the right ---*-.-*a* {ei*hpr boa soon or in the wrong d irection) are assumed to be at lG3)pvLI~z, \CIA*tiUW. least some positive fraction A>O of agents’ expected response in the right direction. That is,
of reaction if too soon, so e&l also that 2: is the expected absolut positive. In addition, since that Zy/!$I is positive since I$1 and 2: are %r and $ necessarily refer to reactions in opposite directions (respectively away and toward a shifting optimal decisio x*(t)), then - x’r/$ is always positive. The basic idea of assumptions Al and A2 is that imperfect agents have a ither too soon or in the positive probability of mistaken initial g direction) at rates which are not small compared to their ons are vir1 reactions toward a shiftin tually implied in the very meaning of imperfect dynamic choice. Thus, the low follow without assuming any special properties inherent in the basic concept of imperfict adjustment alone.
‘T3ic s&on
&z&&es the implications of assumptions Al and A2 for sting to ‘first-o r’ parameter shifts. hese refer to shifts whose dynamic ns with a non-ze re precisely, this means that ur5s thg: initiall optimal re/aa is non-zero by (l), (2), (3) agents will also react at a fore an arbitrarily small delay f;,rst= agents may still not benefit from readng to rection e same
R.A. Weiner, The necessity o/delaying economic adjustment
265
order optimal response rate as their response delay goes to zero (i.e., their ‘immediate’ expected response rate must be ‘secon er’ relative to the initial rate of parameter change). For example, this ies that no matter how rapid the initial rate of parameter change might be short of infinity, agents’ initial respons rate must still drop to zero if their maximum reacfion delay also goes to zer 2.1.
en is immediate but imperfhct reaction benefichl?
When will expected utility rise (instead of fall) from a shifting parameter? At the instant when agen absolute change in expected utility is necessarily zero. Th to deteramine when expected utility will begin rising start to react, conditional on starting before a m parameter begins to shift. To fulfill this objective, define the following expected values of at t= r+& They are conditional either on beginning adjustment not 5) beyond to, or on not adjusting up to S(c) beyond to. That is, for any x?(t)@, let
V(c) = E( V(x?(f+<),a(f+ 5))1f--to I W)), (11) Agents’ expected utility will rise from imperfectly responding (instead of not reacting) at c >O beyond a starting point f that is delayed at most S(c) beyond t” sniy if the instantaneous rate of difference betwee r(c) and V’(c) is positive at {. Consequently, the followi g condition must be satisfied in order for the net difference in expected utility between v(e) and P”(Q to rise immediately after agents start to adjust with at most an infinitesimal delay after a decision parameter starts shifting,
The objective is to determine what (12) implies for any decision rule X% that imperfect!y adjusts according to a_ssumpiiuns inequality for answering this question is described proof are nresented in the appendix Thfj proof shows (a (5) above) that condition (12) is equivalent to a ra first-order effects in which both the numerator and
266
R.A. Heiner, The necessity of delaying economic adjustment
and also define the chance of tzt* later than S(c) after to,
given that agents start responding no
can then show (see the a the ratio of x’:(c) to x,S(c), ondition for any first-order parameter shift,
(12) and (Al), (A2) (c!, must satisfy the
where
(Ma) (15b) (1%) Wd! !lSe!
Wf! ow consider what ns if /I#) does not go to zero when < goes to First recall that {, so that f= t at c =Q which in turn implies ~~S)~~t-t”lb-‘3~ci)=0 as &O. ence, 1@)4 as <4. In additL)n, assumptions A and A2 imply tha k,(tJ converges to a positive limit baklded below y a >O, and w(t) also converges to a positive limit, denoted * r 0. Consequently, lSb, c, d, e, f) imply the n
is necessary to satisfy c consider the ratio j&&j)
! 16)
R.A. Heiner, The necessity of delaying economic tijustvnent
Similar reasoning to that used for /$&) also implies /J c) must to zero in order to satisfy (17). applied to the ratio /II,,(&) =Zr(t Therefore, all three response rati must converge to zero as +Q 0 2.2. The necessity of d&ying
267
converge
rfeci 0djusinoepst:A generai resu
The main result shown above is the following.
must all converge to zero as {-+O in order for conditidri (i2) to **@uSing age.st’s e_xpxtd rate of response to a hold. That is, m i_mpe.r$ecr!, first-order parameter chge (whether too soon, in the wrong & t direction) must become arbitmrily maOi relative to t rate in order for expected utility to rise immediately ufter starting to respond, conditional on beginning adjustment immediately after the parameter starts to shift. $~[)/xf({)
Think of Theorem 1 in terms of a comparison between an agent who responds imperfectly to a shifting parameter, and one who responds optimally. The latter ‘perfect’ agent will thereby benefit from immediate response to a shifting parameter. On the other hand, the imperfect agent cannot benefit from immediate response without severely constraining its expected adjustment rate relative to that of the perfect agent. However, this restriction gradually relaxes as the imperfect agent responds with a longer lag after a parameter starts shifting instead of trying to react with little or no delay. Thus, the imperfect agent can try to react soon but at a much slower rate than the perfect agent; or alternatively, he can start adjusting with a noticeable delay compared to the perfect agent. In either case, the cumulative * a while must lag behind that of t adjustment by the imperfect agem4 cro1
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R.A. Weiner, The necessityof delaying economic adjustment
section, where C now represents t tations (or other subjective beliefs) used to guide be raised to the limit where
plies to imperfect ex
is implication provides a theoretical rationale for distributed lag models ften referred to as ‘adaptive e of expectation adjustme ave been criticized as seemin
Lucas (1981), Sargent (1979)].
owever, once the behavioral effects of
illiams (1987) concerning the
systematic adaptive expectation features3 as indicated in the concluding the general conclusion [is] that rted only as an equilibrium tment process’ [emphasis o rimental study by
al expectations may be a
This conclusion is also relevant to G Sh 0 0
of neoclassical
R.A. Heiner, The necessityof delaying economicadjustment
equilibrium inconsistent
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to be govern
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y
I have constructed a general model to analyze the imperfectly to shifts in decision or ex tation parameters. I ay sometimes means agents tion, or at the wrong rate to a shifting parameter. imperfection exists, agents’ rate of response must become relative to the optimal adjustment rate if the initial rate of par is non-zero and they begin responding immediately after a to shift. This implies that an imperfect agent’s cumulati while must lag behind that of a perfect agent who always reacts optimally, thereby producing a general symptom of inertially delayed response conipared to perfectly adjusting agents. These results are implied irrespective of whether there are positive or zero costs of adjusting decisions or expectations (such as search, information processing, or transactions costs). They also apply without having to assume ‘habit formation’ or related psychological costs of quickly modifying previous decisions or expectations. Instead, such factors can themselves be regarded as havioral mechanisms that arise in order to control the accumulation of dynamic response errors over time. The analytical tools used in the pa r are also compatible with those traditionally used to analyze optimal behavior, and can be applied to a wide variety of other to its where optimal decisions have been assumed. The paper thereby illustrates how we can modify traditional analysis to explicitly model the behavioral effects of both perfect and imperfect decisions.
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