kt+1 =
θf (et´ - et + (1 - δ)et
(1 + n) (1 + e)
(12)
where yet = f ket with f : R+ → R+
amount of labor.
denotes the production function (2) written in per effective
Proposition 1 According to assumptions 1 and 2, the non-trivial steady state (y, c) is a saddle; see
Azariadis 1993.
Proof. The eigenvalues of the Jacobian matrix J of the system (9), (12) are the roots of the following
characteristic polynomial: Q(z) = z2 — TrJz + DetJ = 0, where both the trace and the determinant
of the Jacobian matrix evaluated at the non-trivial steady state (y, c) are TrJ = 1 + βf00 (k) u0(c)∕(1 +
n)2(1 + e)-p+1 u00(c) + 1∕β(1 + e)p+1, DetJ = 1∕β(1 + e)p+1 respectively. Under the assumption that
β(1 + e)p+1 < 1, then TrJ > 2, and DetJ > 1. Since Q(1) = —βf00 (k) u0(c)∕(1 +n)2(1 + e)-p+1u00(c) < 0,
Q(—1) = 2 + 2∕β(1 + e)p+1 + βf00 (k) u0(c)∕(1 + n)2(1 + e)-p+1u00(c) + 1∕β(1 + e)p+1 > 0 and (TrJ¢2 —
4DetJ ≥ (1 — 1∕β(1 + e)p+^2 > 0, we can conclude that the Jacobian matrix of the system (9), (12) has
a pair of positive real eigenvalues namely z1 , z2 with 0 <z1 < 1 and 1 <z2 . Therefore, the non-trivial
steady state (y, c) is a saddle. ■
4 Bounded Rationality
In this section, I present the lifetime constrained optimization problem of a representative boundedly
rational household and analyze the local stability properties of the competitive equilibrium trajectories
around the non-trivial steady state under various characteristics of the functional forms and assumptions
about the underlying information set.
4.1 The Household’s Problem
Let us consider that households have incomplete knowledge about their environment in the sense that
they do not know the actual law of motion for the physical capital stock per capita given by an equation
of the form: kt+1 = g(kt, Bt), and the relationships described by equations (6), (7) between the market
clearing input prices and the current state variables. At the beginning of every time t for t ∈ Z+ ∪ {0},
each household observes his current physical capital holdings: xt0 = xt, the current state of the economy:
(kt,Bt), and sets current input price expectations to be equal to the prices announced by the ‘Walrasian
auctioneer’: wt0 = wt, rt0 = rt . For each of these prices, every household evaluates his rental income
yt0 = yt = wt + Rtxt, derives his planned consumption for the current and the next period: (ct0,ct1) based
on one-period ahead input price forecasts, and two-period ahead and up physical capital holdings and
consumption expectations.
I consider that the representative boundedly rational household derives one-period ahead price fore-
casts using time invariant expectation functions defined over information sets including observations from
time t — M up to time t — h with h ∈ {0, 1} on the variable being predicted:
wt1 = Ψ (wt-h, wt-1,..., wt-M) (13)
rt1 = Φ (rt-h, rt-1, ..., rt-M) (14)
For h =0, new expectations are formed for every different prices announced by the ‘Walrasian auc-
tioneer’. For h =1, expectations are based on realized prices and are not revised for every different
prices announced by the ‘Walrasian auctioneer’. The parameter M denotes a fixed memory length, with
M ∈ Z+ and M ≥ h. I consider in the rest of the paper that individuals have unlimited memory: M = t.
Therefore, the information set expands as time goes by and includes at time t a total number of t +1 — h
observations on input prices.
Assumption 3 The expectation functions Ψ, Φ: R1+-h+M → R+ are continuously differentiable and
satisfy the following conditions at the permanent regime (w,r): w = Ψ (w, ...,w) and r = Φ (r, ...,r).