We present simple general equilibrium models of resource allocation, factor income distribution, and trade among nations that are technologically identical, but different in tastes. Under conditions of autarky, this paper highlights the importance of “tastes” that determine the intra -national factor shares of returns to productive factors. We demonstrate how a stronger taste for specified factor-intensive goods leads to higher returns to factors intensively used. Trade, in our model, however, erases both intra -national and inter -national factor income inequalities caused by taste differences. Trade, therefore, is no good news to those factors and income groups benefitting from autarky. Free trade therefore is a difficult proposal not only in practice, but also in theory.
Main determinants of international trade are known in the literature to include differences in the methods of production (Ricardo), factor endowments (Heckscher- Ohlin), and tastes (or demand patterns a la Linder). While the study of tastes as a determinant of international trade has attracted little attention, we feel that further analysis is warranted. According to Jagdish Bhagwati [ [
Among these three different models the third one is hitherto least known, but not least important. Noting this apparent discrepancy, this paper attempts to narrow it by focusing on taste differences (rather than the similarities that Linder stressed) between otherwise identical trade partners, endowed with both identical technologies and identical sets of productive factors2. Expanding on Linder’s work, we propose to move beyond the focus on similarities in taste, and examine how differences in taste affect international trade. Similar to Linder we will assume that the trade partners are otherwise identical, endowed with both identical technologies and identical sets of productive factors.
Moreover, this paper is also motivated by a beauty of simplicity to present our models of production, distribution, and trade. Proposed accordingly are general equilibrium (GE) models with two factor inputs, two outputs, and two nations. The two factors of production are allocated to two sectors to produce two kinds of goods under alternative conditions of autarky and free trade. Proposed models are therefore referred to as the 2 × 2 × 2 GE models. The simple models are moreover made even simpler by only one CD (Cobb-Douglas) parameter to represent both intra-national differences in technologies and international differences in tastes. Nevertheless, the models are analytically solvable for all the assumed endogenous variables explicitly in terms of the assumed one single parameter.
Our single parameter modeling makes, without much loss of generality, the rich content of GE theory, such as the Stolper-Samuelson, readily available to the general reader as well as specialists in the field.
Our purpose therefore is twofold: 1) to present the simplest possible GE model of production, and resource allocation (pursuant to Ohta [
The original Walras GE model has long remained almost forlorn due to its gigantic structure sometimes referred to as a huge “empty box”. We attempt at making the huge complex model as simple as possible so as to reveal the original importance of the Stolper-Samuelson theory4.
In what follows, Section 2 sets forth the formal mathematical system of equations along with the underlying assumptions to represent our simple 2 × 2 GE model without trade. Section 3 in turn sets forth a 2 × 2 × 2 GE model of trade between two technologically identical nations that, however, have different national tastes. Section 4 discusses the GE solutions (with diagrams) for the autarky economies and the two free trading economies. Section 5 concludes.
We begin with an autarky model predicated on the following assumptions:
(A1-1) A country produces two final homogeneous goods: bananas B and nuts N.
(A1-2) The country is endowed with two primary factors of production: capital K (owned by capitalists) and labor L (owned by workers).
(A1-3) The total factor endowments are normalized to unity: L ¯ = K ¯ = 1 .
(A1-4) The sector production functions are of CD technologies, such that in the sector producing B, the output elasticity of labor is α , and that of capital is ( 1 − α ) , whereas in the sector producing N, the output elasticity of labor is ( 1 − α ) , and that of capital is α .
(A1-5) Consumers (workers and capitalists) in the same country have identical tastes, defined by the same CD utility function, such that in one country, the parameter α is the utility elasticity of good N, and ( 1 − α ) that of good B; in the other country, these utility parameters are reversed.
Pursuant to these assumptions we now present our simplest possible models in what follows.
Let us begin by describing a country, say, the Banana Country, which is producing two kinds of goods, bananas B and nuts N. Their production functions are specifiable subject to Assumption (A1-4) as:
B = g ( L B , K B ) = L B α K B 1 − α (1)
N = f ( L N , K N ) = L N 1 − α K N α (2)
where B ( = g ( . ) ) and N ( = f ( . ) ) are specific CD functions of capital K i and labor L i ( i = B , N ) used in the subscripted sectors, B and N, respectively, the superscript α above represents “output elasticity” of capital in the N sector and also “output elasticity” of labor in the B sector. This is a deliberate contrivance (assumption) to differentiate the two sectors’ methods of production by a single parameter α .
The contrivance is deliberate in the sense that it is intended simply, yet generally, to derive the so-called contract curve CC in the familiar Edgeworth (endowment) box as a rectangular hyperbola in Allen [
Utility functions of workers and of capitalists are defined, by Assumption (A1-5) above, as:
U K ( B K , N K ) = B K 1 − α N K α (3)
U L ( B L , N L ) = B L 1 − α N L α (4)
where U ( . ) s on the left hand side are identical functions of B and N consumed by the factor suppliers K (capitalists) and L (workers) having the same tastes. It is to be noted here that the same parameter α used to represent the production technology above in (1) and (2) is once again used to represent the consumers’ utility functions (3) and (4).
These assumptions lead to the relevant optimization conditions subject to the budget constraints to follow below.
Consumptive Optimum: Equi-marginal utility MU per Dollar
( 1 − α ) ( N K B K ) α p B = α ( B K N K ) 1 − α p N (5)
( 1 − α ) ( N L B L ) α p B = α ( B L N L ) 1 − α p N (6)
where the left hand side numerators are the factor provider K ( L ) ’s MU of banana B, the right hand sides are the corresponding MU of nuts N, and p i ( i = B , N ) is the price of ith good.
Individual Budget Constraints:
r = p B B K + p N N K (7)
w = p B B L + p N N L (8)
where r is rental rate on capital (capitalists’ income), w wage rate (workers’ income), B i bananas consumed, and N i nuts consumed, respectively, by factor owners of K and L.
Related to the consumptive optimum are the productive/allocative optimum conditions, followed in turn by the market equilibrium conditions that must follow consecutively below. Thus,
Productive/Allocative Optimum (Equi-Marginal Rate of Technical Substitution, MRTS):
w r ( = f L f K ) = 1 − α α K N L N (9)
w r ( = g L g K ) = α 1 − α K B L B (10)
where the left-hand sides of equations above represent the relative wage rate to rental rate w / r and the right-hand sides the relative marginal product of factors ( = M P L / M P K ) in the sector i ( = B , N ), respectively. Note that while the left-hand sides, w / r , are identical, the right-hand sides are defined as distinctively different functions of capital/labor ratio K i / L i in sector i ( = B , N ).
Factor market equilibrium:
K B + K N = 1 (11)
L B + L N = 1 (12)
where total factor supplies (endowments), respectively, of K ¯ ( = 1 ) and L ¯ ( = 1 ) , are required to be demanded by (allocated to) two sectors B and N, respectively.
Product market equilibrium:
B = B K + B L (13)
N = N K + N L (14)
where B and N supplied are to be demanded, respectively, by factors K and L in respective sectors.
Finally required for general equilibrium to be reached are the following conditions for income distribution.
Factor Income distribution (Factor Cost):
p B B = w L B + r K B (15)
p N N = w L N + r K N (16)
where the left-hand sides are value outputs (revenues) of B and N, respectively, and the right-hand sides are the corresponding factor incomes distributed to labor L and capital K.
The system of equations above has 16 equations in 16 unknown variables: w , r , p B , p N , K B , L B , K N , L N , B K , B L , N K , N L , B , N , U L , and U K . However, in light of the Walras’ Law only 15 equations are independent. So the system is determinate only if all the prices are treated as relative prices, relative to, say, p N . Given the specific production functions and utility functions of the CD type as assumed above, our autarky GE-equation system can then be solved explicitly for each/all the 15 endogenous variables above in terms of one single parameter, namely α . All these parametric autarky GE solutions― for both Banana (b) and Nuts (n) countries―are given in
The following observations are important:
1) When α = 1 / 2 : The two production functions become identical. Also the consumer tastes are such that both bananas B and nuts N are equally appreciated.
2) When α > 1 / 2 : The output elasticity of N with respect to capital is larger than that with respect to labor. It follows that the N sector becomes capital intensive, and the B sector labor intensive. It also means that the utility elasticity of nuts in Banana country (b), (3)-(4), is higher than that of bananas, implying a stronger taste for N than for B6.
3) The larger the parameter α (exceeding 1/2), the higher the capital intensity of the capital-intensive N sector is, also the higher the labor intensity of the B sector by comparison, and vice versa7. This in effect is a corollary to 2), and an increasingly higher α also requires an increasingly higher taste for N than for B, and vice versa8.
The observations above lead to certain particular relations of optimal factor allocations and output mix known, respectively, as the so-called “contract curve” CC and the “production possibilities frontier” PPF. The CC and PPF, however, are pre-GE outcomes derived from the technology/endowment Assumptions (A1-A4) only. Although the tastes/preferences (A-5) are yet to be introduced for autarky general equilibrium to obtain, note how CC may be related to PPF.
Focusing on autarky, when α > 1 / 2 ( α < 1 / 2 ), we shall see that our contract curves are concave (convex) curves in the factor space ( L , K ). The optimal sectoral factor allocation condition is given by combining (9) and (10):
M R T S B = M R T S N between Sectors B and N . (17)
This equation is further by (9-10) written as:
{ − d K B d L B = K B L B α 1 − α } = { − d K N d L N = K N L N 1 − α α } , (18.1)
Hence we get,
K B L B α 1 − α = K N L N 1 − α α . (18.2)
which combined with the conditions of a given total endowments of L and K, both assumed to be unity (11-12) yields a unique relation between L N and K N (and hence between L B and K B ) in the factor space ( L , K ):
K N = 1 1 − ( 1 − 1 α ) 2 ( 1 − 1 L N ) = L N L N + ( 1 − 1 α ) 2 ( 1 − L N ) (19)
Geometrically, the expression (19) represents concave or convex curves that are rectangular hyperbolas, cf. Allen [ [
Connected to (19) is another curve (always concave), relating the ordered efficient pairs ( L N , K N ) from the CC curve to their ordered pair (B, N) of maximum outputs, shown in a separate quadrant (B, N) in
N = T ( B ; α ) (20)
Regarding (19) and (20), both CC and PPF are uniquely given for any given CD parameter α as aforementioned (See (3) and (4) supra). Thus, given the technologies underscoring α being the same for the two countries, their CCs are the same, and so are the PPFs10.
Consider now a point ( L N 0 , K N 0 ) on CC and another related point ( B 0 , N 0 ) on PPF, both points labeled E n ,
Though not shown explicitly upon
These distinct tastes of the two otherwise clone nations (with α > 1 / 2 ) can thus be represented, on both PPF and CC curves identified either above or below midpoints thereof, by the Nutties’ taste α > 1 / 2 and the Bananans’ taste ( 1 − α ) < 1 / 2 , respectively. Specifically, the partitioned parts (of points on CC and PPF) above their midpoints represent the Nutties’ equilibrium, and the parts below represent the Bananans’ counterpart optimum, to be explained more fully in the next section.
Given the Nutties’ taste parameter α > 1 / 2 , cum the technology parameter α > 1 / 2 , the share of returns to capitalist (rental income) must strictly exceed 1/2 and the share of wage income, by comparison, fall below 1/2 in the Nuts country. It then follows that the Nuttie workers under autarky must be poorer than their capitalist neighbors accordingly11.
Specifically, wage rate given by factor allocation identified at any point along the CC curve above its midpoint (circled in white above) must be strictly lower than unity. At this white-circled point, MRTS is required to be unity12. The corresponding wage rate at this point is unity. Not only relative factor price is unity at this point on CC, but also unity is the corresponding relative product price identified at a point on the PPF on the right13.
Thus, on CC and PPF above one single parameter α > 1 / 2 can be used to identify two points above and two points below the two white circles, respectively. The two such distinct points along PPF show two distinct national tastes and two related points along CC, by contrast, represent distinctive resource allocations of two technologically clone nations. The four points identifiable by just one parameter α > 1 / 2 along PPF and CC above thus point to equilibrium resource allocations and related output mixes of two clone nations, having different tastes, however.
Related observations of particular importance are:
1) Even within the confines of our “ideal types” model, “ruthless outcomes” of income distribution aforementioned are unavoidable unless α = 1 / 2 , which requires two clone nations to be identical in both tastes and technologies. If people have an unbalanced and skewed taste, for either nuts or bananas, the degree of bias will, by our basic assumption, influence the methods of production. The stronger the skewed taste, the greater the differences in the methods of production will be in equilibrium between the two sectors. Thus, the more the α deviates from α = 1 / 2 , the greater the income inequality.
2) Such a ruthless outcome is inevitable despite the larger pie of nuts and bananas, under the conditions of distinctively different methods of production.
3) No ruthless outcomes will arise only if methods of production are assumed identical, α = 1 / 2 .
4) Only under such special (technological) conditions everybody will be equally remunerated, but equally poor as well, regardless of tastes.
The two autarky economies with all their associated endogenous variables are represented by their explicit solutions in terms of the single parameter α in
The simplest possible 2 × 2 × 2 general equilibrium model of trade, once again by the same one single CD parameter, is now in order, but with basically the same assumptions as those introduced in Section 1. Assumptions with an asterisk below, however, should be noted for differences from the preceding assumptions.
(A2-1)* There are two clone nations: Nuts Country and Banana Country, respectively labeled n and b.
(A2-2) Same as (A1-2).
(A2-3) Same as (A1-3).
(A2-4) Same as (A1-4).
(A2-5)* The two countries’ national tastes are different such that n’s taste is represented by the same CD parameter of α for nuts and ( 1 − α ) for bananas, whereas the b’s taste is reversed; the b’s taste for nuts is given by ( 1 − α ) , and that for bananas by α .
The assumed differences, both in methods of production and in tastes, are
| Country | Variable Definitions | Equilibrium Solutions for n | Equilibrium Solutions for b | |
|---|---|---|---|---|
| (1) | L employed in j to produce N | |||
| (2) | L employed in j to produce B | |||
| (3) | K employed in j to produce N | |||
| (4) | K employed in j to produce B | |||
| (5) | N produced in j | |||
| (6) | B produced in j | |||
| (7) | Relative wage rate in j | |||
| (8) | Relative price of B in j | |||
| (9) | B produced in j and consumed by K (capitalists) | |||
| (10) | B produced in j and consumed by L (workers) | |||
| (11) | N produced in j and consumed by K (capitalists) | |||
| (12) | N produced in j and consumed by L (workers) |
NOTE: In the table M R T j ≡ ( − d N j d B j ) and M R S j ≡ ( − d N j d B j ) , then M R T j = M R S j = p B j p N j , ( Country j = b , n ) at the autarky equilibrium. Also note that given the unrestricted domain for α , each country may prefer either one of the two goods to the other. Thus, if α < 1 / 2 , contra the
thus represented by one single CD parameter α , signifying both technology and taste/preference. Pursuant to these assumptions we now proceed to derive 2 × 2 × 2 open general equilibrium conditions of production, related resource allocation/distribution, and consumption.
We begin by defining the two clone countries’ production functions in terms of only one CD parameter α to differentiate the two sectors’ production functions:
The B Sector:
B j = L i j α K i j 1 − α , i = B , N ( Sectors ) ; j = b , n ( Countries ) (21)
The N Sector:
N j = L i j 1 − α K i j α , i = B , N ( Sectors ) ; j = b , n ( Countries ) (22)
where the subscripts i and j stand for the sector i (= B for bananas, N for nuts) and the Country j (= b for Bananans, n for Nutties), respectively. Subject to these production functions the optimization conditions for factor allocations yield a convex production possibilities set with its frontier PPF: concave function from B j to N j .
Given α > 1 / 2 (or < 1 / 2 ), either one of the two identical PPFs for the clone countries b and n may be transposed and made tangent to the other PPF for purposes of international trade. The two such PPFs that are tangent to one another and back-to-back are illustrated by Ohta [ [
Let us next consider the two countries’ indifference curves that are derived from their national utility functions represented by the same single parameter α as follows.
The Bananans:
U ( B b c , N b c ) = ( B b c ) α ( N b c ) 1 − α (23)
The Nutties:
U ( B n c , N n c ) = ( B n c ) 1 − α ( N n c ) α (24)
Note carefully here that although U’s of the two national consumptions B b c and N b c for country b, and B n c and N n c for country n are defined by the identical parameter α , the larger the α > 1 / 2 , the stronger are both the Nutties’ taste for nuts and the Bananans’ taste for bananas14. Total consumptions by individual factor suppliers must add up to total outputs B j and N J produced in both countries. Thus, individual factor consumptions (9) to (16) on
The conditions for consumptive optimum are then given by:
M R S B N j = p B j p N j , j = b , n ( Countries ) (25)
The related conditions for optimizing factor allocation are given by:
M R T S B N j = w j r j , j = b , n ( Countries ) (26)
Given the specific production functions and utility functions of the CD type as assumed above, the system can then be solved for all the 16 endogenous variables above in terms of one parameter, namely α in each country.
Remember that the system of equations set forth in Section 1 applies to either country b or n before trade (autarky). It is now solvable again in terms of α only ( 1 > α > 0 ) for the 12 variables L j N , L j B , K j N , K j B , N j , B j , ( w / r ) j , ( p B / p N ) j , B j K , B j L , N j K , and N j L ( j = b , n ) , are summarized in
Numerical solutions to the model are also derivable.
Thus, if both the methods of production and tastes for goods are different enough ( α < 0.18 or α > 0.82 ), then the greater the differences in taste and technology, the greater is the number of both goods under autarky. This is not to say, however, that a small difference in taste or technology is bad news. Any small deviation from α = 1 / 2 indeed yields a strict increase in national welfare as implied by a solid locus staying above the linear 45 degree line representing PPF in the case α = 1 / 2 ,
This particular set corresponds to, and is also shown by, a U-shaped non-monotonic B A T n ( = N A T b ) curve vis-a-vis a monotonically increasing N A T n ( = B A T b ) curve in
Also depicted here, strictly above the two asymmetric curves N A T n ( = B A T b ) and B A T n ( = N A T b ) , are two symmetric U-shaped curves. Of the two, note the lower curve is a vertical sum of the N A T b + N A T n and B A T b + B A T n , hence labeled N A T n + N A T b ( = B A T n + B A T b ). It depicts the aggregate output either bananas or nuts, produced by the two countries n and b, under conditions of autarky. The higher, topmost U-shaped curve, by comparison, depicts the aggregate world output (of either N or B) still greater under free trade than under autarky.
As a further related note to
numerical values are model-specific15, then any further deviation thereof in α will increase the production of both nuts and bananas in a given clone country, be it the Banana Country or the Nuts Country. However, insofar as α remains within the normal domain in-between 0.18 < α < 0.82 , then there will be strict trade-offs between nuts and bananas produced as α changes in either direction. Thus, for example, if the value of α approaches 0.18 or larger, then the output of bananas must keep declining while nuts start to increase (along the U-shaped curves of B A T n ( = N A T b ) and N A T n ( = B A T b ) , respectively.)
Within this normal domain (reflecting more balanced tastes and technologies), nuts increase with the parameter value α more than bananas decrease and the former output surpasses the latter output as long as α remains to exceed 1/2. The nuts production keeps increasing while bananas keep decreasing until α reaches 0.82. Remember in this connection that the larger α implies not only a higher output elasticity of capital in the production of nuts, but also a stronger preference for nuts. So, more nuts are produced than bananas. This is why the aggregate two-country output (of either N or B, i.e., N A T n + N A T b or B A T n + B A T b ,
It warrants emphasis, however, that despite a strong taste for nuts, along with a high technology parameter α , high enough to approach unity, substantial amounts of bananas are also produced nonetheless. This is due to the high and low technology parameter effect of differences in the methods of production in the two assumed sectors. Related to this, and perhaps more important, is a certain intrinsic property of the CD function. Its factors are essentials as well as substitutes. This is because each factor is indispensable for the other. (Labor is as indispensable as capital is).
However, once again, this is not to say that everybody will be happier if only national welfare increases with α beyond 1/2 (or with smaller α below 1/2). This is because unless α = 1 / 2 any taste differentials tend to bring about unequal income distribution under the market conditions of perfect competition. Behind the locus of ( B n , N n ) as well as of ( B b , N b ), in
The particular factor is that used more intensively than the other factor. Thus in our present example of the nuts sector that uses capital more intensively than labor, the greater the α (implying greater taste for nuts as well as greater output elasticity of capital), the greater the output and hence capital input in the capital-intensive sector. However, the other factor does not increase at all, remains instead its sector allocation constant, (1/2) as shown in
Since the greater α implies the greater taste for bananas, as in the Banana Country, then the same greater parameter α requires the smaller taste for bananas in the Nuts Country, being represented by 1 − α , i.e., lower taste for bananas. If the Nuts Country’s more labor-intensive nuts sector is in greater demand, then more and more labor is needed to produce nuts while capital input remains constant, as in
Related to impacts of α upon total as well as sectoral outputs (
loci. In the case of free trade between countries b and n, note that the system of equations set forth in Section 1 can be basically reapplied to both countries b and n under free trade. It is once again solvable in terms of α only. The free trade GE solutions of our 2 × 2 × 2 CD model are summarized in
In comparing
1) When α = 1 / 2 , the two countries are genuine clones producing both nuts and bananas equally and distributing them equally between workers and capitalists. So, w / r = 1 .
2) For any α > 1 / 2 , w / r is strictly less than unity in one country (Nuts
| Country j | Variable Definitions | Equilibrium Solutions for n | Equilibrium Solutions for b | |
|---|---|---|---|---|
| (1) | L employed in j to produce N | |||
| (2) | L employed in j to produce B | |||
| (3) | K employed in j to produce N | |||
| (4) | K employed in j to produce B | |||
| (5) | N produced in j | |||
| (6) | B produced in j | |||
| (7) | Relative wage rate in j | 1 | 1 | |
| (8) | Relative price of B in j | 1 | 1 | |
| (9) | N produced in j and consumed by K (capitalists) in n | |||
| (10) | B produced in j and consumed by K (capitalists) in n | |||
| (11) | N produced in j and consumed by L (workers) in n | |||
| (12) | B produced in j and consumed by L (workers) in n | |||
| (13) | N produced in j and consumed by K (capitalists) in b | |||
| (14) | B produced in j and consumed by K (capitalists) in b | |||
| (15) | N produced in j and consumed by L (workers) in b | |||
| (16) | B produced in j and consumed by L (workers) in b |
NOTE (1): M R T = M R S = p B j p N j = 1 , ( Country j = n , b ) at the free trade equilibrium between Countries n and b. NOTE (2): (5) = (9) + (11) + (13) + (15) for good N, and (6) = (10) + (12) + (14) + (16) for good B.
Country here) and greater than unity in the other country (Banana Country). This implies that the workers in one country are poorer and those in the other country richer, respectively, than their capitalists.
3) The intra-national income gaps furthermore widen increasingly within each of the two clone countries as α increases.
4) Thus, it warrants emphasizing that no international taste differential is needed for intra-national income inequality to occur. If only a skewed taste for one good over another exists, then intra-national income disparity is inescapable insofar as output elasticity is not equal to 1/2.
5) It is thus not free trade per se that would cause income inequality, inasmuch as a domestic market can create horrendous factor price differentials even with no free trade.
6) Under autarky not only intra-national income gaps but also international income gaps widen increasingly with α . Thus, for example, if α exceeds 1/2 or becomes larger, as seen from
7) Free trade, although often alleged to bring about “ruthless” outcomes, actually favors all deserving individuals with factor price equalization. Here in our model complete equalization of incomes among all equally gifted, not only internationally, but also intra-nationally. All these outcomes are depicted in
Let us summarize before we conclude. Commodity prices are distinctively different before trade and so are factor prices between almost clone nations. One
factor could be very expensive because the preferred commodity had forced both factors to be used for a product that they are not equally suited to produce. The opening of trade permits the two countries to specialize less in production, and more in consumption, in accordance with their different tastes. All this is accomplished by factor reallocations induced by trade, thereby equalizing both commodity and factor prices. Thus, worldwide output gains from trade arise from free trade. And all countries could increase their consumption to a higher level along with a higher national welfare level.
So, everybody ought to be happy with the outcomes of free trade as shown above. They ought to be, but aren’t. The reason is that the different tastes under conditions of autarky, keep a certain factor in each nation scarcer than the other nation’s identical factor even if their physical endowments are the same. Higher wages in one nation and higher rents in the other before trade, however, are nothing but quasi-rent. The market mechanism with free trade would surely wipe it out. Here lies an apparent incentive for rent seeking, so as to protect the quasi-rent as a vested interest granted under autarky government/legislation.
A simple GE model of production, distribution, and trade has shown some interesting rediscoveries of fundamental principles of economics, revealing the importance of intra-national differences not only in technology (even if human capability were alike), but also in taste, both skewed.
Another discovery is the importance of not only the skewed taste within a country, but also international differences in skewed tastes, in creating large income gaps not only intra-nationally but also internationally. Such ruthless outcomes of income distribution (among equally gifted individuals) are inescapable only if no trade is permitted internationally. With trade, factor price equalization necessarily yields a stumbling block (revealed by the Stolper- Samuelson Theorem) to the vested interest group. They benefit from a skewed national taste before trade, but lose after trade, all the more if only the skewed taste cum technology parameter α deviates more from α = 1 / 2 , reflecting greater differences not only in the methods of production within each country but also in tastes internationally.
A related finding is how the output effect of skewed methods of production can outweigh the negative effects of skewed tastes upon a production possibility frontier. Such a net output effect of a larger α , our single parameter assumed, may also sound surprising all the more for its paradoxical net output effect of skewed methods of production, outweighing the negative skewed taste effects upon a production possibility frontier.
A sheer skewed difference in tastes, combined with skewed technology, can aggravate devastating ruthless outcomes in income distribution, as
Needless to say, our research is limited by the fact that we are presenting an “ideal types” model of a complex real world economy. As such, we have simplified our ideal world to include only those factors that we have deemed most salient. It is possible that we have included or excluded factors that may later prove to be significant or insignificant. While we feel confident that we have presented the best possible model, further research will elucidate complexities that we may have yet overlooked.
We are greatly indebted to Murray Kemp, Martin McGuire, Wen-Jung Liang for their expertise critiques and helpful suggestions on earlier versions of this paper. Extensive comments and suggestions by both the Editor and an anonymous referee are also gratefully noted. This research was financially supported by the Japan Society for the Promotion of Science under the Grant-in-Aid for Scientific Research (C) (JSPS KAKENHI) Grant Number 24530306.
Kawano, H., Ohta, H., Jensen, B.S. and Hwang, A.R. (2017) Production, Distribution, and Trade: General Equilibrium Models with One Single Cobb-Douglas Parameter. Modern Economy, 8, 970-993. https://doi.org/10.4236/me.2017.87068
Thus we must here obtain the autarky GE function Ψ and its inverse Ψ − 1 :
k j = K j / L j = Ψ j ( w j / r j ) , w j / r j = Ψ j − 1 ( k j ) ; j = b , n (27)
With CD two-sector technologies and CD consumer preferences, the Walrasian general equilibrium (27) becomes, see Jensen [ [
k = Ψ ( w / r ) = a 2 − α j ( a 2 − a 1 ) 1 − a 2 + α j ( a 2 − a 1 ) ⋅ ( w / r ) ; j = b , n (28)
Our special CD parameter assumptions―sector N = sector 1, sector B = sector 2―were:
a 1 = α ; 1 − a 1 = 1 − α ; a 2 = 1 − α ; 1 − a 2 = α ; α n = α ; α b = 1 − α (29)
Inserting (29) into (28), we get the basic GE formulas for our two autarky economies (n, b):
k n = Ψ n ( w n / r n ) = 1 − 2 α ( 1 − α ) 2 α ( 1 − α ) ⋅ ( w n / r n ) (30)
w n / r n = Ψ n − 1 ( k n = 1 ) = 2 α ( 1 − α ) 1 − 2 α ( 1 − α ) (31)
k b = Ψ b ( w b / r b ) = 2 α ( 1 − α ) 1 − 2 α ( 1 − α ) ⋅ ( w b / r b ) (32)
w b / r b = Ψ b − 1 ( k b = 1 ) = 1 − 2 α ( 1 − α ) 2 α ( 1 − α ) (33)
Relative factor price expressions (31) and (33) are seen in
The CD relative commodity price (unit-cost) functions in our two-sector eco- nomies are given by, cf. Jensen [ [
p 1 p 2 = γ 2 a 2 a 2 ( 1 − a 2 ) 1 − a 2 γ 1 a 1 a 1 ( 1 − a 1 ) 1 − a 1 [ w r ] a 2 − a 1 ; p B p N = p 2 p 1 = [ p 1 p 2 ] − 1 ; ( w / r ) ∈ [ 0 , ∞ ) (34)
With γ 1 = γ 2 = 1 and using (29), the relative price formulas (34) are easily seen to become,
p 1 p 2 = ( w / r ) 1 − 2 α ; p B p N = ( w j / r j ) − 1 ( 1 − 2 α ) ; j = b , n (35)
By relative factor prices (31) and (33) and the relative price (cost) expression (35), we get:
p B n p N n = ( w n / r n ) − 1 ( 1 − 2 α ) = [ 1 − 2 α ( 1 − α ) 2 α ( 1 − α ) ] 1 − 2 α (36)
p B b p N b = ( w b / r b ) − 1 ( 1 − 2 α ) = [ 2 α ( 1 − α ) 1 − 2 α ( 1 − α ) ] 1 − 2 α (37)
The relative commodity price expressions (36) and (37) are seen in
By the Equations ((9), (10)) of section 2.2, and wage-rental expressions (31), (33), we get,
k N n = α 1 − α ⋅ w n r n = α 1 − α ⋅ 2 α ( 1 − α ) 1 − 2 α ( 1 − α ) = 2 α 2 1 − 2 α ( 1 − α ) (38)
k B n = 1 − α α ⋅ w n r n = 1 − α α ⋅ 2 α ( 1 − α ) 1 − 2 α ( 1 − α ) = 2 ( 1 − α ) 2 1 − 2 α ( 1 − α ) (39)
k N b = α 1 − α ⋅ w b r b = α 1 − α ⋅ 1 − 2 α ( 1 − α ) 2 α ( 1 − α ) = 1 − 2 α ( 1 − α ) 2 ( 1 − α ) 2 (40)
k B b = 1 − α α ⋅ w b r b = 1 − α α ⋅ 1 − 2 α ( 1 − α ) 2 α ( 1 − α ) = 1 − 2 α ( 1 − α ) 2 α 2 (41)
The GE sectoral capital-labor ratios, (38-41) are implied by the lines (1-4) in
The fractions of labor (capital) of country (j) employed in sector 1 are given by, cf. Jensen [ [
L 1 j L j = α j ( 1 − a 1 ) 1 − a 2 + α j ( a 2 − a 1 ) , L 2 j L j = 1 − L 1 j L j ; K 1 j K j = α j a 1 a 2 − α j ( a 2 − a 1 ) (42)
Inserting (29) into (42) gives sector fractions of labor (capital) for autarky economies (n,b):
L N n L n = 1 2 = L B n L n ; K N n K n = α 2 1 − 2 α ( 1 − α ) (43)
L N b L b = ( 1 − α ) 2 1 − 2 α ( 1 − α ) , L B b L b = 1 − L N b L b ; K N b K b = 1 2 = K B b K b (44)
which appear in the lines (1-4) of
N n = L N n k N n α = 1 2 1 − α [ α 2 1 − 2 α ( 1 − α ) ] α , N b = L N b k N b α = 1 2 α [ ( 1 − α ) 2 1 − 2 α ( 1 − α ) ] α (45)
B n = L B n k B n 1 − α = 1 2 α [ ( 1 − α ) 2 1 − 2 α ( 1 − α ) ] 1 − α , B b = L B b k B b 1 − α = 1 2 1 − α [ α 2 1 − 2 α ( 1 − α ) ] α (46)
which are seen in lines (5-6) of
We have assumed free trade between our two countries, j = b , n . Thus by absence of frictions of trade, the law of one commodity price, and hence one relative price (p) apply:
p B j = p B , p N j = p N , j = b , n : p b = p B / p N = p n = p (47)
Country trades are here always balanced, and world market equilibrium implies:
X B j = B j − Q B j = − X B n = − ( B n − Q B n ) (48)
where X B j are exports (imports = − X B j ) of good B by country (j); Q B j is domestic demand (absorption) of good B by country (j). The relative price p, (47), (world market equilibrium terms of trade) is derived by solving the trade balance (“reciprocal demand, offer-curves”) equation, (48)―reflecting specific forms of production and utility functions in both countries. With parameters (29) and k b = k n = 1 , it can be proved, cf. (54), that
p b = p B / p N = p n = p = 1 (49)
Then by (49) and (35), we get wage-rental ratios, and next capital-labor ratios by (38-41):
w b / r b = w n / r n = 1 ; k N n = α 1 − α , k B n = 1 − α α , k N b = α 1 − α , k B b = 1 − α α (50)
With L N n / L n = L N b / L b = 1 − α , same technologies and factor prices imply same outputs:
N n = L N n k N n α = ( 1 − α ) [ α 1 − α ] α = α α ( 1 − α ) 1 − α = L N b k N b α = N b (51)
B n = L B n k B n 1 − α = α [ 1 − α α ] 1 − α = α α ( 1 − α ) 1 − α = L B b k B b 1 − α = B b (52)
Remark. For parameter set (29), relative prices (49) are replaced (not proved here) by,
p = p B / p N = [ 2 α ( 1 − α ) k b + [ 1 − 2 α ( 1 − α ) ] k n ] 2 α − 1 = Φ ( k b , k n ) (53)
k b = k n = k ¯ : p = p B / p N = k ¯ 2 α − 1 ; k ¯ = 1 : p = p B / p N = 1 (54)
Thus (49) [independent of α]―with α-expressions (50-52), seen in