Y=F(K*,N)
where Y is output, K * is total physical stock available in the domestic economy, and N is
labor stock. The optimization conditions for the representative firm entail equality between
the marginal products and the factor prices:
(3)
f '( k *) = r
(4a)
(4b)
f ( k *) - k * f ′( k *) = w
In (4), k * is capital per person that exists in a domestic country at a particular time, w is the
real wage rate, and r is the world’s real rate of interest. Capital accumulation function for the
domestic resident is
(5)
where k is capital per person owned by domestic residents, n is the population growth rate,
c is the consumption. If we substitute for w from equation (4b) and for r from equation (4a)
into equation (1), the change in assets per capita can be determined as
k& = f (k*)-r(k* -k) -nk -c
(6)
k * - k represents the sum of foreign investments per capita in the domestic country and
without loss of generality, we assume that k* -k > 0. Note from equation (6) that it would
become the standard equation of motion of Ramsey if the economy were closed, k* -k = 0.
The difference between equation (6) and the macroeconomic budget constraint of Ramsey
model is that the domestic economy is incurring rental cost for the total foreign capital that
t
came in until time t . By definition, it must be true that k* - k = ∫ FDIdt , where FDI is the
0
physical capital inflow from abroad at time t . If we take time derivative of this identity, we
obtain that k& * - k& = FDI . Hence, we may alternatively express equation (6) as follows:
k&* = f(k*)-r(k*-k)-nk-c+FDI
(7)
* . ʃz,^ ɪ, ɪ У y y . _ У f '( k ɔ k * ^ 1 ,ʌʌɪ-
Given that y = f (k ), the growth rate of output is g = — = —*--*. Substituting
y f(k*) k*
k&*
respective value of — from (7), we may express growth rate of output as
k*
= f '(k∙)k∙ Γfτ._r (k*-k) -nk- c_ + fdl
gy Z∕J*∖ 7* ' 7* n 7* 7* 7*
f ( k ) k k k k k