holdings coincides with the physical capital per capita in the economy: kt = xt for any t ∈ Z+ ∪ {0}. In
this framework, future consumption and physical capital holdings plans may not always be carried out:
ctj 6= ct+j , xtj 6= xt+j for j ∈ Z+ , t ∈ Z+ ∪ {0}, price expectations may be wrong: wtj 6= wt+j , rtj 6= xt+j
for j ∈ Z+ , t ∈ Z+ ∪ {0}, and expectations may be revised as time goes by: ctj 6= ctj+-ii , xtj 6= xtj+-ji for
j ≥ i ≥ 1 where i, j, t ∈ Z+ ∪ {0}.
Proposition 3 If the model under perfect foresight has a unique non-trivial steady state, and assumption
3 is satisfied, then the model with boundedly rational households has the same unique steady state.
Proof. Under perfect foresight, the optimality condition at time t for consumption per effective amount
of labor satisfies equation (9). From the first-order necessary conditions associated with the constrained
optimization problem (19), the optimality condition at time t for actual and planned consumption
per effective amount of labor satisfies: u0 (et) /u0 (e1) = β(1 + e)pρ1/(1 — β(1 + e)p+^ where ρ1 =
rt1 - n - e - ne / (1 + n). According to assumption 3, those two optimality conditions imply the same
steady level for capital stock per effective amount of labor. ■
4.2 Competitive Equilibrium Trajectories under Constant Gain Learning
An intertemporal competitive equilibrium for a production economy populated with boundedly ra-
tional households is a sequence of factors prices: {wt ,rt}t+=∞0 , lifetime consumption: {ct}t∞=0 , beliefs:
wt1 ,rt1 ,ct1 t∞=1 , capital holdings: {xt}t∞=0, input demands: ktd,ltd t∞=1 , and output supply: {yts}t∞=1 such
that the following four conditions are satisfied at every time period t for t ∈ Z+ ∪ {0}: i) current con-
sumption and expected next period’s consumption and physical capital holdings ct, ct1, xt+1 solve (19);
ii) given wt , rt , input demands: ktd, ltd, and output supply: yts maximize firms’ profits; iii) wt , rt clear the
labor market and the physical capital market: ltd =1, ktd = xt = kt respectively; iv) given {wt-i, rt-i }iM=h,
one-period ahead price forecasts: wt1 , rt1 are derived from the expectation functions Ψ, Φ respectively.
In the competitive equilibrium, the capital stock per effective amount of labor at time t +1 is a
function of its lagged value and one-period ahead input price expectations formed at time t:
et+1 = G (et,wl,rl´ (23)
where Gι, G2, G3 denote the partial derivatives of function G: R+ → R evaluated in the steady state
(k, w, r) with Gi > 0, G2 < 0 according to the normality of the consumption commodity stated by
assumption 1, and G3 > 0 under the standard hypothesis that the substitution effect outweighs the
income effect. Let us consider that one-period ahead input price expectation functions (13), (14) are
derived from simple constant gain adaptive expectation schemes denoted by weighted averages between
the last observation and prediction:
w11 = . + λ (w— — wi-ə (24)
rt1 = rt1-1 + μ (rt-h- rt1-1) (25)
where λ, μ ∈ (0, 1] can be interpreted as speed of adjustment parameters. For h = 0, the second terms
of equations (24), (25) represent last period’s forecast errors. These two expectation schemes imply that
each new observation is as important as the previous in making next period’s forecasts. In the limit case of
"fast" learning: λ = μ = 1, constant gain adaptive schemes (24), (25) correspond to the situation in which
households have naive expectations: wet1 = wet , rt1 = rt for t ∈ Z+ ∪ {0}. The competitive equilibrium
tra jectories under constant gain adaptive learning are described by a three-dimensional system in kt, wet1,
rt1 given by difference equations (23), (24), (25). To simplify the local dynamic analysis of the model,
I consider that parameters λ, μ are identical and equal to η with η ∈ (0, 1]. I characterize in the rest
of this section the local dynamic properties of the model around its non-trivial steady state. Depending
whether households’ information sets used in making forecasts include the input prices announced at the
beginning of the time period by the ‘Walrasian auctioneer’: h ∈ {0, 1}, I characterize the eigenvalues of
the Jacobian matrix of the system (23), (24), (25) evaluated at (k, w, r) under various possible values
for G1, a, η where G1 > 0, a ≡ —(1 — α)G2f0 (k) — G3f00 (k) > 0 and η ∈ (0, 1].