Annex B
Derivation of FDI Growth Relationship for Foreign Country
Recall we assumed that capital may freely move between borders and that domestic and
foreign capital are perfect substitutes as factor of production and therefore each pay the same rate of
return, r , the world interest rate. Suppose that capital K~* represents physical capital that literally
exists in the foreign country at a particular time. Suppose also that K~ is physical capital that
belongs to residents of foreign country. Hence, K~ - K~* represents the sum of outflow of foreign
investments from the foreign country to the rest of the world (=domestic country in our model).
K~ - K~ * is also called net claims by citizens of foreign country from rest of the world. For matter of
illustration, we assume that K~ - K~* > 0, without loss of generality. The production technology in
the foreign country is represented by
Y~ = F~(K~*,N~) (B.1)
where Y~ is output, K~* is total physical stock available in the foreign economy, and N~ is labor
stock. The optimization conditions for the representative firm entail equality between the marginal
products and the factor prices:
~ . ~ .t.
f '(k k) = ~ (B.2a)
f (k*) -k*f '(~*) = w (B.2b)
In (B.2), k * is capital per person that exists in the foreign country at a particular time, wf is the real
wage rate, rf is the world’s real rate of interest (hence, fr = r ; we use tilde to keep consistency in
notation). Capital accumulation function for the resident of the foreign country is
(B.3)
where k is capital per person owned by domestic residents, nf is the population growth rate, cf is
the consumption per capita. If we substitute for wf from equation (B.2b) into equation (1), the
change in assets per capita can be determined as
rZ r~^ l~ \ _ ~ r~^
fk = ff(kf*)-rf(kf-kf*)-nfkf-cf (B.4)
Note that kf - kf* represents the sum of investments per capita made by foreign country in the
domestic country and that we assume kf - kf* > 0, without loss of generality. Note again that
equation in (C.4) would become the standard equation of motion of Ramsey if the economy were
t
closed, k - k * = 0 . By definition, it must be true that k - k * = ∫ FDIdt , where FDI is the physical
0
capital outflow to domestic country from the foreign country at time t . If we take time derivative of
• •
this identity, we obtain that k - k * = FDI . Hence, we may alternatively express equation (B.4) as
follows: