globalization and income distribution inequality in ...oh6.pdf · i: a specified case traditional...

Globalization and Income Distribution Inequality in Heckscher-Ohlin Model II: A Simulation for Small Country Case with

Non-Traded Good

Toshitaka Fukiharu, JiaNuo Sun(Faculty of Economics, Hiroshima University)

October, 2010

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Heck$Oh6.nb 1

Introduction

As developing countries, such as China and India, attained the economic development, the inequality of incomedistribution has been discussed. The argument of inequality of income distribution is also heard in the developedcountries. On the one hand, it is quite important to investigate whether the inequality has expanded worldwide empiri-cally. On the other hand, it is also quite important to investigate from the theoretical viewpoint of whether the inequal-ity expands through the change of exogenous variables. In the previous papers, Fukiharu [2008, 2009] conductedtheoretical examination: whether the inequality expands through ICT innovations. In Fukiharu [2010], it was examinedtheoretically whether the inequality expands through the globalization of a small country, which possesses smallamount of initial endowments of working hours and capital goods, utilizing Heckscher-Ohlin model. Heckscher-Ohlin model is an application of general equilibrium theory with trading two countries. In Fukiharu[2010], first, Gini coefficient was computed for the income distribution in a small country, A, isolated from the worldeconomy, in the general equilibrium. A modification was made somewhat into the traditional Heckscher-Ohlin model,in such a way that the two production functions are under decreasing returns to scale, so that positive profit accrues tothe entrepreneurs. Thus, there are four consumers of goods, the aggregate workers, the aggregate capitalists, and twoentrepreneurs. In this modified model, supposing that the country A opens its economy to a large country B, whichpossesses large amount of initial endowments of working hours and capital goods, Gini coefficient was computed forthe income distribution for the four economic agents in a country A, in the general equilibrium with trade. If the formeris smaller than the latter, it is defined that the inequality expands through the globalization. Randomly specifying parameters in Cobb-Doulas type production and utility functionsfor two the commoditie andinitial endowments of working hours and capital goods, selected small for country A and large for country B, wecomputed country A's general equilibrium incomes for the four economic agents and compared Gini coefficients. Theconclusion was that out of 1000 simulations, about 65% showed that the country A's inequality of income distributionexpanded through globalization. In the present paper, the third commodity is introduced into the previous model. This commodity is a non-tradedcommodity, while the third production functions is also under decreasing returns to scale, so that positive profit accruesto the third entrepreneur. Thus, there are five consumers of goods, workers, capitalists, and three entrepreneurs. Utiliz-ing the same simulation approach, we examine how the per cent of expanded inequality case among the 1000 simula-tions changes.

Heck$Oh6.nb 2

I: A Specified Case

Traditional Heckscher-Ohlin model is an application of general equilibrium model with two commodities and twofactors of production, labor and capital, for two trading countries country A and country B with two industries. SinceOhlin [1933] explicitly asserted that capital-rich countries export capital intensive commodities, while labor-richcountries export labor intensive commodities, the Heckscher-Ohlin (H-O) theorem has been one of the fundamentaltheorems in international economics. Sufficient conditions for the H-O theorem are as follows:

[i] Production technologies are the same between the two countries.[ii] Constant returns to scale in production.[iii] Perfect competition prevails.[iv] Imperfect specialization prevails.[v] Homothetic utility functions are the same between the two countries.

In this paper, two modifications are made. First, [ii] is modified to be decreasing returns to scale in production.Furthermore, this paper adopts simulation approach, so that production and utility functions are assumed to be ofCobb-Douglas type. By the modification of [ii] the "factor price equalization theorem" does no hold as shown later.Second, it is assumed that there are three commodities (industries or sectors) in country A, where two commodities aretraded, while the third commodity is a non-traded commodity.

Formally, there are two countries, A with three sectors and B with two sectors. The production functions for the twocountries regarding the two industries are assumed to be of the same Cobb-Douglas type, where the one for the firstsector is f1 A = f1 B =L1

a1 K1b1 , the one for the second sector is f2 A = f2 B =L2

a2 K2b2 , and the one for the third sector in

country A is f3 A =L3a2 K3

b2 where ai +bi <1 (i=1, 2, 3). It is assumed that country A is endowed with LeA units of Laborand KeA units of capital, while country B is endowed with LeB units of Labor and KeB units of capital. In this section,specifying parameters, first, we compute general equilibrium, GE, for country A in national isolation with Gini coeffi-cient for the GE income distribution. Next, supposing that country A trades with country B, we compute the Ginicoefficient for the country A's "trade GE" income distribution, examining whether income distribution becomes moreunequal or not.

ü I.1: Decreasing Returns Without Trade

We start with the computation of GE for a country A under national isolation, "no trade GE", proceeding to the oneof "trade GE", in which country A trades with country B. In both cases, parameters are specified by particular values,for the purpose of explaining the structure of the model.

ü Production of country A

Country A is under national isolation. She has three sectors of production, which produces different consumptiongood, utilizing labor, Li , and capital, Ki , y stands for the output of sector 1, x stands for the output of sector 2, and zstands for the output of sector 3. (i=1, 2, 3) Production function of sector 1, y = f1 A , is assumed as follows, witha1 +b1 <1: decreasing returns to scale.

In[2]:= f = L^a ∗ K^b; c1 = 8a → 1ê6, b → 1ê5, L → L1, K → K1<; f1 = f ê. c1

Out[2]= K11ê5 L11ê6

Heck$Oh6.nb 3

Production function of sector 2, x= f2 A , is assumed as follows, with a2 +b2 <1: decreasing returns to scale.

In[3]:= c2 = 8a → 1 ê4, b → 1 ê3, L → L2, K → K2<; f2 = f ê. c2

Out[3]= K21ê3 L21ê4

Production function of sector 3, z= f3 A , is assumed as follows, with a3 +b3 <1: decreasing returns to scale.

In[4]:= c3 = 8a → 1 ê2, b → 1 ê4, L → L3, K → K3<; f3 = f ê. c3

Out[4]= K31ê4 è!!!!!!L3

From the profit maximization of the sector 1, demand for labor, L1 AD , and demand for capital, K1 A

D , are computedas in what follows, where py stands for the price of the consumption good, y, wL , wage rate of labor, and wK , rentalprice of capital.

In[5]:= pi1 = py ∗ f1 − wL ∗ L1 − wK ∗ K1;sol1 = Solve@8D@pi1, L1D 0, D@pi1, K1D 0<, 8K1, L1<D@@2DD

Out[5]= 9K1 →py30ê19

5 56ê19 65ê19 wK25ê19 wL5ê19 , L1 →py30ê19

6 56ê19 65ê19 wK6ê19 wL24ê19 =

In[6]:= demand$L1 = L1 ê. sol1; demand$K1 = K1 ê. sol1;

Thus, supply function of y, yAS , is computed as follows, with py , wL , and wK , as parameters.

In[7]:= supply$A$y = PowerExpand@f1 ê. sol1D

Out[7]=py11ê19

56ê19 65ê19 wK6ê19 wL5ê19

Profit function of sector 1, p1 A , is computed as follows, with py , wL , and wK , as parameters. This profit accrues toentrepreneur 1.

In[8]:= pi10 = PowerExpand@pi1 ê. sol1D

Out[8]=19 py30ê19

30 56ê19 65ê19 wK6ê19 wL5ê19


D , as follows,where px stands for the price of the consumption good, x.

In[9]:= pi2 = px ∗ f2 − wL ∗ L2 − wK ∗ K2;sol2 = PowerExpand@Solve@8D@pi2, L2D 0, D@pi2, K2D 0<, 8K2, L2<D@@1DDD

Out[9]= 9K2 →px12ê5

6 21ê5 34ê5 wK9ê5 wL3ê5 , L2 →px12ê5

8 21ê5 34ê5 wK4ê5 wL8ê5 =


Thus, supply function of x, xAS , is computed as follows, with px , wL , and wK , as parameters.

In[11]:= supply$A$x = PowerExpand@f2 ê. sol2D

Out[11]=px7ê5

2 21ê5 34ê5 wK4ê5 wL3ê5

Heck$Oh6.nb 4

Profit function of sector 2, p2 A , is computed as follows, with px , wL , and wK , as parameters. This profit accrues toentrepreneur 2.


Out[12]=5 px12ê5

24 21ê5 34ê5 wK4ê5 wL3ê5


D , as follows,where px stands for the price of the consumption good, z.

In[13]:= pi3 = pz ∗ f3 − wL ∗ L3 − wK ∗ K3;sol3 = PowerExpand@Solve@8D@pi3, L3D 0, D@pi3, K3D 0<, 8K3, L3<D@@1DDD

Out[13]= 9K3 →pz4

64 wK2 wL2 , L3 →pz4

32 wK wL3 =


Thus, supply function of z, zAS , is computed as follows, with pz , wL , and wK , as parameters.

In[15]:= supply$A$z = PowerExpand@f3 ê. sol3D

Out[15]=pz3

16 wK wL2

Profit function of sector 3, p3 A , is computed as follows, with pz , wL , and wK , as parameters. This profit accrues toentrepreneur 3.


Out[16]=pz4

64 wK wL2

ü Consumption of country A

We proceed to the demand side of country A. She is endowed with LeA =100 and KeA =50.

In[17]:= LeA = 100; KeA = 50;

All the agents in this paper: workers, capitalists, and entrepreneurs, have the same Cobb-Douglas utility function,u[y , x, z]=ya xb zc , which is specified as u[y , x]=y3 x2 z4 .

In[18]:= uA = x^a ∗ y^b ∗ z^c ê. 8a → 2, b → 3, c → 4<

Out[18]= x2 y3 z4

All the consumers maximizes utility subject to income constraint:

max u[y , x , z] s.t. py y+px x+pz z=m (1)

where m is income. Worker's income consists of initial endowment of labor, evaluated by the wage rate: wL LeA . It isassumed that they supply LeA for labor supply. It is assumed that they supply LeA for labor supply. Capitalist's incomeconsists of initial endowment of capital, evaluated by the rental price of capital: wK KeA . It is assumed that they supplyKeA for capital supply. Entrepreneur 1's income consists of profit for the sector 1, p1 A . Entrepreneur 2's incomeconsists of profit for the sector 2, p2 A .Finally, entrepreneur 3's income consists of profit for the sector 3, p3 A .

Heck$Oh6.nb 5

Demand function of workers for commodity y , yLD , that for commodity x , xL

D , that for commodity z , zLD ,demand

function of capitalists for commodity y , yKD , that for commodity x , xK

D , that for commodity z , zKD ,demand function

of entrepreneur 1 for commodity y , yE1D , that for commodity x , xE1

D , that for commodity z , zE1D ,and demand func-

tion of entrepreneur 2 for commodity y , yE2D , that for commodity x , xE2

D , that for commodity z , zE2D ,are derived as

in what follows.

In[19]:= sol3A = Solve@8D@uA, xDê D@uA, yD pxê py,D@uA, zDêD@uA, yD pzê py, px ∗ x + py ∗ y + pz ∗ z m<, 8x, y, z<D@@1DD

Out[19]= 9x →2 m

9 px, y →

m3 py

, z →4 m

9 pz=

In[20]:= demand$L$x = x ê. sol3A ê. m → wL ∗ LeA; demand$L$y = y ê. sol3A ê. m → wL ∗ LeA;demand$L$z = z ê. sol3A ê. m → wL ∗ LeA; demand$K$x = x ê. sol3A ê. m → wK ∗ KeA;demand$K$y = y ê. sol3A ê. m → wK ∗ KeA; demand$K$z = z ê. sol3A ê. m → wK ∗ KeA;demand$E1$x = x ê. sol3A ê. m → pi10; demand$E2$x = x ê. sol3A ê. m → pi20;demand$E3$x = x ê. sol3A ê. m → pi30; demand$E1$y = y ê. sol3A ê. m → pi10;demand$E2$y = y ê. sol3A ê. m → pi20; demand$E3$y = y ê. sol3A ê. m → pi30;demand$E1$z = z ê. sol3A ê. m → pi10; demand$E2$z = z ê. sol3A ê. m → pi20;demand$E3$z = z ê. sol3A ê. m → pi30; 8demand$L$y, demand$L$x, demand$L$z,

demand$K$y, demand$K$x, demand$K$z, demand$E1$y, demand$E1$x, demand$E1$z,demand$E2$y, demand$E2$x, demand$E2$z, demand$E3$y, demand$E3$x, demand$E3$z<

Out[20]= 9 100 wL3 py

, 200 wL9 px

, 400 wL9 pz

, 50 wK3 py

, 100 wK9 px

, 200 wK9 pz

,

19 py11ê19

90 56ê19 65ê19 wK6ê19 wL5ê19 , 19 py30ê19

135 56ê19 65ê19 px wK6ê19 wL5ê19 ,

19 214ê19 py30ê19

135 35ê19 56ê19 pz wK6ê19 wL5ê19 , 5 px12ê5

72 21ê5 34ê5 py wK4ê5 wL3ê5 , 5 px7ê5

108 21ê5 34ê5 wK4ê5 wL3ê5 ,

5 px12ê5

54 21ê5 34ê5 pz wK4ê5 wL3ê5 , pz4

192 py wK wL2 , pz4

288 px wK wL2 , pz3

144 wK wL2 =

Country A's demand for commodity y, yAD , is the sum of yL

D , yKD , yE1

D , yE2D , and yE3

D .

In[21]:= demand$A$y = Simplify@demand$L$y + demand$K$y + demand$E1$y + demand$E2$y + demand$E3$yD

Out[21]=1

43200 py wK wL2 H225 pz4 + 500 24ê5 31ê5 px12ê5 wK1ê5 wL7ê5 +

304 513ê19 614ê19 py30ê19 wK13ê19 wL33ê19 + 720000 wK2 wL2 + 1440000 wK wL3L

Country A's demand for commodity x, xAD , is the sum of xL

D , xKD , xE1

D , xE2D , xE3

D .

In[22]:= demand$A$x =

Simplify@demand$L$x + demand$K$x + demand$E1$x + demand$E2$x + demand$E3$xD;

Country A's demand for commodity z, zAD , is the sum of zL

D , zKD , zE1

D , zE2D , zE3

D .

In[23]:= demand$A$z = Simplify@demand$L$z + demand$K$z + demand$E1$z + demand$E2$z + demand$E3$zD

Out[23]=1

32400 pz wK wL2 H225 pz4 + 500 24ê5 31ê5 px12ê5 wK1ê5 wL7ê5 +


Heck$Oh6.nb 6

ü "No Trade GE"

General equilibrium for country A without trade, "no trade GE", is defined by

yAD =yA

S , (2) xA

D =xAS , (3)

zAD =zA

S , (4) L1 A

D +L2 AD +L3 A

D =LeA , (5) K1 A

D +K2 AD +K3 A

D =KeA . (6)

As is well known, by the Walras law, 4 conditions, (2)~(6), are not independent. Thus, assuming wL =1, we compute"no trade GE" prices: py * , px *, pz *, and wK *, as a solution to (2)~(5).

In[24]:= check1 = 8supply$A$x demand$A$x, supply$A$y demand$A$y,supply$A$z demand$A$z, demand$L1 + demand$L2 + demand$L3 LeA< ê. wL → 1;

sol6 = Solve@check1, 8px, py, pz, wK<D@@4DD

Out[24]= 9px →4 25ê12 è!!!5 171ê3

33ê4 , py →10 25ê6 54ê15 171ê5

37ê30 , wK →6845

, pz →4 è!!!2 851ê4

33ê4 =

It is ascertained that these prices are indeed "no trade GE", by showing that { py *, px *,pz *, wK *} is a solution to (2),(3), (4), and (6).

In[25]:= check2 = 8supply$A$x demand$A$x, supply$A$y demand$A$y,supply$A$z demand$A$z, demand$K1 + demand$K2 + demand$K3 KeA< ê. wL → 1;

solWT = Solve@check2, 8px, py, pz, wK<D@@4DD

Out[25]= 9px →4 25ê12 è!!!5 171ê3

33ê4 , py →10 25ê6 54ê15 171ê5

37ê30 , wK →6845

, pz →4 è!!!2 851ê4

33ê4 =

In[26]:= N@%D

Out[26]= 8px → 13.4674, py → 37.326, wK → 1.51111, pz → 7.5351<

The "no trade GE" income for the (aggregate) workers is computed as follows.

In[27]:= income$A$L0 = wL ∗ LeA ê. wL → 1

Out[27]= 100

The "no trade GE" income for the (aggregate) capitalists is computed as follows.

In[28]:= N@income$A$K0 = wK ∗ KeA ê. sol6 ê. wL → 1D

Out[28]= 75.5556

The "no trade GE" income for the entrepreneur 1 is computed as follows.

In[29]:= N@income$A$E10 = pi10 ê. sol6 ê. wL → 1D

Out[29]= 63.3333


Heck$Oh6.nb 7


Out[30]= 27.7778



Out[31]= 33.3333

Using the "no trade GE" incomes for 5 agents, "no trade" Gini coefficient, gini0, is computed.

In[32]:= giniA@y_D :=

Module@8z1, z2<, z1 = Sort@yD; z2 = Hy@@1DD + y@@2DD + y@@3DD + y@@4DD + y@@5DDLê5; 1 +

H1 ê5L − 2 Hz1@@5DD + 2 ∗ z1@@4DD + 3 ∗ z1@@3DD + 4 ∗ z1@@2DD + 5 ∗ z1@@1DDLêH5^2 ∗ z2LD;gini0 = N@giniA@8income$A$L0, income$A$K0, income$A$E10,

income$A$E20, income$A$E30<DD

Out[33]= 0.248889

ü I.2: Decreasing Returns With Trade

We proceed to the computation of "trade GE" Gini coefficient, in which country A trades with country B. In thiscomputation, the same parameters are used as in the "no trade" Gini coefficient. It is assumed that commodities, y and xare traded, while z is a non-traded commodities.

ü CountryA

Required modification is on the actual production of commodities, since the trade with country B implies that vari-ables such as import and export must be introduced. The country A's actual production of commodity y, yA

P , is the sumof yA

D and export of commodity y, yAE .

In[34]:= pro$A$y = Hdemand$A$y + exp$A$yL

Out[34]= exp$A$y +1

43200 py wK wL2 H225 pz4 + 500 24ê5 31ê5 px12ê5 wK1ê5 wL7ê5 +


The country A's actual production of commodity x, xAP , must be introduced as the difference between xA

D andimport of commodity x, xA

I .

In[35]:= pro$A$x = Hdemand$A$x − imp$A$xL

Out[35]= −imp$A$x +1

64800 px wK wL2 H225 pz4 + 500 24ê5 31ê5 px12ê5 wK1ê5 wL7ê5 +


Heck$Oh6.nb 8

ü CountryB

Country B has two sectors of production, which produces y and x, utilizing labor, Li , and capital, Ki , where y standsfor the output of sector 1, and x stands for the output of sector 2. (i=1,2) Production function of sector 1, y = f1 B , isassumed with parameters specified exactly the same as in country A. From the profit maximization, country B'sdemand for labor, L1 B

D , and demand for capital, K1 BD , supply function of y, yB

S , are computed as in what follows,with px , wLB , and wKB , as parameters. Profit function for the first sector is also computed as the same one for countryA, with px , wLB , and wKB , as parameters. Note that factor prices may well be different between the two tradingcountries since the decreasing returns to scale is assumed.

In[36]:= demand$B$L1 = L1 ê. sol1 ê. 8wK → wKB, wL → wLB<;demand$B$K1 = K1 ê. sol1 ê. sol1 ê. 8wK → wKB, wL → wLB<;supply$B$y = PowerExpand@f1 ê. sol1 ê. 8wK → wKB, wL → wLB<D;pi10B = PowerExpand@pi1 ê. sol1 ê. 8wK → wKB, wL → wLB<D;8demand$B$L1, demand$B$K1, supply$B$y, pi10B<

Out[36]= 9 py30ê19

6 56ê19 65ê19 wKB6ê19 wLB24ê19 , py30ê19

5 56ê19 65ê19 wKB25ê19 wLB5ê19 ,

py11ê19

56ê19 65ê19 wKB6ê19 wLB5ê19 ,19 py30ê19

30 56ê19 65ê19 wKB6ê19 wLB5ê19 =

Production function of sector 2, x= f2 B , is assumed with parameters specified exactly as in country A. From the profitmaximization, country B's demand for labor, L2 B

D , and demand for capital, K2 BD , supply function of x, xB

S , arecomputed as in what follows. Profit function for the second sector is also computed as the same one for country A.

In[37]:= demand$B$L2 = L2 ê. sol2 ê. 8wK → wKB, wL → wLB<;demand$B$K2 = K2 ê. sol2 ê. 8wK → wKB, wL → wLB<;supply$B$x = PowerExpand@f2 ê. sol2 ê. 8wK → wKB, wL → wLB<D;pi20B = PowerExpand@pi2 ê. sol2 ê. 8wK → wKB, wL → wLB<D;8demand$B$L2, demand$B$K2, supply$B$x, pi20B<

Out[37]= 9 px12ê5

8 21ê5 34ê5 wKB4ê5 wLB8ê5 , px12ê5

6 21ê5 34ê5 wKB9ê5 wLB3ê5 ,

px7ê5

2 21ê5 34ê5 wKB4ê5 wLB3ê5 ,5 px12ê5

24 21ê5 34ê5 wKB4ê5 wLB3ê5 =

We proceed to the demand side of country B. She is endowed with LeB =500 and KeB =50. All the agents in thissection: workers, capitalists, and entrepreneurs in country B, have the same Cobb-Douglas utility function, which isspecified as u[y , x ]=y3 x2 . All the consumers maximizes utility subject to income constraint as specified in (1). Work-er's income consists of initial endowment of labor, evaluated by the wage rate: wL LeB . It is assumed that they supplyLeB for labor supply. Capitalist's income consists of initial endowment of capital, evaluated by the rental price ofcapital: wK KeB . It is assumed that they supply KeB for capital supply. Entrepreneur 1's income consists of profit for thesector 1, p1 B . Finally, entrepreneur 2's income consists of profit for the sector 2, p2 B . Demand function of workers for commodity y , yLB

D , that for commodity x , xLBD , demand function of capitalists for

commodity y , yKBD , that for commodity x , xKB

D , demand function of entrepreneur 1 for commodity y , yE1BD , that for

commodity x , xE1BD , and demand function of entrepreneur 2 for commodity y , yE2B

D , that for commodity x , xE2 BD ,are derived as in what follows.

In[38]:= u = y^3 ∗ x^2; sol3 = Solve@8D@u, yDêD@u, xD pyêpx, px ∗ x + py ∗ y m<, 8x, y<D@@1DD;

Heck$Oh6.nb 9

In[39]:= LeB = 500; KeB = 50; demand$B$L$y = y ê. sol3 ê. m → wL ∗ LeB ê. 8wK → wKB, wL → wLB<;demand$B$L$x = x ê. sol3 ê. m → wL ∗ LeB ê. 8wK → wKB, wL → wLB<;demand$B$K$y = y ê. sol3 ê. m → wK ∗ KeB ê. 8wK → wKB, wL → wLB<;demand$B$K$x = x ê. sol3 ê. m → wK ∗ KeB ê. 8wK → wKB, wL → wLB<;demand$B$E1$y = y ê. sol3 ê. m → pi10B ê. 8wK → wKB, wL → wLB<;demand$B$E1$x = x ê. sol3 ê. m → pi10B ê. 8wK → wKB, wL → wLB<;demand$B$E2$y = y ê. sol3 ê. m → pi20B ê. 8wK → wKB, wL → wLB<;demand$B$E2$x = x ê. sol3 ê. m → pi20B ê. 8wK → wKB, wL → wLB<;

Country B's demand for commodity y, yBD , is the sum of yLB

D , yKBD , yE1B

D , and yE2BD .

In[40]:= demand$B$y = Simplify@demand$B$L$y + demand$B$K$y + demand$B$E1$y + demand$B$E2$yD;

Country B's demand for commodity x, xBD , is the sum of xLB

D , xKBD , xE1B

D , and xE2BD .

In[41]:= demand$B$x = Simplify@demand$B$L$x + demand$B$K$x + demand$B$E1$x + demand$B$E2$xD;

The trade with country A implies that variables such as import and export must be introduced. The country B's actualproduction of commodity y, yB

P , is the difference of yABD and import of commodity y, yA

I .

In[42]:= pro$B$y = demand$B$y − imp$B$y;

The country B's actual production of commodity x, xBP , is the sum of xB

D and export of commodity x, xBE .

In[43]:= pro$B$x = demand$B$x + exp$B$x;

ü "Trade GE"

General equilibrium for country A with trade, "trade GE", is defined by

yAP =yA

S , (2A) xA

P =xAS , (3A)

zAD =zA

S , (4) K1 A

D +K2 AD +K3 A

D =KeA . (6) px xA

I =py yAE , (7)

yBP =yB

S , (8) xB

P =xBS , (9)

K1 BD +K2 B

D =KeB . (10) Assuming wL =1, we compute "trade GE" prices: py ** , px **, pz **, wK **, wKB **, wLB **, yA

E , and xAI as a solution

to (2A), (3A), (4), (6), (7), (8), (9), and (10).

In[44]:= ec1 = Hpro$A$x supply$A$xL ê. wL → 1;ec2 = Hpro$A$y supply$A$yL ê. wL → 1; ec3 = Himp$A$x ∗ px exp$A$y ∗ pyL;ec4 = Hdemand$K1 + demand$K2 + demand$K3 KeAL ê. wL → 1;ec5 = Hpro$B$y supply$B$yL ê. wL → 1 ê. imp$B$y → exp$A$y;ec6 = Hpro$B$x supply$B$xL ê. wL → 1 ê. exp$B$x → imp$A$x;ec7 = Hdemand$B$K1 + demand$B$K2 KeBL ê. 8wK → wKB, wL → wLB<;ec8 = Hsupply$A$z demand$A$zL ê. wL → 1;

Heck$Oh6.nb 10

In[45]:=

sol10 = FindRoot@8ec1, ec2, ec3, ec4, ec5, ec6, ec7, ec8<, 8px, 20<, 8py, 100<,8pz, 100<, 8wK, 1<, 8imp$A$x, 1<, 8exp$A$y, 1<, 8wKB, 1<, 8wLB, 1<D

Out[45]= 8px → 12.0907, py → 41.0529, pz → 7.53189, wK → 1.48935,imp$A$x → 1.27867, exp$A$y → 0.376588, wKB → 1.75075, wLB → 0.137539<

It is confirmed that "trade GE" satisfies country A's labor market equilibrium condition.

In[46]:= demand$L1 + demand$L2 + demand$L3 ê. sol10 ê. wL → 1

Out[46]= 100.

It is also confirmed that "trade GE" satisfies country B's labor market equilibrium condition.

In[47]:= demand$LB = Hdemand$B$L1L + Hdemand$B$L2L ê. sol10 ê. wL → 1

Out[47]= 500.

After the globalization, in country A, incomes of capitalists and the second entrepreneur decline, while the one of firstentrepreneur rises and the one of workers remains the same.

In[48]:= N@88income$A$L0, income$A$L1 = wL ∗ LeA ê. wL → 1<,8income$A$K0, income$A$K1 = wK ∗ KeA ê. sol10 ê. wL → 1<,8income$A$E10, income$A$E11 = pi10 ê. sol10 ê. wL → 1<,8income$A$E20, income$A$E21 = N@pi20 ê. sol10 ê. wL → 1D<,8income$A$E30, income$A$E31 = N@pi30 ê. sol10 ê. wL → 1D<<D

Out[48]= 88100., 100.<, 875.5556, 74.4676<,863.3333, 73.9406<, 827.7778, 21.694<, 833.3333, 33.7628<<

After the globalization, the Gini coefficient of income distribution rises, so that the inequality of income distributionbecomes expands.

In[49]:= 8gini0, gini1 =

giniA@8income$A$L1, income$A$K1, income$A$E11, income$A$E21, income$A$E31<D<

Out[49]= 80.248889, 0.259743<

Heck$Oh6.nb 11

II: General Case

Formally, there are two countries, country A with three industries and country B with two industries. The firstindustry in country A and B produces output y, and the second produces x, while the third industry in country Aproduces output z. The commodities y and x are traded commodities, while z is non-traded. The production functionsfor the first and second industries of the two countries are assumed to be of the same Cobb-Douglas type, where the

one for the first industry is y= f1 A = f1 B =L1a1 K1

b1 and the one for the second sector is x= f2 A = f2 B =L2a2 K2

b2 , while

the one for the third industry in country A is z= f3 A =L3a3 K3

b3 . (ai +bi <1, i=1, 2, 3) It is assumed that country A isendowed with LeA units of Labor and KeA units of capital, while country B is endowed with LeB units of Labor andKeB units of capital. In this section, without specifying parameters, first, we compute demand and supply functions.Next, specifying the same parameters as in the previous section, we confirm that the Newton method computes thesame general equilibrium, GE, for country A in national isolation with Gini coefficient for the GE income distributionas in the previous section. Finally, the same approach is adopted in confirming that the Newton method computes thesame general equilibrium, GE, for country A when she trades with country B as in the previous section for the samespecification of parameters as in the previous section. This procedure is applied to the case of random specification ofparameters in examining the income inequality change when a small country A starts trading with a large country B.

ü I.1: Decreasing Returns Without Trade


III: Simulations

In this section, collecting the Mathematica programs for the general case , we examine how the conclusion in section1 is robust. Formally, there are two countries, A and B. The production functions for the two countries are assumed tobe of the same Cobb-Douglas type with respect to industry 1 and industry 2, where the one for the first industry is f1 A =

f1 B =L1a1 K1

b1 and the one for the second industry is f2 A = f2 B =L2a2 K2

b2 . The production functions for country 1's

third industry is assumed to be of the Cobb-Douglas type: f3 A =L1a3 K1

b3 . It is assumed that country A is endowed

with LeA units of Labor and KeA units of capital, while country B is endowed with LeB units of Labor and KeB units ofcapital. First, we compute "no trade GE" for country A in national isolation. Next, supposing that country A trades withcountry B, we compute "trade GE" for country A in the trade with country B.

ü II.1: Decreasing Returns Without Trade—General Case

We start with the computation of "no trade GE" for country A in national isolation.

ü "No Trade GE" forCountryA

General equilibrium for country A without trade, "no trade GE", is computed as in what follows. Given arbitraryparameters, the collection of section 2, check1, derives "no trade GE" in terms of Newton method.

Heck$Oh6.nb 12

In[130]:=

Clear@LeA, KeA, LeB, KeB, f, f1, f2, f3, u, uA, a, b, c, a1, a2, b1, b2, a3, b3, pi1, sol1,sol2, sol3, sol3A, demand$L1, demand$K1, supply$A$y, pi10, pi2, supply$A$x,pi20, demand$L2, demand$K2, demand$L$x, demand$L$y, demand$L$z, demand$K$x,demand$K$y, demand$K$z, pi3, pi30, demand$L3, demand$K3, supply$A$z, demand$E1$x,demand$E1$y, demand$E1$z, demand$E2$x, demand$E2$y, demand$E2$z, demand$E3$x,demand$E3$y, demand$E3$z, demand$A$x, demand$A$y, demand$A$z, check1, check2D;

In[131]:=

f = L^a ∗ K^b; f1 = f ê. 8a → a1, b → b1, L → L1, K → K1<;f2 = f ê. 8a → a2, b → b2, L → L2, K → K2<; f3 = f ê. 8a → a3, b → b3, L → L3, K → K3<;pi1 = py ∗ f1 − wL ∗ L1 − wK ∗ K1;sol1 = Solve@8D@pi1, L1D 0, D@pi1, K1D 0<, 8K1, L1<D@@1DD;demand$L1 = Simplify@Factor@L1 ê. sol1DD;demand$K1 = Simplify@Factor@K1 ê. sol1DD;supply$A$y = Simplify@PowerExpand@Factor@f1 ê. sol1DDD;pi10 = Simplify@PowerExpand@Factor@pi1 ê. sol1DDD;pi2 = px ∗ f2 − wL ∗ L2 − wK ∗ K2;sol2 = Solve@8D@pi2, L2D 0, D@pi2, K2D 0<, 8K2, L2<D@@1DD;demand$L2 = Simplify@Factor@L2 ê. sol2DD;demand$K2 = Simplify@Factor@K2 ê. sol2DD;supply$A$x = Simplify@PowerExpand@Factor@f2 ê. sol2DDD;pi20 = Simplify@PowerExpand@Factor@pi2 ê. sol2DDD;pi3 = pz ∗ f3 − wL ∗ L3 − wK ∗ K3;sol3 = Solve@8D@pi3, L3D 0, D@pi3, K3D 0<, 8K3, L3<D@@1DD;demand$L3 = Simplify@Factor@L3 ê. sol3DD;demand$K3 = Simplify@Factor@K3 ê. sol3DD;supply$A$z = Simplify@PowerExpand@Factor@f3 ê. sol3DDD;pi30 = Simplify@PowerExpand@Factor@pi3 ê. sol3DDD;uA = x^a ∗ y^b ∗ z^c;sol3A = Solve@8D@uA, xDêD@uA, yD pxê py,

D@uA, zDê D@uA, yD pzê py, px ∗ x + py ∗ y + pz ∗ z m<, 8x, y, z<D@@1DD;demand$L$x = x ê. sol3A ê. m → wL ∗ LeA; demand$L$y = y ê. sol3A ê. m → wL ∗ LeA;demand$L$z = z ê. sol3A ê. m → wL ∗ LeA;demand$K$x = x ê. sol3A ê. m → wK ∗ KeA; demand$K$y = y ê. sol3A ê. m → wK ∗ KeA;demand$K$z = z ê. sol3A ê. m → wK ∗ KeA; demand$E1$x = x ê. sol3A ê. m → pi10;demand$E2$x = x ê. sol3A ê. m → pi20; demand$E3$x = x ê. sol3A ê. m → pi30;demand$E1$y = y ê. sol3A ê. m → pi10; demand$E2$y = y ê. sol3A ê. m → pi20;demand$E3$y = y ê. sol3A ê. m → pi30; demand$E1$z = z ê. sol3A ê. m → pi10;demand$E2$z = z ê. sol3A ê. m → pi20; demand$E3$z = z ê. sol3A ê. m → pi30;demand$A$y =

Simplify@demand$L$y + demand$K$y + demand$E1$y + demand$E2$y + demand$E3$yD;demand$A$x = Simplify@demand$L$x + demand$K$x +

demand$E1$x + demand$E2$x + demand$E3$xD;demand$A$z = Simplify@demand$L$z + demand$K$z + demand$E1$z +

demand$E2$z + demand$E3$zD;check1 = 8supply$A$x demand$A$x, supply$A$y demand$A$y,

supply$A$z demand$A$z, demand$L1 + demand$L2 + demand$L3 LeA< ê. wL → 1;

It is confirmed that specifying the same parameters as in section 1, Newton method derives the same "no trade GE"from check1 as in section 1.

Heck$Oh6.nb 13

In[155]:=

FindRoot@check1 ê. 8a1 → 1ê 6, b1 → 1ê5, a2 → 1ê4, b2 → 1ê3, a3 → 1ê2, b3 → 1ê4,a → 2, b → 3, c → 4, a1B → 1ê 6, b1B → 1ê5, a2B → 1ê4, b2B → 1ê3, aB → 2, bB → 3,LeA → 100, KeA → 50, LeB → 500, KeB → 50<, 8py, 1<, 8px, 1<, 8pz, 1<, 8wK, 1<D

Out[155]=

8py → 37.326, px → 13.4674, pz → 7.5351, wK → 1.51111<

In what follows, 1000 pairs of parameters for {a1 , b1 , a2 , b2 , a3 , b3 , a, b, c, LeA , KeA , LeB , KeB } are selectedrandomly, where ai +bi <1, i=1, 2, 3, a, b, and c are integers belonging to [1, 100], LeA and KeA are integers belongingto [1, 100], LeB and KeB are integers belonging to [900, 1000].

In[156]:=

nu = 1000;

In[157]:=

g@x_D := If@x@@1DD < 1 && x@@2DD < 1 && x@@1DD + x@@2DD < 1,x, 8Random@Integer, 81, 1000<DêRandom@Integer, 81, 1000<D,

Random@Integer, 81, 1000<DêRandom@Integer, 81, 1000<D<D;

In[158]:=

d1 := FixedPoint@g, 8Random@Integer, 81, 1000<Dê Random@Integer, 81, 1000<D,Random@Integer, 81, 1000<DêRandom@Integer, 81, 1000<D<D;



d10 = Table@d1, 8nu<D; d20 = Table@d2, 8nu<D; d30 = Table@d3, 8nu<D;data1 = Table@8d10@@i, 1DD, d10@@i, 2DD, d20@@i, 1DD, d20@@i, 2DD, d30@@i, 1DD,

d30@@i, 2DD, Random@Integer, 81, 100<D, Random@Integer, 81, 100<D,Random@Integer, 81, 100<D, Random@Integer, 81, 100<D, Random@Integer, 81, 100<D,Random@Integer, 8900, 1000<D, Random@Integer, 8900, 1000<D<, 8i, 1, nu<D;

In[160]:=

Take@data1, 3DOut[160]=

99 161855

, 5926

, 1093

, 595788

, 341889

, 19

, 44, 63, 91, 44, 26, 934, 978=,

9 89292

,219397

,46

339,

264661

,283983

,119262

, 8, 93, 94, 84, 51, 915, 982=,

9 233452

, 72449

, 59759

, 159275

, 7492

, 287311

, 26, 60, 17, 41, 52, 949, 948==

In what follows, from these 1000 parameters, we attempt to derive 1000 pairs of GE income distributions for 5agents, as data2NT. In order to do so, first, we derive 1000 pairs of candidate for "no trade GE", data2, in terms ofNewton method, where the initial values for the Newton method are py [0]=10, px [0]=5, pz [0]=10, wK [0]=0.1, andwL ª1.

In[161]:=

h@i_D := 8a1 → data1@@i, 1DD, a1B → data1@@i, 1DD, b1 → data1@@i, 2DD,b1B → data1@@i, 2DD, a2 → data1@@i, 3DD, a2B → data1@@i, 3DD,b2 → data1@@i, 4DD, b2B → data1@@i, 4DD, a3 → data1@@i, 5DD,b3 → data1@@i, 6DD, a → data1@@i, 7DD, aB → data1@@i, 7DD, b → data1@@i, 8DD,bB → data1@@i, 8DD, c → data1@@i, 9DD, LeA → data1@@i, 10DD,KeA → data1@@i, 11DD, LeB → data1@@i, 12DD, KeB → data1@@i, 13DD, wL → 1<;

Heck$Oh6.nb 14

In[162]:=

data2 = Table@FindRoot@Hcheck1 ê. h@iDL, 8py, 10<, 8px, 5<, 8pz, 10<, 8wK, 0.1<,AccuracyGoal → 20, WorkingPrecision → 34, MaxIterations → 1000D, 8i, 1, nu<D;

The price trajectory on the Newton method, py [t], px [t], pz [t], and wK [t] might not converge to "no trade GE",which satisfy (2)~(5) starting from py [0]=10, px [0]=5, pz [0]=10, and wK [0]=0.1. Indeed, since not all the 1000 pairssatisfy (2)~(4), we must select the parameters which satisfy (2)~(4).

In[163]:=

data3 = Table@check1 ê. h@iD ê. data2@@iDD, 8i, 1, nu<D;data4 = Table@Count@data3@@iDD, TrueD, 8i, 1, Length@data3D<D;

In[164]:=


88True, True, True, True<, 8True, True, True, True<, 8True, True, True, True<<

In[165]:=


84, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4<

The following set, data5, is the set of positions of 1000 pairs, at which (2)~(4) are satisfied.

In[166]:=

data5 = Flatten@Position@data4, 4DD;

In[167]:=


81, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20<

The following set, giniA00, is the one of 1000 "no trade GE" Gini co-officiate, constructed from data2, which containnon-convergent GE prices.

In[168]:=

data7 = Table@8wL ∗ LeA, wK ∗ KeA, pi10, pi20, pi30< ê. h@iD ê. data2@@iDD, 8i, 1, nu<D;giniA@y_D :=

Module@8z1, z2<, z1 = Sort@yD; z2 = Hy@@1DD + y@@2DD + y@@3DD + y@@4DD + y@@5DDLê5;1 + H1ê5L − 2 Hz1@@5DD + 2 ∗ z1@@4DD + 3 ∗ z1@@3DD + 4 ∗ z1@@2DD + 5 ∗ z1@@1DDLêH5^2 ∗ z2LD;

giniA00 = Table@giniA@data7@@iDDD, 8i, 1, Length@data7D<D;

In[169]:=

Take@giniA00, 3DOut[169]=

80.19804203286852842211977623330440,0.47197378935281829447882006674466, 0.3991857441558381932943484274604<


We proceed to the computation of "trade GE" Gini coefficients, in which country A trades with country B. In thiscomputation, the same 1000 parameters are used as in the computation of "no trade" Gini coefficient.

Heck$Oh6.nb 15

ü Country A

In[170]:=

pi1 = py ∗ f1 − wL ∗ L1 − wK ∗ K1;sol1 = Solve@8D@pi1, L1D 0, D@pi1, K1D 0<, 8K1, L1<D@@1DD;demand$L1 = Simplify@Factor@L1 ê. sol1DD;demand$K1 = Simplify@Factor@K1 ê. sol1DD;supply$A$y = Simplify@PowerExpand@Factor@f1 ê. sol1DDD;pi10 = Simplify@PowerExpand@Factor@pi1 ê. sol1DDD;pi2 = px ∗ f2 − wL ∗ L2 − wK ∗ K2;sol2 = Solve@8D@pi2, L2D 0, D@pi2, K2D 0<, 8K2, L2<D@@1DD;demand$L2 = Simplify@Factor@L2 ê. sol2DD;demand$K2 = Simplify@Factor@K2 ê. sol2DD;supply$A$x = Simplify@PowerExpand@Factor@f2 ê. sol2DDD;pi20 = Simplify@PowerExpand@Factor@pi2 ê. sol2DDD;pi3 = pz ∗ f3 − wL ∗ L3 − wK ∗ K3;sol3 = Solve@8D@pi3, L3D 0, D@pi3, K3D 0<, 8K3, L3<D@@1DD;demand$L3 = Simplify@Factor@L3 ê. sol3DD;demand$K3 = Simplify@Factor@K3 ê. sol3DD;supply$A$z = Simplify@PowerExpand@Factor@f3 ê. sol3DDD;pi30 = Simplify@PowerExpand@Factor@pi3 ê. sol3DDD;uA = x^a ∗ y^b ∗ z^c;sol3A = Solve@8D@uA, xDêD@uA, yD pxê py,

D@uA, zDê D@uA, yD pzê py, px ∗ x + py ∗ y + pz ∗ z m<, 8x, y, z<D@@1DD;demand$L$x = x ê. sol3A ê. m → wL ∗ LeA; demand$L$y = y ê. sol3A ê. m → wL ∗ LeA;demand$L$z = z ê. sol3A ê. m → wL ∗ LeA;demand$K$x = x ê. sol3A ê. m → wK ∗ KeA; demand$K$y = y ê. sol3A ê. m → wK ∗ KeA;demand$K$z = z ê. sol3A ê. m → wK ∗ KeA; demand$E1$x = x ê. sol3A ê. m → pi10;demand$E2$x = x ê. sol3A ê. m → pi20; demand$E3$x = x ê. sol3A ê. m → pi30;demand$E1$y = y ê. sol3A ê. m → pi10; demand$E2$y = y ê. sol3A ê. m → pi20;demand$E3$y = y ê. sol3A ê. m → pi30; demand$E1$z = z ê. sol3A ê. m → pi10;demand$E2$z = z ê. sol3A ê. m → pi20; demand$E3$z = z ê. sol3A ê. m → pi30;demand$A$y =

Simplify@demand$L$y + demand$K$y + demand$E1$y + demand$E2$y + demand$E3$yD;demand$A$x = Simplify@demand$L$x + demand$K$x +

demand$E1$x + demand$E2$x + demand$E3$xD;demand$A$z = Simplify@demand$L$z + demand$K$z + demand$E1$z + demand$E2$z + demand$E3$zD;

Heck$Oh6.nb 16

ü Country B

In[191]:=

f = L^a ∗ K^b; f1 = f ê. 8a → a1, b → b1, L → L1, K → K1<;f2 = f ê. 8a → a2, b → b2, L → L2, K → K2<; f3 = f ê. 8a → a3, b → b3, L → L3, K → K3<;pi1 = py ∗ f1 − wL ∗ L1 − wK ∗ K1;sol1 = Solve@8D@pi1, L1D 0, D@pi1, K1D 0<, 8K1, L1<D@@1DD;pro$A$y = Hdemand$A$y + exp$A$yL; pro$A$x = Hdemand$A$x − imp$A$xL;demand$B$L1 =

Simplify@PowerExpand@Factor@L1 ê. sol1 ê. 8a1 → a1B, b1 → b1B, wK → wKB, wL → wLB<DDD;demand$B$K1 = Simplify@PowerExpand@

Factor@K1 ê. sol1 ê. 8a1 → a1B, b1 → b1B, wK → wKB, wL → wLB<DDD;supply$B$y = Simplify@PowerExpand@Factor@

f1 ê. sol1 ê. 8a1 → a1B, b1 → b1B, wK → wKB, wL → wLB<DDD;pi10B = Simplify@PowerExpand@Factor@pi1 ê. sol1 ê.

8a1 → a1B, b1 → b1B, wK → wKB, wL → wLB<DDD;pi2 = px ∗ f2 − wL ∗ L2 − wK ∗ K2; sol2 = Solve@8D@pi2, L2D 0, D@pi2, K2D 0<,

8K2, L2<D@@1DD;demand$B$L2 = Simplify@PowerExpand@

Factor@L2 ê. sol2 ê. 8a2 → a2B, b2 → b2B, wK → wKB, wL → wLB<DDD;demand$B$K2 = Simplify@PowerExpand@

Factor@K2 ê. sol2 ê. 8a2 → a2B, b2 → b2B, wK → wKB, wL → wLB<DDD;supply$B$x = Simplify@PowerExpand@Factor@

f2 ê. sol2 ê. 8a2 → a2B, b2 → b2B, wK → wKB, wL → wLB<DDD;pi20B = Simplify@PowerExpand@Factor@pi2 ê. sol2 ê.

8a2 → a2B, b2 → b2B, wK → wKB, wL → wLB<DDD;u = x^aB ∗ y^bB; sol3 = Solve@8D@u, yDêD@u, xD pyêpx, px ∗ x + py ∗ y m<, 8x, y<D@@1DD;demand$B$L$y = y ê. sol3 ê. m → wL ∗ LeB ê. 8wK → wKB, wL → wLB<;demand$B$L$x = x ê. sol3 ê. m → wL ∗ LeB ê. 8wK → wKB, wL → wLB<;demand$B$K$y = y ê. sol3 ê. m → wK ∗ KeB ê. 8wK → wKB, wL → wLB<;demand$B$K$x = x ê. sol3 ê. m → wK ∗ KeB ê. 8wK → wKB, wL → wLB<;demand$B$E1$y = y ê. sol3 ê. m → pi10B ê. 8wK → wKB, wL → wLB<;demand$B$E1$x = x ê. sol3 ê. m → pi10B ê. 8wK → wKB, wL → wLB<;demand$B$E2$y = y ê. sol3 ê. m → pi20B ê. 8wK → wKB, wL → wLB<;demand$B$E2$x = x ê. sol3 ê. m → pi20B ê. 8wK → wKB, wL → wLB<;demand$B$y = Simplify@demand$B$L$y + demand$B$K$y + demand$B$E1$y + demand$B$E2$yD;demand$B$x = Simplify@demand$B$L$x + demand$B$K$x + demand$B$E1$x + demand$B$E2$xD;pro$B$y = demand$B$y − imp$B$y; pro$B$x = demand$B$x + exp$B$x;ec1 = Hpro$A$x supply$A$xL ê. wL → 1;ec2 = Hpro$A$y supply$A$yL ê. wL → 1; ec3 = Himp$A$x ∗ px exp$A$y ∗ pyL;ec4 = Hdemand$K1 + demand$K2 + demand$K3 KeAL ê. wL → 1;ec5 = Hpro$B$y supply$B$yL ê. wL → 1 ê. imp$B$y → exp$A$y;ec6 = Hpro$B$x supply$B$xL ê. wL → 1 ê. exp$B$x → imp$A$x;ec7 = Hdemand$B$K1 + demand$B$K2 KeBL ê. 8wK → wKB, wL → wLB<;ec8 = Hsupply$A$z demand$A$zL ê. wL → 1;checkE1 = 8ec1, ec2, ec3, ec4, ec5, ec6, ec7, ec8<;

In what follows, we derive 1000 pairs of candidate for "trade GE", data10, in terms of Newton method, where theinitial values of prices for the Newton method are those derived in data2, and imp$A$x[0]=1, exp$A$y[0]=1, andwLB [0]=1/2, and wL ª1.

Heck$Oh6.nb 17

In[203]:=

data10 = Table@FindRoot@HcheckE1 ê. h@iDL, 8px, Hpx ê. data2@@iDDL<,8py, Hpy ê. data2@@iDDL<, 8pz, Hpz ê. data2@@iDDL<, 8wK, HwK ê. data2@@iDDL<,8imp$A$x, 1<, 8exp$A$y, 1<, 8wKB, HwK ê. data2@@iDDL<, 8wLB, 1 ê2<,AccuracyGoal → 20, WorkingPrecision → 34, MaxIterations → 1000D, 8i, 1, nu<D;

FindRoot::njnum : The Jacobian is not a matrix of numbers at8px, py, pz, wK, imp$A$x, exp$A$y, wKB, wLB< = 81.86161, 0., 3.72323, 0., 3.72323, 0., 1.82649, 0.<. More…

In[204]:=


88px → 0.5255882695512903172715450348150774,py → 50.52547382170233194318897608777439, pz → 14.23539061258310230898910606903765,wK → 0.3328578535116872364314623770980912,imp$A$x → 66.64728792830395266831328324078636,exp$A$y → 0.6932944925194896021942242394823991,wKB → 0.1382013386433161200458228418630121,wLB → 0.05461732046377808648737190147936345<<

Note that data10 contain non-convergent GE prices. The following set, data13, is the set of positions of 1000 pairs, atwhich (2A), (3A), (4), (6), (7), (8), (9), and (10) are satisfied.

In[205]:=

data11 = Table@HcheckE1 ê. h@iDL ê. data10@@iDD, 8i, 1, nu<D;data12 = Table@Count@data11@@iDD, TrueD, 8i, 1, nu<D;data13 = Flatten@Position@data12, 8DD;

In[206]:=


88, 8, 8, 8, 8, 8, 8, 0, 8, 8, 8, 8, 8, 8, 8, 8, 8, 0, 8, 8<

The following set, data5A, is the intersection of data5 and data13: the set of positions of 1000 pairs, at which (2)~(4)and (2A), (3A), (4), (6), (7), (8), (9), and (10) are satisfied.

In[207]:=

data5A = Intersection@data5, data13D;

In[208]:=

Take@data5A, 10DOut[208]=

81, 2, 3, 4, 5, 6, 7, 9, 10, 11<

The following set, giniA0T, is the one of 1000 "trade GE" Gini co-officiate, constructed from data10, which containnon-convergent GE prices.

In[209]:=

data14 = Table@8wL ∗ LeA, wK ∗ KeA, pi10, pi20, pi30< ê. h@iD ê. data10@@iDD, 8i, 1, nu<D;giniA0T = Table@giniA@data14@@iDDD, 8i, 1, nu<D;

In[210]:=

Take@giniA0T, 3DOut[210]=

80.43720573960324733664347367044704,0.40457406923254087309919153861537, 0.3989275542107535251642483207979<

Heck$Oh6.nb 18

The following set, data15, is the set of giniA00-giniA0T. The set data15 contains such values as r1 +r2 10-36 for somereal numbers, r1 and r2 , which is actually equal to r1 . The following procedure modifies such r1 +r2 10-36 to r1 . Evenif such procedure is applied to data15, not all of them are real numbers. The set, data16, is the set of positions, at whichthe elements are real numbers.

In[211]:=

data15 = Table@HginiA00 − giniA0TL@@iDD, 8i, 1, nu<D;kk@x_D := If@0 < Abs@Im@xDD < 10^H−30L, Re@xD, xD;data15A = Table@kk@data15@@iDDD, 8i, 1, nu<D; kk1@x_D := If@Im@xD =!= 0, n, gD;data15AA = Table@kk1@data15A@@iDDD, 8i, 1, nu<D;data16 = Flatten@Position@data15AA, gDD;

In[212]:=


81, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22<

In[213]:=

Length@data16DOut[213]=

839

The following set, data17, is the intersection of data5A and data16.

In[214]:=

data17 = Intersection@data5A, data16D;

In[215]:=

Length@data17DOut[215]=

771

In what follows, the share of "expanded inequality" cases in data17.

In[216]:=

data18 = Table@data15A@@data17@@iDDDD, 8i, 1, Length@data17D<D;

In[217]:=

Length@Select@data18, # ≤ 0 &DDêLength@data18DOut[217]=

431771

In[218]:=

N@%DOut[218]=

0.559014

It is approximately 55%. Other simulations using 1000 random parameters each show almost the same results;0.570332.

After the globalization, we may conclude that the Gini coefficient of income distribution rises, so that the inequalityof income distribution becomes expands. Compared with the results in Fukiharu and Choi [2009], the share of

Heck$Oh6.nb 19

"expanded inequality" cases among the GE solutions is somewhat lower than in Fukiharu and Choi [2009] with theshare approximately 65%, which does no contain non-traded commodity.

Conclusion

The purpose of the present paper was to examine theoretically whether the inequality expands through the globaliza-tion of a small country, utilizing Heckscher-Ohlin model. Heckscher-Ohlin model is an application of general equilib-rium theory with trading two countries. In the present paper, first, Gini coefficient was computed for the incomedistribution in a small country, A, in the general equilibrium: "no trade GE". Thus, we start with the case of Country Ain national isolation. Next, supposing the country A opens its economy to a large country B, Gini coefficient is com-puted for the income distribution in a country A, in the general equilibrium with trade: "trade GE". If the country A'sGini coefficient at "no trade GE" is smaller than the one at "trade GE", it is defined that the inequality expands throughthe globalization of country A. In the traditional Heckscher-Ohlin model, it is assumed that, trading two countries have the same production andutility functions, where the production functions for the two industries are under constant returns to scale. In thepresent paper, the production functions are assumed to be of Cobb-Duals type under decreasing returns to scale, sothat positive profit accrues to the entrepreneurs, while the two countries have the same production and utility functions.Thus, Fukiharu and Choi [2009] assumed that there are four consumers of goods; aggregate workers, aggregatecapitalists, and two entrepreneurs. It adopted a simulation approach: i.e. randomly selecting 1000 pairs of parameters inproduction and utility functions and initial endowments of working hours and capital goods where country A is a smallcountry compared with country B. In the repeated simulation using this Mathematica program, about 65% of the"result" showed that the income distribution of country A becomes more unequal after the globalization. In the present paper, a third industry was introduced, whose output is a non-traded commodity and its productionfunctions is assumed to be of Cobb-Duals type under decreasing returns to scale, so that positive profit accrues to thethird entrepreneur. Thus, in the present paper, there are five consumers of goods; aggregate workers, aggregate capital-ists, and three entrepreneurs. Adopting the same simulation approach: i.e. randomly selecting 1000 pairs of parametersin production and utility functions and initial endowments of working hours and capital goods where country A is asmall country compared with country B. In the repeated simulation using this Mathematica program, about 55% of the"result" showed that the income distribution of country A becomes more unequal after the globalization. Thus, we mayconclude that there is still a tendency that, when a small country open its economy due to the globalization the incomedistribution becomes more unequal.

References

Fukiharu, T. [2004], "A Simulation of the Heckscher-Ohlin Theorem", Mathematics and Computers in Simulation 64, pp.161-168.

Fukiharu, T. [2008], "Information and Communication Technologies and the Income Distribution: A General Equilibrium Simulation", Miquel Sànchez-Marrè, Javier Béjar, Joaquim Comas, Andrea E. Rizzoli, Giorgio Guariso (Eds.) Proceedings of the iEMSs Fourth Biennial Meeting: International Congress on Environmental Modelling and Software (iEMSs 2008). International Environmental Modelling and Software Society, Barcelona, Catalonia, July 2008, vol1, pp.240-247.

Fukiharu, T. [2009], "Information and Communication Technologies and the Income Distribution: A Simulation 　　　through Inequality Measures", Anderssen, R.S., R.D. Braddock and L.T.H. Newham (eds) 18th World IMACS

Heck$Oh6.nb 20

　　 Congress and MODSIM09 International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand and International Association for Mathematics and Computers in Simulation, July 2009, pp. 1404-1410. ISBN: 978-0-9758400-7-8. http://www.mssanz.org.au/modsim09/F12/kragt.pdf

Fukiharu, T. , and I.J. Choi [2009], "Globalization and Income Distribution Inequality in Heckscher-Ohlin Model I: ASimulation for Small Country Case", http://home.hiroshima-u.ac.jp/fukito/index.htm.

Layard, P.R.G. and A.A. Walters [1978], Microeconomic Theory, McGraw-Hill Book, U.K.

Heck$Oh6.nb 21

globalization and income distribution inequality in ...oh6.pdf · i: a specified case traditional...

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