For simplicity, all variables have the same unit, say, the Japanese yen. The variables evaluated in the US-dollar unit are to be converted to the Japanese-yen unit variables by multiplying the exchange rate(yen/US-dollar).
The Japanese multi-national firms are assumed to face the following dynamic optimization problem:
where 
is a revenue of home firms at time t,
is a cost of domestic production by home firms at time t,
r is a discount rate,
is a revenue of host firms at time t,
is a cost of foreign production by host firms at time t.
Revenues and costs in the equation (3-2-1) are:
where 
is home firms' product price(deflator),
is home firms' aggregated output,
is a nominal wage rate in Japan,
is an aggregated labor input of home firms,
is the Japanese FDI flows from home firms to host firms,
is an exogenous investment constraint put forward in (A6),
and 
are adjustment cost functions
.
The asterisk(*) represents the variables of foreign production(host firms
in North America).
Two adjustment functions, 
are assumed to be
convex:
The constraint conditions in the optimization problem (3-2-1) are:
where 
is a production function and ¦Ä is a
rate of capital retirement(depreciation).
In order to solve the dynamic optimization problem(3-2-1) under the constraints(3-2-4), we make the following Hamiltonian:
In order to be simple enough to be handled empirically, two adjustment functions are specified as:
where 
are parameters and 
.
The first-order conditions to maximize the Hamiltonian(3-2-5) are:
By substituting (3-2-6-a) and (3-2-6-b) into (3-2-7-a), we get:
By eliminating 
,
from (3-2-7-b),(3-2-7-c) and (3-2-8),
we get the
following equation that determines the optimal FDI:
where
The equation (3-2-9) shows that the optimal FDI is determined by the
difference between the marginal products of capital invested for domestic
production(
) and
the marginal products of capital invested for foreign production
(
).
That is, the larger the relative marginal products of capital invested for
foreign production, the larger the FDI. Equivalently,
the larger the relative marginal products of capital invested for
domestic production, the smaller the FDI.