endogenous wealth and knowledge in heckscher …this study integrates endogenous knowledge with the...

ENDOGENOUS WEALTH AND KNOWLEDGE IN HECKSCHER-OHLIN THEORY

PROF. WEI-BIN ZHANG*Ritsumeikan Asia Pacific University,

Japan

This study introduces endogenous knowledge in the two-country free-trade model with endogenous capital recently proposed by Zhang (2015). We examine the role of knowledge utilization efficiencies, learning by doing efficiencies, total factor productivities, and preferences in influencing dynamics of capital stocks, knowledge, and pattern of trade on the basis of the generalized Heckscher-Ohlin (H-O) theory of international trade. The paper constructs a free trade model with wealth and knowledge accumulation as growth determinants. Labor and capital distributions between countries and between sectors are determined in perfectly competitive markets. The model is an integration of the Oniki-Uzawa trade model, the H-O model, the Solow-Uzawa neoclassical growth model, Arrow’s learning-by-doing model, and Zhang’s treatment of knowledge as international public stock. The model is simulated. The existence of equilibrium point is confirmed. The motion of the dynamic system is plotted. We carry out comparative dynamic analysis with regard to the knowledge utilization efficiencies, population expansion, the propensity to consume capital good, the propensity to consume countries’ specified goods, and the propensity to hold wealth.

Keywords: O-H model; learning by doing; knowledge utilization efficiency; global economic growth; knowledge accumulation

1. Introduction

This study integrates endogenous knowledge with the generalized Heckscher-Ohlin (H-O) model recently proposed by Zhang (2015). As one of the core models in modern international trade theory, the H-O model is originally constructed for two countries (Heckscher, 1919; Ohlin, 1933; and Heckscher and Ohlin, 1991). The original model is basically static. The global economy produces two goods by employing two fixed input factors (labor and capital). Markets in national economies and trade between countries are perfectly competitive with constant returns to scale. The two strict assumptions in the model are that countries share the same technology and input factors of production are immobile between countries. Moreover, it is assumed that production factors are exogenously fixed. We relax these assumptions by introducing endogenous capital,

International Journal of Development and Conflict7(2017) 119–137

*Email: [email protected]

Copyright © Wei-Bin Zhang Licensed under the Creative Commons Attribution-NonCommercial License 3.0

120 Wei-Bin Zhang

endogenous knowledge and free trade. The importance of examining differences in saving and knowledge utilization is illustrated by what Chen (1992: 923-4) describes when reviewing the literature of O-H theory and international trade: “It appears to be the general consensus in this body of literature that the main determinant of long-run comparative advantage is the countries’ savings rates. The question of what has caused the difference in savings rates among countries, however, is rarely explicitly discussed in the literature. The models that do endogenize savings rates (e.g., Stiglitz 1970) attribute the difference in savings rates and hence long-run comparative advantage to a difference in preferences; in particular, a difference in agents’ time discount factors among countries. Yet explaining trade in terms of differences in preferences is no longer in the spirit of the Heckscher-Ohlin model in which trade arises because of differences in relative factor proportions.”

Many studies are conducted in generalizing and extending the H-O theory. For instance, Chen (1992) examines open economies in the H-O modelling framework. Chen introduces endogenous labor supply and savings. Baxter (1992) applies a model structurally similar to Chen’s model and includes different tax rates across countries. Ventura (1997) examines trade issues within an Ramsey growth modelling framework. A dynamic H-O model is proposed by Bajona and Kehoe (2010). The model integrates a static standard H-O trade model with a two-sector growth model of infinitely lived consumers with borrowing and lending. A recent extension of the H-O model to a dynamic neoclassical framework is given by Zhang (2015). The model by Zhang deals with trade patterns with different propensities to save and to consume by applying the approach to describing household behavior proposed by Zhang (1993). This study generalizes Zhang’s study by introducing endogenous knowledge through Arrow’s idea of learning by doing.

This study deals with a dynamic interdependence between global economic growth, economic structural change, wealth accumulation, knowledge accumulation, and international trade. Productivity differences and technological changes are important determinants of variations in incomes across countries (Manasse and Turrini, 2001; Agénor, 2004; Aghion et al. 2009; Gersbach et al. 2013). The model is built on the basis of neoclassical growth theory with endogenous capital and neoclassical growth theory with endogenous knowledge. It is a synthesis of the H-O theory, the Solow growth model (Solow, 1956), the Uzawa two-sector model (Uzawa, 1961, 1963), and the Oniki-Uzawa trade model (Oniki and Uzawa, 1965). We model technological change on the basis of literature of endogenous economic growth (Arrow, 1962, Uzawa, 1965, Romer, 1986; Lucas, 1988; Grossman and Helpman, 1991; Aghion and Howitt, 1992). There are different ways of introducing endogenous knowledge into trade theory (e.g., Chari and Hopenhayn, 1991; Martin and Ottaviano, 2001; Brecher et al. 2002; Nocco, 2005; Hinloopen, 2014). In this study knowledge is assumed to be an international public good. All countries have free access to knowledge and knowledge utilization by any country does not influence that by others (Zhang, 1992). It should be noted that the model in this study is Ricardian. It assumes differences between countries in technology and labor productivity. IThe traditional Ricardian theory does not explains endogenous determinants of the differences. It should be noted that except endogenous knowledge (Zhang, 1992), most of our model

Endogenous Wealth and Knowledge in Heckscher-Ohlin Theory 121

based on Zhang (2015). The rest of the paper is developed in the following way. The basic model is cinstructed in Section 2. The dynamic analysis and simulation of the global economy are carried out in Section3. Comparative dynamic analysis is conducted in Section 4. In Section 5 we conclude the study. The main results in Section 3 are checked in the Appendix.

2. The Model

This study synthesizes the models developed by Zhang (1992, 2015). The trade model by Zhang (2015) is influenced by the H-O model and neoclassical trade growth theory. The global economy is composed of two national economies. They are indexed by j = 1, 2 No tariff on any good exists in the global economy. A national economy supplies three goods. The two countries produce a homogeneous capital good. Capital good is used as capital and consumer goods. The sector which supplies capital good is called the capital goods sector. The capital goods sector has almost the same features as the production sector in traditional neoclassical trade theory with a homogeneous good and endogenous capital (e.g., Ikeda and Ono, 1992). There is internationally free mobility of capital goods. Each country provides service. Services is only consumed by domestic consumers. Each national economy also supplies a country specified-good. It is produced only by the country and is internationally tradable. Specified goods are pure consumption goods. Hence, each country has the capital goods sector, specified goods sector, and service sector, indexed by i, j, s. Each sector uses two input factors, capital and labor. All markets are perfectly competitive. Input factors are fully employed at every moment in time. There is no international mobility in labor markets. Labor is freely mobile within each national economy. Capital is freely domestically and internationally mobile. Saving is made only by households. All earnings of firms are paid to input factors of production. There is no possibility of hoarding output in the form of non-productive inventories. Let all the prices be measured in terms of capital good. We use pj(t) to represent the price of country j’s service. The price of country j’s specified good is pj(t). No international migration implies difference in wage rates wj(t) between countries. Free movement of capital implies equalization of rate of interest r(t). Capital depreciation rates,

,jkδ are fixed. Let Njq(t) and Kjq(t) represent labor force and capital stocks employed by sector q in country j We use Fjq(t) to represent output level of sector q in country j

Production functions

Following Zhang (1992), we assume that knowledge is an international public good. Knowledge can be applied by any sector without affecting others. Knowledge is a public good which can be freely used by different producers. The production functions are neoclassical and homogeneous of degree one with the inputs. We take the production function of section q in country j on the following form

( ) ( ) ( ) ( ), , , 0, 1,m jq jq jq

jq jq jq jq jq jq jqjq jqF t A Z t K t N t Aα β α β α β= > + = (1)

122 Wei-Bin Zhang

in which Z(t) (> 0) is the knowledge stock at time t, and Ajq, αjq, and βjq are positive parameters. Here, mjq is called (j, q)’s knowledge utilization efficiency parameter. If

( ) ( )/m jq jqjZ t N t

β is interpreted as (j, q)’s qualified labor force, we see that the production function is a neoclassical one and homogeneous of degree one with the inputs. It should be noted that the Richardian trade theory is built with technological differences between countries, while the H-O theory is built for differences in capital and labor endowments but identical technology between countries. This paper takes account of not only differences in capital and labor endowments but also differences in technologies between countries.

The marginal conditions for the capital goods sectors

All prices are determined by freely competitive markets. Each individual firm has rate of interest, wage rate, and prices as given. The capital goods sector selects Kji(t) and Nji(t) The marginal conditions for maximizing profit are

( ) ( )( ) ( ) ( )

( ), , 1, 2.ji ji ji ji

jk jji ji

F t F tr t w t j

K t N t

α βδ+ = = = (2)

The marginal conditions for country j’s specified goods sectors

The marginal conditions imply

( ) ( ) ( )( ) ( ) ( ) ( )

( ), , 1, 2.jj j jj jj j jj

jk jjj jj

p t F t p t F tr t w t j

K t N t

α βδ+ = = = (3)

The marginal conditions for service sectors

The marginal conditions for service sectors imply

( ) ( ) ( )( ) ( ) ( ) ( )

( ), , 1, 2.js js js js js js

jk jjs js

p t F t p t F tr t w t j

K t N t

α βδ+ = = = (4)

The current income and disposable income

This study applies an alternative approach to household behavior developed by (Zhang, 1993). Consumers choose consumption levels of goods and saving. We use ( )jk t to represent the value of wealth of household in country j. Per household's current income from the interest payment ( ) ( )jr t k t and the wage payment wj(t) is

( ) ( ) ( ) ( ).j j jy t r t k t w t= +

It is assumed that selling and buying wealth can be conducted instantaneously without any transaction cost. The per capita disposable income is the sum of the current income and the value of the wealth

( ) ( ) ( ) ( )( ) ( ) ( )ˆ 1 .j j j j jy t y t k t r t k t w t= + = + + (5)The disposable income is distributed between saving and consumption.


The budgets and utility functionsWe use cjq(t) to represent consumption level of consumer good q in country j Let sj(t) stand for the representative household’s saving. Consumer j’s budget constraint implies

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 2 ˆ .ji j j js js j jc t p t c t p t c t p t c t s t y t+ + + + = (6)

Consumers’ utility is dependent on the consumption levels of goods and services, and the saving as follows

( ) ( ) ( ) ( ) ( ) ( )0 10 20 0 00 10 20 0 01 2 , , , , , 0,ji j j js j

j ji j j js jji js jj jU t c t c t c t c t s tξ ξ ξ γ λ ξ ξ ξ ξ λ= > (7)

where 0jiξ is the propensity to consume capital good, 0jjξ the propensity to consume country j’s specified good, and 0jλ the propensity to save. Maximizing (7) subject to (6) yields

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

1 1 1 2 2 2ˆ ˆ ˆ, , ,

ˆ ˆ, ,

ji ji j j j j j j j

js js js j j j j

c t y t p t c t y t p t c t y t

p t c t y t s t y t

ξ ξ ξ

ξ λ

= = =

= = (8)

where

0 1 10 2 20 0 0

0 10 20 0 0

, , , , ,

1.

ji j ji j j j j j j js j js j j j

jji j j js j

ξ ρ ξ ξ ρ ξ ξ ρ ξ ξ ρ ξ λ ρ λ

ρ ξ ξ ξ ξ λ

≡ ≡ ≡ ≡ ≡

≡+ + + +

Wealth accumulationThe change in wealth is the saving minus the dissaving. The definitions of sj(t) implies the following equation for the wealth change of the representative household

( ) ( ) ( ).j j jk t s t k t= − (9)

Factors being fully employedThe capital stocks employed by country j is denoted by Kj(t) The full employment of national capital stock implies

( ) ( ) ( ) ( ), 1, 2.ji jj js jK t K t K t K t j+ + = = (10)

Fully employment of national labor force implies

( ) ( ) ( ) , 1, 2.ji jj js jN t N t N t N j+ + = = (11)

Market clearing for two tradable goods We have

( ) ( ), 1, 2.js j jsc t N F t j= = (12)

Market clearing in each specified good markets We have

( ) ( ) ( )1 1 2 2 , 1, 2.q q qqc t N c t N F t q+ = = (13)

124 Wei-Bin Zhang

Market clearing in capital marketsThe global net savings equals the global capital production

( ) ( ) ( )( ) ( )

2 2

1 1.j j j j kj j ji

j js t N k t N K t F tδ

= =− + =∑ ∑

(14)

Wealth balanceAll the households’ wealth equals the total global wealth

( ) ( ) ( ) ( ) ( )1 1 2 2 1 2 , 1, 2.k t N k t N K t K t K t j+ = + = = (15)

Knowledge creationFollowing Arrow’s idea of learning-by-doing (Arrow, 1962), we have the following equation to model knowledge growth (Zhang, 1992)

( ) ( )( )

( )( )

( )( )

( )2

1,

ji ji jj jj js jszji jj jsj

F t F t F tZ t Z t

Z t Z t Z tε ε ε

τ τ τδ

=

= + + −∑

(16)

where ( 0)zδ ≥ is the fixed rate of knowledge depreciation, and jqε and jqτ are parameters. The parameters jqτ are non-negative. The term ( ) ( )/ jq

jqF t Z tετ is the contribution of

country j’s capital goods sector to knowledge growth through learning by doing.We thus built the dynamic model with endogenous wealth accumulation.

3. Dynamic Analysis

The system we built to describe motion of global economy are nonlinear differential equations. It is usually not possible to solve such nonlinear equations. We simulate the model. Before we provide a computational procedure to follow the movement of the global economy, we introduce the following variable

( ) ( )( )

11

1.kr t

z tw t

δ+≡

The following lemma is checked in the Appendix.

Lemma

We describe movement of 3 variables z1(t), ( )2 ,k t and Z(t) by the following three differential equations

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )

1 1 2

2 2 1 2

1 2

, , ,

, , ,

, , ,

z t z t k t Z t

k t z t k t Z t

Z t z t k t Z t

= Λ

= Φ

= Λ

(17)


where ( ),tΛ and ( )2 ,tΦ and ( )tΛ are functions of Z1(t), ( )2 ,k t and Z(t) defined in the Appendix. The values of the other variables are given as functions of Z1(t), ( )2 ,k t and Z(t) at any point in time as follows: r(t) by (A2) → wj(t) by (A2) → pj(t) by (A4) →pjs(t) by (A5) → ( )1k t by (A15) → K(t) by (A14) → Njj(t) by (A12) →Nji(t) by (A8) → Njs by (A7) → Kji(t), Kjj(t), Kjs(t) by (A1)→ Fjq(t) by (1) → ( )ˆ jy t by (A5) → Kj(t) by (10) →cj1(t), cj2(t) and sj(t) by (8).

This lemma enables us to follow the motion of the system. For simulation, we specify values of the parameters as follows

1 2 1 2 11 22 1 2

1 2 11 22 1 2 1

1 2 2 11 11 22 22 1

10, 20, 0.05, 1.2, 1.3, 1.3, 1.2, 1, 1,

0.4, 0.3, 0.3, 0.2, 0.2, 0.1, 0.06, 0.02,

0.01, 0.01, 0.4, 0.02, 0.01, 0.01, 0.4, 0.32,

jk i i s s

i i s s z i

i i i i

N N A A A A A A

m m m m m m δ τε τ ε τ ε τ ε α

= = = = = = = = == = = = = = = == = = = = = = =

2 11 22 1 2 1 0 110 1200.3, 0.31, 0.31, 0.29, 0.3, 0.04, 0.05, 0.03,i s s iα α α α α ξ ξ ξ= = = = = = = =

1 0 1 0 2 0 210

220 2 0 2 0

0.03, 0.75, 0.05, 0.03,

0.05, 0.04, 0.7.s s i

s s

ξ λ ξ ξξ ξ λ

= = = == = =

Country 1’s population is smaller than country 2's population. Depreciation rates of physical capital are equal between the two countries and between the sectors. The total factor productivities vary between the two countries. Country 1’s knowledge utilization efficiencies are higher than country 2’s. Country 1 is more efficient in learning by doing than country 2. The total factor productivity of country 2’s industrial sector is lower than country 1’s. The propensity to save of country 1’s representative household is higher than that of country 2’s. The initial conditions are

( ) ( ) ( )1 20 0.0055, 0 55, 0 76.z k Z= = =

We plot the motion of the system in Figure 1 in which the national incomes and the global income are defined as follows:

1 2, .j ji js js j jjY F p F p F Y Y Y≡ + + ≡ +

Before the global economy becomes stationary, the global income, national incomes, and wealth rise in association with falling in knowledge. In the initial stage country 1 is in trade deficit and in the long term country 1 is in trade surplus. Economic structural changes take place in the two national economies. The output of country 1’s capital sector rises and the output of country 2’s capital sector alls. The output levels of the two countries’ domestic goods sectors are increased. The output of country 1’s service sector rises and the output of country 2’s service sector falls. The rate of interest falls and the wage rates rise slightly. The representative household of country 1 has more wealth and consumes more goods and service. The wealth and consumption levels of the goods and service of the ntative household of country 2 are slightly changed.

126 Wei-Bin Zhang

0 30 60 90

2300

2000

17000 30 60 90

75.2

75.7

76.2

0 30 60 90

580

605

630

0 30 60 90

284

297

310

0 30 60 90295

310

325

0 30 60 90620

920

1220

0 30 60 90860

1030

1200

0 30 60 90128

144

160

0 30 60 904.4

6.3

8.2

0 30 60 90362

431

500

0 30 60 90

65

75

85

0 30 60 903.6

5.8

8

0 30 60 90280

390

500

0 30 60 90

10

15

20

0 30 60 90

1.5

2.9

4.3

0 30 60 9084

152

220

0 30 60 900.035

0.045

0.055

0 30 60 90

12

17

22

0 30 60 901.5

1.55

1.6

0 30 60 903.1

3.19

3.28

0 30 60 90

60

90

120

0 30 60 90

4

5.5

7

0 30 60 901.8

2.5

3.2

0 30 60 901.8

3.6

5.4

0 30 60 90

0.9

1.2

1.5

t t t t

t t t t

t t t t

t

K

1K 2K

12c

iN1 iF1

iN2

iF2

iK2

22F

sp2

sp1

11c

sc2

iK1

11K

sc1

22c

2K 1K

Y

2k

11F 11N

22N 22K

1Y 2Y

sK2

sN2 sF2

1k

1w

2w

21c

t

t

t

t

t t t

1p

2p

r

1sF

1sK

1sN

t t t t t

1ic

2ic

Z

Figure 1. The Motion of the Global Economy

Figure shows that the system tends to be stationary as time passes. The procedure in the lemma enables us calculate the equilibrium values of the variables as follows

1 2 1 1

2 2 1 2 1 2

1 2 1 2 11 21 12

22 1

631.5, 74.9, 2342.5, 308.6, 322.9, 1157.8, 1272.3,

1184.7, 1070.2, 0.033, 21.3, 11.3, 1.49, 1.60,

3.28, 3.08, 6.79, 3.82, 5.7, 1.54, 3.18,

2.38, 1.5

s s i i

s

Y Z K Y Y K K

K K r w w p p

p p c c c c c

c c

= = = = = = == = = = = = == = = = = = =

= = 2 1 25, 0.99, 127.2, 53.5,sc k k= = =

1 2 11 22

1 2 11 22

1 2 11 22

1 2

1 2

1

127 134.4 87.7 79.5

4.06 , 8.37 , 4.24 , 7.82 ,

491 486.8 489 476.5

15.5

1.7 ,

178.2

i i

i i

i i

s s

s

s

F F F F

N N N N

K K K K

F F

N N

K

= = = =

= 2

19.84

3.81 .

221.5s

sK

=

We calculate the three eigenvalues

{ }0.17, 0.14, 0.03 .− − −

All the eigenvalues are negative. This implies that the world economy is stable. The stability guarantees comparative dynamic analysis.


4 Comparative Dynamic Analysis

The previous section simulated the motion of the dynamic system. We now conduct comparative dynamic analysis. Basing on the results in the previous section, we can examine transitory and long-term effects of changes in any parameter. A variable ( )x t∆ introduced to represent the change rate of the variable x(t) in percentage due to changes in the parameter value.A rise in the knowledge utilization efficiency of country 1’s capital goods sectorFirst, we study the impact on the movement of the global economy when the knowledge utilization efficiency of country 1’s capital goods sector is enhanced as follows:

1 : 0.4 0.41.im ⇒ The simulation results are plotted in Figure 2. Sector (1, i)’s output is augmented. The wage rate in country 1 rises. The rate of interest falls. Sector (1, i) increases labor and capital inputs. Sector (1, 1) reduces labor input. Sector (1, s) initially reduces labor input and raise labor input in the long term. Sectors (1, 1) and (1, s) initially produce less and in the long term produce more. Sectors (1, 1) and (1, s) employ more capital. In country 2 the wage rate is reduced. Sector (2, i) reduces its output and employs less labor and capital. Sector (2, 2) augments its output and employs more labor. The capital stock employed by sector (2, 2) is slightly changed. The prices of country 1’s specified good and service in country 1 are lowered and the prices of country 2’s specified good and service in country 2 are enhanced. The household in country 1 owns initially less wealth and in the long term more wealth; initially consumes less the capital good, country 1’s specified good and service and in the long term re; consumes more country 2’s specified good. The household in country 2 owns less wealth, consumes less the capital good, country 1’s specified good, and consumes more service and country 2’s specified good.

30 60 902

1

4

0 30 60 901

4.5

8

30 60 900.4

2.2

4

0 30 60 905

7.5

10

0 30 60 90

5

3.4

1.8

30 60 906

2

1030 60 90

6

2.5

1

30 60 9010

2

14

30 60 90

9

2

5

30 60 90

15

2

11

30 60 906

1

8

30 60 906

0

6

30 60 906

1

8

30 60 90

8

1.5

5

30 60 908

1.5

5

30 60 90

10

0

10

0 30 60 904.6

6.3

8

30 60 906

2

10

30 60 90

6

1

4

30 60 906

0

6

30 60 906

2

10

30 60 904

3

10

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4

10

16

30 60 908

1.5

5

30 60 90

8

1.5

5

t t

t t

t t t t

t t t t

t

K∆

1K∆ 2K∆

12c∆

iN1∆ iF1∆

iN2∆ iF2∆ iK2∆

22F∆

sp2∆

sp1∆

11c∆ sc2∆

iK1∆

11K∆

sc1∆

22c∆

2K∆ 1K∆

Y∆

2k∆

11F∆ 11N∆

22K∆ 22N∆

1Y∆ 2Y∆

sK2∆

sN2∆ sF2∆

1k∆

1w∆

2w∆

21c∆

t

t

t

t

t t t

1p∆

2p∆

r∆

1sF∆

1sN∆

1sK∆

t t

t

t t 1ic∆

2ic∆

Z∆

Figure 2. A Rise in the Knowledge Utilization Efficiency of Country 1’s Capital Goods Sector

128 Wei-Bin Zhang

An improvement in the total factor productivity of country 1’s specified goods sectorWe now deal with the effects of an improvement in the total factor productivity of country 1's specified goods sector. The total factor productivity is shifted as follows 11 :1.3 1.4.A ⇒ We plot the simulation results in Figure 3. Country 1's specified goods sector expands. The sector increases output, raises capital input, and reduce labor input. Country 1’s service sector produces more, raises capital input and reduces labor input. Country 1’s capital goods sector raises output and raises labor and capital input factors. Country 2’s capital and service sectors raise output and employ more capital. In the long term the two sectors employ less labor. Country 2’s specified goods sector produces more and employs more the two input factors. The global wealth and total product are augmented. Knowledge is increased. The two countries’ national products are enhanced. The rate of interest rises initially and does not change in the long term. The wage rates are augmented. The prices of the services in the two countries are increased. The price of country 1’s specified good falls and the price of country 2’s specified good rises. All the households in the global economy own more wealth, consume more goods and services.

0 30 60 900

1.3

2.6

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2.5

5

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1.3

2.6

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1.5

3

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1.1

2.2

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3

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1.1

2.2

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1.7

3.4

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0

0.3

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3.4

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10

30 60 900.3

0

0.3

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1.3

2.6

30 60 900

1

230 60 90

0.2

0.1

0

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3

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0.3

0.6

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1.5

330 60 90

6

3

0

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0.6

1.2

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1.5

3

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1.5

3

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1.2

2.4

0 30 60 907.9

9.2

10.5

30 60 900

1

2

t t t t

t

t t

t

t

t

t t

t

K∆

1K∆ 2K∆

12c∆

iN1∆ iF1∆


22F∆

sp2∆ sp1∆

11c∆ sc2∆

iK1∆

11K∆

sc1∆

22c∆

2K∆ 1K∆

Y∆

2k∆

11F∆

11N∆

22K∆ 22N∆

1Y∆ 2Y∆

sK2∆

sN2∆ sF2∆

1k∆

1w∆

2w∆

21c∆

t

t

t

t

t t

t

1p∆

2p∆

r∆

1sF∆

1sN∆

1sK∆

t t t t t

1ic∆ 2ic∆

Z∆

Figure 3. A Rise in the Total Factor Productivity of Country 1’s Specified Goods sector

Country 2’s population rises

We now deal with the impact on the motion of the global economy when country 2’s population rises as follows: 2 : 20 22.N ⇒ We have the simulation results in Figure 4. There is little impact on the knowledge stock. There are increases in the global product and wealth. Country 1’s total output level falls and country 2’s total output level rises. From the figure we also see the economic structural changes in the two economies. In the long term the per households’ wealth and consumption levels are slightly affected. We see that


in the long term the change in country 2’s population has little impact on the households’ wealth and consumption levels.

0 30 60 904.6

4.8

5

0 30 60 900

0.11

0.22

0 30 60 905.04

5.07

5.130 60 90

0.07

0.035

0

0 30 60 90

9.92

9.95

9.98

30 60 90

4

2

0

0 30 60 90

9.8

10.1

30 60 902

5

12

30 60 902

5

12

30 60 905

3

11

0 30 60 903.5

5

6.5

0 30 60 903.5

5

6.5

0 30 60 903.5

5

6.5

30 60 904

3

10

30 60 904

3

10

30 60 904

3

10

0 30 60 900

0.35

0.7

30 60 90

0.06

0.03

0

0 30 60 900

0.013

0.026

0 30 60 900

0.02

0.04

30 60 90

4.2

2.1

030 60 90

3.4

1.7

030 60 90

3.4

1.7

030 60 90

3.4

1.7

030 60 90

3.4

1.7

0

t

t t t

t t t t

t t

t

t

t

K∆

1K∆

2K∆

12c∆

iN1∆ iF1∆


22F∆

sp2∆ sp1∆

11c∆

sc2∆

iK1∆

11K∆

sc1∆

22c∆

2K∆ 1K∆

Y∆

2k∆

11F∆ 11N∆ 22K∆

22N∆

1Y∆ 2Y∆

sK2∆

sN2∆ sF2∆

1k∆

1w∆ 2w∆

21c∆

t

t

t

t

t t t

1p∆ 2p∆ r∆

1sF∆ 1sN∆

1sK∆

t t t t t

1ic∆

2ic∆

Z∆

Figure 4. Country 2’s Population Rises

Country 1’s population rises

We show that a change in country 1’s population has little impact on the households’ wealth and consumption levels. We now examine the impact on the motion of the global economy when country 1’s population rises as follows: 1:10 11.N ⇒ We have the simulation results in Figure 5. Comparing es 5 and 6, we observe difference in the effects of changes in the two countries’ population. A rise in country 1’s population benefits all the households in the global economy. The difference comes from that country 1 uses knowledge more effectively and is more efficient in learning by doing than country 2. Combining Figures 4 and 5, we see that if some of country 2’s population is migrated to country 1 and the immigrants can adapt to country’s social and economic conditions, the global economy benefits.

130 Wei-Bin Zhang

0 30 60 90

6

11

16

0 30 60 900

10

20

0 30 60 90

6

10

14

0 30 60 9010

16

22

0 30 60 900

4

8

0 30 60 90

12

17

22

0 30 60 900

4

8

30 60 900

13

26

30 60 905

5

15

30 60 900

15

30

0 30 60 904

9

14

0 30 60 903.5

5

6.5

0 30 60 90

5

11.5

18

30 60 900

8

16

30 60 90

2

8

14

30 60 900

10

20

30 60 900.5

0.5

1.5

0 30 60 900

5

10

0 30 60 900

1

2

0 30 60 900

2

4

0 30 60 900

5

10

0 30 60 900

5

10

0 30 60 900

4

8

0 30 60 900

4

8

30 60 900

3.5

7

t t

t t

t t t t

t t t t

t

K∆

1K∆ 2K∆

12c∆

iN1∆ iF1∆


22F∆

sp2∆ sp1∆

11c∆ sc2∆

iK1∆

11K∆

sc1∆

22c∆

2K∆ 1K∆

Y∆

2k∆

11F∆ 11N∆

22K∆ 22N∆

1Y∆ 2Y∆

sK2∆

sN2∆ sF2∆

1k∆

1w∆

2w∆

21c∆

t

t

t

t

t t t

1p∆ 2p∆

r∆

1sF∆ 1sN∆

1sK∆

t t t t t

1ic∆ 2ic∆

Z∆

Figure 5. Country 1’s Population Rises

Country 1 raises the propensity to consume its specified good

We now deal with the effect of a rise in in country 1’s propensity to consume the country’s specified good: 110 : 0.05 0.06.ξ ⇒ We have the results in Figure 6. Country 1’s household raises the consumption of the country’s specified good. The household’s wealth and consumption levels of the other goods and services fall. Country 2’s household has less the wealth level and consumes less the s and service. We see that except that the household in country 1 consumes more country 1’s specified good, all the micro variables for all the households are reduced. Except that country 1 produces more country 1’s specified good, the output levels of all the other sectors are reduced. The global wealth, knowledge and global product are reduced. The two countries’ total products fall.


30 60 90

5

2.5

030 60 90

3

1.5

030 60 90

2.6

1.3

030 60 90

3

1.5

030 60 90

2.5

1.25

0

30 60 90

8

4

030 60 90

5

2.5

030 60 90

9

4.5

030 60 90

7

3

130 60 90

12

6

0

30 60 904

2

8

30 60 902

4

10

30 60 906

0

6 30 60 90

8

4

0

30 60 90

6

2.5

1

30 60 90

12

6

0

0 30 60 900

3.5

730 60 90

3

1.5

030 60 90

0.34

0.17

030 60 90

0.7

0.35

0

30 60 90

8

4

030 60 90

8

4

030 60 90

8

4

0

30 60 900

8

1630 60 90

8

4

0

t t t t

t t

t t

t t t t

t

K∆

1K∆ 2K∆

12c∆

iN1∆ iF1∆

iN2∆ iF2∆

iK2∆

22F∆

sp2∆ sp1∆

11c∆

sc2∆

iK1∆

11K∆

sc1∆ 22c∆

2K∆ 1K∆

Y∆

2k∆

11F∆ 11N∆

22K∆ 22N∆

1Y∆ 2Y∆

sK2∆

sN2∆ sF2∆

1k∆

1w∆ 2w∆

21c∆

t

t

t

t

t t t

1p∆

2p∆

r∆

1sF∆ 1sN∆

1sK∆

t t t t

t 1ic∆

2ic∆

Z∆

Figure 6. Country 1 Raises the Propensity to Consume Its Specified Good

Country 1 augmenting the propensity to consume country 2’s specified good

We now examine the impact of a rise in country 1’s propensity to consume the other country’s specified good: 120 : 0.03 0.04.ξ ⇒ We have the simulation results in Figure 7. Country 1’s household consumes more country 2’s specified good. The household’s wealth and consumption levels of the other goods and service fall. Country 2’s household slightly changes the wealth and the consumption levels of goods and service. The global wealth, global product, and the two countries’ total products are reduced. The knowledge stock rises. The rate of interest and the wage rates rise. Country 1’s trade balance is deteriorated and country 2’s trade balance is improved. Sector (1, i) produces more and employs more the two input factors and sector (2, i) produce less and employs less the two input factors. Sector (1, s) supplies less and employs less the two input factors. The output level of sector (2, s) is slightly affected. Sector (1, 1) supplies less and employs less the two input factors. Sector (2, 2) supplies more and employs more the two input factors.

132 Wei-Bin Zhang

30 60 90

4

2

0

0 30 60 900

0.4

0.830 60 90

1

0.5

030 60 90

1

0.5

030 60 90

1

0.5

0

30 60 90

7

3.5

030 60 90

4

2

0

30 60 90

11

3

5

30 60 90

10

2

630 60 90

15

7

1

30 60 904

3

10

30 60 904

3

10

30 60 906

0

6 30 60 90

7

3.5

0

30 60 90

6

2.5

1

30 60 90

10

5

0

0 30 60 900

4

830 60 90

1.2

0.6

0

30 60 900.02

0.05

0.12

30 60 900.05

0.05

0.15

30 60 90

7

3.5

030 60 90

7

3.5

0

0 30 60 900

15

3030 60 90

7

3.5

030 60 90

7

3.5

0

t t

t

t

t t

t t

t t

t t

t

K∆

1K∆ 2K∆

12c∆

iN1∆ iF1∆


22F∆

sp2∆

sp1∆

11c∆

sc2∆

iK1∆

11K∆

sc1∆ 22c∆

2K∆

1K∆

Y∆

2k∆

11F∆ 11N∆

22K∆ 22N∆

1Y∆ 2Y∆

sK2∆

sN2∆ sF2∆

1k∆

1w∆ 2w∆

21c∆

t

t

t

t

t t t

1p∆ 2p∆ r∆

1sF∆ 1sN∆

1sK∆

t t

t

t t

1ic∆

2ic∆

Z∆

Figure 7. Country 1 Augmenting the Propensity to Consume Country 2’s Specified GoodCountry 1 raises the propensity to save

We now examine the impact of a rise in country 1’s propensity to save: 10 : 0.75 0.77.λ ⇒ We have the simulation results in Figure 8. The global total income, global wealth, and the two countries’ total incomes are increased. The knowledge stock rises. Country 1’s household increase the wealth over time. The consumption levels of the two goods and service of country 1’s household fall initially and rise in the long term. The wealth and the consumption levels of the two goods and service in country 2 rise. The wage rates rise. The rate of interest falls. The prices of the two countries’ specified goods and services are increased. Country 1’s trade balance is improved and country 2’s trade balance is deteriorated.


0 30 60 900.2

1.2

2.2

0 30 60 900

0.6

1.2

0 30 60 900

0.5

1

0 30 60 900

0.6

1.2

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0.5

1

0 30 60 900

2

4

0 30 60 900

1

2

0 30 60 900.4

1

1.6

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0.7

1

0 30 60 900.6

1.6

2.6

30 60 901

0.2

0.6

0 30 60 901

0.6

0.2

30 60 901

0.5

2

30 60 90

2

0.5

130 60 90

2

1

0

30 60 902

0

230 60 90

3

1.5

0

0 30 60 900

0.6

1.2

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0.07

0.14

0 30 60 900

0.15

0.3

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2

4

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2

0.5

1

30 60 90

2

0.5

1

30 60 90

2

0.5

1

30 60 90

2

0.5

1

t t t t

t t

t t

t t t t

t

K∆

1K∆ 2K∆

12c∆

iN1∆ iF1∆


22F∆

sp2∆ sp1∆

11c∆ sc2∆

iK1∆

11K∆

sc1∆ 22c∆

2K∆ 1K∆

Y∆

2k∆

11F∆ 11N∆ 22K∆

22N∆

1Y∆ 2Y∆

sK2∆

sN2∆ sF2∆

1k∆

1w∆ 2w∆

21c∆

t

t

t

t

t t t

1p∆

2p∆ r∆

1sF∆ 1sN∆

1sK∆

t

t t t t 1ic∆

2ic∆

Z∆

Figure 8. Country 1 Raises the Propensity to Save

5. Concluding Remarks

This study introduced endogenous knowledge in the two-country free-trade model with endogenous capital recently proposed by Zhang (2015). We examined the role of knowledge utilization efficiencies, learning by doing efficiencies, total factor productivities, and preferences in influencing dynamics of capital stocks, knowledge, and pattern of trade on the basis of the generalized H-O theory of international trade. The paper constructs a free trade model with wealth and knowledge accumulation as growth determinants. Labor and capital distributions between countries and between sectors are determined in perfectly competitive markets. The model is an integration of the Oniki-Uzawa trade model, the H-O model, the Solow-Uzawa neoclassical growth model, Arrow’s learning-by-doing model, and Zhang’s treatment of knowledge as international public stock. The model was simulated. The existence of equilibrium point was confirmed. The motion of the dynamic system was plotted. Comparative dynamic analysis with regard to the knowledge utilization efficiencies, population expansion, the propensity to consume capital good, the propensity to consume countries’ specified goods, and the propensity to hold wealth was conducted. We may extend the model in different directions. We may, for instance, further simulate the model with different types of utility and production functions. We may also conduct more comparative dynamic analysis with regard to other parameters. As the model is built on a few core models in the literature of economics and there is a vast literature on generalizing and extending each model, we may generalize the model on the basis of the rich literature.

134 Wei-Bin Zhang

Appendix: Proving the Lemma

The appendix checks the lemma. Equations (2)-(4) imply

,jk ji ji jj jj js js

jj ji jj js

r N N Nz

w K K K

δ α α α+≡ = = = (A1)

where / .jq jq jqα α β≡ From (A1), (1), and (2), we get

( ) ( ), , , .

m ji jiji ji j jk

j jk j jji jji

A Z z rr z Z w z Z

z

β

βα δ

δα

+= − = (A2)

From (A2) we have

( )1/ 2

22 1 22

2 2

, .

ik

im ii i

rz z Z

A Z

βδ α

α

+=

(A3)

From (A1), (1), and (3), we have

( )1 , .jj

j jj m jj jj

jj jj jj

w zp z Z

A Z

α

αβ α= (A4)

Equations (1), (4) and (A1) imply

( )1 , .js

j jjs m js js

js js js

w zp z Z

A Z

α

αβ α= (A5)

From (A1) and (10) we get

.ji ji jj jj js js j jN N N z Kα α α+ + = (A6)

Equations (12) and (8) imply

,js j j js js jN n k Nβ ξ= + (A7)

where

( )11

, .j js js jj

rn z Z N

wβ ξ

+≡

Substitute (A7) into (11)

, 1, 2,ji j j j jjN n n k N j= − − = (A8)

where ( )1 .j js js jn Nβ ξ≡ − Substitute (A7) and (A8) into (A6)

( ) ( )1

.jj j j js js js j ji j js ji j jjj ji

N z K N n n kα β ξ α α αα α

= − − − − − (A9)


Substitute (8) into (13)

1 1 1 2 2 2ˆ ˆ , 1, 2.j j j jjy N y N p F jξ ξ+ = = (A10)

Substitute (3) into (A10)

1 1 1 2 2 2ˆ ˆ, 1, 2.

jj j jj jjj

j j

y N y NN j

w w

β ξ β ξ= + = (A11)

Substitute (5) into (A11)

1 1 2 2 , 1, 2.j j j jjw k w k w N j+ + = = (A12)

where

( ) ( )1 1 2 2 1 1 1 2 2 2

1 2

1 1, , .

jj j jj j jj j jj jj j j

j j j j

r N r N w N w Nw w w

w w w w

β ξ β ξ β ξ β ξ+ +≡ ≡ ≡ +

Equaling (A12) and (A9) yields

( ) ( ) ( )

( )

( ) ( ) ( )

( )

1 2111 1 11 1 1 1 11 1 2

1 1

11 1 1 1 1 1 1 1 11

1

12 222 2 1 22 2 22 2 2 2

2 2

22 2 2 2 2 2 2 2 22

2

,

.

i s i i

i s s s i

i i s i

i s s s i

k ww n k

z z

w N nK

z

w kk w n

z z

w N nK

z

α α α α α α

α α α β ξ α

α α α α α α

α α α β ξ α

− + − + − +

− + +=

− + − + − +

− + +=

(A13)

Add the two equations in (A13)

1 1 2 2 0 ,a k a k a K+ + = (A14)

where

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

121 11 1 11 1 1 1 22 2

1 2

212 11 1 22 2 22 2 2 2

1 2

11 1 1 1 1 1 1 1 1 22 2 2 2 2 2 2 2 20

1 2

1,

1,

.

i s i i

i i s i

i s s s i i s s s i

wa w n

z z

wa w n

z z

w N n w N na

z z

α α α α α α

α α α α α α

α α α β ξ α α α α β ξ α

≡ − + − + −

≡ − + − + −

− + + − + +≡ +

From (15) and (A14) we solve

( ) 2 2 01 1 2 2

1 1 1 1, , .

N a ak z k Z k

a N a N− = Φ ≡ − − − (A15)

136 Wei-Bin Zhang

Hence, we show that all the variables are represented as functions of z1, 2k and Z as follows: r by (A2) → wj by (A2) → z2 by (A3) → pj by (A4) → pjs by (A5) → 1k by (A15) → K by (A14) → Njj by (A12) → Nji by (A8) → Njs by (A7) → Fjq by (1) → ˆ jy by (5)→ cj1, cj2 and sj by (8) → Kji, Kjj, Kjs by (A1) → Kj by (10). From this procedure and (8), we have

( )1 0 1 2 1, , ,k z k Z s ϕ= Φ ≡ −

(A16)

( )2 2 1 2 2 2, , ,k z k Z s k= Φ ≡ −

( ) 2

1 21

, , .ji ji jj jj js js

zji jj jsj

F F FZ z k Z Z

Z Z Zε ε ε

τ τ τδ

=

= Λ ≡ + + −∑ (A17)

We taking derivatives of (A15) in t

1 1 21 2

,k zz Z k

∂Φ ∂ Λ ∂Φ= + Λ + Φ∂ ∂ ∂

(A18)

In which (A17) is used. Substitute (A18) into (A16)

( )1

1 1 2 0 212

, , .z z k ZZ zk

− ∂ Λ ∂Φ ∂Φ = Λ ≡ Φ − Λ − Φ ∂ ∂ ∂ (A19)

The procedure in the lemma results the dynamics of the economic system. The lemma is checked.

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