where At is a technology parameter and Yt, ht and Kt are all in per-capita (household) units.
Assume first that capital can adjust instantly without investment costs. Then equating the
marginal product of labour with the real wage and the marginal product of capital with its
the cost (given by the real interest rate plus the depreciation rate, Rt + δ), we have:4
Fh,t = αY = Wt
Fκ,t = (1 — α ) -j½- = Rt + δ
(9)
Kt
(10)
Let investment in period t be It . Then capital accumulates according to
Kt+1 = (1 — δ ) Kt + It
(11)
The RBC model is then completed with an output equilibrium equating supply and demand
Yt = Ct + It + Gt
(12)
where Gt is government spending on services assumed to be formed out of the economy’s
single good and by a financial market equilibrium. In this model, the only asset accumulated
by households as a whole is capital, so the latter equilibrium is simply Bt = Kt . Substituting
into the household budget constraint (5) and using the first-order conditions (9) and (10),
and (11) we end up with the output equilibrium condition (12). In other words, equilibrium
in the two factor markets and the output market implies equilibrium in the remaining
financial market, which is simply a statement of Walras’ Law.
Now let us introduce investment costs. It is convenient for the later development of
the model to introduce capital producing firms that at time t convert It of output into
(1 — S(Xt))It of new capital sold at a real price Qt and at a cost (that was absent before) of
S(Xt). Here, Xt ≡ -j-ι and the function S(■ ) satisfies S0, S00 ≥ 0 ; S(1 + g) = S0(1 + g) = 0
where g is the balanced-path growth rate. Thus, investment costs are convex and disappear
along in the balanced growth steady state. They then maximize expected discounted profits
Et Dt,t+k [Qt+k (1 — S (It+k/It+k-1))It+k — It+k]
k=0
where Dt,t+k ≡ β—CÇ+k is the real stochastic discount rate and
Kt+1 = (1 — δ)Kt + (1 — S(Xt))It
(13)
(14)
This results in the first-order condition
Qt(1 — S(Xt) — XtS0(Xt)) + Et Dt,t+1 Qt+1S(Xt+1) ∣21 = 1
4Thus Yt = Wtht + (Rt+ δ), the zero-profit condition for the competitive firm.