In the following empirical study, what are called macro-production functions that aggregates each firm's production function are introduced. Macro production functions of both home firms and host firms are assumed to be Cobb-Douglas functions with no constant-returns-to-scale assumptions. That is:
where A is a technological factor(level), ¦Ñ is a rate of capital
utilization,
K is an aggregated capital stock, h is a working hours, L is an
aggregated number of employees, ¦Á is a rate of capital share, ¦Â is a
rate of labor share and asterisk(*) represents
the variables of foreign production by host firms in North America
The assumption of constant-returns-to-scale,
that is put forward in Takenaka et al.(1989):
is not assumed in this paper.
As one can see from (5-1-1), the value of
a rate of capital utilization times the capital stock 
is used as a capital
input, and the value of working hours times the number of employees
(
)
is used as a labor input in the Cobb-Douglas production function.
Under this assumption, marginal productivity of capital invested for domestic and foreign production are:
Since technologies of home firms in Japan and host firms in North America are assumed to be identical as mentioned in (A4), parameters of production function(¦Á and ¦Â), the rate of capital utilization(¦Ñ) and technological factor(level) in the production function(A) are assumed to be the same between the home firm's macro production function and the host firm's macro production function:
So all we have to do is to build the data-set of the
following five variables:
.
Conceptually, the variable 
in (5-1-2) must be equal to
the sum of all home firms' outputs
and the variable 
in (5-1-2) must be
the sum of all home firms' capital stocks.
Likewise, the variable 
must be the sum of
all host firms' outputs and the
variable 
must be the sum of all host firms' capital stocks.
Therefore, ideally, the variables
estimated in the following way should be used in (5-1-2) and (5-1-3).
where a subscript 
denotes the identity of the Japanese MNC,
is the output of the home firm of the Japanese MNC ,
is the capital stock of the home firm
of the MNC , 
is the output
of the host firm of the MNC , 
is the
capital stock of the host firm of the MNC , and
is the number of MNCs.
But unfortunately, it is hard to use this method due to
the statistical shortages. the data from
Annual Report on National Accounts(Economic Planning Agency,
Government of Japan) is used instead in this paper.
More specifically,
the data of the real(base year=1985) GDP in manufacturing sector
is used as in (5-1-2) and the data of the real
capital stock in manufacturing sector is used as in
(5-1-2).
Since the variables 
are not available
statistically, we must estimate and build the
data-set of these variables( ) in the following way.