Learning and Endogenous Business Cycles in a Standard Growth Model

Yt =θF(Kt,BtLt)=θLtF(kt,Bt)                              (2)

where θ represents the total factor productivity, B_t denotes a labor augmenting technical progress, and
kt = Kt /Lt is the utilized physical capital per employed worker.

Assumption 2 The production function F : R²₊ → R+ is concave, twice continuously differentiable,
positive, increasing, strictly concave, and satisfies Inada conditions.

Population and labor augmenting technical progress are assumed to grow at constant and known
geometric rates n and e respectively:

Nt = (1 + n)Nt-1                                      (3)

Bt = (1 + e)Bt-1                                         (4)

where N0 , B0 are given. I consider that these two laws of motion are part of the individual knowledge.
I assume that the capital stock used in the production process depreciates from any time period to the
next at a constant and known fraction δ . If at every time t, households can borrow or lend consumption
commodities to themselves at the real interest rate r_t , and under the hypothesis that all assets are perfect
substitutes, then the real rental rate of physical capital is always equal to the real interest rate plus the
depreciation rate: Rt = rt + δ . Under this non-arbitrage condition, households are indifferent between
lending consumption commodities to themselves or renting capital stock to firms. If the real interest rate
is positive: rt > 0 for any t ∈ Z+ ∪ {0}, then the entire stock of physical capital owned by households will
be rented to firms. The expected individual holdings of physical capital in j +1 period(s) ahead depends
on the expected individual investment and undepreciated physical capital holdings in j period(s) ahead:
(1 + n)x_t^j+1 = i_t^j +(1- δ) x_t^j . Thus, the expected budget constraint of a household in j period(s) from
time t may be rewritten as follows:

c_t^j +(1 + n)x^j_t ⁺¹ = w_t^j+ 1+R_t^j - δ x_t^j + π_t^j                           (5)

I consider that the economy consists of a large number of identical firms using the same constant returns
to scale production technology given by (2). At every time t, each of them derives its capital demand k_t^d,
labor demand l_t^d , and output supply y_t^s to maximize its current profits: Πt = θF k_t^d, Btl_t^d - Rtk_t^d - wt l_t^d.
From the first-order necessary condition to the representative firm’s time period optimization problem,
each production factors is paid its marginal product: R_t = θF1 k_t^d ,B_tl_t^d and w_t = θB_tF2 k_t^d ,B_tl_t^d under
perfect competition. Since technology exhibits constant returns to scale, each firm realizes zero profits:
Πt = πt =0for t ∈ Z+ ∪ {0}. In equilibrium, the aggregate level of output, utilized physical capital
stock, and employment are equal to their respective aggregate supply and demand: Yt = Nt (ct + it) = Y_t^s,
Kt = Ntxt = K_t^d , Lt = Nt = L_t^d . Therefore at any given time period, the market clearing real wage and
real interest rate depend on the current state of the economy:

wt = W (kt,Bt)                                          (6)

rt =Γ(kt,Bt)                                                (7)

where k₀ and B₀ are given. Forecasting input prices implies knowing the functional forms for W , Γ:
R²₊ → R+ , and the laws of motion for the state variables kt , Bt . The aggregate physical capital stock
at time t depends on both its last period’s undepreciated level, and last period’s aggregate investment:
Kt+1 = It +(1- δ) Kt . In the competitive equilibrium, the law of motion for the physical stock per
capita will be given by: k_t+1 = g(k_t, B_t).

Variables per effective amount of labor are denoted by ye_t^j = y_t^j /B_t+j , xe_t^j = x_t^j /B_t+j , ce_t^j = c_t^j /B_t+j ,
ei_t^j = I_t^j /B_t+j , we_t^j = w_t^j /B_t+j , and are assumed to stay unchanged along with the real interest rate in the
permanent regime: e_t = y, et = x, et = c, et = i, wt = w, e_t = r for t ∈ Z+ U {0}.

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