A Consideration of the Comparative Cost Model Using Three-Dimensional Diagrams

This study uses three-dimensional diagrams to express the export function, total labor demand function of individual countries, and the world-market supply function, and describes a method to specify the trade equilibrium point. The trade model is constructed as a multi-commodity, multi-country general equilibrium model, but actual simulation is conducted by a twocountry, two-commodity model. The specific factors model is adopted when assuming the production function has a decreasing return to scale and only one variable (i.e., labor input), and wage is assumed to be an exogenous variable. First, the total labor demands of each country are derived as functions of the price of each good, and these total labor demand functions are expressed by three-dimensional diagrams. Second, the “price possibility frontier” domain for each country is defined by the total labor demand function and labor endowment. Next, the equilibrium trade point is specified by the “price possibility frontier” and the restriction on the balance of supply and demand of each good. Last, the normative equilibrium trade point is simulated, using actual data from the Japan–U.S. international input–output table for the year 2000. The simulated normative trade follows the same direction as actual trade. This result suggests the usefulness of this method for analyzing how the internalization of public benefits from forestry or agriculture impacts the pattern of international specialization.


Introduction
The goal of this study is to investigate the impact of the internalization of public benefits from forestry on the pattern of international specialization, using a multi-commodity, multi-country comparative cost model, including intermediate goods as a linear programming problem. Improvements were necessary because the original version of this model does not consider price (Ejiri, 2005). As a preliminary step toward such improvement, this paper presents a method to reveal the process leading to trade equilibrium; the method involves three-dimensional diagrams devised by the author, as well as export and other functions.
Only two countries, Japan and the U.S., are considered here. Two sectors (goods), agriculture (wheat) and manufacturing (cloth), are considered in the first Ricardian model and in the following example using the specific factors model. In the last simulation, which includes actual data, two aggregated sectors are considered. The specific factors model is used as the trade model. The specific factors model describes the production function as follows (Kimura, 2000): In the specific factors model, L in the production functions is supposed to be mobile within a country but not between countries. The variables a K and m K are considered immobile both between and within countries. Here, a K and m K are supposed to be constant during the entire specialization period. Under this assumption, the production function ( , ) There is some evidence that differences in specialization patterns may be explained by the difference in labor productivity alone (Krugman 2003). The correlation between labor productivity and export and import rates is also examined  1996). Figure 1 shows the correlation between r q (Japan/U.S. relative labor productivity) and r e (Japan/U.S. relative export rate), and between r q and r m (import rate), with respect to 11 goods (sectors) chosen from the 13 tradable goods listed in the Japan-U.S. international input-output table for 1995. The excluded goods are "textiles" and "petroleum products" (Note 1). Figure 1. Correlation between Japan/U.S. labor productivity and Japan/U.S. export and import rates Note: / r J A q q q ≡ (Japan/U.S. labor productivity), where, / D : employees, gross value added, output, export, import, and total domestic demand for each sector or good in Japan in 1995, respectively. Although the capital intensities differ among the sectors, Fig. 1 (a), (b) shows fairly high correlations between the variables. Furthermore, although the plots are limited and the two sectors mentioned above are excluded, the figure suggests the validity of explaining differences in the specialization pattern using only the difference in labor productivity.
In this study, the "flow approach" principle is used to decide the exchange rate.
Because foreign currency supply and demand are supposed to occur only through the export or import of merchandise, and because of the capital account, foreign currency reserves are assumed to be constant, and the exchange rate is decided solely so as to maintain the trade balance. Hereafter, "present" or "before specialization" refer to stable state economies before trade, and "future" or "after specialization" indicate the stable state economies after trade. Additionally, "world" indicates the total market consisting only of Japan and the U.S.

Ricardian Model
Here, the Ricardian model is explained as the foundation of the specific factors model, using simple parameters. Table 1 (1) and (2) show the outputs and allocated labor forces of each sector in each country, respectively. Table 1 (3) shows the average labor productivities calculated from these values, while Figure 2 (a) through (c) illustrate the production possibility frontiers of Japan, the U.S., and the world, respectively, for this case (Shinkai, 1973). Table 1 (4) indicates how the output of each good in the world changes if Japan specializes in manufacturing and the U.S. specializes in agriculture. This table and Figure 2 (c) reveal that if each country specializes as in Table 1 (4), the world outputs of wheat and cloth increase by 25 (10 8 t) and 10 (10 8 ㎡ ) , respectively. If these increased goods are allocated "properly" to each country, the economic welfare of each country is expected to increase.
In this model, the range of π (i.e., the yen/dollar exchange rate), in which mutual trade between Japan and the U.S. may occur, is specified as follows (Ito et al., 1994). , q : labor productivity of the sector) holds for each sector in each country, the price of each good is decided as in Table 1 (6). Therefore, the range of π in which mutual trade may occur between Japan and the U.S. is specified as 62.5<π<250 (yen/dollar), as shown in Figure 3.

Three-dimensional Diagrams of the Process toward Trade Equilibrium
By the Ricardian model described above, the outputs of each good in the world increase by the appropriate specialization, and the range of π in which mutual trade may occur is specified. However, by that model, it is not possible to determine the equilibrium amounts of exports and imports or inspect the process leading to trade equilibrium. Therefore, these problems are considered by the specific factors model, making use of three-dimensional diagrams.

Study framework
The framework of this study is described below, with k ( 1, , k m = ) as the country index and i ( 1, , i n = ) as the sector (good) index (that is common to each country) (Allen, 1967;Henderson et al., 1971;Kimura, 2000;Komiya et al., 1979).
(1) Production function (for each country concerned; the situation is the same hereafter): ).

Derivation of the export function and other values
The production possibility frontier, supply function, and other values derived from the above equations are described by equations [7] through [12] below.

Three-dimensional graphs
after the "production possibility frontier." The situation is the same as in Figure 9. Next, to construct a more general framework, each function is referred to by a general type. Each function and condition necessary to decide the trade equilibrium point by the general equilibrium model are described in subsections (1) through (8) below.
(1) Supply function (expressed by the currency of country K) k n X X : output of each good in country K. (2) Function for income (expressed by the currency of country K) [14]   Therefore, the inner domain of OQR shows the domain of underemployment for Japan, as described above. Curve OU represents the Japanese "line of self-sufficient equilibrium" of wheat; that is, the set of ( , ) x y p p , which brings about Jx E (export of wheat from Japan) = 0 in Figure 16. The curve OU also shows Japan's line of selfsufficient equilibrium for cloth; that is, the set of ( , ) x y p p , which brings about Jy E (export of cloth from Japan) = 0 in Figure 17.    Figure 25 is the same as Figure 22, except for the case of π = 107 (yen/$), that is, the case of trade equilibrium. Figure 26 is the same as Figure 25, except for the case in which actual data are used, as stated below.

Setting up the sectors
Here, Japan and the U.S. are again the focus, and two sectors are considered.
The first sector is comprised of four sectors ("agriculture," "forestry," "fishing," and "food"; Note 7), and the second sector is comprised of three sectors ("general machinery," "electric machinery," and "transportation equipment"). The first is set as the sector of Japanese comparative disadvantage (i.e., U.S. comparative advantage). The second is set as the sector of Japanese comparative advantage (i.e., the U.S. comparative disadvantage). The first integrated sector is simply called "agriculture" and the second is simply called "machinery" hereafter. Simulation is carried out by the two-sector, two-country model using actual data.  1)Includes the "forestry", "fishing" and "food" sectors.
2) "o" indicate that the value is the "present" value.
3) ROW; rest of the world. 4) Caluculated using the values in each "Total" column.

Specification of equilibrium outputs before specialization
Outputs (physical amount) of each good are specified by adopting a "dollar value unit" (abbreviated by "$vu" hereafter) as the unit of physical output. The "dollar value unit" is defined as the physical amount of goods that could be bought with one dollar (more precisely, by one US dollar in 2000, the year when the above actual data were collected; Niida 1978). When $vu is adopted, outputs in physical units coincide with outputs in the monetary units in Table 4.
For outputs (i.e., demands) in equilibrium states before specialization, total domestic demands (D o ) in Table 4 are adopted. In general, outputs before specialization should be calculated using the Leontief inverse matrix (Morishima, 1956); however, for convenience, a simpler method is used here.

Specification of α in the production function
If production functions are assumed as in equation [1 below: X : price or output of each good at any time during the process of specialization) Values of α are specified by equation [23].

Specification of λ in utility functions
In the case where the utility function is defined as equation [2], if either 1 λ or 2 λ is specified, then the other is inevitably decided by the relation noted in Table 2.
Insofar as this relation holds, the difference of 1 λ , 2 λ has no effect on exports and imports (Table 5(1)) or prices (Table 5(2)) in the equilibrium state, although this difference does affect the level of utility (Table 5(3)). Therefore, a completely arbitrarily value of 1 λ = 0.5 is set in each country.

Specification of w
As mentioned above, w is supposed to be an exogenous variable and constant during the process of specialization. No difference in wages between the sectors in the same country is assumed, and w is calculated as  Table 4 is adopted for the Japanese wage, and the value in ［10 2 $/person/year］units (i.e., 380.6) is adopted for the U.S.

Results
Table 5 (1) -(3) shows the results from the above parameters. As a result of specialization, agriculture outputs decrease by several tens of billions of $vu and machinery outputs increase by over one hundred billion $vu in Japan; additionally, the former increases by several tens of billions of $vu, and the latter decreases by over one hundred billion $vu in the U.S. This result coincides with the direction of actual trade. 2) Includes the "forestry", "fishing" and "food" sectors.

Conclusions
The method presented in this study offers several novel and advantageous features. First, use of this model enables an intuitive understanding of the process leading to the state of trade equilibrium. Second, because this model approaches the Ricardian model when α →1, a simple comparison can be made between the two models. Finally, if the production function has the fundamental property of decreasing return to scale, other restrictions on the function can be fairly loose. For example, there is no need to assume the same production function in each country.
The previous model (Ejiri, 2005, Note 9) focused on how evaluation of public benefit from forestry affects the pattern of internal specialization. The main goal of the present study is to clarify the direction of future research-it improves on the previous model by incorporating price into general equilibrium trade theory. The results imply that the method presented here, which reveals the principle of comparative cost theory using three-dimensional diagrams, may be helpful for this purpose. It will also be helpful when analyzing how subsidies for public benefits from forestry or agriculture affect the pattern of international specialization (Ejiri, 1999a(Ejiri, , 1999b. In addition, this method will be useful when econometrically analyzing the trade of forest products within a global market (Yukutake et. al., 2003(Yukutake et. al., , 2006(Yukutake et. al., , 2007Yoshimoto et al., 2002), especially when production functions have been set and the proposed model is evaluated in the general equilibrium theory. It could also be useful to evaluate the process of constructing a spatial equilibrium model (Shimamoto, 2002).