CENTRE FOR DYNAMIC MACROECONOMIC ANALYSIS WORKING PAPER SERIES

* For helpful comments, I thank Herbert Dawid, Martin Ellison, Christian Haefke, Morten Ravn, Michael Reiter, Martin Summer, Peter Tinsley, Stephen Wright and seminar participants at the Bank of England, Birkbeck College London, the CDMA conference in St Andrews and the Institute for Advanced Studies, Vienna. † Department of Economics, University College London, Gower Street, London WC1E 6BT, UK.

Liam.Graham@ucl.ac.uk

CASTLECLIFFE, SCHOOL OF ECONOMICS & FINANCE, UNIVERSITY OF ST ANDREWS, KY16 9AL TEL: +44 (0)1334 462445 FAX: +44 (0)1334 462444 EMAIL: cdma@st-and.ac.uk

www.st-and.ac.uk/cdma

CDMA11/12

Individual rationality, model-consistent

expectations and learning*

Liam Graham†

20 AUGUST 2011

ABSTRACT

To isolate the impact of the assumption of model-consistent expectations, this paper

proposes a baseline case in which households are individually rational, have full information

and learn using forecast rules specified as in the minimum state variable representation of the

economy. Applying this to the benchmark stochastic growth model shows that the economy with

learning converges quickly to an equilibrium very similar to that with model-consistent

expectations. In other words, if households are individually rational, the assumption that they

can also form model-consistent expectations does not seem a strong one. The mechanism by

which learning affects the model is considered in detail and the implications of relaxing the

assumptions of the baseline case are explored.

JEL Classification: D83; C62; E30.

Keywords: adaptive learning; rational expectations; bounded rationality; expectations

formation.

Individual rationality, model-consistent expectations

and learning∗

Liam Graham†

20 August 2011

Abstract

To isolate the impact of the assumption of model-consistent expectations, this

paper proposes a baseline case in which households are individually rational, have

full information and learn using forecast rules specified as in the minimum state

variable representation of the economy. Applying this to the benchmark stochastic

growth model shows that the economy with learning converges quickly to an equi-

librium very similar to that with model-consistent expectations. In other words,

if households are individually rational, the assumption that they can also form

model-consistent expectations does not seem a strong one. The mechanism by

which learning affects the model is considered in detail and the implications of

relaxing the assumptions of the baseline case are explored.

JEL classification: D83; C62; E30.

Keywords: adaptive learning; rational expectations; bounded rationality; ex-

pectations formation.

1 Introduction

The macroeconomic learning literature assumes agents are unable to formmodel-consistent

("rational") expectations. The question is then how to model the formation of expec-

tations and whether a particular model of expectation formation will mean the economy

converges to the same equilibrium as an economy with model-consistent expectations or

whether learning adds new dynamics.

∗For helpful comments, I thank Herbert Dawid, Martin Ellison, Christian Haefke, Morten Ravn,Michael Reiter, Martin Summer, Peter Tinsley, Stephen Wright and seminar participants at the Bank ofEngland, Birkbeck College London, the CDMA conference in St Andrews and the Institute for AdvancedStudies, Vienna.†Department of Economics, University College London, Gower Street, London WC1E 6BT, UK.

Liam.Graham@ucl.ac.uk

1

Models of learning need to make assumptions in three further areas. Firstly, how

rational are individuals conditional on their expectations of the macroeconomy (Adams

andMarcet, 2011, refer to this as internal rationality)? Some work (e.g. Evans et al, 2009,

also see the discussion in Evans et al, 2011) assumes individuals are rational regarding

their individual decisions (i.e. they are fully forward-looking and know their budget

constraints). Others (e.g. Bullard and Mitra, 2002; Carceles-Poveda, and Giannitsarou,

2007; Evans et al, 2011) adopt the "Euler equation learning" approach which assumes

that agents are boundedly rational with regard to their individual decisions only looking

forward a single period and ignoring their first-order conditions and budget constraints

beyond that.

Secondly, what information is at agents’ disposal? Some work (e.g. Evans and

Honkapohja, 2001) assumes that agents learn from aggregate consumption, some from

the states (e.g. Carceles-Poveda, and Giannitsarou, 2007) and some from other variables

(e.g. Evans et al, 2009).

Thirdly, given this information, how are learning rules specified? Some papers assume

learning rules are specified in the same way as the minimum state variable solution of the

economy with model-consistent expectations (e.g. Carceles-Poveda, and Giannitsarou,

2007, such learning rules are also the "saddlepath learning" of Ellison and Pearlman,

2011); some add an intercept to such a rule (e.g. Milani, 2011) and many others make

plausible but apparently arbitrary assumptions about which variables are in the learning

rules.

There is a wide diversity of such assumptions across the literature and it is often

diffi cult to see the extent to which results arise from the central question of the learning

literature, the inability to form model-consistent expectations, or from assumptions in

other areas. To address this issue, this paper proposes a baseline case in which the

only assumption relaxed is that of model-consistent expectations. This case consists of

individuals who are rational conditional on their expectations; who have full information

about the macroeconomy (in the sense that they observe relevant aggregates without

noise) and who have learning rules specified in terms of the minimum state variable

(MSV) representation of the economy under model-consistent expectations.

Such a baseline case isolates the impact of the assumption of model-consistent expec-

tations and is applicable to any learning model. This paper uses it to investigate the

impact of learning in the stochastic growth model. Then the assumptions of the baseline

case are relaxed in two directions. Firstly, the degree of individual rationality is allowed

to vary by adding the household’s forecast horizon as a parameter. Secondly, different

specifications of the learning rule are investigated1.

The main results are as follows:1The third assumption of full information is relaxed in a companion paper, Graham (2011).

2

1. The degree to which households are forward looking has a dramatic effect on the

speed of convergence. If households look forward only one period (which is similar

to "Euler equation learning" ) the model can take many tens of thousands of periods

to converge and may not converge at all. On the other hand, if households have

an infinite horizon, convergence is fast and robust - in the case of ordinary least

squares learning a standard theorem can be applied to show the rate of convergence

is at least√t.

2. Constant gain learning has only very small effects on the business cycle properties

of the model. Further, under most parameterizations the effect of learning is to

mute the response of output to technology shocks.

3. If an intercept is included in the learning rules, the effect of learning is somewhat

stronger, but still small and the response of output is still muted.

The first set of results relates to the speed of convergence. This matters for two

reasons. Firstly, fast convergence gives a justification for the hypothesis of model-

consistent expectations. If convergence is fast, a model-consistent expectations equi-

librium (MCEE) can be interpreted as the outcome of a learning process that has already

converged (Grandmont, 1998) without the need for strong assumptions on households’

cognitive ability. Secondly, studies of the implications of learning for business cycle dy-

namics typically initialize the model at the MCEE to avoid arbitrary transition dynamics

contaminating the results. If convergence is fast, this seems justified. If it is slow, this

is endowing households with exactly what they would have diffi culty learning.

This paper shows that households’forecast horizon is the key variable that determines

the speed of convergence under both ordinary least squares and constant gain learning.

With infinite horizons, convergence is fast; with short horizons, it is very slow and may

not occur at all. Some previous work (Dawid, 2005; Branch et al 2010) has examined

the impact of such horizons for macroeconomic dynamics. This paper shows a further

way in which the horizon matters.

The second set of results relates to business cycle dynamics. With the baseline set

of assumptions, the effects of learning are quite small. A natural metric is the standard

deviation of aggregate consumption, and learning increases this by at most 2% over

its value at the MCEE. While learning makes consumption more volatile, its impact

on labour supply and investment means output is less volatile than at the MCEE. This

stands in contrast to the simple intuition that by increasing the volatility of expectations,

learning increases the volatility of the macroeconomy (it turns out that this intuition can

be recovered only in the case of very short forecast horizons, which correspond to "Euler

equation learning"). Understanding these results requires a careful consideration of

the complicated mechanism by which constant gain learning affects the dynamics of this

3

economy. To elucidate this mechanism, the paper considers a simple univariate example

with exogenous income and fixed capital and labour before turning to the model economy.

One way of interpreting results (1) and (2) is that if households are individually ratio-

nal, the assumption of model-consistent expectations is not all that important. Individual

rationality and constant gain learning is suffi cient for the economy to converge quickly

to an equilibrium which would in practice be indistinguishable from the equilibrium with

model-consistent expectations.

This all applies to the baseline case in which learning rules are specified as in the

MSV solution. While there are many other plausible specifications for learning rules

(for example, including aggregate consumption or more lags) a number of recent papers

(e.g. Milani, 2011) have added an intercept to the learning rule, often interpreted as

representing uncertainty about the steady state. This is shown to strengthen the effect

of learning on the standard deviation of consumption, the maximum effect being an

increase of around 4% in the standard deviation, but output is still less volatile than

under model-consistent expectations.

The paper proceeds as follows. Section 2 presents the model and section 3 discusses

its general properties under learning. The properties of the model under least squares

learning are discussed in section 4 and under constant gain learning in section 5. Section

6 concludes. Detailed derivations are provided in the Appendices.

2 The model

This section presents the standard stochastic growth model. Rather than starting from a

representative household, a large number S of identical households are considered. While

in equilibrium households will be identical, it is important to carefully distinguish between

aggregate and individual quantities when modelling an individual household’s decision.

Upper case letters represent levels; lower case letters their linearized equivalents.

2.1 Households

The problem of household s is to choose paths for consumption (Cst ) and labour supply

(N st ) to maximize expected lifetime utility given by

Est

∞∑i=0

βi

[lnCs

t+i + θ

(1−N s

t+i

)1−η

1− η

](1)

where 1ηis the intertemporal elasticity of labour supply and β the subjective discount

rate. The expectations operator is written as Est since in the general case households will

have model-inconsistent ("non-rational") expectations and expectations will differ across

4

households. The maximization is subject to a budget constraint

RsktK

st +W s

t Nst = Cs

t + Ist (2)

where W st is wage, R

skt the aggregate return to capital, I

st is investment and K

st capital

which evolves according to

Kst+1 = (1− δ)Ks

t + Ist (3)

where δ is the depreciation rate.

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

1

Cst

= βEst

[Rst+1

Cst+1

](4)

where Rst = Rs

kt + 1 − δ is the gross return to a one-period investment in capital, and alabour supply relation

θ (1−N st )−η =

W st

Cst

(5)

2.2 Firms

There are also a large number of identical firms which use aggregate capital and labour

in their production function:

Yt = (Kt)1−α (AtNt)

a (6)

where At is an aggregate productivity shock. The first-order conditions are

Rkt = (1− α)YtKt

,Wt = αYtNt

(7)

2.3 Aggregates

Aggregates are defined explicitly, for example aggregate consumption is

ct =1

S

S∑s=1

cst (8)

2.4 Linearisation

As is standard in the learning literature, this paper will study a linearised version of the

above model2. The household’s Euler equation (4) is

Est∆c

st+1 = Es

t rst+1 (9)

2More details can be found in Appendix A.

5

and labour supply (5) is

nst = ς [wt − cst ] (10)

where ς = 1−NNη

and N is steady state labour. Household capital evolves according to

kst+1 = (1− δ) kst + δist (11)

and the budget constraint (2) is

$cst + (1−$) ist = α (wst + nst) + (1− α) (rskt + kst ) (12)

where $ is the steady state consumption - output ratio.

The first-order conditions for firms are

wt = yt − nt (13)

rkt = yt − kt (14)

and the production function is

yt = (1− α) kt + α (nt + at) (15)

To close the model, specify a process for exogenous technology

at = ρat−1 + εt (16)

where εt is drawn from N (0, σ2).

2.5 Equilibrium

Definition 1. (Equilibrium) A competitive equilibrium for the above economy is a

sequence of plans for

• allocations{cst , n

st , k

st+1

}s=1:S

t=1:∞

• prices {rt, wt}t=1:∞

• aggregate factor inputs {kt, nt}t=1:∞

such that

1. Given prices, the allocations solve the utility maximization problem for all house-

holds.

2. {rt, wt} are the marginal products of aggregate and individual capital and labouri.e. rst = rt, w

st = wt ∀s.

6

3. All markets clear

2.6 Calibration

Values for most of the parameters are chosen following Campbell (1994): δ = 0.025,

α = 0.6, β = 0.99, N = 0.2. The intertemporal elasticity of labour supply 1ηis chosen

to be 5. The aggregate productivity shock is given standard RBC values, ρ = 0.9,

σ = 0.7% per quarter. This is only a benchmark calibration. Sensitivities are given to

all important parameters.

3 Learning and optimal decisions

With model-consistent expectations, the model presented above is fully specified. With

learning, further modelling choices have to be made in three areas. Firstly, what degree

of individual rationality do we assume? Secondly, what variables are in households’

information set? Thirdly, how are households’learning rules specified?

The existing literature makes various choices. The "Euler equation" approach (see

Evans et al, 2011) assumes households are boundedly rational in that they use only their

Euler equation to implement consumption, forecasting their own consumption and the

return just one period ahead. This implies that consumption decisions will not satisfy

expected budget constraints (since the household does not look forward more than one

period), though the budget constraints themselves always hold. The learning rules

are written in terms of aggregate consumption3 and the states. Households have full

information on aggregate consumption (or at least they know all households are identical

so their consumption is the aggregate), the return and the states. This is generalised in

Branch et al (2010) to allow households to look forward an arbitrary number of periods.

In contrast, Carceles-Poveda and Giannitsarou (2007) implement a model in which

households only look one period ahead but forecast capital using a learning rule specified

as in the MSV solution (their equation 45). The process for technology is assumed to be

known. They do not make the distinction between individual and aggregate quantities

which is equivalent to assuming that households know they are identical i.e. the forecast

of aggregate capital is taken to also be a forecast of individual capital.

Eusepi and Preston (2011) take a different approach. They assume households are

rational (having infinite horizons and using both budget constraints and Euler equations).

In terms of information, households observe aggregate states and prices. Learning rules

contain the same variables that appear in the minimum state variable representation of

the economy, with the addition of an intercept. They further assume that the process

3In this sense, "Euler equation learning" has agents learning about their own consumption decisions.Forecasting choice variables seems a somewhat odd way to model bounded rationality.

7

for technology is known (this is necessary for households to be able to detrend by it)

and that while the innovation to technology is observed by households for the purposes

of implementing consumption, it is not used in the learning process (see the discussion

in their section II). A further example is Evans et al (2009). Here households are

individually rational and the learning rule is specified in terms of prices not states.

These four examples illustrate some of the diversity of assumptions to be found across

the learning literature. This paper proposes a baseline case in which households:

1. Are rational conditional on their expectations of the macroeconomy

2. Have an information set consisting of all aggregates

3. Forecast using rules of the same form as the minimum state variable representation

of the economy with model-consistent expectations

Comparing such an economy with one in which households have model-consistent

expectations gives the cleanest answer to the basic question of the learning literature -

how important is the assumption of model-consistent expectations for the properties of

the macroeconomy?

3.1 Optimal consumption

Assume the household looks forward T periods when making its consumption decision

(Dawid, 2005, refers to this as the "planning" horizon). Clearly T =∞ is the standard

infinite horizon case. With T = 1 the structural model is identical to that of Carceles-

Poveda and Giannitsarou (2007) and is closely related to the "Euler equation learning"

of Evans et al (2011)4.

To solve for consumption, substitute for labour from (10) and for capital from (11) in

the budget constraint (12) to obtain

kst = βkst+1 +1

γ1

[γ2cst − γ3w

st − γ5r

st ] (17)

where the constants are defined as part of the derivation in Appendix B.

Iterate this forward T periods, take expectations at time t then rearrange to give

γ2Est

T∑j=0

βjcst+j = [γ1kst + γ3w

st + γ5r

st ] + Es

t

T∑j=1

βj(γ3w

st+j + γ5r

st+j

)− γ1β

T+1Est k

st+T+1

4There are cases in which one-period forecasts can be optimal e.g. risk neutrality.

8

Then the Euler equation (9) can be used to substitute for expected future consumption

in terms of the return to give

cst = γck [γ1kst + γ3w

st + γ5r

st ] + γcwE

st

T∑j=1

βj(γcww

st+j + γcrr

st+j

)(18)

+γcsEst

T∑j=1

rst+j − γ1γckβT+1Es

t kst+T+1 (19)

where the constants are again defined in Appendix B.

The term in square brackets is consumption out of current wealth consisting of the

households’capital in their current factor income (the constants on the prices arise from

substituting out for quantities). The other terms represent consumption out of expected

future income. The presence of the final term is a reminder that the problem is that of

an infinitely lived household with a finite forecast horizon.

3.2 The perceived law of motion

Following the "saddlepath learning" of Ellison and Pearlman (2011), learning rules are

assumed to be specified in terms of the variables in the MSV representation of the economy

with model-consistent expectations, which in this model means the aggregate state vector

is

Xt =[kt at

]′Define matrices Tk and Ta such that kt = TkXt, at = TaXt.

Households are assumed to estimate a first-order VAR in the aggregate state vector

Xt = φstXt−1 + εφt (20)

Since aggregate states do not appear in (19), households also need to estimate the relation

between prices and states

Zst = ϕstXt + εϕt (21)

where Zt =[wst rst

]′.

Then

Estw

st+i = Twϕ

st (φst)

iXt (22)

Est rst+i = Trϕ

st (φst)

iXt (23)

where Tw and Tr are matrices such that wst = TwZst ; r

st = TrZ

st .

A number of papers (e.g. Carceles-Poveda and Giannitsarou, 2007 and Evans et al,

2011) omit this step. This is equivalent to assuming that (a) households know the

9

relations between prices, states and consumption (28) and (29) below (or can estimate

exactly such relations) and (b) households know that they are identical. In this case

there is no need to estimate processes for the prices, and (19) reduces to an expression

in the aggregate states and expectations thereof.

In the infinite horizon case, the final term in (19) can be dropped using the transversal-

ity condition. With finite forecast horizons, households need to forecast this term, their

own future capital. This is odd, and is analogous to households with "Euler equation

learning" needing to forecast their own consumption (see footnote 3). Since households

are only identical in equilibrium, the most consistent way of addressing this would be

to have households have a separate learning rule for their own capital. But to simply

things, and to allow comparison with Carceles-Poveda and Giannitsarou (2007), this pa-

per assumes that for the purposes of forecasting the final term households know that their

capital will always be equal to aggregate capital and so they can use (20) to forecast it.

Then the consumption function (19) can be written

cst = γck (γ1kt + γ3wst + γ5r

st ) + γscXXt (24)

where the expectational terms are captured in

γscX = (γcwTw + γcrTr)ϕstβφ

st

(I − (βφst)

T)

(I − βφst)−1 +

γcsTrϕstφt

(I − (φst)

T)

(I − φst)−1 − γ1γckTk (βφst)

T+1 (25)

In the case of T =∞, this expression reduces to

γcX = (γcwTw + γcrTr)ϕstβφ

st (I − βφst)

−1 (26)

and in the case of T = 1

γcX = (γcwTw + γcrTr)ϕstβφ

st + γcsTrϕ

stφ

st − γ1γckTk (βφst)

2 (27)

3.3 The actual law of motion

The derivations that follow are from the modeler’s perspective. No agent in the economy

has suffi cient knowledge to carry them out (which is another way of saying that they are

unable to form model-consistent expectations).

In equilibrium, all households are identical i.e. for any variable x, xst = xt ∀s. If

markets clear, prices are:

wt = λwkkt + λwaat + λwcct (28)

rt = λrkkt + λraat + λrcct (29)

10

and labour is

nt = ν ((1− α) kt + αat − ct) (30)

Expressions for the coeffi cients are given in Appendix A.3. Note that in the case of fixed

labour supply (η →∞) prices are independent of aggregate consumption, λwc = λrc = 0.

Given households are identical, (24) is also an expression for aggregate consumption

and substituting for current prices in terms of states from (28) and (29) gives

ct =γck

1− γck (γ3λwc + γ5λrc)

([γ1 + γ3λwk + γ5λrk

γ3λwa + γ5λra

]′+ γcX

)Xt (31)

where γcX is defined in (25).

Substituting this in the aggregate capital evolution equation allows the economy to

be written in the form

Xt+1 = T (Φ)Xt +

[0

1

]εt (32)

where εt is the innovation to the process for aggregate technology and

Φ =

[φ

ϕ

](33)

3.4 The model-consistent expectations equilibrium

The model-consistent equilibrium is a fixed point of

Φ = T (Φ) (34)

As in the standard case, there is no closed-form expression for the MCEE so it has to

be calculated numerically5. As would be expected, the model-consistent equilibrium is

independent of the forecast horizon. For the baseline calibration the law of motion for

the states is:

φ∗ =

[0.9635 0.0585

0.0000 0.9000

](35)

and for prices

ϕ∗ =

[0.4550 0.4886

−0.0237 0.0267

](36)

3.5 Learning rules

A general updating rule for φ can be written

5In practice, this is done by using a numerical equation solver (Matlab’s fsolve function) to find azero of T (Φ)− Φ.

11

Rt = Rt−1 + γt(Xt−1X

′t−1 −Rt−1

)(37)

φt = φt−1 + γtR−1t Xt−1

(X ′t −X ′t−1φt−1

)(38)

where γt is the gain. This paper will consider two cases, ordinary least squares learning

(γt = 1t) and constant gain learning (γt = γ). An updating rule of the same form is

specified for ϕ. Stacking the rules as in (33) gives an updating rule for Φ.

3.6 E-stability and learnability

Will the model-consistent expectations equilibria found in section 3.4 be e-stable and

learnable? A standard result (Evans and Honkapohja, 2001) is that the stability of a

system consisting of a PLM, (20), and ALM, (32) and a learning rule (37) and (38) is

related to the stability of an associated ordinary differential equation (ODE)

dΦ

dτ= h (Φ) , where h (Φ) = lim

t→∞E (T (Φ)− Φ) (39)

The economy with ordinary least squares learning (γt = 1t) will converge to Φ only if Φ

is a locally stable fixed point of the associated ODE i.e. the eigenvalues of the Jacobian

of h (Φ) have negative real parts. An analytical expression is only available for these

eigenvalues in the case of T = 1 (see Appendix C.2); in other cases they must be obtained

numerically.

Under constant gain learning (γt = γ > 0), things are more complex but Evans and

Honkapohja (2001, p162) show that if the gain is suffi ciently close to zero the PLM will

converge to a limiting normal distribution around the MCEE.

3.7 Projection

In the baseline case with an infinite horizon, the consumption function (31) is only de-

fined when (I − βφ) is invertible, see (26). Since the term comes from computing the

discounted sum of the expected future path of prices, the invertibility condition is the

same as requiring the sum to be bounded. This is summarised in the following definition

Definition 2. (stable PLM). A given φs is stable if it results in consumption being

bounded. This will be the case if the eigenvalues of φst are less than β−1 > 1 in ab-

solute value.

Theorem 4 of Ljung (1977, p. 557), which forms the basis of many convergence

results in the learning literature employs a "projection facility" constraining estimates to

remain in a region around the MCEE. This has been widely criticized (e.g. Grandmont

12

and Laroque, 1991 and Grandmont, 1998) since it involves endowing households with

knowledge of what they are supposed to be learning. Even though a projection facility

has been shown not to be necessary to proofs of convergence and stability in models

with a unique MCEE (Bray and Savin, 1986) or more generally (Evans and Honkapohja,

1998), it is crucial for any numerical implementation of learning. To see this note that

with a non-zero gain there is always a finite probability that particular sequence of shocks

will lead to a household estimating a PLM that is unstable in the sense of definition 2,

leading forecasts to grow without limit and consumption to be undefined.

The form of the consumption function (31) gives a natural way to define a projection

algorithm which escapes the critiques of Grandmont and Laroque.

Definition 3. (projection facility). After estimating the PLM households check the eigen-

values of φst . If they are greater than q the household discards the estimated φst and chooses

a different one.

If the projection facility is used there are many ways to pick a φst which are do not

involve endowing households with knowledge of the RPE. The simplest way is to use the

value from the previous period6.

In the remainder of the paper, q is taken to be unity which can be interpreted as

endowing households with the knowledge that the macroeconomy is stationary. There

are two justifications for this. Firstly, estimating a VAR of the form (20) is problematic

with non-stationary variables. Secondly, the consumption function is strongly non-linear

for PLMs with eigenvalues greater than unity (recall that as eig (φs) → β−1, cs → +∞)and allowing beliefs to enter this range means arbitrary amounts of volatility can be

generated in the macroeconomy (see the discussion in section 5.6).

Projection is rarely discussed in the context of numerical analysis. Williams (2003)

and Eusepi and Preston (2011) both mention in footnotes that they discard explosive

values though it is not clear if this includes rational bubble paths, though in the latter

paper at least the very small gains used means that such paths will be rare events.

With "Euler equation learning" (Evans et al, 2011), there is no infinite forward sum

in the consumption function so the issue does not arise although Carceles-Poveda and

Giannitsarou (2007, p2673) explicitly exclude non-stationary paths.

4 Ordinary least squares learning

The results on e-stability and learnability discussed in section 3.6 are local and asymp-

totic. To investigate the convergence properties of the model it is necessary to turn to

6From a Bayesian perspective, projection is equivalent to having a truncated prior. When a drawtriggers the projection facility, the response of a Bayesian would be to move the posterior in the directionof the non-stationary solution rather than simply ignoring the information. In practice, the method ofchoosing the “projected”value makes no difference to the properties of the model.

13

simulations. This section takes the case of OLS learning (something of a benchmark in

the learning literature); the next section deals with constant gain learning.

4.1 The speed of convergence

Figure 1 shows the convergence of the model in the two extreme cases of households who

only look forward one period (T = 1) and those with an infinite horizon (T = ∞). A

uninformative prior is chosen setting all the elements of φ0 and ϕ0 to zero. The model

is then simulated many times and the figures show the the mean path of each element of

φ− φ∗ (the difference to the value at the MCEE) along with the range in which 99% of

paths lie.

[FIGURE 1 HERE]

Comparing the two panels of the figure is striking - if households have infinite horizons

(T = ∞), the model converges very quickly, within a 100 periods or the elements of φare very close to their value at the MCEE. In contrast, if households only look forward

one period (T = 1) the economy has not converged within the thousand periods shown

on the diagram and in fact doesn’t converge at all.

To understand this result, take a simple example in which households have learnt a

PLM which implies no persistence for the aggregate states i.e. φ = 02x2. In this case,

given the baseline calibration, with T = 1 the consumption function (31) is approximately

ct ≈ 0.5 (kt + λ2at) (40)

and with T =∞ct ≈ 0.01 (kt + λ2at) (41)

This is a direct consequence of the limited forecast horizon - in the first instance house-

holds spread their total wealth (the term in parenthesis, which since households believe

there is no persistence consists simply their holdings of capital and the output arising

from current technology) over two periods; in the second instance they spread it over

their infinite horizon.

The resulting ALMs are with T = 1

kt+1 = 0.53kt + 0.04at (42)

and with T =∞kt+1 = 1.01kt + 0.07at (43)

14

Recall from (35) that under model-consistent expectations, the law of motion of capital

is

kt+1 = 0.96kt + 0.06at (44)

Comparison of these shows that with T = ∞ the ALM is very close to the MCEE even

with a PLM so far from the MCEE7.

In the next period, the PLM is updated so will move further towards the MCEE in

the case of T = ∞ than in the case of T = 1 and hence convergence is faster. With

OLS learning the gain falls as time passes and with T = 1 the economy gets stuck away

from the MCEE (see section 5.1 below for a similar case with constant gain learning). In

fact, with T = 1 convergence only occurs if the economy is initialized very close to the

MCEE.

To summarise, the speed of convergence is increasing in the forecast horizon because

given a PLM the higher the forecast horizon, the more the resulting ALM resembles the

MCEE so the faster the PLM is updated towards the MCEE.

Plotting the elements of φ provides a useful illustration of the speed of convergence

but doesn’t say much about how the economy along the convergence path compares with

that at the MCEE since in general, different elements of φ will have different impacts

on the equilibrium (and households are also learning ϕ). Figure 2 instead plots impulse

response functions along the convergence path. These are computed by using the same

data as for figure 1 then for each draw running an impulse response function assuming

the law of motion in the economy is fixed at its value at a particular point along the

convergence path.

[FIGURE 2 HERE]

They tell the same story - convergence in terms of the behaviour of the economy is

much quicker for the infinite horizon case - but also show another interesting feature.

The confidence intervals for the T = ∞ case are much wider than with T = 1. This is

because the consumption in the infinite horizon case is much more sensitive to φ than in

the case of T = 1, a simple consequence of the infinite sum in the consumption function.

So a given volatility of φ results in a higher volatility of consumption with T =∞ than

with T = 1. This forms the basis of the formal result used in the following section.

4.2√t convergence

Theorem 3 of Benveniste et al (1990, p110)8 studies a system of the form of (20) and (32)

under OLS learning (γt = 1t). It states that if the derivative of h (Φ) defined in (39) has

7The ALM for capital with T =∞ is actually explosive but recall that this is not a steady state butinstead just a point along the convergence path.

8Also used by Marcet and Sargent (1995) and Ferrero (2007).

15

all eigenvalues with real parts less than −0.5 then

√t (Φt − Φ∗)

D→ N (0, P ) (45)

where the matrix P satisfies the Lyapunov equation

[I/2 + hΦ (Φ∗)]P + P [I/2 + hΦ (Φ∗)]′ + EH (Φ∗, Xt)H (Φ∗, Xt)′ = 0 (46)

As pointed out by Marcet and Sargent (1995), this means that if the conditions are satis-

fied, there is root - t convergence, although the formula for the variance of the estimators

is modified from the classical case. As the eigenvalues become larger, convergence is

slower in the sense that the variance covariance matrix of the limiting distribution P is

larger.

An analytical expression is available for the eigenvalues only in the case of T = 1 (see

Appendix C.2) so in the general case they are calculated numerically. The Jacobian will

have two eigenvalues equal to −1. For T = 1 the other two are −0.074 and −0.042; for

T = ∞, −2.56 and −1.54, so the theorem holds for the latter case and not the former.

Figure 3 plots the largest eigenvalue for a range of forecast horizons. Under the baseline

calibration the theorem holds for T > 12.

[FIGURE 3 HERE]

Eigenvalues were then calculated for around 15,000 calibrations over a wide grid9.

With T = ∞, the largest eigenvalue increases with α, η, ρ and decreases with β, δ.

The only cases where the theorem is not satisfied are with very persistent of aggregate

technology, ρ = 0.99. On the other hand, with T = 1 there were no cases which satisfy

the theorem.

There are a number of ways convergence could be further studied. One would be to

calculate the variance-covariance matrix P of the limiting distribution in (45). Another

would be to follow Marcet and Sargent (1995) who propose a statistic that allows the

speed of convergence to be studied (Ferrero, 2007 is a more recent application). They

define the rate of convergence δ as

tδ ‖φt − φ∗‖D→ F (47)

for some non-degenerate and well-defined distribution F . This has the desirable property

that it captures convergence in a single statistic.

9The ranges were chosen to encompass values commonly used in the literature. The grid isnot particularly fine, but experimentation showed no evidence of any non-linear effects. δ ∈[0.001, 0.01,0.025, 0.10, 0.50] ;α ∈ [0.4, 0.5,0.6, 0.7, 0.8] ; ; β ∈ [0.96, 0.97, 0.98,0.99, 0.999] ; η ∈[0, 1,5, 10,∞] ; ρ ∈ [0.7,0.9, 0.95 , 0.97, 0.99]; σ ∈ [0.01, 0.5,0.7, 1, 10]. The bold figure represents thebaseline calibration.

16

A drawback is that in general different elements of the PLM will have different effects

on the dynamics of the economy, the statistic may not give much useful information about

how close the behaviour of the economy along the convergence path is to that at the

MCEE. An alternative approach would look at the convergence of the impulse response

functions Figure 3 shows how the response of the economy to a positive technology

shock changes along the convergence path, the dotted line is the response at the REE,

the solid line is the mean response and 99% of responses lie within the shaded area. A

similar statistic to (47) could be defined in terms of the impulse-response function, and

this would give a more economically meaningful measure of convergence.

Carrying out such exercises for the model of this paper gives little interesting informa-

tion. The impact of different calibrations on the rate of convergence is modest and the

results of the kind reported in figures 1 and 2 are robust to all the calibrations studied.

The forecast horizon is the dominant determinant of the speed of convergence.

5 Constant gain learning

Constant gain learning is often used to study business cycle dynamics since it captures

the idea that learning is perpetual and allows households to respond to changes in the

structure of the economy. The gain parameter can be chosen in various ways. Milani

(2007, 2011) estimate it along with the other parameters of the model. Eusepi and

Preston (2011) use survey data. Evans and Ramey (2006) allow households to choose

it optimally. This paper will study gain parameters in the range [0.001 0.05] which

encompasses all the values commonly used. A baseline value of 0.01 is chosen.

A simple way to interpret the gain is by noting that the weight on the forecast error

from τ periods ago relative to the weight from the most recent forecast error is given by

(1− γ)s. So a gain of 0.02 (as estimated in Milani, 2007) implies data from around 34

quarters ago is given approximately half the weight of current data. On the other hand,

a gain of 0.002 (the baseline value of Eusepi and Preston, 2011) means households put

half as much weight on data from 84 years ago as they do on current data.

5.1 The speed of convergence

Figure 4 shows how φkk, the autoregressive term on capital in the PLM, converges for

different values of the gain10. Panel A shows the case of T = 1, panel B the case of

T =∞. Again the figures in panel A are drawn 1,000 periods, whereas those in panel Bare over 100 periods.

[FIGURE 4 HERE]

10Other elements of φ are not shown to save space, but tell a similar story.

17

These figures reinforce the message of the previous section. With T =∞ convergence

is fast for all values of the gain. With T = 1, convergence is much slower.

5.2 The distribution of beliefs

Evans and Honkapohja (2001, Theorem 7.8, p165) show that under certain conditions

beliefs converge to a limiting normal distribution with mean at the MCEE and standard

deviation increasing in the gain. In practice, the conditions for the theorem may not be

satisfied and the distributions of beliefs needs to be investigated numerically.

Beliefs in the model economy are characterized by 8 variables defined by (20) and

(21). Figure 5 shows the distribution of the elements of φ (similar graphs can be drawn

for the elements of ϕ but they do not add much to the intuition given here). It is

important to remember that the variables are not independent and in fact the stationary

distribution is an 8-diminesional object. For low gains, the mean of the distribution is the

same as the value of the PLM at the MCEE, and the distribution is symmetric. As the

gain increases, there are three effects. Firstly, the standard deviation of the distribution

increases; secondly, the mean of the distribution falls and thirdly the distribution becomes

more skewed.

[FIGURE 5 HERE]

The first of these is a direct result of the increasing gain and as expected from the

theorem of Evans and Honkapohja (2003). Higher gains mean more weight in the learning

rule on forecast errors, so the PLM becomes more volatile. The second and third are a

consequence of the interaction between increasing gain and projection. The higher the

gain, the higher the standard deviation of beliefs so the more likely they will be unstable

(in the sense of Definition 2) and so be eliminated by the projection algorithm11. This

truncates the distribution to the right, so reducing the mean and making the distribution

more skewed to the left12.

Table 1 shows the first three moments of these distributions and, in addition to con-

firming the observations made in the last paragraph also shows a further feature, that

the mean of the distribution is lower than at the MCEE even for low values of the gain

for which the projection algorithm is not used.

[TABLE 1 HERE]

11It might be thought that the likelihood of the projection algorithm being used would depend onthe volatility of the driving process for technology. However this is not the case since the weightingmatrix R in (20) corrects for this. At the MCEE, R is simply the variance covariance matrix of capitaland technology, so when the technology shock has small standard deviation the inverse of the weightingmatrix is large.12This argument applies to the autoregressive parameters φkk and φaa. It is a priori unclear what

effect projection will have on the cross terms φka and φak or on the elements of ϕ.

18

Understanding the effects of these stationary distributions in the model economy is

complex, so first consider a simple example.

5.3 A simple example

To understand the effect of a stationary distribution of beliefs on the macroeconomy, it

is helpful to consider a simple univariate example13 in which capital and labour are fixed

and income follow an exogenous AR (1) process:

yt = ρyt−1 + εt (48)

Beliefs are parametrized by a scalar φ such that

Etyt+i = φiyt (49)

then the consumption function is

ct =r

1 + r

[(1 + r) bt +

1

1− φt (1 + r)−1yt

](50)

where bt is current wealth and the second term represents expectations about future in-

come. Note the first and second derivatives to φ of the second term are positive capturing

the positive and increasing effect of income persistence on consumption. Although only

the infinite forecast horizon case will be considered here, the second derivative of f is

positive as long as T > 0.

When beliefs are model-consistent, i.e. φt = ρ ∀t consumption will be a random walkand the standard deviation of the first difference of consumption is

σ∗∆c =r

1 + r

1

1− ρ (1 + r)−1σε (51)

Beliefs are updated by a simplified constant gain learning algorithm

φt+1 = φt + γ (yt − φtyt−1) (52)

How does the stationary distribution of φ affect the economy? Firstly, assume that

the distribution has a mean of ρ (the value of beliefs at the MCEE); non-zero standard

deviation and is symmetric. To understand the impact of this distribution on the un-

conditional properties of consumption consider the response of consumption to a positive

innovation to income. Taking ρ = 0.9, figure 6 shows the response in the three cases of

φ0 = φ∗ = ρ; φ0 = 0.95 > ρ and φ0 = 0.85 < ρ.

13Full details are in Appendix D.

19

[FIGURE 6 HERE]

5.3.1 Case 1: φ = ρ

If households’beliefs are correct, then the impact response of consumption will be exactly

that at the MCEE. In the second period, beliefs will be revised upwards. This will mean

consumption in period 2 is higher than it would be in at MCEE since households believe

income will be more persistent than it actually is. In the third period, there are two

effects. Firstly, beliefs will be revised downward towards the MCEE. Secondly, household

wealth will be lower than expected. Both of these tend to reduce consumption. As time

passes, these two effects continue, and at some point consumption will fall below its

value at the MCEE and remain there for the rest of history (as is required to satisfy the

intertemporal budget constraint).

To summarise, learning has no impact effect but causes consumption to rise above

its value at the MCEE for a number of periods after the initial one, then fall below this

value for the rest of time.

Proposition 1. If beliefs are initialized at the MCEE, the impulse response function withlearning implies a higher volatility of consumption growth than without learning

Proof. See Appendix D.1

5.3.2 Case 2: φ > ρ

In this case households believe that income is more persistent that it is at the MCEE so

on impact increase their consumption by more than with correct beliefs. In subsequent

periods there are two effects. Firstly, households wealth will be lower than expected

which will tend to reduce consumption. Secondly, beliefs will be revised, in the second

period upward and in subsequent periods downward back towards the MCEE. In the

second period the second effect dominates so consumption increases further, in subsequent

periods both effects go in the same direction and as time passes, consumption will fall

below its value at the MCEE and stay there for the rest of time. So the overall effect is

higher consumption than at the MCEE for some initial periods, then consumption lower

than at the MCEE for the rest of time.

5.3.3 Case 3: φ < ρ

The intuition for this case is simply the mirror image of that with φ > ρ. However

note the difference in magnitude. Since the derivative of the consumption function is

increasing in φ, the response is much smaller to a lower value of φ than to the higher one

of the previous section.

20

Given these three cases, the unconditional properties of consumption will be the aver-

age of the three cases weighted by the stationary distribution of φ. Since the distribution

is assumed to be symmetric, the larger impact of case 2 will dominate the smaller one of

case 3 and the volatility of consumption will increase.

So the distribution of beliefs will unambiguously increase the volatility of consump-

tion. The higher the gain, the higher will be the standard deviation of beliefs so the

higher will be the standard deviation of consumption. This is the simple intuition that

"learning increases aggregate volatility" though note that the volatility it increases is that

of consumption. As we shall see below, in general labour supply and investment effects

of this mean a reduction in the volatility of output.

There is a further effect. The theorem of Evans and Honkapohja (2001) that states

the mean of the distribution will be at the MCEE only holds for small values of the gain.

In practice, the mean will often be different from the MCEE. Since the distribution of

beliefs causes the mean response of consumption to be different from that at the MCEE,

the response of capital will also be different (if consumption responds by more capital

would be expected to be less persistent) and hence the mean ALM will be different from

the MCEE. So the mean of the distribution will be different from the MCEE, in this

case lower.

How do the properties of consumption change if the mean of the distribution is lower

than at the MCEE (either for the reason given in the previous paragraph or due to the

projection facility, as will be discussed in the next section)? If the mean is lower, draws

of φ from case 3 are more likely than those from case 2, and if the it is suffi ciently low

this will result in the standard deviation of consumption falling below its value at the

MCEE. Similarly, if the distribution is suffi ciently skewed to the left this will result in

the standard deviation of consumption falling.

To summarise, this simple example suggests that the stationary distribution of beliefs

will have the following effects:

1. If it is symmetrical, the non-linearity of consumption to beliefs will mean consump-

tion responds by more on impact and be more volatile. This will imply the mean

of the distribution is slightly lower than at the MCEE.

2. If the mean of the distribution is lower, this will offset the effects in (1) and make

consumption respond by less on impact and be less volatile

3. If the distribution is skewed to the left, this will further offset the effects.

5.4 Inspecting the mechanism

Returning to the model presented in section 2, the statistic used to study the effects of

learning will be the ratio of the standard deviation of a variable in the model with learning

21

to the standard deviation in the model under model-consistent expectations. Table 2

shows how this statistic varies with the learning gain. First note the key features: learning

increases the standard deviation of consumption, labour and investment but reduces that

of output. Overall the effects are small, at most a few percent. The impact on output

is in contrast to the conventional intuition which says learning increases volatility.

[TABLE 2 HERE]

The final column of the table shows that percentage of periods in which the projection

facility was used is increasing in the gain. This is because increasing gain implies in-

creasing volatility of beliefs so a higher likelihood that beliefs will correspond to "rational

bubble" or explosive paths for which the projection facility is invoked.

To understand these statistics, and the impact of learning it is useful to look at the

impulse response functions to a positive innovation to the process for technology in figure

714. These show the difference between the impulse response under learning and the

impulse response at the MCEE. As with all RBC-type models, the dynamics of all the

variables are driven by the response of consumption so start by focussing on this variable.

[FIGURE 7 HERE]

First take the case of a very low gain γ = 0.001. This eliminates the effect of

projection (see the final column of table 1) so the mean of the distribution is very close

to the MCEE (it will be slightly different due to the effect discussed in the previous

section). Then, as discussed in the previous section, because of the non-linearity of the

consumption function to beliefs the impact response of consumption will be higher than

at MCEE. This means labour supply responds by less than at the MCEE (see 30) so

output and investment also respond by less on impact.

In subsequent periods the intuition is similar to that given in the previous section.

Beliefs are updated and then adjust back to their value at the MCEE which tends to

lead to higher consumption than at the MCEE. Secondly, household wealth falls below

its value at the MCEE due to lower investment and this tends to reduce consumption.

As time passes, the first effect becomes weaker and the second effect comes to dominate:

consumption remains above its value at the MCEE for the first ten or so periods; then

the effect of lower wealth causes it to fall below the MCEE value. Investment and labour

supply remain below their value at the MCEE for fifteen or so periods; output is always

below its value at the MCEE, explaining its lower standard deviation.14Calculating impulse responses is complicated in a model with learning for two reasons. Firstly, the

size of the forecast error resulting from a shock (and hence the extent to which beliefs are updated andthe form of the IRF) will depend on the current state of the economy i.e. on Xt−1 in the perceived lawof motion (20). Secondly, the effect of beliefs on the subsequent evolution of the economy is non-linearso the economy can not be understood as sum of the IRFs in each period as is the case when everythingis linear. Impulse responses are therefore obtained by simulating a large amount of data and estimatingthe impulse response function as a high order MA.

22

Now consider the case of a higher gain γ = 0.02. Now the projection algorithm

will come into play (see the final column of table 2) and, as discussed in section 5.1, this

implies that mean beliefs correspond to less persistent expectations than at the MCEE.

So on impact consumption responds by less than at the MCEE. In the next period beliefs

are updated, but note the much larger effects of the updating due to the higher gain. In

subsequent periods, the intuition is the same as with a lower gain with the stronger effect

on labour supply and investment meaning output is further below its value at the MCEE.

5.5 Forecast horizon and dynamics

Table 3 shows the impact of varying the forecast horizon, holding the gain constant at

0.01. The table again shows the standard deviations of the variables relative to the

MCEE. The volatility of consumption is increasing in the forecast horizon, whereas the

volatility of output, labour and investment is falling. Note that for low values of the

forecast horizon we recover the simple intuition that learning unambiguously increases

volatility, but the effects are still small.

[TABLE 3 HERE]

To help understand these results, figure 8 shows the difference between the impact

response of consumption for T = 1 and T = ∞. The response with a forecast horizon

of 1 differs in two important respects: firstly, the impact effect is larger; secondly the

"kink" in the second period caused by learning is much smaller, so much so that con-

sumption remains below its value at the MCEE for the first 10 periods. These are both a

consequence of the observation made in section 4.1 that the shorter the forecast horizon,

the less sensitive is consumption to beliefs. The two effects arise because with T = 1

consumption is less affected by mean beliefs being lower (so the impact effect is larger);

and less affected buy the updating of beliefs in period 2 (so the kink is smaller).

[FIGURE 8 HERE]

In the case of T = 1, the fact that investment is higher than its value at the MCEE

means the capital stock is higher and so output is higher than at the MCEE in all

periods i.e. the response of output is unambiguously amplified and the simple intuition,

that learning increases volatility is recovered. Note this only happens in the special case

of a low forecast horizon.

5.6 The impact of projection

The results in the previous sections assumed the projection facility is implemented to

prevent households learning non-stationary "rational bubble" paths for the states. This

23

is a key part of the mechanism by which learning affects the dynamics of the economy.

Table 4 shows the relative standard deviations if this assumption is relaxed, and instead

the projection facility is set to allow PLMs with eigenvalues in the range(1, β−1

)i.e. non-

stationary "rational bubble" paths can be learned, but explosive paths are excluded.

[TABLE 4 HERE]

The first two rows are the same as those of table 2 since for small values of the gain

projection is not used. Larger values of the gain can result in explosive paths for the

endogenous variables (recall that both Williams, 2003 and Eusepi and Preston, 2011 write

that they discard such explosive solutions) and the final column of the table shows how

often such explosive paths occur. This issue arises because as the eigenvalues of the PLM

approach β−1, consumption asymptotes to infinity. In a suffi ciently long sample of a

model with a suffi ciently high gain, PLMs with such eigenvalues will be learnt resulting

in a response of consumption which is arbitrarily large.

This emphasizes the importance of always simulating learning models for very large

samples. In a short sample, particularly if the gain is in the middle of the range used

in this paper, it is possible that PLMs with eigenvalues close to β−1 will not be drawn,

so the economy appears to be stable. A potential way of dealing with this is to choose

to invoke the projection facility at some point within the range(1, β−1

). However this

allows any level of consumption volatility to be generated, and there seems to be no way

to support a particular choice.

5.7 Other sensitivities

Table 5 shows how the relative standard deviation varies with η, the intertemporal elastic-

ity of labour supply. As η increases labour supply becomes less elastic and the volatility

of consumption increases. Recall that, other things equal, labour supply varies inversely

with consumption and hence if consumption increases, labour supply falls which reduces

current period output causing investment to fall and thus reducing output and consump-

tion in the next period. The less elastic is labour supply, the weaker is this effect so the

volatility of consumption rises with rising η. The more volatile is consumption the less

volatile will be investment and hence the capital stock, so the volatility of output will

also fall.

[TABLE 5 HERE]

Table 6 shows the effect of ρ, the persistence of the technology process. As ρ falls the

volatility of both consumption and output rises. The reason for this is straightforward.

The lower is ρ, the less likely it is that the projection facility will be used, so the closer

is the mean of the distribution of the PLM to its value at the MCEE. This mutes the

24

effect described in section 5.4 so increasing the volatility of the variables relative to the

MCEE.

[TABLE 6 HERE]

5.8 An intercept in the learning rule

The previous sections assumed the learning rules were specified in terms of the variables

in the MSV representation of the economy with model-consistent expectations. Many

other specifications are possible, for example including other aggregates on the right hand

side of (20) such as consumption or returns; or including more lags. Such changes to the

learning rule will change the effects of learning on the economy, but it is not clear how

any particular learning rule can be justified.

A number of recent papers (Milani, 2011, Eusepi and Preston, 2011) include an in-

tercept in the learning rule, interpreted as capturing households’uncertainty about the

steady state. It is straightforward to augment the model of this paper with an intercept

by changing the state vector in the PLM (20) to Xt =[

1 kt at

]′. Table 7 shows the

effect of this change on relative standard deviations. Comparing this table with table 2

shows that including an intercept strengthens the effect of learning. For example, con-

sidering a gain of 0.01, with the intercept consumption is 4% more volatile than at the

MCEE compared to 2% without the intercept; and output is 4% less volatile, compared

to 2% without the intercept. The final table of the column shows the percentage of runs

that are unstable (in which consumption exceeds machine limits). For a gain of 0.02,

around 10% of runs are unstable, for a gain of 0.05 all runs are unstable.

[TABLE 7 HERE]

Why should uncertainty about the intercept translate into high volatility and insta-

bility? To answer this, modify the simple example of section 5.3 to include an intercept.

Consumption is then

ct =r

1 + r

[(1 + r) bt +

1

1− (1 + r)−1φ1 +1

1− φt (1 + r)−1yt

](53)

where the second term picks up the effect of the intercept, a discounted forward sum of

a constant. For the discount factor of the baseline calibration, 11−(1+r)−1

≈ 100 which, if

the persistence of income is 0.9 is around 10 times higher than the coeffi cient on income.

Take a case where the household has estimated a positive value of φ1. This means

consumption will be high and labour supply low and so the wage will be higher than the

household expected. This will mean in the next period φ1 is higher, and so on until

the economy explodes. With small values of the gain, deviations in φ1 are small so this

25

mechanism is dominated by the shocks. With larger gain, the economy becomes unstable.

Note that the instability only if households are very forward looking - in models that

take the "Euler equation learning" approach (for example Milani, 2011, which uses an

intercept) it is not an issue since the amplification of the effect of learning a non-zero

intercept is much smaller.

6 Discussion

This paper described a baseline case which allows the impact of the assumption of model-

consistent expectations to be studied in isolation. If households are individually rational,

the assumption of model-consistent expectations does not seem important in the sense

that if households are endowed with simple learning rules, the economy with learning

is very similar to that with model-consistent expectations. The response of output

is actually muted under learning, which shows that the simple intuition that learning

increases volatility does not hold in this benchmark model. The mechanism by which

learning affects the economy is in fact quite complex.

The degree of individual rationality, captured by parametrizing households’forecast

horizons, has significant effects on the properties of the model, in particular it is the

key parameter which affects the speed of convergence. Also only in the case of very

short forecast horizons (similar to "Euler equation learning") can learning increase the

volatility of all variables, in keeping with simple intuition.

One way of reading the results of this paper is as support for the assumption of

model-consistent expectations. However there are a number of implicit assumptions in

the paper which mean that this reading should be taken with a pinch of salt.

The first is that if households are endowed with suffi cient cognitive ability estimate

VARs and solve infinite horizon problems, it seems strange to assume that they do not

realize they are identical and that solving their consumption problem is the same as

solving the aggregate economy. However a previous version of this paper studied a model

with heterogeneity among households (created by a household specific productivity shock)

and while this greatly complicated the model it did not have any dramatic effect on the

results (though it did mitigate the instability problem with an intercept). So perhaps

the assumption that households do not know they are identical is not as important as it

might initially seem.

The second is that the results are derived under the assumption of full information.

A large literature (see Hellwig, 2006 for a review) argues that this is a very strong as-

sumption and Graham and Wright (2010) solve a model similar to this paper under

incomplete information and model-consistent expectations and find that imperfect infor-

mation has a dramatic effect on the properties of the model. Graham (2011) extends

this to a model with learning but finds that while information matters, the assumption

26

of model-consistent expectations is not important.

Thirdly, the assumption of linearity means that the household’s problem is a relatively

simple one. Embedding learning rules into a non-linear model (of the style of Krusell

and Smith, 1997) seems an interesting avenue for future research.

References

Adam, K., Marcet, A., 2011. Internal Rationality, Imperfect Market Knowledge and Asset

Prices. Journal of Economic Theory, 146, pp. 1224-1252.

Benveniste, A., Metivier, M., Priouret, P., 1990. Adaptive Algorithms and Stochastic

Approximations. Springer, Berlin.

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28

Figure 1A:Convergence under OLS learning with T = 1 over 1,000 periods

φkk φkaConfidence intervals for phi(k,k)

Periods

Dev

iatio

n fro

m M

CE

100 200 300 400 500 600 700 800 900 1000­1.2

­1

­0.8

­0.6

­0.4

­0.2

0

0.2Confidence intervals for phi(k,a)

Periods

Dev

iatio

n fro

m M

CE

100 200 300 400 500 600 700 800 900 1000­0.07

­0.06

­0.05

­0.04

­0.03

­0.02

­0.01

0

0.01

0.02

0.03

φak φaaConfidence intervals for phi(a,k)

Periods

Dev

iatio

n fro

m M

CE

100 200 300 400 500 600 700 800 900 1000­4

­3

­2

­1

0

1

2

3Confidence intervals for phi(a,a)

Periods

Dev

iatio

n fro

m M

CE

100 200 300 400 500 600 700 800 900 1000­2

­1.5

­1

­0.5

0

0.5

x-axis is time; y-axis is the deviation of the element of the PLM from its value at the MCEE,

95% of responses lie within the shaded areas. Graphs taken from 25,000 repetitions.

29

Figure 1B:Convergence under OLS learning with T =∞ over 100 periods

φkk φkaConfidence intervals for phi(k,k)

Periods

Dev

iatio

n fro

m M

CE

10 20 30 40 50 60 70 80 90 100­1.2

­1

­0.8

­0.6

­0.4

­0.2

0

0.2Confidence intervals for phi(k,a)

Periods

Dev

iatio

n fro

m M

CE

10 20 30 40 50 60 70 80 90 100­0.1

­0.05

0

0.05

0.1

0.15

0.2

0.25

φak φaaConfidence intervals for phi(a,k)

Periods

Dev

iatio

n fro

m M

CE

10 20 30 40 50 60 70 80 90 100­4

­3

­2

­1

0

1

2

3Confidence intervals for phi(a,a)

Periods

Dev

iatio

n fro

m M

CE

10 20 30 40 50 60 70 80 90 100­2

­1.5

­1

­0.5

0

0.5

x-axis is time; y-axis is the deviation of the element of the PLM from its value at the MCEE,

95% of responses lie within the shaded areas. Graphs taken from 25,000 repetitions.

30

Figure 2A:Convergence under OLS learning with T = 1 , impulse responses

t = 20 t = 50Confidence intervals for IRF at time 20

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7Confidence intervals for IRF at time 100

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t = 50 t = 500Confidence intervals for IRF at time 200

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7Confidence intervals for IRF at time 500

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t = 700 t = 1000Confidence intervals for IRF at time 700

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7Confidence intervals for IRF at time 1000

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x-axis shows number of from impulse; y-axis the deviation of the consumption from its steady

state. The solid line is the impulse response at the MCEE, the dotted line the mean response

under learning. 99% of the responses under learning lie within the shaded areas. Graphs taken

from 25,000 repetitions.

31

Figure 2B:Convergence under OLS learning with T =∞ , impulse responses

t = 10 t = 20Confidence intervals for IRF at time 10

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Confidence intervals for IRF at time 20

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t = 50 t = 100Confidence intervals for IRF at time 50

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7Confidence intervals for IRF at time 100

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t = 200 t = 500Confidence intervals for IRF at time 200

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Confidence intervals for IRF at time 500

Periods

Dev

iatio

n fro

m s

tead

y st

ate

5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x-axis shows number of from impulse; y-axis the deviation of the consumption from its steady

state. The solid line is the impulse response at the MCEE, the dotted line the mean response

under learning. 99% of the responses under learning lie within the shaded areas. Graphs taken

from 25,000 repetitions.

32

Figure 3: The largest eigenvalue for different forecast horizons

0 100 200 300 400 500 600­3

­2.5

­2

­1.5

­1

­0.5

0

Forecast horizon, T

Max

imum

 eig

enva

lue

33

Figure 4A:Convergence of φkk under constant gain learning with T = 1 over1,000 periods for different values of the gain

γ = 0.001 γ = 0.002Gain 0.001: Confidence intervals  for phi(k ,k )

Periods

Dev

iatio

n fro

m M

CE

100 200 300 400 500 600 700 800 900­1

­0.8

­0.6

­0.4

­0.2

0

0.2

0.4Gain 0.002: Confidence intervals  for phi(k ,k )

Periods

Dev

iatio

n fro

m M

CE

100 200 300 400 500 600 700 800 900­1

­0.8

­0.6

­0.4

­0.2

0

0.2

0.4

γ = 0.01 γ = 0.02Gain 0.01: Confidence intervals  for phi(k ,k )

Periods

Dev

iatio

n fro

m M

CE

100 200 300 400 500 600 700 800 900­1

­0.8

­0.6

­0.4

­0.2

0

0.2

0.4Gain 0.02: Confidence intervals  for phi(k ,k )

Periods

Dev

iatio

n fro

m M

CE

100 200 300 400 500 600 700 800 900­1

­0.8

­0.6

­0.4

­0.2

0

0.2

0.4

x-axis is time; y-axis is the deviation of the element of the PLM from its value at the MCEE,

95% of responses lie within the shaded areas. Graphs taken from 25,000 repetitions.

34

Figure 4B:Convergence of φkk under constant gain learning with T =∞ over100 periods for different values of the gain

γ = 0.001 γ = 0.002Gain 0.001: Confidence intervals  for phi(k ,k )

Periods

Dev

iatio

n fro

m M

CE

10 20 30 40 50 60 70 80 90­1.2

­1

­0.8

­0.6

­0.4

­0.2

0

0.2Gain 0.002: Confidence intervals  for phi(k ,k )

Periods

Dev

iatio

n fro

m M

CE

10 20 30 40 50 60 70 80 90­1.2

­1

­0.8

­0.6

­0.4

­0.2

0

0.2

γ = 0.01 γ = 0.02Gain 0.01: Confidence intervals  for phi(k ,k )

Periods

Dev

iatio

n fro

m M

CE

10 20 30 40 50 60 70 80 90­1.2

­1

­0.8

­0.6

­0.4

­0.2

0

0.2Gain 0.02: Confidence intervals  for phi(k ,k )

Periods

Dev

iatio

n fro

m M

CE

10 20 30 40 50 60 70 80 90­1.2

­1

­0.8

­0.6

­0.4

­0.2

0

0.2

x-axis is time; y-axis is the deviation of the element of the PLM from its value at the MCEE,

95% of responses lie within the shaded areas. Graphs taken from 25,000 repetitions.

35

Figure 5: The stationary distribution of φ

(best viewed in colour; if viewed in monochrome, note that as the gain rises, the mean

of the distributions fall)

φkk φka

0.95 0.955 0.96 0.965 0.97 0.9750

2000

4000

6000

8000

10000

120000.020.010.0050.0020.0010.0001

0.05 0.055 0.06 0.0650

2000

4000

6000

8000

10000

12000

14000

16000

180000.020.010.0050.0020.0010.0001

φak φaa

­0.2 ­0.15 ­0.1 ­0.05 0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5x 10 4

0.020.010.0050.0020.0010.0001

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 10

2000

4000

6000

8000

10000

12000

14000

160000.020.010.0050.0020.0010.0001

36

Figure 6: A simple example: impulse responses of consumption withdifferent beliefs

0 5 10 15 20 25 30

0.04

0.045

0.05

0.055

0.06

0.065

0.07phi0=phi*; no learning

phi0<phi*

phi0=phi*

phi0>phi*

37

Figure 7: Changing gain: difference between impulse responses withlearning and those at the MCEE

consumption output

0 10 20 30 40 50­0.02

­0.01

0

0.01

0.02

0.03

0.04

0.05

0.060.0010.02

0 10 20 30 40 50­0.025

­0.02

­0.015

­0.01

­0.005

0

0.005

0.010.0010.02

labour investment

0 10 20 30 40 50­0.035

­0.03

­0.025

­0.02

­0.015

­0.01

­0.005

0

0.005

0.010.0010.02

0 10 20 30 40 50­0.25

­0.2

­0.15

­0.1

­0.05

0

0.05

0.10.0010.02

38

Figure 8: Changing forecast horizon: difference between impulse responseswith learning and those at the MCEE

consumption output

0 10 20 30 40 50­0.02

­0.015

­0.01

­0.005

0

0.005

0.01

0.015

0.02

0.025

0.031inf

0 10 20 30 40 50­0.015

­0.01

­0.005

0

0.005

0.011inf

labour investment

0 10 20 30 40 50­0.02

­0.015

­0.01

­0.005

0

0.005

0.01

0.0151inf

0 10 20 30 40 50­0.1

­0.08

­0.06

­0.04

­0.02

0

0.02

0.04

0.06

0.081inf

39

Table 1: Properties of the stationary distribution of the economy withconstant gain learning

φkk (φ∗kk = 0.964) φka (φ

∗ka = 0.058)

Gain Mean SD Skewness

0.0001 0.964 0.000 0.000

0.001 0.963 0.001 0.000

0.002 0.963 0.002 0.007

0.005 0.962 0.003 0.026

0.01 0.962 0.005 0.093

0.02 0.962 0.005 0.317

Gain Mean SD Skewness

0.0001 0.059 0.000 0.000

0.001 0.059 0.001 0.000

0.002 0.059 0.001 −0.157

0.005 0.059 0.002 −0.735

0.01 0.058 0.003 −1.049

0.02 0.058 0.004 −0.944

φak (φ∗ak = 0.000) φaa (φ

∗aa = 0.9000)

Gain Mean SD Skewness

0.0001 0.000 0.004 0.000

0.001 −0.001 0.014 0.000

0.002 −0.002 0.020 −0.012

0.005 −0.004 0.033 −0.077

0.01 −0.007 0.050 −0.106

0.02 −0.014 0.077 −0.159

Gain Mean SD Skewness

0.0001 0.900 0.004 0.000

0.001 0.899 0.011 −0.210

0.002 0.898 0.016 −0.283

0.005 0.897 0.025 −0.351

0.01 0.893 0.035 −0.432

0.02 0.886 0.054 −0.558

40

Table 2: Changing gain: ratio of moments of key variables in the economywith learning to their value at the MCEE

Gain c y n x Proj

0.001 1.002 0.999 1.003 1.001 0.00%

0.002 1.003 0.997 1.004 1.000 0.00%

0.005 1.008 0.993 1.005 0.997 0.00%

0.01 1.017 0.984 1.007 0.993 0.14%

0.02 1.030 0.977 1.019 0.990 1.05%

0.05 1.085 0.975 1.029 0.989 4.12%

Table 3: Changing forecast horizon: ratio of moments of consumption in theeconomy with learning to their value at the MCEE

T c y n x Proj

1 1.001 1.002 1.009 1.006 0.33%

2 1.001 1.001 1.007 1.005 0.33%

5 1.001 0.999 1.002 1.000 0.31%

10 1.002 0.996 0.997 0.996 0.29%

50 1.007 0.990 0.998 0.995 0.21%

∞ 1.017 0.984 1.007 0.993 0.14%

Table 4: Changing gain with rational bubble paths: ratio of moments of keyvariables in the economy with learning to their value at the MCEE

Gain c y n x Errors

0.001 1.002 0.999 1.003 1.001 0%

0.002 1.003 0.997 1.004 1.000 0%

0.005 1.011 0.990 1.011 1.000 0%

0.01 1.078 0.983 1.120 1.045 9%

0.02 2.023 1.031 2.685 1.985 31%

0.05 n/a n/a n/a n/a 100%

41

Table 4: Changing elasticity of labour supply, η : ratio of moments of keyvariables in the economy with learning to their value at the MCEE

η c y n x

0 1.000 0.981 1.024 1.005

1 1.006 0.981 1.016 0.997

2 1.010 0.982 1.012 0.994

5 1.017 0.984 1.007 0.993

10 1.021 0.986 1.005 0.990

∞ 1.028 0.991 1.003 0.989

Table 5: Changing persistence of technology, ρ : ratio of moments of keyvariables in the economy with learning to their value at the MCEE

ρ c y n x Proj

0.5 1.024 0.998 1.005 1.002 0.03%

0.6 1.024 0.996 1.004 1.000 0.04%

0.7 1.023 0.994 1.003 0.998 0.05%

0.8 1.022 0.992 1.003 0.996 0.06%

0.9 1.017 0.984 1.007 0.993 0.14%

0.95 1.010 0.977 1.034 0.991 0.37%

Table 6: Changing gain with intercept in learning rule: ratio of moments ofkey variables in the economy with learning to their value at the MCEE

Gain c y n x Errors

0.001 1.009 0.995 1.009 1.000 0%

0.002 1.019 0.983 1.021 0.998 0%

0.005 1.025 0.974 1.028 0.995 0%

0.01 1.041 0.960 1.031 0.986 4%

0.02 1.063 0.942 1.046 0.979 9%

0.05 n/a n/a n/a n/a 100%

42

A Linearisation

A.1 The steady state

From the Euler equation (4):

R = β−1 (A.1)

From (7)

rk = (1− α)y

k(A.2)

so

(1− α)Y

K= β−1 + (1− δ) (A.3)

K

Y=

1− αr − 1 + δ

(A.4)

Then from the capital evolution equation (3)

I = δK (A.5)

Then from the resource constraint

C

Y= 1− I

Y(A.6)

A.2 Linearisation

The Euler equation (4):

Est∆c

st+1 = Es

t rst+1 (A.7)

Definition of return to capital

rt = κrkt (A.8)

κ =R− 1 + δ

R(A.9)

FOC for labour (5):

nst = ς (wst − cst) (A.10)

where

ς =1− nNη

(A.11)

The household’s budget constraint (2):

Cs

Y scst +

(1− Cs

Y s

)ist = α (wst + nst) + (1− α) (rskt + kst ) (A.12)

43

and the capital evolution equation (3)

kst+1 = (1− δ) kst + δist (A.13)

The firm FOCs (7)

wt = yt − nt (A.14)

rt = yt − kt (A.15)

The production function (6)

yt = αat + αnt + (1− α) kt (A.16)

and the aggregate resource constraint is

yt =C

Yct +

(1− C

Y

)it (A.17)

A.3 Market clearing prices

For the next two sections, the derivation is from the modeler’s perspective i.e. with the

knowledge that in equilibrium all households are identical.

To get an expression for the wage, use (A.10) and (A.14) to write

wt =αat + (1− α) kt + (1− α) ςct

1 + (1− α) ς(A.18)

rkt = αat − αkt + αnt (A.19)

= (αat − αkt + να (αat + (1− α) kt − ct)) (A.20)

Then using (A.8)

rt = κ [(ν (1− α)− 1) kt + (1 + να) at − νct] (A.21)

Write these as

wt = λwkkt + λwaat + λwcct (A.22)

rt = λrkkt + λraat + λrcct (A.23)

44

where

λwk = ν (1− α) (A.24)

λwa = να (A.25)

λwc = ν (1− α) ς (A.26)

λrk = λ3 (ν (1− α)− 1) (A.27)

λra = λ3 (1 + να) (A.28)

λrc = −λ3ν (A.29)

Then (A.22) can be combined with (A.10) to give a relation between labour supply and

consumption

nt = ς

(αat + (1− α) kt + (1− α) ςct

1 + (1− α) ς− cst

)(A.30)

= ς

(αat + (1− α) kt − ct

1 + (1− α) ς

)(A.31)

A.4 A useful representation

This follows Campbell (1994) to write the capital evolution in terms of states and con-

sumption. Combining (A.13) and (A.17) gives:

kt+1 = (1− δ) kt +i

k

(yiyt −

c

xct

)(A.32)

= λ1kt + λ2 [at + nt] + λ4ct (A.33)

where

λ1 = (1− δ) +y

k(1− α) =

1

β(A.34)

λ2 =y

kα (A.35)

λ4 = − ck

(A.36)

Then substituting (A.16) into (A.15) gives

rkt = αat − αkt + αnt (A.37)

and using (A.8)

rt = κ (at − kt + nt)

45

Next substitute in (A.10) to get

kt+1 = λ1kt + λ2 [at + nt] + λ4ct (A.38)

= λ1kt + λ2 [at + ς (wt − ct)] + λ4ct (A.39)

and finally use the (A.22) to give

kt+1 =1

γ

(λ1kt + λ2

[at + ς

(αat + (1− α) ςct + (1− α) kt

1 + (1− α) ς− ct

)]+ λ4ct

)(A.40)

= λ1kt + λ2at + λ4ct (A.41)

where

λ1 = λ1 + λ2 (1− α) ν (A.42)

λ2 = λ2 (1 + αν) (A.43)

λ4 = λ4 + λ2ς [1− (1− α) ν] = λ4 + λ2ν = 1− λ1 − λ2 (A.44)

ν =ς

1 + (1− α) ς(A.45)

B Optimal consumption given prices

Substituting for investment from (A.13) and labour from (A.10) into (A.12) gives

c

ycst +

k

y

(kst+1 − (1− δ) kst

)= α (wst + ς (wst − cst)) + (1− α) (rkt + kst ) (B.1)

rearranging, and substituting for the return from (A.8)

kst =1

γ1

k

ykst+1 +

1

γ1

[γ2cst − γ3w

st − (1− α) rt] (B.2)

where

γ1 = (1− δ) ky

+ (1− α) (B.3)

γ2 =c

y+ ας (B.4)

γ3 = α (1 + ς) (B.5)

γ5 =(1− α)

κ(B.6)

and note from (A.4)1

γ1

k

y=

1

(1− δ) + (1− α) yk

= β (B.7)

46

so

kst = βkst+1 +1

γ1

[γ2cst − γ3w

st − γ5r

st ] (B.8)

Then solving T periods forward gives

γ1kst = γ2

T∑j=0

βjcst+j − γ3

T∑j=0

βjwst+j − γ5

T∑j=0

βjrst+j + γ1βT+1kst+T+1 (B.9)

Rearranging, then taking expectations gives

γ2Est

T∑j=0

βjcst+j = γ1kst + γ3w

st + γ5r

st + Es

t

T∑j=1

βj(γ3w

st+j + γ5r

st+j

)− γ1β

T+1Est k

st+T+1

(B.10)

From the Euler equation (A.7)

Est cst+j = cst + Es

t

j∑i=1

rst+i (B.11)

so

Est

T∑j=0

βjcst+j =1− βT+1

1− β cst + Est

T∑i=0

βji∑

j=1

rst+i (B.12)

The second term of this is

i∑j=1

rst+i = βrt+1 + β2 (rt+1 + rt+2) + β3 (rt+1 + rt+2 + rt+3) + ...

= β(rt+1 + βrt+2 + ...+ ..βT rt+T .) + β2(rt+1 + βrt+2 + ...βT−2rt+T−1

)+ ...+ ...βT rt+1

= β(1 + β + ...+ βT−1

)rt+1 + β2

(1 + β + ...+ βT−2

)rt+2 + ...

=β(1− βT

)1− β rt+1 +

β2(1− βT−1

)1− β rt+2 + ...

=β

1− β (rt+1 + βrt+2 + ...)− βT+1

1− β (rt+1 + rt+2 + rt+3 + ....)

=1

1− β

T∑j=1

βjrst+j −βT+1

1− β

T∑j=1

rst+j (B.13)

then (B.12) becomes

γ2Est

T∑j=0

βjcst+j = γ2

1− βT+1

1− β cst + γ2

1

1− β Est

T∑j=1

βjrst+j − γ2

βT+1

1− β Est

T∑j=1

rst+j (B.14)

47

Substituting this into (B.10):

γ2

1− βT+1

1− β cst + γ2

1

1− β Est

T∑j=1

βjrst+j − γ2

βT+1

1− β Est

T∑j=1

rst+j =

γ1kst + γ3w

st + γ5r

st + Es

t

T∑j=1

βj(γ3w

st+j + γ5r

st+j

)− γ1β

T+1Est k

st+T+1

and rearranging gives

cst =1− β

γ2

(1− βT+1

)

γ1kst + γ3w

st + γ5r

st + γ3E

st

T∑j=1

βjwst+j+(γ5 − γ2

11−β

)Est

T∑j=1

βjrst+j + γ2βT+1

1−β Est

T∑j=1

rst+j − γ1βT+1Es

t kst+T+1

or

cst = γck [γ1kst + γ3w

st + γ5r

st ] + γcwE

st

T∑j=1

βjwst+j + γcrEst

T∑j=1

βjrst+j (B.15)

+γcsEst

T∑j=1

rst+j − γ1γckβT+1Es

t kst+T+1 (B.16)

where

γck =1− β

γ2

(1− βT+1

) (B.17)

γcw = γ3γck (B.18)

γcr = γck

(γ5 − γ2

1

1− β

)(B.19)

γcs = γckγ2

βT+1

1− β (B.20)

C Learning

The state vector is vector is

Xt =[kt at

]′and the PLM for states

Xt = φstXt−1 + εφt (C.1)

48

and prices

Zst = ϕstXt + εϕt (C.2)

where Zt =[wst rst

]′.

Then

Estw

st+i = Twϕ

st (φst)

iXt (C.3)

Est rst+i = Trϕ

st (φst)

iXt (C.4)

Tw and Tr are matrices such that wst = TwZst ; r

st = TrZ

st .

Then the expectational terms in (B.15) are

Et

T∑j=1

βjwt+j = Twϕt

T∑j=1

βiφitXt (C.5)

= Twϕtβφt(I − βTφTt

)(I − βφt)

−1Xt (C.6)

similarly

Et

T∑j=1

βjrt+j = Trϕtβφt(I − βTφTt

)(I − βφt)

−1Xt (C.7)

so can write consumption function (B.15) as

cst = γckγ1kst + γck (γ3wt + γ5rt) + (γcwTw + γcrTr)ϕtβφt

(I − βTφTt

)(I − βφt)

−1Xt(C.8)

+γcsTrϕtφ(I − φTt

)(I − φt)

−1Xt − γ1γckβT+1Tkφ

T+1t Xt (C.9)

= γck (γ1kt + γ3wt + γ5rt) + γcXXt (C.10)

where

γcX = (γcwTw + γcrTr)ϕtβφt(I − βTφT

)(I − βφ)−1+γcsTrϕtφt

(I − φTt

)(I − φt)

−1−γ1γckTk (βφt)T+1

C.1 Special cases

T =∞

cst =1− βγ2

(γ1kt + γ3wt + γ5rt) + γcXXt (C.11)

γcX = (γcwTw + γcrTr)ϕtβφ (I − βφ)−1 (C.12)

49

γcw =γ3 (1− β)

γ2

(C.13)

γcr =γ5 (1− β)

γ2

− 1 (C.14)

T = 1

cst =1

γ2 (1 + β)(γ1kt + γ3wt + γ5rt) + γcXXt (C.15)

γcX = (γcwTw + γcrTr)ϕtβφt +β2

1− β2Trϕtφt −γ1

γ2 ((1 + β))Tk (βφt)

2 (C.16)

γck =1

γ2 (1 + β)(C.17)

γcw =γ3

γ2 (1 + β)(C.18)

γcr =1

γ2 (1 + β)

(γ5 − γ2

1

1− β

)(C.19)

γcs =β2

1− β2 (C.20)

C.2 Comparison with Carceles-Poveda and Giannitsarou (2007)

Rearranging (A.41) gives

λ4ct = kt+1 − λ1kt − λ2at (C.21)

λ4Etct+1 = Etkt+2 − λ1Etkt+1 − λ2Etat+1 (C.22)

and using (A.23)

Etrt+1 = θrkEtkt+1 + θraEtat+1 + θrcEtct+1 (C.23)

Substituting these into the Euler equation (A.7) gives

kt+1 = µ1kt + µ2at + µ3Etat+1 + µ4Etkt+2 (C.24)

50

where

µ1 =λ1

1 + λ1 (1− θrc) + λ4θrk(C.25)

µ2 =λ2

1 + λ1 (1− θrc) + λ4θrk(C.26)

µ3 =λ2 (1− θrc) + λ4θra

1 + λ1 (1− θrc) + λ4θrk(C.27)

µ4 =1− θrc

1 + λ1 (1− θrc) + λ4θrk(C.28)

(this is equivalent to equation (20) in Carceles-Poveda and Giannitsarou, 2007).

If the PLM is

kt = φkkkt−1 + φkaat−1 (C.29)

at = φaktkt−1 + φaatat−1 (C.30)

and

φ =

[φkk φka

φakt φaa

](C.31)

So

Etkt+1 = φkktkt + φkatat (C.32)

Etkt+2 = φkktEtkt+1 + φkatEtat+1 (C.33)

= φkkt (φkktkt + φkatat) + φka (φaktkt + φaaat) (C.34)

=(φ2kk + φkaφak

)kt + (φkkφka + φkaφaa) at (C.35)

Substituting this in the ALM (C.24) gives

kt+1 = µ1

[(φ2kk + φkaφak

)kt + (φkkφka + φkaφaa) at

]+ µ2kt + µ3 [φaktkt + φaaat] + µ4at

=[µ1

(φ2kk + φkaφak

)+ µ2 + µ3φak

]kt + [µ1φka (φkk + φaa) + µ3φaa + µ4]µat

T (φ) =

[µ4

(φ2kk + φkaφak

)+ µ1 + µ3φak µ4φka (φkk + φaa) + µ3φaa + µ2

0 ρ

](C.36)

T (φ)−φ =

[µ4

(φ2kk + φkaφak

)+ µ1 + µ3φak − φkk µ4φka (φkk + φaa) + µ3φaa + µ2 − φka−φak ρ− φaa

]

51

J =

2µ4φkk − 1 µ4φak µ4φka + µ3 0

µ4φka µ4 (φkk + φaa)− 1 0 µ4φka + µ3

0 0 −1 0

0 0 0 −1

(C.37)

µ4φkt + ρµ4 − 1 (C.38)

2µ4φkt − 1 (C.39)

While it would take many pages of algebra to prove that the consumption function

implies by (C.24) is equivalent to (27), it is easy to check numerically that the eigenvalues

(and indeed the coeffi cients in (C.24)) are identical to those for the model of this paper.

C.3 Market clearing

The modeler, using the knowledge that in equilibrium all households are identical, can

sum (C.10) over households to give an expression for aggregate consumption

ct = γck (γ1kt + γ3wt + γ5rt) + γcXXt (C.40)

Substituting for market clearing prices from (A.22) and (A.23) gives

ct = γck (γ1kt + γ3 (λwkkt + λwaat + λwcct) + γ5 (λrkkt + λraat + λrcct)) + γcXXt (C.41)

or

ct = γXXt (C.42)

where

γX =1

1− γck (γ3λwc + γ5λrc)[γck (γ1 + γ3λwk + γ5λrk)Tk + γck (γ3λwa + γ5λra)Ta + γcX ]

C.4 The actual law of motion

Substitute(C.42) into the capital evolution equation (A.41) to give

kt+1 =([

λ1 λ2

]+ λ4γX

)Xt (C.43)

stack this in top of the process for aggregate technology ( to give

Xt+1 =

[ λ1 λ2

]+ λ4γX

0 ρ

Xt +

[0

1

]εt (C.44)

52

D Simple example

Take a representative household maximizing expected discounted lifetime utility

max{ct+i}∞i=0

Et

∞∑i=0

1

(1 + r)iln ct+i (D.1)

subject to a budget constraint

bt+1 = (1 + r) bt + yt − ct (D.2)

where yt is an exogenous process

yt = ρyt−1 + εt (D.3)

and the innovation is drawn from N (0, σ).

Let the household have belief φt about the persistence of the income process, so

Etyt+i = φit yt (D.4)

Note in the case of model-consistent expectations φt = ρ

The first-order condition for consumption is

ct = Etct+1 (D.5)

and, using the transversality condition limt→∞

1(1+r)t

bt = 0, optimal consumption is

ct =r

1 + r

[(1 + r) bt +

1

1− φt (1 + r)−1yt

](D.6)

Let beliefs be updated according to a simple constant gain algorithm

φt+1 = φt + γ (yt − φtyt−1) (D.7)

Note that at the MCEE consumption is a random walk (taking initial wealth to be zero)

∆c∗t =r

1 + r

[1

1− ρ (1 + r)−1

]εt (D.8)

and

σ∗∆c =r

1 + r

[1

1− ρ (1 + r)−1

]σ (D.9)

53

D.1 Proof of proposition 1

Starting from φ0 = ρ, in response to an innovation ε0 beliefs in period 1 are φ1 = ρ+ γε0

and then are revised back towards the MCEE i.e. φt > ρ ∀t > 2. Since f ′ (φ) > 0 this

implies c > c∗ for a number of initial periods then c < c∗ for the rest of history. If

we define the impulse response of consumption as a function IRF then the standard

deviation of the first difference of consumption is given by

σ∆c = σ

∞∑0

∆IRF 2t (D.10)

Since φ0 = ρ = φ∗, ∆IRF0 = ∆IRF ∗0

σ∆c = σ∗∆c + σ

∞∑1

∆IRF 2t > σ∗∆c (D.11)

54

www.st-and.ac.uk/cdma

ABOUT THE CDMA

The Centre for Dynamic Macroeconomic Analysis was established by a direct grant from the

University of St Andrews in 2003. The Centre facilitates a programme of research centred on macroeconomic theory and policy. The Centre is interested in the broad area of dynamic macroeconomics but has particular research expertise in areas such as the role of learning and expectations formation in macroeconomic theory and policy, the macroeconomics of financial globalization, open economy macroeconomics, exchange rates, economic growth and development, finance and growth, and governance and corruption. Its affiliated members are Faculty members at St Andrews and elsewhere with interests in the broad area of dynamic macroeconomics. Its international Advisory Board comprises a group of leading macroeconomists and, ex officio, the University's Principal.

Affiliated Members of the School

Dr Fabio Aricò. Dr Arnab Bhattacharjee. Dr Tatiana Damjanovic. Dr Vladislav Damjanovic. Prof George Evans (Co-Director). Dr Gonzalo Forgue-Puccio. Dr. Michal Horvath Dr Laurence Lasselle. Dr Peter Macmillan. Prof Rod McCrorie. Prof Kaushik Mitra (Director). Dr. Elisa Newby Dr Geetha Selvaretnam. Dr Ozge Senay. Dr Gary Shea. Prof Alan Sutherland. Dr Kannika Thampanishvong. Dr Christoph Thoenissen. Dr Alex Trew.

Senior Research Fellow

Prof Andrew Hughes Hallett, Professor of Economics, Vanderbilt University.

Research Affiliates

Prof Keith Blackburn, Manchester University. Prof David Cobham, Heriot-Watt University. Dr Luisa Corrado, Università degli Studi di Roma. Prof Huw Dixon, Cardiff University. Dr Anthony Garratt, Birkbeck College London. Dr Sugata Ghosh, Brunel University. Dr Aditya Goenka, Essex University. Dr Michal Horvath, University of Oxford. Prof Campbell Leith, Glasgow University. Prof Paul Levine, University of Surrey. Dr Richard Mash, New College, Oxford. Prof Patrick Minford, Cardiff Business School. Dr Elisa Newby, University of Cambridge. Prof Charles Nolan, University of Glasgow.

Dr Gulcin Ozkan, York University. Prof Joe Pearlman, London Metropolitan University. Prof Neil Rankin, Warwick University. Prof Lucio Sarno, Warwick University. Prof Eric Schaling, South African Reserve Bank and

Tilburg University. Prof Peter N. Smith, York University. Dr Frank Smets, European Central Bank. Prof Robert Sollis, Newcastle University. Prof Peter Tinsley, Birkbeck College, London. Dr Mark Weder, University of Adelaide.

Research Associates

Mr Nikola Bokan. Mr Farid Boumediene. Miss Jinyu Chen. Mr Johannes Geissler. Mr Ansgar Rannenberg. Mr Qi Sun. Advisory Board

Prof Sumru Altug, Koç University. Prof V V Chari, Minnesota University. Prof John Driffill, Birkbeck College London. Dr Sean Holly, Director of the Department of Applied

Economics, Cambridge University. Prof Seppo Honkapohja, Bank of Finland and

Cambridge University. Dr Brian Lang, Principal of St Andrews University. Prof Anton Muscatelli, Heriot-Watt University. Prof Charles Nolan, St Andrews University. Prof Peter Sinclair, Birmingham University and Bank of

England. Prof Stephen J Turnovsky, Washington University. Dr Martin Weale, CBE, Director of the National

Institute of Economic and Social Research. Prof Michael Wickens, York University. Prof Simon Wren-Lewis, Oxford University.

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