JSPS Grant-in-Aid for Scientific Research (S)Central Bank Communication Design

working paper
No.048 (March 2024)

Rich by Accident: the Second Welfare Theorem with a
Redundant Asset Under Imperfect ForesightShurojit Chatterji
Atsushi Kajii

Research Project on Central Bank Communication
702 Faculty of Economics, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Tel: +81-3-5841-5595 E-mail: watlab@e.u-tokyo.ac.jp
http://www.centralbank.e.u-tokyo.ac.jp/en/

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Rich by accident: the second welfare theorem with aredundant asset under imperfect foresight∗
Shurojit Chatterji

Atsushi Kajii†

Singapore Management University

Kwansei Gakuin University

March 07, 2024

Abstract
We consider a multiperiod (T -period) model with no uncertainty where shortterm bonds co-exist with a long term bond. Markets are complete with just the
short term bonds so that under the usual hypothesis of perfect foresight, the longterm bond is redundant by no arbitrage in that it has no allocational implications.
We dispense with perfect foresight, derive appropriate no arbitrage conditions andshow that the presence of the long term bond has significant allocational implications. Specifically, in the model with just the short term bond, we show that a T
dimensional subset of efficient allocations can arise as Walrasian equilibria whereas
the dimension of efficient allocations is one less than the number of households (assumed to be much larger than T ). In the model with the both types of bonds,
essentially all efficient allocations might arise as Walrasian equilibria; minute errors
in forecasting prices might generate all income transfers that are consistent withefficiency. We argue that the beneficiaries of such unanticipated income transfers
are determined not by the superiority of forecasts but rather by accident. (JEL
classification numbers: D51, D53, D61)
Keywords:General equilibrium; Efficient temporary equilibrium; Endogenous priceforecasts; Redundant Assets∗

This paper subsumes the earlier version entitled “On the Welfare Role of Redundant Assets with

Heterogeneous Forecasts,” Working Papers on Central Bank Communication, No. 046, March 2023.
†
Kajii is a visiting Professor of Singapore Management University. Kajii acknowledges support from
JSPS Grant-in-aid for scientific research (S)18H05217.

1

1Introduction

What allocational role might a redundant financial asset play in an intertemporal Walrasian setting? Traditional wisdom would suggest none, since by definition, a redundantfinancial asset can be replicated by trading other assets dynamically at market prices
so that any trader is indifferent between holding it and ignoring it, and so its presence
in no way alters the possibilities of income transfers across periods/states. But noticethat this conclusion might not be valid if the market prices are not correctly anticipated.
That is, this conclusion relies entirely on the feature that the axiom of perfect foresightis built into the particular equilibrium concept, Radner equilibrium, used in the analysis.
We dispense with perfect foresight and show that essentially all intertemporally efficient
allocations can arise as Walrasian equilibria when a redundant asset is traded.
To fix ideas, consider a discount bond which matures in a few years. If there is no
uncertainty about the fundamentals of the economy at all, then its yield must be givenby the compound one year interest rates, since the long term bond can be replicated byan iterative one year saving. Then the bond is a redundant asset which does not add
any new saving opportunity and the law of one price, or the no-arbitrage principle, will
immediately determine its market value. However, notice that this argument implicitly
assumes that the one year rates are known, which in particular means that the forecasts
are perfectly aligned and thus homogenous across agents. Even though there is no
uncertainty, the one year rates that will prevail in the future years are not realized
yet, leaving some room for heterogenous forecasts. The long term bond might not be
redundant even in an idealized world of frictionless trading, unless perfect foresight isexogenously imposed.
Imposing that forecasts be perfectly aligned (as is implied by perfect foresight), however, does not sit well with the Walrasian paradigm since decentralized households cannot
be expected to coordinate on prices that are not commonly observed.1 One might still
think that if the degree of heterogeneity of forecasts is small enough, the market outcome
will be close to the one predicted in a perfect foresight model, and the presence of thelong term bond like above will not add much qualitatively. We will argue that such a
1

See Radner [1982] and Grandmont [1988] for instance. Various kinds of evidence against alignmentof forecasts can be supplied even in more restrictive contexts of asset pricing (see e.g., Bossaerts [2002]).

2

conjecture is incorrect, and that there are profound welfare effects which are completely
missed out in rational expectations models.
We formulate our question in a one good economy which lasts for T periods, andwhich has two nominal assets. The first of these is a discount bond traded in periods
t = 0, 1, ..., T − 1, which matures in one period, i.e., the bond traded in period t pays
out $1 in period t + 1. The second is a discount bond with a longer maturity, which will
be called the L-bond: the L-bond can be traded in every period, but pays $1 in period
T , and nothing in other periods. In every period, the good and the bonds are traded
competitively. We shall compare two models which only differ in their asset structures:
In Model I, the discount bond is traded in every period while in Model II, in addition to
the bond, the L-bond is traded in every period.
As is known, if perfect foresight (rational expectation) is assumed, the markets are
already complete with bonds with one period maturity, and the payoffs of the L-bondcan be replicated by a plan of dynamic transactions of the bonds. It means that themarket for the L-bond has no additional implication on the allocation of the good except
for indeterminacy of asset trade, which arises since the L-bond and the dynamic plan are
perfect substitutes at any time. In this sense, models I and II are equivalent in terms of
the allocations they generate.
We allow for heterogeneous forecasts, however, which leads us to inspect temporary
equilibria. To avoid the myriad of temporary equilibrium allocations that emerge whenallowing for heterogenous forecasts, and to bring out the allocative implications in a
sharper manner, one needs to put some more discipline into the analysis.2 One way of
doing so would be to put restrictions directly on the sort of forecasts agents are allowed
to hold.3 We advocate instead an allocation based approach that induces restrictions
2

Allocative implications did not receive attention in the earlier literature on temporary equilibrium(Grandmont [1988]), which was focussed on existence (achieved by restricting the dependence of forecasts
on current period prices). Subsequently, the literature examined learning behavior wherein forecasts were
updated in a structured way (using OLS or Bayesian updating for instance (Chatterji [1995])) with aview to investigating the stability of a perfect foresight equilibrium. Our interest is in scenarios where
heterogeneity of forecasts persists and we investigate its allocational implications.
3
Recent literature in macroeconomics and finance explores implications of various kinds of forecasts.
For instance, Woodford and Xie [2022] and Woodford, [2013]) consider bounded rationality in the designof fiscal and monetary policy and inflation targeting. Adam et al [2016] considers the CAPM model to3

on forecasts implicitly: in so doing we are motivated by the concerns that underlie thefundamental welfare theorems, namely characterizing the parts of the Pareto set thatare attainable via Walrasian markets when allowing heterogeneity in forecasts. These
allocations are then a natural benchmark for any sort of welfare analysis in these markets
to say the least. Thus, we postulate that markets allocate resources in an intertemporally
efficient manner and characterize the set of efficient temporary equilibrium allocations(ETE) that can arise in Model I or Model II.
Taking an allocation based approach to its extreme, we choose to impose no restrictions on forecasts a priori. However our methodology can in principle allow one toincorporate additional restrictions in forecasts. Specifically, we separate the analysis into
two parts. The first part identifies efficient allocations that are “price supportable” inthat they satisfy the budget and feasibility conditions necessary for a Walrasian equilibrium. The second part investigates which of these allocations are “justifiable”, that is,
arise as solutions to optimization for some specification of forecasts. If there are a priori
reasons to restrict the sort of forecasts that are admissible in the model, these restrictions will only impinge on the justifiability part of our analysis and can be incorporated
therein.
We show that in Model I, the dimension of ETE allocations is at most T , one less than
the number of trading periods, while, as is well known, the set of efficient allocationsis H − 1 dimensional. While the case H < T appears to be of interest in modellingscenarios where trading opportunities arise frequently, it is at odds with the spirit ofperfect competition since it means in effect that there are more markets than traders.
Our emphasis will therefore be on the case H > T , where ETE induces discipline on
allocations, in that while accommodating heterogeneous forecasts does expand the set of
efficient market outcomes, significant parts of the Pareto set can never be realized.
Moving to Model II, recall that the L-bond is redundant under perfect foresight.
Even when perfect foresight is not assumed, if a trader finds an arbitrage opportunity
with its price forecasts, the markets cannot be in a temporary equilibrium: Hence atshow that small deviations from the rational expectations generate realistic amounts of stock price volatility. This literature allows some sorts of deviations of forecasts from the rational expectations hypothesisthat cause inefficiencies and studies the scope of policy interventions in making welfare improvements.

4

every trading opportunity, every household must have price forecasts which do not admit
any arbitrage opportunity in the markets to take advantage of. It means that the two
types of the bonds are deemed perfect substitutes at any time by any household, i.e., ahousehold will never find it advantageous to trade the L-bond in equilibrium. A plausible
conjecture might then be that the set of ETE allocations is the same as in Model I.
But surprisingly, the set of ETE allocations is of dimension H − 1, which includes all
intertemporally efficient allocations in the vicinity of the Arrow Debreu allocation. When
H > T in particular, the set of ETE expands rather dramatically with the additionalbond since without it the set of ETE is at most of dimension T .
Roughly speaking, the result holds since in perfectly competitive markets, any income
transfer that is consistent with efficiency as well as individual optimization can occur
owing to ex post forecasting errors that lead to ex post arbitrage opportunities. Note thatunless forecasts are perfectly correct, such ex post errors are inevitable, although theymight be small for sophisticated traders. Interestingly enough, for the result in ModelII, the errors might be arbitrarily small: the ex post prices that sustain an ETE different
from the perfect foresight allocation, may resemble arbitrarily closely the sequence ofperfect foresight prices to an observer. This brings into question the interpretation of
the determinacy of the perfect foresight allocation, at least in Model II, since as far as weare aware of, (publicly known) asset pricing models about derivative securities are built
upon an arguably stringent assumption that the assumed price processes of underlying
assets are correct.
Notice that an ETE by definition yields an efficient outcome and in this sense themarkets function ideally, fulfilling their mandates. Every household’ mathematical model
explains the bond prices perfectly at any time under its forecast prices. In our simple
world, each household is no worse than a financial firm which uses very sophisticatedmodel to find out the correct valuations of redundant assets in every trading opportunity.
In spite of these aspects, our analysis concludes that any kind of income transfer canarise implicitly in competitive markets, i.e., there will be winners and losers. We argue
that the beneficiaries of such unanticipated income transfers are determined not by thesuperiority of forecasts but rather by accident.
The point will be seen clearly if the households are identical, and they might trade5

only when their forecasts differ from each other. One can view this special case as anadequate setup to explore a version of purely speculative trade. We show that the notrade outcome is a unique ETE in Model I, but any efficient allocation with some lower
bound can arise as an ETE in Model II; that is, it is the presence of redundant assets
which bring about income transfers of all sorts.
The remainder of the paper is organized as follows. Section 2 introduces the model,
the ex post budget constraints and the no arbitrage condition. Sections 3 studies wealthtransfers using the key intermediary concept of price supportability. Section 4 studies
conditions for justifiability and characterizes the set of efficient allocations that are obtained in the model. Section 5 briefly explores the possibility of speculative trade in thisset up while Section 6 concludes.

2

The Model and Temporary Equilibrium

2.1

Set up

We consider a very simple market economy whose properties are well-known, but wenonetheless summarize its key properties for completeness. Let there be T + 1 periods
starting with period 0, where T > 0. A single non-storable good is available in every
period. There is no uncertainty in the economy.
There are H households, labeled by h = 1, 2, ..., H. Household h is endowed with eth
units of the good in period t, t = 0, 1, ..., T , which is known to household h. To avoid triviality and zero income, we assume H > 1 and eh >> 0 for every h. We shall write xth ≥ 0
for the consumption of household h in period t, and xh = · · · , xth , · · · for the sequenceHof consumption. An allocation of the goods, x = (· · · , xh , · · · ) ∈ RT++1, is feasible ifPHtt
h=1 xh − eh = 0 for t = 0, 1, ..., T . Household h’s preferences are represented by an
additively time separable utility function uh (xh ) = u0h x0h + u1h x1h + · · · + uTh xTh .
The additive structure allows us to provide a clean analysis of the efficient allocations
which arise as temporary equilibria. It also eliminates the conceptual issues about continuation utility which might appear as a potential hazard for the intertemporal generalequilibrium analysis. However, it will become clear that the additive structure is notneeded mathematically for the general point about induced income transfers.

6

We will consider two kinds of assets in the economy. The first of these is a discountbond traded in periods t = 0, 1, ..., T − 1, which matures in one period. That is, thebond traded in period t pays out $1 in period t + 1. The second is a discount bond witha longer maturity, called the L-bond, which can be traded in every period. The L-bondpays $1 in period T , nothing in other periods. Note that the payout is fixed in units ofaccount, not in units of good. The net supply of any of these bonds is zero. In every
period, the good and the bonds are traded competitively. By assumption no default
occurs.
We shall compare two models which only differ in their asset structures: In ModelI, the discount bond is traded in every period while in Model II, in addition to thebond, the L-bond is traded in every period. In both models, we write pt and q t for therespective prevailing prices of the good and the bond in units of account in period t.
Write bth for the amount of the bond household h holds at the end of period t. Write qLt
for the prevailing price of the L-bond in period t and lht for the amount of the L-bondheld by household h at the end of period t. Thus lht − lht−1 is the amount traded in periodt, which costs qLt lht − lht−1 . Unlimited short sales are allowed, so bth and lht are possibly
any negative number, but recall that default is not allowed. We write bt = (..., bth , ...)and lt = (..., lht , ...) for allocations of the bond and the L-bond in period t, respectively,and write b = · · · , bt , · · · and l = · · · , lt , · · · for a sequence of such allocations. SincePt
the bonds are in zero net supply, we say b (resp. l) is feasible if H
h=1 bh = 0 (resp.
PH t
h=1 lh = 0) holds in every period t.
As is known, if perfect foresight (rational expectation) is assumed, the markets are
already complete with bonds with one period maturity, and the L-bond is a redundantasset since its payoffs can be replicated by a plan of dynamic transactions of the bonds:
buy q t · · · q T −1 units of the bond in period t − 1, which costs q t−1 q t · · · q T −1 , and use
the payout of the bond to buy q t+1 · · · q T −1 units of the bond in the next period t.
This trading plan has exactly the same yield as a unit of the long term bond in period
T , and hence by the no arbitrage principle or the law of one price, the market priceof the L-bond in period t, qLt , must be the same as the cost of the plan. If they are
different, an arbitrarily large amount of profits will be extracted with no cost, i.e., there
is free lunch in the bond markets. It can be readily verified that the converse holds,

7too: the bond markets admit no free lunch if and only if qLt = q t q t+1 · · · q T −1 holds for
t = 0, 1, ..., T − 1.4 Thus we shall use the following convention:
Definition 1 For a sequence of bond prices starting in period t, q t , q t+1 , ..., q T −1 and
qLt , qLt+1 , , , , qLT −1 , the no arbitrage condition is satisfied in period t if qLs = q s+1 · · · q T −1
holds for s = t, t + 1, ..., T − 1.
In other words, the no arbitrage condition holds in period t if and only if the Lbond and the bond are perfect substitutes under those prices in period t as well as thefollowing periods. Notice in particular that qLT −1 = q T −1 holds since the two bonds
are indistinguishable perfect substitutes in period T − 1 by construction. Consequently,
under perfect foresight, the market for the L-bond has no additional implication on the
allocation of the good except for indeterminacy of asset trade, which arises since theL-bond and the dynamic plan are perfect substitutes in Model II: the two models areequivalent in terms of the allocations they generate.
To ask to what extent the conclusion above depends on perfect foresight, we shallfirst define the temporary equilibrium which accommodates heterogeneous forecasts ineach model. In both models, household h will trade with some forecast prices in mind,
which are not necessarily correct ex post, given market prices prevailing in period t,i.e., pt q t and qLt . It means in particular that households might anticipate differentTrate of real returns of the bond. For t, t = 0, 1, ..., T , write p̂h|t = p̂t+1
,
...,
p̂
h|t ,
h|t




T −1
t+1
t+1
T −1
for the forecast prices of the good and, ..., q̂Lh|t
q̂h|t = q̂h|t, ..., q̂h|tand q̂Lh|t = q̂Lh|trespectively the prices of the short and the L-bond, where the subscript h|t indicates that
it is the forecast of household h made in period t. As mentioned in the introduction, in
what follows we do not restrict forecasts a priori; thus in particular we do not impose any
specific learning procedure at this point, though requiring markets to be in a temporary
equilibrium will dictate that forecasts cannot admit arbitrage opportunities.
4

It is clear that the bonds have linearly independent payoffs and so any (positive) prices are consistent.

Then no free lunch is equivalent to the L-bond being properly priced, which is exactly the condition
states. More generally, as is well known (e.g., Lemma 19.E.1 of Mas-Collel et al [1995]), in the rational
expectations model, no free lunch is equivalent to the existence of state prices, and asset prices arederived using the state prices. In our context, the (normalized) state prices are discounted values offorecast future bond prices.

8

In Model I, in every period t, t = 0, 1, ..., T , household h optimizes given prices pt
and q t as well as the outstanding bond holdings bt−1
(where b−1hh = 0 by convention)
under the following constraints,
pt xt + q t bt ≤ pt eth + bt−1
h ,

(1)

t+1 t+1
t+1
t+1
t
p̂t+1
+ q̂h|tb̂≤ p̂t+1
h|t x̂
h|t eh + b ,..
.
T −1 T −1
T −1 T −1p̂h|tx̂+ q̂h|tb̂≤ p̂Th|t−1 eTh −1 + b̂T −2 ,p̂Th|t x̂T ≤ p̂Th|t eTh + b̂T −1 ,with variables xt , bt , x̂t+1 , ..., x̂T , and b̂t+1 , ..., b̂T −1 . Note that the choice variables withhats are also forecasts (where these variables are written without the subscript h|t for
convenience) that are yet to be realized at the time household h trades xt and bt , andthey do not necessarily coincide with the actual trades carried out in future. Moreover,
forecasts need not constitute feasible allocations.
The inequalities in (1) exhibits some homogeneity, and one might wonder if theprice of the good each period can be normalized to 1 without loss of generality. Indeed
if all future prices were perfectly anticipated, it would be natural to set pt and p̂t+s
h|t ,s = 1, ..., T − t equal to one. In our setting, the price forecasts are heterogeneous andpresumably updated each period. Setting the price of the good to one each period,
though technically feasible, is somewhat problematic conceptually as it does not sit wellwith the heterogeneity of forecasts and their updating across households. We therefore
choose to proceed without normalization.
In Model II, in every period t, t = 0, 1, ..., T , household h optimizes given prices pt ,q t and qLt as well as the outstanding bond holdings bt−1
and lht−1 (where lh−1 = b−1
h
h = 0

9

by convention) under the following constraints,pt xt + q t bt + qLt lt − lht−1 ≤ pt eth + bt−1h ,
t+1 t+1t+1 ˆt+1
t+1
t+1
t
p̂t+1
+ q̂h|tb̂+ q̂Lh|tl− lt ≤ p̂t+1
h|t x̂
h|t eh + b ,

(2)

..
.


T −1 T −1
T −1 T −1
T −1 ˆT −1p̂h|tx̂+ q̂h|tb̂+ q̂Lh|tl− ˆlT −2 ≤ p̂Th|t−1 eTh −1 + b̂T −2 ,p̂Th|t x̂T ≤ p̂Th|t eTh + b̂T −1 + lT −1 ,with variables xt , bt , lt , x̂t+1 , ..., x̂T , b̂t+1 , ..., b̂T −1 and ˆlt+1 , ..., ˆlT −1 . Like in Model I, thechoice variables with hats are also forecasts and yet to be realized at the time householdh trades xt , bt and lt . Here too, forecasts do not necessarily coincide with the actual
trades carried out in future, and they need not constitute feasible allocations. For thesame reason as in Model I, we proceed without normalizing forecast prices.
We say that choice variables in period t are justifiable if there are forecasts such thatthe chosen value of the variables is part of an optimal trade, given period t market prices.
That is, the current choice is justifiable if they constitute household h’s demand for thegood and the bonds in period t for some forecasts. In Model I, xt , bt is justifiable at
prices pt , q t (and bt−1h ) in period t for household h if there exist forecasts p̂h|t and q̂h|tabout future prices such that household h’s utility is maximized at xt , bt , x̂t+1 , ..., x̂T , andb̂t+1 , ..., b̂T −1 for some x̂t+1 , ..., x̂T , and b̂t+1 , ..., b̂T −1 . In Model II, xt , bt , lt is justifiable
t−1
at prices pt , q t , qLt (and bt−1
h , lh ) in period t for household h if there exist forecastsp̂h|t , q̂h|t , and q̂Lh|t about future prices such that household h’s utility is maximized at
xt , bt , lt , x̂t+1 , ..., x̂T , b̂t+1 , ..., b̂T −1 and ˆlt+1 , ..., ˆlT −1 for some x̂t+1 , ..., x̂T , b̂t+1 , ..., b̂T −1and ˆlt+1 , ..., ˆlT −1 .
Sequential markets are in temporary equilibrium if the demand meets the supply inevery market. In our models, a temporary equilibrium (TE) is defined as follows:
Definition 2 Model I: Prices pt and q t , t = 0, 1, ..., T , and an allocation (x, b) constitutea temporary equilibrium if (1) x, and b are feasible, (2) for every household h, in every

period t = 0, 1, ..., T , xth , bth is justifiable at prices pt , q t (and bt−1h ).
Model II: Prices pt , q t and qLt , t = 0, 1, ..., T , and an allocation (x, b, l) constitute a
temporary equilibrium if (1) x, b, and l are feasible, (2) for every household h, in every
10



t−1
period t = 0, 1, ..., T , xth , bth , lht is justifiable at prices pt , q t , qLt (and bt−1
h , lh ).
The prices pt , q t in Model I and pt , q t and qLt , t = 0, 1, ..., T in Model II, will be referred
to as ex post temporary equilibrium (ex post TE) prices.
Remark 3 The justifiability requirement (2) above has two implications. First, household h’s forecasts must not allow any arbitrage opportunity to itself at any time, or elsethe utility maximization problem has no solution since there is no limit on the volume oftrade. Thus in particular, each household must have forecasts for which the L-bond is aredundant asset at every trading opportunity, or else it will provide itself with a free lunch
at some point. Secondly, even though justifiability is an individualistic exercise, since
the ex post prices are common across the households, the households will end up being
subject to some common constraints induced by the budget constraint with the ex post
prices. The constraints take different forms in the two models, which will be scrutinizedin the following subsection.

2.2

Ex post budget constraint and no arbitrage condition

For a sequence of prices pt , q t and a feasible allocation (x, b) to constitute a TE in ModelI, a necessary condition is that the constraint (1) must hold in any period t with theseprices and allocations. With the monotonicity of the utility function, we therefore havethe following system of equations that is required to sustain TE allocations for every
household h. We shall refer to this system as the ex post budget constraint for Model I.
p0 x0h + q 0 b0h = p0 e0h(3)

p1 x1h + q 1 b1h = p1 e0h + b0h..
.
pt xth + q t bth = pt eth + bt−1h..
.
pT −1 xhT −1 + q T −1 bTh −1 = pT −1 eTh −1 + bTh −2
pT xTh = pT eTh + bTh −1
Although households do not necessarily anticipate the prices correctly, the well known
technique for the rational expectation models can be adopted to show that these dynamic
11

constraints can be reduced to a single equation. Specifically, multiplying the period t
equation by q 0 q 1 · · · q t−1 and summing up, we obtain a single ex post budget constraint:TXp̃t xth − eth = 0

(4)

t=0

where p̃t is the discounted period t price, i.e., p̃t = q 0 q 1 · · · q t−1 pt . It can be readily seenthat a feasible allocation x of the goods satisfies (4) for every h if and only if there exista feasible allocation b of bond such that (3) holds for every h.
We emphasize that constraint (4) holds in any TE ex post. Therefore, if perfectforesight is assumed in addition, constraint (4) holds ex ante with forecast prices, i.e.,household h plans to choose a utility maximizing xh given constraint (4), and trades
the bonds to finance, i.e., to satisfy (3). If the markets of the good clear with a streamof discount prices, we have an Arrow Debreu equilibrium (AD equilibrium). This isof course the reason why a perfect foresight equilibrium (PFE) is equivalent to an AD
equilibrium in this model.
But even without perfect foresight, constraint (4) means that household h’s income is
the market value of the endowments ex post. Thus in any TE, the wealth of a household
is determined by the discounted market prices only, and in this sense, there is no income
transfer among the households except for implicit ones induced by market prices.
In Model II, for prices pt , q t and qLt and a feasible allocation x, b, and l to arise as
TE, since the total expenditure must be equal to the total income in every period, thefollowing budget equations, which we shall refer to as the ex post budget constraint for
Model II, must hold for xh , bh , lh for every household h:p0 x0h + q 0 b0h + qL0 lh0 = p0 e0hp1 x1h + q 1 b1h + qL1 lh1 − lh0 = p1 e0h + b0h..
.
pt xth + q t bth + qLt lht − lht−1 = pt eth + bt−1
h
..
.


pT −1 xhT −1 + q T −1 bTh −1 + qLT −1 lhT −1 − lhT −2 = pT −1 eTh −1 + bTh −2
pT xTh = pT eTh + bTh −1 + lhT −1
12(5)

We ask if these equations can be reduced to a single constraint as in Model II. Recall
that in a TE, the forecasts must not allow any free lunch, which means that in every
period, the forecast prices must respect the no arbitrage condition (Definition 1). We
k is household h’s
shall report these observations as a Lemma below:5 recall that q̂h|t
T = 1.
forecast period k bond price in period t, where k > t. By convention, we set q̂h|t

We have:
Lemma 4 Consider a TE. (1) if q t , qLt , t = 0, 1, 2..., T − 1, arise in the TE, then qLT −1 =q T −1 must hold. For period t = 0, 1, ..., T − 2, and household h = 1, ..., H, forecasts
t+1
T −1
satisfy the no arbitrage condition in period t if and only if (a) qLt = q t q̂h|t· · · q̂h|t
fort+jt+j t+j+1
T −1, t = 0, ..., T − 2, j = 1, ..., T − 1 − t.
t = 0, 1, ..., T − 1 and (b) q̂Lh|t= q̂h|tq̂h|t· · · q̂h|t(2) for any household, in every period, the bond and the L-bond are perfect substitutes.
Proof. (1) qLT −1 = q T −1 must hold since the two bonds are perfect substitutes in period
T −1 in any TE. Conditions (a) and (b) are exactly the no arbitrage condition (Definition
1) applied to the sequence of forecasts.
(2) Since a household must have forecasts which satisfies the no arbitrage condition(Definition 1) in period t, it must find that the L-bond and the synthetic trading strategyof the bond are an identical saving instrument. Hence in particular, forecasts must be
formed in such a way that the bond and the L-bond in period t markets are perfect
substitutes.
Note that condition (a) in Lemma 4 can be written as
qLt
t+1
T −1
= q̂h|t· · · q̂h|t
for t = 0, 1, ...T − 1,qt
which means in any TE, every household must forecast future bond prices in a way whichis consistent to the relative L-bond price commonly observed in the markets. That is,
the no arbitrage condition requires that households’ forecasts are aligned to the ex post
q T −1

relative price. Setting t = T − 1, we have qLT −1 = 1, which is a condition we have alreadypointed out. One might wonder if there are more constraints about ex post prices owing
to the no arbitrage condition of the forecasts.
5

In Model I, there is no effective constraint like the no arbitrage condition. The reason is, as long as

the forecast prices are positive, the maximization problem is well defined and has a solution.

13

Suppose that for the price sequence pt , q t and qLt under consideration, the no arbitrage
condition qLt = q t q t+1 · · · q T −1 holds for t = 0, 1, ..., T − 1. Then multiplying the period t
equation in (5) by q 0 q 1 · · · q t−1 and summing them up, we get the same equation (4) asfor Model I, and hence Model II does not add anything to Model I as far as the allocationof the good is concerned. But note that although the L-bond must be a perfect substituteof the synthetic strategy at the forecast prices, it need not be so at the ex post TE prices,
i.e., the no arbitrage condition qLt = q t q t+1 · · · q T −1 of ex post TE prices is not warranted.
Indeed, we argue that qLT −1 = q T −1 is the only binding implication of the no arbitrage
condition on ex post prices under heterogeneous forecasts. Indeed, for t < T −1, conditionqt(a) says qLt might be any number, since the forecasts about future prices are free variables.
Clearly, for any given stream of forecast bond prices, there is a unique L-bond price givenby the no arbitrage condition, satisfying condition (b) in Lemma 4. In this argument, it
is important that the forecasts are updated completely freely.
In conclusion, ex post prices of a TE tend not to satisfy the no arbitrage condition. It
turns out that the failure of the no arbitrage condition for ex post prices has significant
implications on income transfers, which in our view are completely overlooked in rational
expectations models, and which we shall explore in the next section.
Remark 5 Since qLT −1 = q T −1 is a common requirement in both models, we shall automatically assume qLT −1 = q T −1 whenever ex post TE prices are considered.
Remark 6 A TE in Model I can be naturally seen as a TE in Model II where households never trade the L-bond: in a TE in Model I, by definition, in every period, every
household has forecasts which justify their choices in period t. By Lemma 4, the bond andthe L-bond are perfect substitutes in the view of household h throughout the remaining
periods and hence in particular it is optimal to choose no trade of the L-bond in periodt, and to trade the bond for desired saving and borrowing. In this sense, justifiability in
Model I implies justifiability in Model II. This observation in particular shows that any
TE in Model I can be identified with a TE in Model II where the L-bond is never traded.

14

3

Price supportability and Wealth Transfer by the Redundant Asset

3.1

Set upSince TE can accommodate a large class of feasible allocations, the aforementioned
redistribution role associated with the L-bond, a redundant asset, will be hard to identify
if all TE allocations are considered. An obvious modelling choice is to focus on theefficient allocations; if two distinct efficient allocations are compared, one can be deemed
as a result of an (efficient) wealth transfer operated on the other. Thus we are primarily
interested in an efficient temporary equilibrium (ETE) in the following analysis.
It will then be useful to work with an economy where the set of efficient allocations
has a simple structure so that the purely distributional effects can be observed in a transparent manner. Therefore, in addition to the additively time separable utility function00
uh x0h + uh x1h + · · · +uh xTh , with u0h > 0, uh < 0 and u0h (0) = +∞, we assume thatPt
the total endowment is one in every period, i.e., H
h=1 eh = 1 for t = 0, 1, ..., T . Consequently, a feasible allocation of goods is efficient intertemporally if and only if assigns atime invariant consumption to every household.6 An efficient allocation can therefore bePparameterized by a tuple of positive numbers ξ1 , ξ2 , · · · , ξH with H
h=1 ξh = 1, whereξh is the time invariant consumption level of household h. We shall identify an efficient
allocation with a tuple (· · · , ξh , · · · ) of H positive numbers summing up to one, whichmight be viewed as a wealth distribution, which will help us to identify the redistributionrole.
H
As a first benchmark, consider an AD equilibrium (p, x) ∈ RT +1 × RT +1of this
economy, where p = · · · , pt , · · · are positive prices of the goods and x = (· · · , xh , · · · )is the associated allocation of the goods; that is, each household is maximizing utility at
xh given prices p and income p · eh , and x is feasible. As is discussed earlier, if perfectforesight is assumed in Model I or Model II, the sequential budget constraint is in effectgiven by a single constraint (4), and hence an AD equilibrium allocation constitutes an
6Indeed, it can be readily confirmed that a time invariant feasible allocation is efficient. Conversely,for any consumption path x0 , x1 , ..., xT , every household prefers the path which provides its averagePTt1
consumption T +1
t=0 x every period, so an efficient allocation must be time invariant.

15

equilibrium allocation in a TE with perfect foresight, a PFE. As far as the allocational
property is concerned, we will use AD and PFE interchangeably. Recall that in a PFE
in Model II, the L-bond is redundant, and the set of PFE allocations remains the same if
the L-bond markets are all closed (i.e., only the bonds are available), which is effectively
the set up of Model I.
The allocation x in an AD equilibrium is of course efficient by the first fundamental
theorem of welfare economics. From utility maximization and the additive time separability of the utility function, prices p must be proportional to the gradient vector u0h (xh ) =
u0h x0h , u0h x1h , · · · , u0h xTh for every household h. The observation about time invariance of the efficient allocations above implies that u0h x0h , u0h x1h , · · · , u0h xTh =(· · · , u0h (ξh ) , · · · ), which is also time invariant. Thus the AD equilibrium price system
must also be time invariant.
By the homogeneity of equilibrium prices, the AD equilibrium price of the good can
1
be normalized to be T +1
in each period. With the normalized AD equilibrium prices,

the value of the total endowments (one in every period) is one, and the market value
1 PTtof household h’s endowments is T +1
t=0 eh , and the market value of the consumption
consuming ξh in every period is ξh . So from the budget constraint, we conclude that theAD equilibrium allocation is unique and household h consumes ξh =PTt
t=0 ehT +1

in every

period.
Example 7 H = 4 and T = 2. The endowments are given as in the following table:
h\t

t=0

t=1

t=2

h=1

1
4 + 2η

1
4 −η

1
4 −η

h=2

1
4 −η

1
4 + 2η

1
4 −η

h=3

1
4 −η

1
4 −η

1
4 + 2η

h=4

1
4

1
4

1
4

where 0 ≤ η < 41 . That is, household h, h = 1, 2, 3, has a high endowment in periodt = h − 1, and a low endowment in the other periods, whereas household h = 4 has aconstant endowment 41 in every period. Notice that e0h + e1h + e2h = 34 for all households.
Thus in a unique AD equilibrium, every household consumes 14 in every period.

16

3.2

Price Supportable Allocations

To address the issue of wealth transfer with heterogeneous forecasts in ETE, it willturn out to be convenient to first examine allocations which satisfy the ex post budget
constraints with some stream of ex post market prices. Such an allocation will be called
price supportable (PS allocation). Since the ex post budget must hold in any TE, it
will certainly identify the largest set of allocations which can arise in equilibrium. Pricesupportability of an efficient allocation is especially important since it is necessary for aETE. If the justifiability requirement is met for all households with some forecasts, thenwe have a ETE.7
We shall first study the structure of price supportable allocations for models I and
II in this section, and the issue of justifiability will be taken up in the next section.
It will become evident that keeping the justifiability requirement separate from pricesupportability in this way makes the analysis more tractable and transparent. The
concept of price supportability has little to do with efficiency, but we will see thatthe set of efficient and price supportable allocations has a simple structure in Model I.
In contrast, the implication of price supportability will be very different in Model II,
indicating that the L-bond might induce an additional channel of wealth transfers which
does not exist in Model I, let alone under rational expectations.
3.2.1

Price supportable efficient allocations of Model I

Recall that in Model I, the ex post sequential budget constraints can be reduced to asingle budget: an allocation is price supportable if and only if there are (discounted)
ex post prices for which (4) is satisfied. Hence an allocation x is feasible and pricesupportable if and only if there are positive prices p0 , p1 , ...pT which are the discounted
7

As we have seen, the ex post budget constaint is implied by the justifiability requirement of TE, andso there is a bit of logical repetition in separating only the budget constraints. Furthermore, since there
is no requirement that the forecast values of future variables that are used in justifying an allocation
actually clear markets, there could be many different profiles of forecasts that may justify the same TE
allocation.

17

prices of the goods, satisfying the following equations:

PT
t
t
t
= 0 for h = 1, 2, .., H
t=0 p xh − ehP
Ht
= 0 for t = 0, 1, .., T
h=1 xh − 1(6)

The first set of H equations implies that the ex post budget balances for all households: mathematically, the net trade vector is perpendicular to the price vector for eachhousehold, and a familiar geometric intuition can be given as in the standard theory ofcompete markets. The second set of T equations implies feasibility.
When restricted to efficient allocations, i.e., time invariant allocations, the systemof equations (6) can be significantly simplified, and hence the set of efficient and pricesupportable (EPS) allocations under the maintained assumptions has a clean structure.
Notice that the last feasibility equation is redundant in (6) since the first H equations P T
PT
P
t t
t x −
imply Tt=0 pt xh − eth =pht=0 p eh = 0 for every h, and hence
t=0
H
X

H PT
X
pt eth

Pt=0
xh =
T
t
p
h=1
h=1
t=0
T
X

1

= PT

t
t=0 p t=0

1
= PT

t=0

T
X

pt

p

t

H
X

!
eth

h=1

pt

t=0

= 1.
Therefore, the system of equations (6) is equivalent toPTt t
t=0 p eh

= xh for h = 1, 2, .., H .

(7)Or equivalently, let E be a (T + 1) × H matrix given by the rule
E = [e1 , e2 , ..., eH ]where eh is the column vector of endowments for household h, and then (7) can be
written as
pE = x,(8)where p = p0 , p1 , · · · , pT and x = (x1 , x2 , · · · , xH ) are row vectors. Notice that if (7) isP
PH PTt et =
satisfied, prices are automatically normalized, since H
x
=
1,
1
=
p
h=1 h
h=1
t=0
h
P
 P
PT
H
T
t
t
t
t=0 p
h=1 eh =t=0 p .
18

To sum up, a row vector of an efficient time invariant consumption allocation, x =(· · · , xh , · · · ) ∈ RH , is a EPS allocation if and only if the simultaneous equations (7), orequivalently (8), have a positive solution p = p0 , p1 , ..., pT ∈ RT .
It means that the set of EPS allocations is the image of the set of normalized discounted positive prices for the linear function p 7→ pE. The dimension of this set isthe rank of matrix E minus one. More specifically, notice that pE can be expressed asa convex combination of row vectors of initially endowed goods among households inperiod t:
pE =

T
X

pt et

t=0

where et =· · · , eth , · · ·

is the row vector of initially endowed goods among households

1
in period t. Recall that a unique AD equilibrium normalized prices are pt = T +1
fort = 0, 1, ..., T , which certainly satisfies the equation above. Thus we obtain the following
result:Proposition 8 In Model I, the set of EPS allocation is
nP
o
PT
T
t = 1, pt > 0, for t = 0, 1, ..., T . That is, the set of EPS allocations
t et :
pp
t=0
t=0
is the relative interior of the convex hull of T + 1 vectors e0 , e1 , ..., eT in RH .
Since the total resource is one for any t, the dimension of the convex set in question is
at most min (T, H − 1). If any choice of min (T + 1, H) vectors among e0 , e1 , e2 , · · · , eT
are affine independent, then the dimension is exactly min (T, H − 1). Since such affine independence is a generic property, we conclude that generically in endowments (with total
resource equal to one in every period), the dimension of EPS allocation is min (T, H − 1).
In comparison with the set of PFE, which is a singleton set, we see that heterogeneityof forecasts alone might create a great deal of wealth transfers among the households,
potentially. Also notice that if T < H − 1, which is the case we focus on, there are
efficient allocations which are not price supportable. Thus efficiency and price supportability do have some explanatory power on market outcomes even in the absence of perfectforesight.
Example 9 In the economy of Example 7, in any EPS allocation, household 4 consumes 41 . For the other households, (x̄1 , x̄2 , x̄3 ) is contained in the relative interior of
19

1
1
1
4 + 2η, 4 − η, 4 − η



,

1
1
1
4 − η, 4 + 2η, 4 − η , and111
4 − η, 4 − η, 4 + 2η , which is a twodimensional set. More explicitly, with the corresponding normalized discounted prices
P
p0 , p1 , p2 with p0 + p1 + p2 = 1, consumption x̄h can be written as x̄h = 2t=0 pt eth =


1
h
h
η = 41 + 3ph − 1 η, for h = 1, 2, 3. Of course if those prices are
4 + 2p − 1 − pequal, x̄h = 14 , i.e., it is the PFE allocation.
3.2.2

Price supportable allocations in Model II

Fix any ex post prices p0 , p1 ..., pT , q 0 , q 1 , ..., q T −1 , and qL0 , qL1 , ..., qLT −1 such that q T −1 =qLT −1 and any (possibly not constant) stream of consumption xh for a household h. The
discounted prices are written as p̃t := q 0 q 1 · · · q t−1 pt , q̃ t := q 0 q 1 · · · q t−1 q t , and q̃Lt :=q 0 q 1 · · · q t−1 qLt . Recall that the no arbitrage condition of these bond prices is equivalent
to the discounted price q̃Lt being invariant of time t and is equal to q 0 q 1 · · · q T −1 = q̃LT −1 .
Our goal here is to show that if the no arbitrage condition above is not satisfied, for
any household h, there is a dynamic transaction of bonds such that the consumption
stream xh is budget feasible. Notice that the prices as well as the consumption arecompletely arbitrary. This in particular shows that any allocation is price supportable
in Model II, let alone efficient ones, in stark contrast to Model I.
Assume that there is a period when the no arbitrage condition fails, and so there is a∗period T ∗ , 0 ≤ T ∗ < T −1 where q̃LT −1 − q̃LT 6= 0. Consider the following dynamic trading
plan, which we shall refer to as a canonical trading plan, where the L-bond is traded
only in period T ∗ : lht = 0 for t < T ∗ , and then keep household h from any additionaltransaction of the L-bond after period T ∗ , so that lht − lht−1 = 0 for t = T ∗ +1, ..., T ∗ −1.
Hence by construction, qLt lht − lht−1 = 0 holds for t = 1, ...T − 1, except t = T ∗ when∗∗lhT units of the L-bond is bought. The amount lhT will be specified later. For t =0, 1, ..., T ∗ − 1, set bth iteratively to meet period t budget, pt xth + q t bth = pt ett + bt−1
h , where
∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

T −1
T
∗
T
T
T T
T T
T T
b−1
,
h = 0. Let bh solve the period T budget, p xh + q bh + qL lh = p eh + bh∗∗

where bTh −1 and lhT have already been determined as above. For t = T ∗ + 1, ..., T − 1,
set bth iteratively to meet period t budget, pt xth + q t bth = pt e0t + bt−1h .
Then by construction the budget in each period t = 0, 1, ..., T − 1 is satisfied, i.e., theplan finances the given stream of consumption at the given prices up to period T − 1,20∗regardless of the choice of lhT :p0 x0h + q 0 b0h = p0 e0h(9)p1 x1h + q 1 b1h = p1 e0h + b0h
..
.
∗

∗

∗

∗

∗

∗

∗

∗

∗

pT xTh + q T bTh + qLT lhT = pT eTh + bTh −1
∗

∗

∗

∗

pT +1 xTh +1 + q T +1 bTh +1 = pt eth + bTh∗..
.
pT −1 xhT −1 + q T −1 bTh −1 = pT −1 eTh −1 + bTh −2
Consequently, the constructed trading plan b0h , b1h , ..., bTh −1 and lh0 , lh1 , ..., lhT −1 is budget
feasible if the period T budget equation is satisfied in addition, which is∗

pT xTh = pT eTh + bTh −1 + lhT ,∗since lht = lhT for t > T ∗ .
Assuming that the period T budget equation holds, multiply the period t budget
with q 0 q 1 · · · q t−1 for t = 1, 2, ..., T , summing them from period 0 to T , we have
T
X


∗
∗
p̃t xth − eth = (q̃LT −1 − q̃LT )lhT ,

(10)

t=0

and hence the period T budget equation is satisfied as well if and only if (10) holds.
∗∗

Since q̃LT −1 − q̃LT 6= 0 by assumption, (10) holds for lhT given by the rule:

PT
t
t
t
t=0 p̃ xh − eh
T∗
lh =
∗
q̃LT −1 − q̃LT

(11)

To sum up, we have established the following result:
Proposition 10 In Model II, for any ex post prices, any stream of consumption is budget
feasible for any household, with a canonical trading plan. In particular, in any efficient
allocation, a household’s consumption is budget feasible for any ex post prices.
Compare this result with Proposition 8: the set of EPS allocations is expressed as
the convex hull of the endowment vectors, and so in particular, an efficient allocation
which assigns to household h an amount x̄h smaller than the minimum of e0h , e1h , ..., eTh is21not a EPS in such an environment. On the other hand, Proposition 10 shows that any
feasible allocation can arise in a TE, let alone efficient allocations.
The construction of the trading plan used in the proof of Proposition 10 might become
clearer if one recalls the AD budget constraint: notice that equation (11) says that
consumption xh is AD budget feasible if household h is provided with an extra income of∗∗(q̃LT −1 − q̃LT )lhT , which might be negative of course. If allocation (· · · , xh , · · · ) is feasible,



PH PT
t xt − et = 0, and so q̃ T ∗ − q̃ T −1 lT ∗ , h = 1, 2, ..., H, constitute income
p̃
h=1
t=0
L
h
h
h
L
transfers among households. Recall that the L-bond and the bond are perfect substitutes
under its forecasts, and household h chooses this particular amount by accident in periodT ∗ . That is, the ones who are “rich by accident”, are subsidized by the ones who are“poor by accident”, and there is no particular link to the quality of their forecasts. From
this viewpoint, the essence of Proposition 10 is that the failure of no arbitrage conditionin any one period ex post, no matter how minor it might be, is consistent with any
amount of income transfer. Therefore, the L-bond, which is redundant under perfectforesight, can have a significant distributional role.
One might wonder why there is no constraint other than failure of the no arbitrage
condition on ex post TE bond prices. Indeed, if the L-bond is not traded as in Model I, tosustain a particular efficient allocation, these prices must be configured in a certain way
depending on the allocation. But in Model II, required income transfers effectively occur
through trade of the L-bond in only one period when the no arbitrage condition fails ex
post. Intuitively, if the no arbitrage condition fails ex post in a period, the law of one
price is broken ex post in that period: if the same object has two prices simultaneously,
any kind of transfers can be established in competitive markets.
We emphasize that the existence of such income transfer does not depend on themagnitude of failure, i.e., the prescribed income transfer might arise if q t q t+1 · · · q T −1
differs from qLt by any slightest amount ex post. It is not owing to households’ improper
forecasts either: in Model II, in every moment, each household behaves rationally witha perfectly sensible forecast which does not give itself any free lunch. The canonicaltrading strategy proposed here is not the unique one that would work for establishing
the Proposition; we chose it as it will be convenient to work with it for the subsequentjustifiability argument.

22

4

Justifiability and Structure of ETE

This section studies justifiability in Models I and II. We maintain the assumption ofhistory-free updating of forecasts, and hence there is no ad hoc learning procedure.8

4.1

Justifiability in Model I

Given a price supportable allocation, we discuss if and when it is justifiable for all households. This is more than we need to do for Model I where our primary interest is on EPS
allocations where consumption is time invariant. However, the idea of justifiability itself
is individualistic and hence it does not depend on efficiency directly. More importantly, aslightly modified idea will be used in Model II where we will argue any price supportableallocation is justifiable.
We will not formally demonstrate the justifiability of EPS allocations in Model I here
as it is not the central finding of this paper. We provide below an informal discussion of
how justifiability is obtained in this framework.

To begin with, we shall ask if xt , bt is justifiable in period t; that is, xt , bt maximizes (continuation) utility with some forecast prices and trading plan subject to the(continuation) dynamic budget (1) in period t, where bt−1
h , the bond purchased in periodt − 1, is taken as a exogenously fixed parameter9 . Recall that maximization implies
forecast prices must not allow any arbitrage opportunity, and hence the dynamic budget
constraint (1) can be reduced to the AD budgetTXp̃t xth − eth +p̂s (x̂sh − esh ) = q̃ t−1 bt−1
h

(12)

s=t+1

where p̂s and x̂s are forecasts of the discounted price of the good and consumption
respectively, for period s. In principle we need to describe the forecast bond prices, butthey can be readily identified from the discount prices with the no arbitrage condition,
8

We refer to Chatterji-Kajii [2023] for general techniques and issues therein. In particular, stochastic

forecasts would make justifiability easier to satisfy, and hence the set of ETE might be larger than what
we shall report. But we restrict attention to point forecasts since it suffices to make our point, besides
expositional simplicity.
9
Since the saving decision in period t − 1 might be done with very incorrect forecasts, it is possiblethat the household is practically bankrupt. However, as long as endowments are positive, there always
exist (very optimisitic) forecasts with which household can repay the debt in future.

23and hence we omit them. In short, the utility maximization can be solved by first finding P
a consumption bundle which maximizes utility uh xt + Ts=t+1 uh (x̂s ) under the AD
budget (12), and then determining bond transactions which finances the consumption
xt , x̂t+1 , ..., x̂T . Then xth , bth of household h is justified in period t if and only if
xth is the quantity demanded of the good in period t under the AD budget (12) forsome forecasts about the discounted prices, p̃t+1 , ..., p̃T , and the associated bth is foundto satisfy the dynamic budget.
Suppose that the demand function under the AD budget is responsive to forecasts
in the sense that as a function of price forecasts, the demand changes in any direction.
This property will be generically true under some mild and plausible conditions on utility
functions and endowments, at least if xth is the AD (thus PFE) consumption in periodt. We therefore contend that any price supportable feasible allocation close enough to
the AD equilibrium allocation is justifiable in any period. Therefore, a fortiori, any EPS
allocation near the AD equilibrium allocation is an ETE allocation.
For a specific case of additive log function, i.e., uh (z) = ln (z), we can give a moreconstructive argument. In this case, in period t, t < T , household h spends its AD
income evenly in the remaining periods including period t, and the quantity demand inperiod t is found by dividing the intended expenditure by the prevailing market priceplus the forecastof the good. From (12) we see that the AD income is p̃t eth + q̃ t−1 bt−1
hPT
(discounted) income m̂ := s=t+1 p̂s esh , which depends on price forecasts, but not time tmarket variables. Thus the problem of justifiability is reduced to find a forecast incomet t

t−1 t−1b+m̂
m̂ such that p̃t xth = p̃ e +q̃
.
(T −t+1)

It is clear that m̂ can be made arbitrarily small by forecasting low prices of the good,
and also arbitrarily large by forecasting high prices of the good as long as endowments
are strictly positive. Then by continuity, the existence of a suitable m̂ is warranted if
and only if(T − t + 1) p̃t xth > p̃t eth + q̃ t−1 bt−1

(13)

The condition above does not take advantage of price supportability: suppose thatthe consumption plan meets the budget with ex post prices, i.e., (3) holds. Summing up
equations from period t to period T with the appropriate weights, we have p̃t xth − eth +PTssst−1 bt−1 . Note that from the budget equations till t − 1, we alsos=t+1 p̂ (xh − eh ) = q̃
h
24

have

Pt−1

s
s
s
t−1 bt−1 = 0; in words, since transactions before t are alreadys=0 p̃ (xh − eh ) + q̃hcompleted and they meet the budget equation (3), bt−1
is equal to the total net savinghaccumulated before t. This relation can also be readily confirmed since by summing up
 P
P
s
s
s
all equations we have p̃t xth − eth + Ts=t+1 p̂s (x̂sh − esh ) + t−1
s=0 p̃ (xh − eh ) = 0. Using
this relation, we see that (13) is equivalent totXp̃s (xsh − esh ) + (T − t) p̃t xth > 0

(14)

s=0

Inequality (14) is trivially satisfied ifPt

s=0 p̃s (xs − es ) ≥ 0, i.e., the discounted value ofhhhousehold h’s consumption is non-negative, which means household h is a borrower atthe end of period t. In general the inequality will hold if household h’s lending is notexcessively large.
Summing up the discussion above, we have,
Lemma 11 Assume uh (z) = ln (z) and eth > 0 for every t. Then consumption xth withoutstanding bond holding bht−1 is justifiable in period t if and only if (13) holds. A PS
consumption stream x0h , x1h , ..., xTh is justifiable if and only if inequality (14) is satisfied
for every t = 0, 1, ...T .
Note that this result is not restricted to being in a neighborhood of the AD equilibrium consumption.

4.2

Justifiability in Model II

Fix a consumption bundle which is budget feasible under given ex post prices, i.e., (5)holds, and fix a portfolio of the bond and the L-bond associated with the consumption.
Thus (5) is satisfied.
We shall first ask if xt , bt , lt is justifiable in period t; Household h maximizes (continuation) utility with some forecast prices and trading plan subject to the (continuation)
dynamic budget (2) in period t, given bt−1
and lht−1 . Recall from remark 3 that in ahTE household h’s forecast prices must satisfy the no arbitrage condition, which meansthat the dynamic continuation budget equation can be reduced to a single AD budget
constraint under the forecasts. Therefore, just like in Model I, with such forecasts in25

mind, household h should be able to reduce the dynamic budget constraint (2) to the
AD budget
t

p̃ x

t

− eth



+

T
X

t t−1
p̂s (x̂s − esh ) = q̃ t−1 bt−1
h + q̃L lh

(15)

s=t+1

where p̂s are forecasts of the discounted price of the good for period s, which arise with
forecasts of the bond prices consistent with the no arbitrage condition from period t + 1
and after.
The utility maximization can be solved by first finding a consumption bundle which
 P
maximizes utility uh xt + Ts=t+1 uh (x̂s ) under the AD budget (15), and then determin
ing bond transactions which finances the consumption xt , x̂t+1 , ..., x̂T . Then xth , bth , lht
of household h is justified in period t if and only if xth is the quantity demanded of thegood in period t under the AD budget (15) for some forecasts about the discounted
prices, p̃t+1 , ..., p̃T , and the associated bth , lht is found to satisfy the dynamic budget (2)for period t.
In principle we need to describe the forecast prices of the bond and the L-bond, butsince the forecasts must satisfy the no arbitrage condition, the two types of bonds mustbe perfect substitutes from the viewpoint of household h when solving the utility maximization problem. It means that when consumption xt is demanded, any combinationbth , lht which satisfies the period t budget can be demanded. Thus if we want to induce
the household to choose a particular position on the L-bond in period t, the period tbudget is a necessary and sufficient condition, and it is not necessary to fix the details of
the forecast bond prices. To sum up, all we need to show is xt is the quantity demanded
under the AD budget (15) for some forecast p̂s , s = t + 1, ..., T . Hence the issue ofjustifiability is essentially the same as in Model I.
Suppose that the demand function under the AD budget is responsive to forecasts
in the sense that as a function of price forecasts, the demand changes in any direction.
This property will be generically true under some mild and plausible conditions on utility
functions and endowments, at least if xth is the AD (thus PFE) consumption in periodt and lhs is zero for t = 0, 1, ..., t − 1, and given that the ex post prices are PFE prices.
We therefore contend that any price supportable allocation close enough to the AD
equilibrium is a TE in Model II if the underlying trade of the L-bond is kept small

26

enough.
In general, when the position of the L-bond is large, it is hard to tell if the demand
function has a desired responsiveness property. We therefore choose to illustrate it for a
specific case of additive log function, i.e., uh (z) = ln (z). Just as in Model I, from (15) wet t−1 plus the forecast (discounted) income
see that the AD income is p̃t eth + q̃ t−1 bt−1
h + q̃L lhPm̂ := Ts=t+1 p̂s esh , which depends on price forecasts, but not time t market variables.

Thus the problem of justifiability is reduced to find a forecast income m̂ such that p̃t xth =t t−1
p̃t et +q̃ t−1 bt−1
h +q̃L lh +m̂
, and a necessary and sufficient condition for the existence is(T −t+1)(T − t + 1) p̃t xth > p̃t eth + q̃ t−1 bt−1 + q̃Lt lht−1 ,(16)which corresponds to (13) in Model I.
In summary, we have
Lemma 12 When uh (z) = ln (z) and eth > 0 for every t, a budget feasible consumption
stream x0h , x1h , ..., xTh is justifiable if and only if inequality (16) is satisfied for everyt = 0, 1, ..., T − 1.
However, since the ex post prices might allow arbitrage between the two types of thebonds, the dynamic budget constraint is not reduced to a single equation. Thus there
is no general counterpart to condition (14) in Model II, which makes the justifiability
issue more complex in Model II. Nonetheless we attempt to make some connection whichis specific to the canonical trading strategy: consider the canonical trading strategy weconstructed to support an arbitrarily given consumption in section 3.2.2. Recall thatwith the canonical strategy, the dynamic budget equations (5) is simplified to (9). Fort with t ≤ T ∗ , the problem of justifiability is identical to Model I, hence (16) can be
reduced to (14) for t ≤ T ∗ . For t > T ∗ , multiplying period s equation in (9) with q̃ s−1 :=Psssq 0 q 1 · · · q s−1 , and summing them up from period 0 to t − 1, we get t−1
s=0 p̃ (xh − eh ) +Pt−1 s sT∗ T∗
t−1 bt−1 = −s
T ∗ T ∗ , where lT ∗q̃ t−1 bt−1
s=0 p̃ (xh − eh ) + q̃L lhhh + q̃L lh = 0, and hence q̃
hwas found in (11). Then (16) readstXp̃s (xsh − esh ) + (T − t) p̃t xth + lhT

∗




∗
q̃LT − q̃Lt > 0

(17)

s=0

The condition above still contains an endogenous variable lhT∗

and so it does notnecessarily give a definitive answer for justifiability. But for a constant consumption27

∗

stream it can be decided since lhT can be explicitly computed. We shall provide here an
example of T = 2 where household h consumes a constant amount x̄h in every period,
which will be required in ETE.
Fix ex post prices, and assume that the period when the no arbitrage condition fails∗

is T ∗ = 0, so that the condition q̃LT − q̃LT −1 6= 0 becomes q 0 q 1 − qL0 6= 0. We shallconsider a canonical trading strategy, where the L-bond is traded only in T ∗ = 0. To
ensure price supportability of the constant consumption x̄h at the ex post prices, we set
b0h , b1h in accordance with the iterative procedure10 specified earlier, and specify lh0 (= lh1 )according to (11); that is,
p0 + p̃1 + p̃2 x̄h − p0 e0h + p̃1 e1h + p̃2 e2h
,q 0 q 1 − qL01b0h = 0 p0 e0h − p0 x̄h − qL0 lh0 ,q
1
b1h = 1 p1 e1h + b0h − p1 x̄h .
q
lh0 =Now we shall examine the justifiability for each period.
Period 2: This holds automatically from the period 2 budget feasibility.
Period 1: Applying condition (16) with t = 1 and T = 2, we find that the necessary
and sufficient condition for justifiability is 2p̃1 x̄h > p̃1 e1h + q̃ 0 b0h + q̃L1 lh0 . Substituting b0h
found above (recall q̃ 0 = q 0 ), it is equivalent to p0 + 2p̃1 x̄h > p0 e0h + p̃1 e1h + q̃L1 − qL0 lh0 .
Substituting l0 found above (recall q̃L1 = q 0 qL1 by definition, and q 1 = qL1 since thetwo bonds are identical in period 1), it is equivalent to p0 + 2p̃1 x̄h > p0 e0h + p̃1 e1h +( p0 + p̃1 + p̃2 x̄h − p0 e0h + p̃1 e1h + p̃2 e2h ), which is reduced to:p̃1 x̄h > p̃2 x̄h − e2h .

(18)Period 0: Applying condition (17) with T = 2 and t = T ∗ = 0, we find that thenecessary and sufficient condition for justifiability is p̃0 x̄h − e0h +2p̃0 x̄h +lh0 q̃L0 − q̃L0 >0, which is readily simplified as
3x̄h > e0h(19)

In summary, constant consumption x̄h is justifiable in every period if and only if
conditions (18) and (19) are satisfied.
10−1
Recall we had set bth iteratively to meet period t budget, pt xth + q t bth = pt ett + bt−1
h , where bh = 0.

28

Notice that condition (18) holds if p̃1 = p̃2 , which is true at PFE where the discounted
prices are constant. Therefore, if the ex post prices are close enough to the PFE prices,then any x̄h > 0 can be justified in period 1. Recall that at PFE, household h consumesx̄h = 13 e0h + e1h + e2h , and so (19) is satisfied.
Remark 13 Assuming the common log utility function, in conditions (16) and (17),forecasts are implicitly determined. Although we take forecasts as purely individualistic,
it is curious if they might be aligned across the household to justify an efficient allocation:e.g., every household correctly forecasts period 2 price in period 0, so that the wealthtransfers induced in the ETE are driven only by forecasting errors of period 1 prices.
Indeed, it is possible to construct such forecasts in the log example, which is demonstrated
by direct computation in Appendix.

4.3

Structure of ETE

As is discussed in the previous section, we contend that a EPS allocation is, at least in
the vicinity of a PFE, tends to be justifiable for all household in Model I, and then it isa ETE allocation. As is pointed out earlier (Remark 6), a TE in Model I can be seen asa special case of a TE in Model II where households choose not to trade the L-bond inany period. Also under its forecasts the L-bond is a perfect substitute of the bond, andhence any position on the L-bond is consistent with utility maximization. The proximity
to a Model I ETE allocation assures that it can be generated with a low volume of tradein the L-bond. Therefore, a EPS allocation in Model II which is close enough to a ETE
allocation in Model I is also justifiable, and hence it is a ETE allocation.
Recall that the EPS allocations in Model I constitute a T dimensional set when H islarge (Proposition 8) whereas the set of EPS allocations in Model II is H − 1 dimensional(Proposition 10). The argument above shows that at least around a PFE, EPS allocations
are also ETE allocations. Then, the set of ETE in Model I is T dimensional, whereas
the set of ETE in Model II is H − 1 dimensional set, revealing the distributional effectof a redundant asset.
We believe that a formal analysis can be carried out with differential topology technique to verify the assertions above, but we also expect that it will be technically involved.

29

We therefore choose to focus on the log utility case we explored in the previous subsections where we can clearly see the structure of ETE explicitly with more elementary
technique. So assume uh (z) = ln (z) from now on and fix an efficient allocation andwrite x̄h for the time invariant consumption of household h. Recall that the allocation
is an ETE allocation if it is an EPS allocation and justifiable for every h.
In Model I, an EPS allocation can be expressed as a convex combination of (normalized) discounted prices, p0 , p1 , ..., pT (Proposition 8). When condition (14) is applied to an EPS consumption where household h’s consumption is a constant stream ofPx̄h = Tt=0 pt eth it reads:
t
X
s=0

s

t

p + (T − t) p

! T
X
t=0

pt eth −tX

ps esh > 0

(20)

s=0

where the first term is the value of the whole consumption stream assuming that thediscounted prices stays the same as p̃t after period t. Therefore, we have
Proposition 14 In Model I, an efficient allocation (· · · , x̄h , · · · ) is an ETE allocation
P
if and only if there are Tt=0 pt = 1, pt > 0, for t = 0, 1, ..., T , such that for every h,Px̄h = Tt=0 pt eth and (20) are satisfied for every t = 0, 1, ...T .

Example 15 In the economy of Example 7, as was seen in Example 9, household hPconsumes x̄h = 2t=0 pt eth = 41 + 3ph − 1 η, h = 1, 2, 3. Household 4 consume x̄4 = 14 .
For household 1, conditions (20) for t = 0 and 1 are





0 1
0
0 1
3p
+ 3p − 1 η − p+ 2η > 0
4
4
  




 1

0
1
0
0 1
1 1
p + 2p+ 3p − 1 η − p+ 2η + p
−η
>0
4
4
4
and the set of EPS for which household 1’s consumption is justifiable can be found bysolving the simultaneous (quadratic) inequalities above. Notice that when η is smallenough, both inequalities are satisfied for any prices. A similar computations show thatthe other households’ consumption streams are justifiable if η is small enough. Thus
when η is small enough, the set of ETE is identical to the set of ETE in this economy.

30

In Model II, any feasible allocation arises as a PS allocation by Proposition 10, hence
an efficient allocation (· · · , x̄h , · · · ) is a EPS allocation. The justifiability condition forthe log case follows from Lemma 12. Therefore we have:
Proposition 16 In Model II, an efficient allocation (· · · , x̄h , · · · ) is an ETE allocation
with given ex post prices if and only if inequality (16) is satisfied for every t = 0, ..., T −1.
Remark 17 Recall that in the two period case, if the given ex post prices are closeenough to the PFE prices, then any efficient allocation where (19) holds for every household h is justifiable, and hence it is an ETE allocation. Thus the set of ETE allocations
is a H − 1 dimensional set containing the PFE allocation.
Example 18 In the economy of Example 7 with the common log utility function, using(19), it can be shown that the set of ETE contains all positive x̄1 , x̄2 , x̄3 and x̄4 with
P
1
1
1
1
h x̄h = 1 such that 3x̄1 > 4 + 2η, 3x̄2 > 4 − η, 3x̄3 > 4 − η, and 3x̄4 > 4 . In particular,household 4’s consumption may be different from 41 by a large amount whereas it must
consume x̄3 = 14 in any of ETE in Model I (Example 15). Hence the predictability
retained in Model I is lost in Model II. In the Appendix we compute the forecasts thatjustify the ETE identified here.
Since an ETE in Model I can be seen as an ETE of Model II (Remark 6), there
are at least two types of ETE in Model II: the first involves no trading of the L-bond,and the ex post prices other than the L-bond prices must be configured so that theallocation is budget feasible for all households, and the ex post prices of the L-bond is
induced from the bond prices by the no arbitrage condition. The second rests on theexistence of an ex post arbitrage opportunity between the two kinds of bonds in someperiod, and the configuration of the ex post prices matters only to the extent that thediscounted (continuation) income must be kept positive for the justifiability argument.
This observation has the following implication about the predictability: an analyst who
uses Model I would find some relation between ex post prices and the consumption
allocation. On the other hand, an analyst who uses Model II cannot relate ex post prices
to the underlying allocation at all: the analyst who observed prices which are very close
to PFE prices cannot infer that the allocation is close to that in PFE.
31

Finally, we comment on the accuracy of forecasts and welfare. Since the constructionof a EPS and its justifiability issue can be established separately, it can be readily
inferred that those who are benefitted from the implicit transfers do not necessarily haveforecasts which are accurate ex post; the former is determined by the construction of an
EPS, whereas the latter is related to the issue of justifiability. This observation is indeed
valid, even in the common log utility model where the quality of a forecast might appear
to be the only source for an advantageous trade.
Example 19 Consider Model I for the economy of Example 7 with the common log
utility function. Household 4 is induced to consume 14 in every period in any ETE,
which can be sustained by time invariant price forecasts, just as in the unique PFE (seeRemark 13). In this sense, household 4’ forecasts are always on the right track. But it
does not imply household 4’s forecasts are necessarily correct ex post. Indeed, as is seenin the Appendix, there are ETE where discounted ex post prices p̃1 and p̃2 are different
from p0 , and then household 4’s forecasts cannot be correct ex post.

5

On Speculative Trade

The so called no-trade theorem asks if a purely speculative trade based on private information is possible in a rational expectation equilibrium. Since it is hard to distinguishspeculative motive from other genuine motives based on perceived gains from trade, workin this literature typically start with an ex ante efficient allocation and ask if there isan equilibrium where trade takes place. If there is one, it can be regarded as a result ofpure speculation. A general conclusion in this literature is that there tends not to be any
purely speculative trade, which is referred to as the no (speculative) trade theorem.11
We can carry out the following exercise with a similar motivation in spirit in ourframework. Suppose that there are many, identical households. The initial allocation isefficient by construction, and a unique perfect foresight equilibrium occurs with no trade.
The question is whether or not there is a non-trivial ETE where households trade in this11

The literature was initiated by Milgrom and Stokey [1982], and a clean, efficiency based formulation

was given in Morris [1995]. See Kajii and Ui [2009] for its relation with the fundamental theorems ofwelfare economics.

32

economy. If there is, one might interpret that the trade is driven by heterogeneous (andincorrect) forecasts, i.e., lack of rational expectation.
Of course, there are many inefficient TE, i.e., households might choose trades that
distort intertemporal efficiency. One might think that there might also be trades basedon heterogeneity of forecasts that preserve efficiency: households whose price forecasts
disagree seem to find (incorrectly) that they have mutually beneficial trading opportunities. Even if the initial endowments are efficient, a household which thinks the pricewill be very low is willing to sell the good today to another household which thinks
the price will be very high. This process might induce effective income transfers amonghouseholds from the ones with good forecasts to the ones with bad forecasts, without
distorting efficiency. But in general this is not straightforward even in Model I since
good forecasts are not necessarily recipient of transfers, as is seen in Example 19.
It turns out that there is no ETE other than PFE for Model I. To see this recall thecharacterization result Proposition 8: The set of EPS allocation is
o
nP

PT
T
t = 1, pt > 0, for t = 0, 1, ..., T , where et = · · · , et , · · · . Since ett et :pp
t=0
t=0
is equal to a constant vector ē for t = 0, 1, ..., T , and since the total resource is timePPTt ē = ē for any element ofinvariant, it can be readily seen that Tt=0 pt et =pt=0this set; that is, it is a singleton set consisting of the initial, no trade allocation, and it isexactly the set of ETE allocations consisting of the PFE allocation. In conclusion, lackof rational expectation does not necessarily invoke trade that leads to another efficient
allocation.
The conclusion, however, is sensitive to the structure of the market: to be more
specific, let T = 2 and consider Model II. There are H identical households, and eachgets H1 in every period. Assume that the common utility function is uh (x) = ln (x).
The analysis of Model II in the previous section has shown that any efficient allocation
1
can arise as an ETE, as reported
where each household’s consumption is greater than 3Hin Proposition 16 and Remark 17. That is, lack of rational expectation might invoke
income transfers among the households with the L-bond which is a redundant asset under
perfect foresight.
The analysis of this section so far suggests first that heterogeneity of forecasts alone
might not constitute motives for market trade without distortion of efficiency. Secondly,33

it suggests that heterogeneity of forecasts combined with ex post arbitrage opportunity
might generate market trades which preserve efficiency. However, households are engaged
in trade not because they see arbitrage opportunities for easy profits. Recall that by
the construction of TE each household holds forecasts which do not permit itself any
arbitrage opportunity. Each household thinks, at any time, there is no such thing as
free lunch in the markets, but nevertheless their trade creates income transfers from thewinners to the losers who are determined “by accident”. We conclude this section withan example to illustrate this point.
Example 20 In the economy of Example 7, set η = 0, and assume the log utility function
for all households. In Model I, the set of ETE coincides with PFE, but in Model II any
1efficient allocation which gives more than 13 × 14 = 12
to every household can arise as

an ETE. When qL0 < 1, i.e., where the long term bond is inexpensive according to the expost prices, those who enjoy consumption higher than 41 every period, i.e., the winners,
buy the long term bond in period 0 as in (11), whereas those who sell the long term bondconsume less than 14 to be the losers. Details about forecasts that justify these allocations
can be found in the Appendix.

6

Final Remarks

We first comment on the generality of our findings. For Model II, the key observation
about the distributional role of redundant assets is very general, as it does not rely onanything but budget equations. It therefore holds in a model with more goods, a longer
period, or with general utility functions. The justifiability problem can be more complex,
because it delicately relies on the property of the demand functions. When the demand
functions can be explicitly calculated, we will be able to derive sufficient conditions forjustifiability analogous to the ones we found for the log utility case, but the conditions
will not be clean and attractive. For general time-separable utility functions, we expectthat generically around a PFE, any PS allocation can be justified for all household, andwe shall leave it for future research.
The message from the speculative trade seems very general. The works in this literature typically asks possibility of speculative trade under rational expectations, and34

thus in particular, the way individual forecasts might be related to private informationis common knowledge. One can view the forecasts in our model being related to some
unmodeled private information, and from this perspective our finding can be understoodas possibility of speculation without common knowledge of forecasts: trade occurs not
only by lack of common knowledge about forecasts but also the structure of (redundant)
assets.
The analysis can be extended to an infinite-horizon setup to show that the main
message is general in this respect as well. In Model I, with a suitable requirement, thedynamic budget constraint can still be reduced to a one-shot AD budget constraint,
and hence the idea of price supportability remains the same. Although we no longer
obtain a clear-cut result about the dimension of PS allocations, it can be verified that
not all efficient allocations around the AD equilibrium are price supportable. In ModelII, by requiring that the forecast budget is reduced to a one-shot AD budget, whichholds automatically from no arbitrage condition in the finite horizon setup, one candemonstrate that any kind of income transfers can be generated if the ex-post prices
do not satisfy the no-arbitrage condition, utilizing suitably modified canonical trading
strategies.
A natural and important extension is to accommodate uncertainty in the model. It
is especially important in the context of our interest in studying the role of redundantassets. In particular, derivative securities, which constitute a rich class of financial assets,
can only be studied in models that explicitly incorporate uncertainty.12 It would beinteresting to investigate whether our observation regarding the indeterminacy of wealthtransfers under efficiency can be made by studying a model with uncertainty whichaccommodates a wider variety of financial assets. We expect that under our formulation,
the presence of assets which are equivalent under rational expectations would provide
some channels of income transfers, and thus would expand the set of attainable (ex ante)
12

The classical Black-Scholes option pricing formula finds the theoretical price of an option contract asa derivative asset assuming the relevant price processes are rationally expected. It has also been argued(first by Ross [1976] and subsequently elaborated by Polemarchakis and Ku [1990], Krasa and Werner[1991], Kajii [1997], among others) that under rational expectations, the presence of options mightcomplete the markets, and consequently a rational expectations equilibrium with options is efficient anddeterminate.

35efficient allocations beyond the set of rational expectations allocations.
We make some observations on the quality of forecasts in our framework.13 For
instance, in both Model I and Model II, an ex post correct forecast might not be agood forecasts from the view point of individual utility level. In particular, in Model IIwith log utilities, households forecasts about period 2 can be set to be ex post correct.
In fact, one can write an example with longer periods, where households’ forecasts are
ex post correct from period 2 and after (so they learn to be perfect forecasters, but
welfare shifts takes place before they become perfect). Thus the forecasts are correct formost periods for all households, and in this sense there is little difference in the qualityof forecasts, but any kind of income transfer might occur in a ETE, and winners andlosers will emerge by accident. While this is an artifact of the log form of utilities, it isnonetheless interesting that efficiency preserving transfers can arise due to short term
forecasting problems which resolve themselves over the longer term. It suggests thatthe phenomenon we have identified need not be incompatible with some sort of learning
behavior that leads to improvements in forecasts over time. This (in)dependence of thequality of forecasts to the quality of life appears to be very general, but more researchis needed to articulate this phenomenon.
We are agnostic about forecasts and provide a methodology that will allow learningand forecasting rules to be incorporated in the justifiability part without impinging on
PS allocations. One can readily accommodate stochastic forecasts, which enlarge the setof justifiable allocations while the structure of PS allocations is unaltered, and hence ourmain points about the wealth transfers remain unchanged. One can choose to require
specific learning and forecasting rules as part of TE, which changes the set of justifiable
allocations, but it does not alter the set of PS allocations.
Finally, we point out that the model we use can be seen as a multiperiod extensionof an earlier model (Chatterji and Kajii [2023]) where we have studied the structure ofintertemporally efficient allocations which can arise as a sequence of temporary equilibria(ETE) in a two period set up with multiple non-storable goods with one nominal asset
13

We have observed in earlier work (Chatterji et al [2018]) that households whose forecasts turn out

more accurate need not be the beneficiaries in the ETE induced by heterogeneous forecasts. The same
remains true here.

36

(bond), and shown that there is a one dimensional set of ETE allocations around each
perfect foresight equilibrium (PFE) allocation, generically in endowments. In this one
good set up, Proposition 14 generalizes the indeterminacy finding of that paper to a
multi-period set up and shows the dimension of the set of ETE is generically one lessthan the number of periods. As we wrote above, we expect this finding to hold moregenerally with multiple goods and general utilities. But for such a generalization, as ispointed out our the earlier model, there are additional complications about justifiability
even in a two period model when utilities are non time-separable.

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38

Appendix
1. We first provide the details of the forecasts used in the example in subsection 4.2,
mentioned in Remark 13 where we assumed a common log utility function:
Fix a household h ∈ {1, 2, 3, 4}. We construct a canonical trading strategy to justify
the constant consumption x̄h where household h trades the L-bond only in period 0.
Note that from the ex post budget constraint we have:p0 x̄h + q 0 b0h + qL0 lh0 = p0 e0h
p1 x̄h + q 1 b1h = p1 e1h + b0h
p2 x̄h = p2 e2h + b1h + lh0
First, we explicitly find the forecast p̂2h , which satisfies
p1 x̄h =p1 e1h + q 1 p̂2h e2h + b0h + q 1 lh0,
2so that x̄h is justified in period 1 given p1 and q 1 . Multiply both sides by 2q 0 and
substituting q 1 lh0 = q 1 p2 x̄h − q 1 p2 e2h + b1h (period 2 budget) and q 1 b1h = p1 e1h + b0h −p1 x̄h (period 1 budget), we have
2q 0 p1 x̄h − (q 0 p1 e1h + q 0 b0h + q 1 lh0 ))p̂2h =
q 0 q 1 e2h

2q 0 p1 x̄h − (q 0 p1 e1h + q 0 p1 x̄h − e1h + q 0 q 1 p2=q 0 q 1 e2h

q 0 p1 x̄h − q 0 q 1 p2 x̄h − e2h=q 0 q 1 e2hx̄h − e2h )(21)

To construct an example of an ETE where the forecast about period 2 price are all
correct ex post, we will normalize q 0 = p1 = p2 = 1 and p̂2h = 1 to simplify calculation
from now on throughout the appendix. Then we find q 1 e2h = x̄h − q 1 x̄h − e2h from (21),that is, (21) holds if q 1 = 1. So set q 1 = qL1 = 1, and then the consumption in period 1
is justifiable. Of course, unless we are at PFE, the forecast p̂1h about period 1 price heldin period 0 might not be correct ex post.
2. Next, for illustration, we explicitly compute forecasts about period 1 price which
justify the ETE found in Example 18 where we assumed a common log utility function:

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p̂1h

h
1

1
1−4η

2

1
1+8η

3

1
1−4η

4


−qL0 + 12x̄1 − 8η + 4qL0 η − 1
−qL0 + 12x̄2 + 4η + 4qL0 η − 1
−qL0 + 12x̄3 + 4η − 8qL0 η − 1(22)

2 − qL0

With the price normalization, a perfectly accurate period 1 forecast is p̂1h = 1. Household
4, who must consume 41 in every period, has a correct forecast 1 only when qL0 = 1, i.e.,there is no ex post arbitrage opportunity. Observe that forecast p̂1h is increasing in x̄h
for h = 1, 2, 3, and thus a household who is benefited from the implicit income transfer
tends to forecast a higher price, not an accurate forecast 1.
3. Finally, we provide the computation of forecasts for Example 20 where we assumed
a common log utility function for each household h. With the price normalization, we
can compute period 1 forecasts explicitly as in (22) by setting p̂1h = −qL0 + 12x̄h − 1 forhousehold h. Since p̂1h is increasing in x̄h , for these two households,
1. if qL0 < 1 (thus 2 − qL0 > 1), a household with x̄h > 41 must have p̂1h > 2 − qL0 > 1,
and a household with x̄h < 14 must have p̂1h < 2 − qL0 . Thus a household who isworse off than in the PFE may be ex post correct, but a household who is betteroff than in the PFE never has a correct forecast.
2. if qL0 > 1 (thus 2 − qL0 < 1), a household with x̄h > 41 must have p̂1h > 2 − qL0 , anda household with x̄h < 41 must have p̂1h < 2 − qL0 < 1. Thus a household who isbetter off than in the PFE may also be ex post correct, but a household who isworse off than in the PFE never has a correct forecast.

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