Information, heterogeneity and market

incompleteness
Liam Graham! and Stephen Wright†

17 December 2009‡

Abstract

Information is “market-consistent” if agents only use market prices to infer the

underlying states of the economy. This paper applies this concept to a stochastic
growth model with incomplete markets and heterogeneous agents. The economy
with market-consistent information can never replicate the full information equi-
librium, and there are substantial di!erences in impulse responses to aggregate
productivity shocks. These results are robust to the introduction of a noisy public
signal and aggregate Þnancial markets. We argue that the principle of market-
consistent information should be applied to any model with incomplete markets.
JEL classiÞcation: D52; D84; E32.
Keywords: imperfect information; higher order expectations; Kalman Þlter; dy-
namic general equilibrium

1 Introduction

A number of recent papers (notably Krusell and Smith, 1998) argue that market incom-
pleteness does not much matter for the aggregate properties of DSGE models. This paper
shows that, once account is taken of its informational consequences, market incomplete-
ness does matter.
!
Corresponding author: Department of Economics, University College London, Gower Street, London
WC1E 6BT, UK. Liam.Graham@ucl.ac.uk
†
Department of Economics, Birkbeck College, University of London, Malet Street, London W1E 7HX,
UK. s.wright@bbk.ac.uk
‡
We are grateful to Robert King (the Editor) and an anonymous referee for comments that substan-
tially improved the paper. We also thank Kristo!er Nimark for his advice, and numerous colleagues and
participants in seminars for their input.

1
 Most DSGE models assume complete information, i.e. knowledge of the current state
vector. In the stochastic growth model this consists of aggregate capital and aggregate
technology, quantities which are intrinsically di"cult to measure directly. Radner (1979)
shows that complete markets resolve this measurement problem. If markets are complete
the information set consisting of market prices is invertible i.e. it reveals the states. The
assumptions on markets and information are consistent with each other.
In contrast, models with incomplete markets typically simply assume the states are
directly observable. We propose that the concept of “market-consistent information”
should be applied to models with incomplete markets. This paper analyses a linearised
stochastic growth model with heterogeneous agents in which households have heteroge-
neous information sets arising from the limited set of markets they trade in. Average
expectations then have persistent e!ects via aggregate capital, leading to a “hierarchy
of expectations” (Townsend, 1983, Woodford, 2003, Nimark 2007). The paper extends
this methodology to allow for endogenous states and market-consistent information. This
extension is a non-trivial exercise and our novel solution techniques allow the discipline
of market-consistent information sets to be applied to any linearised DSGE model with
incomplete markets.
This paper shows that a market-consistent rational expectations equilibrium will have
the following three properties. First, the equilibrium will never replicate the full infor-
mation solution. As a direct consequence, the good news of a positive productivity shock
must always be interpreted at least in part as the bad news that the capital stock has been
over-estimated in the past. Second, for virtually all empirically plausible calibrations
this bad news response dominates the short and medium term response to a productivity
shock. Third, this marked contrast with the complete markets response is robust to the
inclusion of a noisy public signal and aggregate Þnancial markets
There is a long tradition of the study of imperfect information in macroeconomics, a
review of which can be found in Hellwig (2006). The approach taken in this paper is
distinct from recent work in explicitly drawing the link between information and markets;
showing how the hierarchy of expectations can be modelled when there are endogenous

2
states; and Þnding that imperfect information has signiÞcant e!ects in the stochastic
growth model, without the need for noisy indicators as in Lorenzoni (2010) or strategic
complementarities as in Angelotos and L’ao (2009).
The remainder of this paper is structured as follows. Section 2 presents the model.
Analytical results are in section 3, and numerical results in section 4 Section 5 discusses
the robustness of these results, and section 6 concludes. Proofs are in Appendix A,
linearizations and supplementary material in Appendix B

2 The model

This section presents a model of the type that is becoming standard in the dynamic gen-
eral equilibrium literature1 . There are a large number of households and a large number
of Þrms, divided into ! islands, on each of which there many Þrms and households. There
are shocks to aggregate and island-speciÞc labour productivity.
Upper case letters denote levels, lower case letters denote log deviations from the
steady state growth path. A superscript " indicates a variable relating to a typical
household or Þrm on island ". Without the superscript the variable is an aggregate.

2.1 Households

A typical household on island " consumes (#"! ) and rents capital ($"! ) and labour (%"! )
to Þrms. Household labour on each island has idiosyncratic productivity (&"! ) whereas
capital is homogenous, so households earn the aggregate return ('#" ) on capital but an
idiosyncratic wage (("! ) on their labour. Apart from the idiosyncratic shock, households
on di!erent islands are unconditionally identical.
The problem of a household on island " is to choose paths for consumption, labour
supply and investment ()"! ) to maximize expected lifetime utility given by

!
" ¡ ¢1"% #
X 1 " % !
"+$
*"! +$ !
ln #"+$ +, (1)
$=0
1"-
1
Examples of papers which use similar models include Krusell and Smith (1998) and Lorenzoni (2010).

3
 1
where %
is the intertemporal elasticity of labour supply, and + the subjective discount
rate, subject to a resource constraint

'#" $"! + ("! %"! = #"! + )"! (2)

and the evolution of the household’s holdings of capital

!
$"+1 = (1 " .) $"! + )"! (3)

The expectations operator for an individual household is deÞned as the expectation given
the household’s information set !!" , i.e. for some variable /"

*"! /" = */" |!!" (4)

The household’s Þrst-order conditions consist of an Euler equation

! ¸
1 ! '"+1
= +*" (5)
#"! !
#"+1

where '" = '#" + 1 " . is the gross return to a one-period investment in capital, and a
labour supply relation
("!
, (1 " %"! )"% = (6)
#"!

2.2 Firms

The production function of a typical Þrm on island " is

0"! = (1"! )1"& (2" &"! %"! )' (7)

where 2" is an aggregate productivity shock and 1"! is the capital rented by the Þrm:
in general, 1"! 6= $"! 3 since capital will ßow to more productive islands. The Þrst-order

4
conditions of this Þrm are
0"! ! 0"!
'#" = 3( = ! (8)
1"! " %"

2.3 Aggregates

Aggregate quantities are sums over household or Þrm quantities, and for convenience are
calculated as quantities per household. For example aggregate consumption is given by
X(
1
#" = ( #"! . The economy’s aggregate resource constraint is then
!=1

0" = #" + )" (9)

2.4 Markets

Firms on island " only rent labour from households on island ", and the wage on island
", ("! , adjusts to set labour supply (6) equal to labour demand (8). In contrast, capital
is homogenous and tradeable between islands, so ßows to islands with more productive
labour. The aggregate return to capital, '" , adjusts to clear the capital market, making
the demand for capital for each Þrm (8) consistent with each household’s Euler equation
(5) and the aggregate resource constraint (9).

2.5 Shocks

For both the aggregate and idiosyncratic productivity shocks assume autoregressive processes
in log deviations:

/" = 4' /""1 + 5" (10)

6"! = 4) 6""1
!
+ 5!" (11)

where 5" and 5!" are i.i.d mean-zero errors, and *52" = 7 2' ; * (5!" )2 = 7 2) . The
(
X
innovation to the idiosyncratic process satisÞes an adding up constraint, 5!" = 0
!=1

5
which implies
(
X
6"! = 08 (12)
!=1

2.6 Linearisation

The remainder of the paper works with the log-linear approximation to the model in
order to be able to use a linear Þlter to model the household’s signal extraction problem2 .
Appendix B1. shows that the features of the economy relevant to a household on island
" can be written as an Euler equation

*"! "9!"+1 = *"! :"+1 (13)

and a law of motion for the economy that is symmetric across islands:

!
;"+1 = <* ;"! + <+ 9" + <! 9!" + =!" (14)

! ¸0
where ;"! = is a vector of underlying states relevant to a household on island
> 0" ?!0
"
! ¸0
" comprising aggregate states > " = @" /" and states speciÞc to the household, given
! ¸0 ! ¸0
! ! !
by ?" = @"! " @" 6"! . The innovation vector for ;" is =" = 0 5" 0 5!" (both
@" and @"! are pre-determined). The structural coe"cient matrices <* 3 <+ and <! are
deÞned in Appendix B1.

2.7 The economy with full information

If complete markets exist, risk-sharing implies that consumption is perfectly correlated

across households so each household also knows aggregate consumption and the idiosyn-
cratic component of its labour income. It is then easy to show that the information
set is “instantaneously invertible” - full information can be recovered using only A-dated
information.

2
Although non-linear Þlters exist, incorporating one in this type of model and dealing with the
hierarchy of expectations in a non-linear world would present a formidable challenge.

6
DeÞnition 1. (Full information) Full information, denoted by an information set !#" ,
is knowledge of the aggregate states in the economy > " , the idiosyncratic states ?!" of all
households and the time-invariant parameters and structure of the underlying model #.

h i
!#" = > " 3 {?!" }(!=1 3 # (15)

On the other hand, if markets are incomplete but full information is simply assumed,
it is straightforward to show that the economy is identical at an aggregate level to the
complete markets economy, but di!ers markedly at a household level3 . This is related to
Krusell and Smith’s (1998) result that an economy with incomplete markets can closely
resemble one with complete markets (the resemblance is exact in the model presented
here because it is linearised). However, as what follows will show, this result is in
general dependent on a market-inconsistent assumption of complete information.

3 Market-consistent information

With only capital and labour markets the market-consistent information set of a house-
hold on island " at time A is

£ ¤
!!" = {:$ }"$=0 3 {B$! }"$=0 3 # (16)

where # contains the parameters and structure of the underlying model4 . DeÞne a
! ¸0
!
measurement vector C" = :" B"! such that the information set evolves according to
!!"+1 = !!" # C!"+1 8 Using the structural equations of the economy,
3
The permanent income response to idiosyncratic shocks implies that the idiosyncratic component of
consumption is a random walk as in Hall (1978). However, the adding-up constraint across idiosyncratic
shocks (12) means that such permanent shifts in idiosyncratic consumption cancel out in the aggregate.
In general, this form of uninsurable income uncertainty would be expected to cause precautionary saving
which would change the steady state of the model, but these e!ects are precluded by linearizing.
4 # #"1
Households also have knowledge of the history of their own optimising decisions, {!!" }"=0 " {#!" }"=0 ,
#
{$"! }"=0 however, since each of these histories embodies the household’s own responses to the evolution
of !!# " it contains no information not already in !!# . Note that while $#! is directly observable, the
idiosyncratic component of capital, $#! " $# " contained in the vector of underlying states, %#! " is not.

7
 C!" = %*
0
;"! + %+ 9" (17)

where the matrices %* and %+ are deÞned in Appendix B1.

Since aggregates are not directly observable, households are unable to distinguish be-
tween aggregate and idiosyncratic productivity shocks. They must therefore make errors
(and will know that they must make errors) in estimating the true values of the states.
Thus innovations in the observable variables could be caused either by true innovations
to the exogenous processes, or by households’ estimates of the aggregate states being
incorrect. But the informational problem for each household is not restricted to forming
estimates of the states, ;"! , since each household knows that the aggregate capital stock
depends on the average expectation of these states. This in turn must imply that the
average expectation of the average expectation also matters, and so on - hence the true
problem has an inÞnite dimensional state vector

3.1 The hierarchy of expectations

Since in general the market-consistent information set will di!er across households, the
state vector relevant to household on island "3 D"! 3 can be shown to consist of the non-
expectational states ;"! , deÞned after (14), and an inÞnite hierarchy of average expecta-
tions of ;"! (Townsend, 1983, Woodford, 2003, Nimark, 2007, 2008)5

! ¸0
D"! = ;"! ;"
(1)
;"
(2)
;"
(3)
8888 (18)

(1)
where the Þrst-order average expectation ;" is an average over all households’ expec-
tations of their non-expectational state vector

(
(1) 1X ! !
;" = * ; (19)
! !=1 " "
5
Appendix B2 provides a heuristic argument that demonstrates how the hierarchy of expectations
arises from the problem of inÞnite regress, and illustrates why the Law of Iterated Expectations cannot
be used to simplify this problem.

8
and higher-order expectations are given by

(
(#) 1 X ! (#"1)
;" = * ; ;@ E 1 (20)
! !=1 " "

3.2 The household’s Þltering problem

To implement optimal consumption a typical household on island " must form estimates
of the state vector D"! by using the information !!" available to it. The optimal linear
Þlter is the Kalman Þlter, however this problem di!ers from the standard Kalman Þlter
in three ways. The Þrst di!erence is that there is no extraneous noise. The second
is that the states depend on the household’s choice variable 9!" . Baxter et al (2009)
describe this endogenous Kalman Þlter in detail, and give conditions for its stability and
convergence which are satisÞed here. Thirdly, since the aggregate states depend on
aggregate consumption, and hence the behaviour of all other households, an assumption
is needed about what a household on one island knows about the behaviour of households
on all other islands. We follow Nimark (2007) in assuming that each household applies
the Kalman Þlter to the entire model on the assumption that each other household is
behaving in the same way.
Assumption 1: It is common knowledge that all households’ expectations are rational
(model consistent).
This is essentially a generalization of the full information rational expectations as-
sumption given idiosyncratic information sets.
Given Assumption 1, Appendix A1 shows that each household faces a symmetric
endogenous Kalman Þlter problem of the form

!
D"+1 = F9!" + GD"! + H=!"+1 (21)

C!" = % 0 D"! (22)

9
where F3 G3 H and % are matrices yet to be determined, =!" is the structural state
innovation in (14) and C!" is the measurement vector of household "3 deÞned in (17).

3.3 Equilibrium with market-consistent information

DeÞnition 2. Rational expectations equilibrium with market-consistent infor-

mation: a competitive equilibrium in which the law of motion of the economy is con-
sistent with each agent solving a decentralized optimizing and informational problem i.e.
© ª!=1:(
a sequence of plans for allocations of households 9!" 3 I!" 3 @"+1
!
"=1:!
; prices {:" 3 B"! }!=1:(
"=1:! ;

and aggregate factor inputs {@" 3 I" }"=1:! such that

1. Given prices and informational restrictions, the allocations solve the utility maxi-
mization problem for each consumer

2. {:" 3 B"! }!=1:(

"=1:! are the marginal products of aggregate capital and island-speciÞc labour.

3. All markets clear

Appendix A1. shows how an equilibrium satisfying Assumption 1 and DeÞnition 2

can be derived. Although in general this problem can only be solved numerically, the
following analytical result describes a key characteristic of the equilibrium.

Proposition 1. Non-Replication of Full Information. In a non-homogenous econ-

omy (7, E 0) with the market-consistent information set (15) an equilibrium satisfying
Assumption 1 and DeÞnition 2 never replicates the full information equilibrium

Proof : See Appendix A2.

This result contradicts the simple intuition that arises from counting shocks and
observables. Each household needs to identify two underlying shocks, 5" and 5!" 3 and
has two observables, their (idiosyncratic) wage and the (aggregate) return on capital.
Since the latter is only a!ected by the aggregate shock, 5" 3 it might appear that 5"
could be recovered simply from innovations to returns. The proof shows that there is
indeed a Þxed point of the Þltering problem in which this happens; but that this Þxed
point is unstable. In time series terms there is a reduced form ARMA representation

10
of :" in which 5" is the innovation, but this representation is “non-fundamental” (Lippi
and Reichlin, 1994), and hence 5" cannot be recovered from the history of :" .
The stability condition that is violated here is mathematically identical to that given
by Fernandez-Villaverde et al (2007) which must be satisÞed for an econometrician to
be able to infer true structural shocks and impulse responses from an estimated vector
autoregressive representation of the economy in which the number of underlying shocks
equals the number of observable variables. If this condition is not satisÞed, Fernandez-
Villaverde et al show that the true shocks and impulse responses will be those to a "non-
fundamental" representation of the economy. Given that this condition is not satisÞed
in the model economy, the typical household is faced with exactly the same inference
problem as Fernandez-Villaverde et al’s econometrician.
A corollary to this result helps understand the mechanism which makes the responses
di!er from those under full information. This corollary can be proved for the case of
Þxed labour supply (- = $) and also in the neighbourhood of this case i.e. for - E -̄,
where -̄ is Þnite. Numerically, the corollary appears always to hold.6

Corollary 1. Impact e!ects of aggregate productivity shocks. A positive aggregate

technology shock leads households to reduce their estimate of aggregate capital

Proof: See Appendix A3.

This result arises from the general property of optimal Þlters that state estimates have
lower variance than states. The only source of aggregate ßuctuations in the model is the
technology shock. The solution to the Þltering problem implies that aggregate technology
must have lower variance than actual technology, so when such a shock hits, the estimate
of technology must rise by strictly less than actual technology i.e. starting from a steady
state in A = 0, *1! /1 J /1 . Assuming Þxed labour supply to simplify things, the linearised
return is, as in Campbell (1994) :" = K (/" " @" ) and since estimates must be consistent
with the information set, *1! :1 = :1 = K (/1 " @1 ), so *1! @1 " @1 = *1! /1 " /1 Since capital
is predetermined, @1 = 0 so *1! @1 = *1! /1 " /1 J 0. Thus each household’s estimate of
6
Appendix A3. provides a su"cient condition for the average estimate of capital to be negative, which
appears numerically always to be satisÞed.

11
capital (and hence the average estimate) must fall on the impact of a positive innovation
to aggregate productivity. What is unambiguously good news under full information
appears to the average household to be a mixture of good and bad news.

4 The response to aggregate productivity shocks

While the previous section shows qualitative di!erences from the full information equi-
librium, this section investigates their quantitative implications.

4.1 Calibration

The key parameters are the persistence and innovation variance of the aggregate and idio-
syncratic productivity processes7 . Aggregate productivity is calibrated with the bench-
mark RBC values for persistence of 4' = 089 and an innovation standard deviation
7' = 087% per quarter (Prescott, 1986). The calibration of idiosyncratic technology
draws on the empirical literature on labour income processes. A calibration that sets
idiosyncratic persistence equal to aggregate persistence (i.e. 4) = 4' = 089) appears
consistent with Guvenen’s (2005, 2007) recent estimates using US panel data. There
is however strong evidence that idiosyncratic technology has a much higher innovation
standard deviation. A Þgure of 489% per quarter is consistent with Guvenen’s results8 .

4.2 The nature of impulse response functions

The response proÞles discussed in this section di!er from standard impulse response func-
tions under full information in that they show the impact of a shock to an underlying
stochastic process, /" 3 that would be unobservable to any agents in the economy. These
impulse response functions could not therefore be observed contemporaneously and the
7
The values for the standard parameters & = 0'025; ( = 0'667; ) = 0'99; * = 0'005 are chosen
following Campbell (1994). The choice of log utility has already restricted the coe"cient of relative risk
aversion to unity. Card (1994) estimates the intertemporal elasticity of labour supply, $1 to be between
0.05 and 0.5. The calibration sets + = 5, in the middle of this range. Steady state labour is taken to
be , = 0'33 which implies the weight of labour in the utility function is - = 3'5.
8
See Appendix B3. for a full discussion. Our calibration technique takes account of households’
observing their own labour supply, but it can be argued that some idiosyncratic innovations to labour
productivity may also be directly observable. This issue is discussed in Section 5.1

12
stochastic properties of the model are crucial in determining the nature of impulse re-
sponse functions.
In contrast, under full information, after the initial shock has taken place, the remain-
der of the impulse response is equivalent to a perfect foresight path, and is thus known
in advance to both observer and agents in the model. The agents in the model are con-
tinuously making inferences from new information as it emerges, and thus are uncertain
not only about the value of future shocks, but also about their own future behaviour in
response to past shocks9 .

4.3 Response to an aggregate productivity shock

Figure 1 shows the e!ect of a 1% positive innovation in the process for aggregate produc-
tivity on aggregate consumption in the baseline model and in the case of full information10 .
Under full information consumption increases on impact. In contrast, with incomplete
markets and market-consistent information, the response of aggregate consumption is
signiÞcantly negative on impact of a positive productivity shock.
This response is driven by the nature of the Þltering problem. Households do not
observe the aggregate technology shock directly, but only the associated positive innova-
tions to the aggregate return and the idiosyncratic wage. They then use these observed
innovations to update their estimates of the states, and it is these state estimates which
determine their consumption decision.
Innovations to the observed variables can occur either because of structural innova-
tions, or because households’ past state estimates were incorrect. For example, if there
is a positive innovation to the return, this could either be caused by a positive aggregate
technology shock, or because households had overestimated aggregate capital in the past.
This is the basis for Corollary 1: households interpret what is actually good news about
aggregate productivity as bad news about aggregate capital. This implies that tech-
9
It is not the form of impulse responses that are unobservable, but the shocks that feed into them.
The assumption of common knowledge of rationality means that any household could draw Figure 1,
but no household would be able to identify contemporaneously that a productivity shock had actually
occurred.
10
Appendix B4 gives details of the numerical solution method.

13
nology must have also been overestimated in the past11 which, given the persistence of
technology, means that the sign of the response of the estimate of aggregate technology
is ambiguous. In the base case the response is positive but quite small, around 20% of
the size of the true shock. If technology is slightly more persistent than in the base case
(4' = 0893)3 the two e!ects perfectly o!set and estimates of aggregate technology never
change.
The response of the state estimates are shown in Figure 2. The predominantly bad
news about the aggregate economy is o!set by good news on both the idiosyncratic
component of capital, @"! " @" 3 since @"! itself is observed, and idiosyncratic technology.
But the pure permanent income response to estimates of idiosyncratic states is small,
and so the overall response is dominated by the response to the estimate of aggregate
capital12 .
Households base their consumption decisions on estimates of the state variables. The
accuracy of these estimates can be assessed by the covariance matrix of one-step ahead
forecasts of the states. For the baseline calibration, under full information the quarterly
standard deviations of one-step ahead forecast errors for aggregate technology and ag-
gregate capital would be 0.7% and zero respectively (since capital is pre-determined). In
the base case with incomplete information the corresponding Þgures increase to 1.6% and
2.2%. It is striking that what seems to be a quite modest degree of uncertainty about the
true value of the capital stock should be enough to cause such a signiÞcant change in the
dynamics of the system, especially so, given that recent debates about the true size of the
capital stock (see, for example, Hall, 2001 or the discussion of intangibles in Laitner and
Stolyarov, 2003) have suggested measurement errors by statistical o"ces that are many
orders of magnitude larger than this. The relative accuracy of households’ estimates in
our simple model suggests we may well be considerably understating the informational
11 !
To see this recall that estimates must be consistent with the information set so .#"1 = /#"1 .#"1 .
! !
Again assuming Þxed labour for clarity, this implies 0#"1 "/#"1 0#"1 = $#"1 "/#"1 $#"1 . If the household
! !
now believes /#"1 $#"1 was too high this implies /#"1 0#"1 was also too high. Since technology is
!
persistent this will reduce the response of /# 0# .
12
This discussion is implicitly framed in “certainty-equivalent” terms, ie it assumes that the typical
household’s consumption only depends on estimates of the underlying states, %#! , whereas consumption
also depends on the entire hierarchy of expectations. However the sign of the actual consumption
response is determined by the certainty-equivalent response.

14
problem households face.

5 Robustness

This section discusses the robustness of the results to changes in the key parameters, and
to the introduction of a noisy public signal and additional Þnancial markets.

5.1 Sensitivities

The informational problem is due to the market-consistent information set being insuf-
Þcient for the household to distinguish between aggregate and idiosyncratic productiv-
ity processes. The key parameters of the model are therefore the properties of these
processes. Changing the other parameters does not a!ect the nature of the informa-
tional problem.
While there is strong evidence in the data that the idiosyncratic economy is much more
volatile than the aggregate, an important question is the extent to which idiosyncratic
shocks are observable. In informational terms, observing some part of idiosyncratic
shocks is equivalent to their having a lower variance. Table 1 shows the sensitivity of
the impact response of consumption to changes in the persistence and variance of the
idiosyncratic shock.
Moving from left to right across the table, the degree of heterogeneity in the economy
progressively falls, with the Þnal column being the limiting homogenous case. As the
relative standard deviation of the idiosyncratic shock decreases, the information problem
becomes less acute, so the impact response of consumption becomes less negative.
Moving up the table, as the persistence of idiosyncratic technology rises, its uncon-
ditional variance rises so the informational problem becomes more acute with an in-
creasingly negative impact on the response of consumption. However there is a second
o!setting e!ect. As the process becomes more persistent, an estimated innovation to
idiosyncratic productivity has a greater e!ect on expected wealth, so, other things being
equal, the response of consumption becomes less negative. These o!setting e!ects re-

15
sult in a non-monotonic relation between the persistence of persistence of idiosyncratic
technology and the impact response of consumption.

5.2 A noisy public signal

The model presented so far has an informational structure which is internally consistent
- in decentralised equilibrium the only source of information about aggregates are market
prices - and this will never replicate the full information solution, so giving rise to welfare
costs compared to full information. This could provide a rationale for the existence of
other sources of information such as government statistical o"ces etc. The impact of such
additional sources of information can be analyzed by extending the measurement vector
(22) to include a public signal of output which di!ers from true output by a white-noise
error13 . Figure 3 shows how this signal a!ects the response of aggregate consumption in
the model with noise in the public signal with a standard deviation ranging from 1% to
3%.
Recall that without a public signal (Figure 1) the impact response of consumption
was negative. With a standard deviation of the noise in the output measure at the top
of the range, the impact response becomes very close to zero. As the accuracy of the
signal increases, the response of consumption approaches the full information case.
Although there is currently a lively debate on the empirical e!ect of technology shocks,
see for example Christiano et al (2003), there seems to be some agreement that a range of
variables, including consumption, respond more sluggishly in the data than in a standard
RBC model. Theoretical explanations for such sluggishness (for example Francis and
Ramey, 2005) are usually couched in terms of nominal or real rigidities, or habit formation.
The result of this section shows that informational imperfections can generate such a
sluggish response of consumption without additional rigidities.
13
How noisy are real-time estimates of output? Orphanides and Norden (2002) attempt to quantify
the extent of uncertainty by calculating the di!erence between real-time and Þnal estimates. Their table
2 shows standard deviations of the di!erence ranging from 1% to 3% per quarter. However, they note
that their method "...overestimate[s] the true reliability of the real time estimates since it ignores the
estimation error in the Þnal series", which given the issues involved in measuring output, is likely to be
large but is by its nature unquantiÞable.

16
5.3 Financial markets

Our baseline model is one without Þnancial markets. However the techniques of e.g.
Lettau (2003) can be extended to price Þnancial assets in this model. In contrast to
the shadow prices normally derived under full information, in any model with incom-
plete markets and market-consistent information, these prices will have an informational
content and so will change the equilibrium.
As an example, consider the case of a risk-free bond. Some algebra (see Appendix
B5.) shows its return, :- " is given by

(1)
:- " = L0 GD" (23)

where L is the vector that relates the return on capital to the non-expectational aggregate
(1)
states, G is as in (21), and D" is the Þrst-order average expectation of the full state
vector. So the risk-free rate reveals a linear combination of the hierarchy of expecta-
tions,. However in practice adding this to households’ information sets barely changes
the response in Þgure 1. The intuition for this is as follows. In the baseline calibration,
with its large variance of the idiosyncratic shock, the wage conveys essentially no useful
information about aggregates: the return on capital is e!ectively the only observable
that gives information about aggregates, and all households know this to be the case. So
even without a risk-free asset the hierarchy of average expectations of aggregate states is
close to being common knowledge. Hence the risk-free rate conveys very little additional
information, and the equilibrium is little changed. There are very similar results when a
stock market is introduced14 .
What additional assets would change these results? The Þrst type would be assets
that help households insure against, and hence identify, idiosyncratic shocks. The more
of these assets there are, the easier the Þltering problem until in the limit with complete
14
To price the stock assume a dividend process given by the one period return on capital as in Lettau
(2003). The pricing equation is complicated by the need to allow for the hierarchy of expectations:
the law of iterated expectations cannot simply be applied in solving forward. Instead substitute out for
increasingly higher orders of average expectations (along very similar lines to the methodology used by
Nimark, 2007). A closed form expression for the stock price, and hence the return, can be derived in
terms of the full hierarchy, which for numerical solutions can again be truncated as outlined in Appendix
B4. Precise details of the methodology are available from the authors.

17
markets the economy collapses to the full information case (Radner, 1979). The second
type are assets which give direct information about aggregate ßows. For example, in
our economy gross proÞts are a linear function of output, so observing these would be
equivalent to observing output and would reveal full information. Since capital is owned
by households, a stock market in this model would not reveal this information. Actual
stock markets provide information about dividend ßows, but in practice dividends and
output are far from being perfectly correlated. We leave further examination of this issue
to future work.
The relative unimportance of aggregate assets has an interesting parallel with the
results of Athanasoulis and Shiller (2000), who Þnd that, in a model of missing markets
where new asset markets are added incrementally according to the magnitude of the
resulting increase in welfare, the last asset to be added is the market portfolio itself, and
that all preceding assets that are added are pure swaps, since these provide the crucial
rule of risk pooling. In our framework, similarly, the assets that would provide the crucial
information would be those that span the distribution of idiosyncratic shocks. Aggregate
markets thus appear to play a marginal role both in terms of welfare and in terms of
information15 .

6 Conclusions

In marked contrast to the conclusions of recent research, we have shown that market
incompleteness matters due to its informational consequences. It remains to be seen
how robust this contrast will be to further modiÞcations.
On the one hand it might be argued that we are overstating the informational im-
plications of incomplete markets. Evidently markets are not so incomplete as in our
model. While aggregate Þnancial markets make very little di!erence, the existence of
insurance and other risk-pooling markets would push these results closer to those under
15
King (1983) introduces a nominal bond into a Lucas-style monetary economy (without capital) and
Þnds it mitigates informational problems. This is consistent with the present paper since the baseline
model already has a common aggregate signal in the return on capital. The point in this section is that
additional aggregate markets have little e!ect in a model with capital.

18
full information.
On the other hand, it is very easy to argue that we may be signiÞcantly understating
the extent of the informational problem. Our model is highly simpliÞed, with only
a single source of idiosyncratic uncertainty; symmetry across households; and a single
aggregate endogenous state variable. Other models have more shocks and more states
(for example Smets and Wouters, 2007, has seven shocks and four states) which will make
the Þltering problem of the household more complex, but may also have more sources of
information. A striking feature of our model is how well households can estimate the
capital stock, in stark contrast to the observed wide variation in estimates within the
academic debate (e.g., Hall, 2001), which, it might be argued, reßect the much greater
complexity of the true inference problem.
While our results are speciÞc to the stochastic growth model, the principle of market-
consistent information is much more general, and should be applied to any model of
incomplete markets.

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21
Figure 1: Response of consumption to a 1% positive innovation to aggregate
productivity
0.4

0.3

0.2

0.1
Consumption
0
0 10 20 30 40 50 60 70 80 90
Full
-0.1 information
consumption
-0.2

-0.3

-0.4

-0.5

x-axis shows periods; y-axis shows percentage deviations from steady state

Figure 2: Response of state estimates to a 1% positive innovation to

aggregate productivity
1.1

0.9

0.7

0.5 Estimate of….

0.3 Aggregate
capital
0.1
Aggregate
-0.1 0 10 20 30 40 50 60 70 80 90 technology

Idiosyncratic
-0.3
capital
-0.5 Idiosyncratic
technology
-0.7

-0.9

-1.1

x-axis shows periods; y-axis shows percentage deviations from steady state

22
Figure 3: Response of aggregate consumption to a 1% positive innovation to
aggregate productivity with a noisy public signal of output
0.3

0.25

0.2

0.15 Full
information
0.1
1% noise

0.05
2% noise

0
3% noise
0 10 20
-0.05

-0.1

x-axis shows periods; y-axis shows percentage deviations from steady state

Table 1: Impact e!ect of aggregate technology shock on aggregate

consumption: sensitivity to properties of idiosyncratic shock
7 ) M7 '
4) $ 10 5 2 1 0
0895 "08352 "08345 "08338 "08273 "08113 08183
089 "08440 "08425 "08410 "08276 08022 08183
0885 "08474 "08448 "08424 "08211 08058 08183
087 "08510 "08438 "08376 "08009 08126 08183
085 "08526 "08365 "08245 080763 08160 08183
Base case shown in bold

On the x-axis is the ratio of the standard deviation of the idiosyncratic shock to that of the
aggregate shock, with the limiting heterogenous and homogenous cases at the left and
right-hand sides. On the y-axis is the persistence of the idiosyncratic shock. The base case is
shown in bold

23