Learning and Endogenous Business Cycles in a Standard Growth Model

3PerfectForesight

In this section, I describe the lifetime constrained optimization problem of a representative household
endowed with complete and perfect knowledge about his environment. Then I present the local stability
properties of the perfect foresight competitive equilibrium trajectories around the non-trivial steady state.

3.1 The Household’s Problem

If each household is endowed with complete and perfect knowledge about his environment, then he knows
for any t ∈ Z+ ∪ {0} the relationships between equilibrium input prices and state variables given by
functions W, Γ, the actual law of motion for the capital stock per capita: kt+1 = g(kt, Bt), and both the
non-arbitrage and zero-profit conditions: Rt = rt + δ , πt = 0 respectively. As results, input price and
state variable expectations made by the representative agent are always accurate: w_t^j = w_t+j , r_t^j = r_t+j ,
k_t^j = k_t+j , B_t^j = B_t+j for any t, j ∈ Z₊ ∪ {0}, and his lifetime consumption and capital holdings plans
are always carried out: c_t^j = c_t+j , x_t^j = x_t+j for any t, j ∈ Z+ ∪ {0}. The constrained lifetime utility
maximizing problem of a representative perfect foresight household born at time t may be written as
follows:

V(xt,kt,Bt)=Max {u (ct) + βV (xt+1, kt+1, Bt+1)}

ct,xt+1

s.t. ct+(1 + n)x_t+1= W (kt,Bt) + (1 + Γ(kt,Bt)) xt

kt+1 =g(kt,Bt)                                            ⁽⁸⁾

Bt+1= (1 + e)Bt
given xt , kt , Bt

where V : R³₊ → R denotes the value function. In equilibrium: k_t = x_t for any t ∈ Z+ ∪ {0}. The
formulation of the representative household’s lifetime optimization problem given by (8) is standard in
optimal growth literature. Since all future input prices and state variable levels are known, lifetime
decisions are set once and for all at the beginning of time t.Thefirst-order necessary condition for
consumption expressed in per effective amount of labor may be written as:

₀                   (1 + n)         ₀

^u (ct+1) = FTTT----W₁ . pp ^u (ct)                               (9)

β(1 +rt+1)(1+e)

At every time period, the utility maximizing levels for current consumption and next period’s physical
capital holdings solution to the household’s lifetime optimization problem (8) are functions of the current
realized values for the states variables:

ct = c (xt,kt,Bt)                                             (10)

xt+1 = x (xt, kt, Bt)                                              (11)

Remark 1: For a Cobb-Douglas production technology F, a log-linear instantaneous utility function
u, and a full depreciation of physical capital: δ =1, the constrained lifetime utility maximizing problem
(8) under perfect foresight has explicit solutions given by equations (10), (11).

Remark 2: If constrained lifetime utility maximizing problem (8) cannot be solved analytically, log-
linearizing its first-order condition around a permanent regime is frequently used to approximate equations
(10), (11); see King, Plosser and Rebelo (1988).

3.2 Competitive Equilibrium Trajectories

A competitive equilibrium trajectory for this production economy populated with perfect foresight house-
holds is a sequence of factors prices: {w_t,r_t}_t⁺₌^∞₀ , consumption: {c_t}_t^∞₌₀, physical capital holdings: {x_t}_t^∞₌₀,
input demands: k_t^d, l_t^d _t^∞₌₁ , and output supply: {y_t^s}_t^∞₌₁ such that the following three conditions are sat-
isfied at every time period t for t ∈ Z+ ∪ {0}: i) consumption {ct}_t^∞₌₀ , and capital holdings: {xt }_t^∞₌₀ solve
(8) given {wt ,rt}_t⁺₌^∞₀ ; ii) input demands: k_t^d, l_t^d, and output supply: y_t^s maximize firms’ profits given wt,
rt; iii) wt , rt clear the labor market and the physical capital market: l_t^d =1, k_t^d = xt = kt respectively.

At time t +1, the competitive equilibrium capital stock per effective amount of labor is a function of
both its lagged level and last period’s utility maximizing consumption per effective amount of labor:

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