Skip to main content

Advertisement

SpringerLink
Log in

Convergence and Overtaking in a Dynamic two Country Model

  • Research Article
  • Published:
Open Economies Review Aims and scope Submit manuscript

Abstract

In two-sector infinite-horizon trade models with factor–price-equalization, convergence of aggregate capital-labor ratios and incomes does not occur because the Euler equations imply equal growth rate of consumption in all economies. In a two-country dynamic specific factors model, we show that factor–price-equalization occurs only in the long run. Per capita incomes and consumptions do not necessarily converge. These depend on the endowments of the primary factors. Depending on these endowments, an initially poorer economy may end up as the richer economy in the steady state, overtaking the initially richer one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The world economy, of course, is a closed economy and hence a decline in the return to capital accompanies any accumulation of capital.

  2. It is not even a candidate to explain trade between similar economies or intra-industry trade.

  3. See Oniki and Uzawa (1965) and Stiglitz (1970) for early contributions with saving being a constant proportion of income. Stiglitz (1970) also considers optimal savings but with the rate of time preference differing between economies. Woodland (1982) uses duality and sketches a dynamic model.

  4. For a discussion of finite lives in dynamic trade models, see e.g. Bajona and Kehoe (2006)—who have a many-period overlapping generations models, (they also discuss capital flows), Bianconi (1995) who has a two period overlapping generations structure and Kaneko (2006) with a continuous time uncertain lifetimes structure.

  5. Below we will see that tampering with this assumption (of only two factors that are mobile across the two sectors) that will result in very different predictions.

  6. See Chen (1992); Atkeson, and Kehoe, (2000); Bajona and Kehoe (2006 and 2010); Chatterjee and Shukayev (2012); see also Baxter (1992); Bianconi (1995) and Kaneko (2006).

  7. This relies on the evidence on the factor content of trade (Trefler (1993, 1995), Davis and Weinstein (1998)) and on the pattern of production (Harrigan (1997)).

  8. See Brecher et al. (2002); Chatterjee and Shukayev (2012))

  9. The case of uniform Hicks-neutral superiority has received some empirical support from Trefler (1995) and Davis and Weinstein (1998).

  10. Ventura (1997) had assumed that there was incomplete specialization and obtained conditional convergence. Bajona and Kehoe (2010) showed that if in a Ventura-type model complete specialization is allowed, then there are other possibilities—e.g., that an economy could decumulate capital and specialize in the labor-intensive good in the new steady state. Atkeson, and Kehoe, (2000) had showed that a “late-comer” small open economy specializing in the labor-intensive consumption good, accumulates capital until it reaches the (lowest) capital-labor ratio of the world economy (the latter is assumed to be in a steady-state).

  11. Jones (1971) revived the literature; this was because of the observation that the Stolper-Samuelson theorem gave predictions on protection that seemed to fly in the face of casual empiricism—namely, in any sector the interests of labor and capital are implacably opposed to one another. The specific factors model, on the other hand, suggests (some) convergence of interest among all factors in the industry demanding protection. Add to this the fact that in the early empirical implementation of the Heckscher-Ohlin model (the Leontief Paradox), there was a feeling that the two-factor framework was too much of a straitjacket (and land needed to be added as a third factor). For the US, all private land constituted 31 % of total wealth in 1900 and 16 % in 1958. For the UK, it was 55 % in 1798, 18 % in 1885 and 4 % in 1927. These figures are taken from Laitner (2000).

  12. A small open economy model is discussed in Sen (2013). In a small open economy model factor-price equalization follows—in our two-country model, this happens only in the steady state.

  13. See Arnold and Trepl (2015); Fedotenkov et al. (2014), and Tadesse and White (2010).

  14. Thus there is no need to identify one of the specific factors as land and other as labor, as is done below. I do this to fix ideas. Later on (in section 3) I give other interpretations. Clearly, the framework is richer than the limited interpretations given in this paper. I thank a referee for emphasizing this.

  15. Chen (1992) and Atkeson, and Kehoe, (2000) are examples. A previous version of our paper used this specification.

  16. Ventura (1997); Chatterjee and Shukayev (2012) and Francois and Shiells (2008) use this structure. Atkeson, and Kehoe, (2000) discuss this briefly.

  17. See Brecher et al. (2002) and Bajona and Kehoe (2006, 2010) for this structure.

  18. The specific factors L and M could be thought of as two kinds of labor. The interpretation that suits the analysis in section 2 is to think of L as labor and M as some endowment of fruits. In section 3.2, we introduce the valuation of the trees that bear these fruits (as in Eaton (1987, 1988). In the earlier dynamic specific factors models, the capitals in the sectors were specific in the short run, while labor was mobile across sectors (Neary (1978)). In the long run, capital was also mobile across sectors, and the model collapsed into the familiar Heckscher-Ohlin model.

  19. Where there is no chance of confusion, we do not explicitly write the time index.

  20. Constant growth rates for L and M can easily be incorporated, as can exogenous technical progress.

  21. Note, while it would be interesting to look at an internationally mobile factor whose return is equalized internationally, such an assumption with free trade in the two intermediates and constant returns to scale would make all factor prices determined internationally. It would be possible to pursue this with one intermediate being non-traded and/or decreasing returns to scale technologies.

  22. This is essentially reproducing the analysis of Dixit and Norman (1980), chapter 5.

  23. Since there is no borrowing or lending internationally, capital is the only store of value. If one of the specific factors was, say, land, then savings would also be allocated to a change in the value of land—see section 3.2 below. In an overlapping context, this can be crucial (see Eaton (1987), (Eaton 1988)).

  24. We are going to use the relationships given in the Appendix A.

  25. Since capital is now used exclusively in the X sector, its accumulation causes the supply of the Y good to fall, causing excess demand for the latter good and its price to rise. This is the big difference in the details of this example over the set-up where capital was the mobile factor. Note that this does not change the stability of the model, or its (qualitative) dynamics.

  26. The issue of land availability on the development path of a small open economy is discussed in Sen (2013).

References

  • Arnold LG, Trepl S (2015) A North-South Trade Model of Offshoring and Unemployment. Open Econ Rev 265:999–1039

    Article  Google Scholar 

  • Atkeson A, Kehoe, PJ (2000) Paths of Development for Early- and Late-boomers in a Dynamic Heckscher-Ohlin Model Research Staff Report No. 256, Federal Reserve Bank of Minneapolis

  • Bajona C and Kehoe TJ (2006) Demographics in Dynamic Heckscher-Ohlin Models: Overlapping Generations versus Infinitely Lived Consumer Research Staff Report No. 377, Federal Reserve Bank of Minneapolis

  • Bajona C, Kehoe TJ (2010) Trade, Growth, and Convergence in a Dynamic Heckscher-Ohlin Model. Rev Econ Dyn 13:487–513

  • Barro RJ, Sala-i-Martin X (2003) Economic Growth. MIT Press, Second Edition

    Google Scholar 

  • Baxter M (1992) Fiscal Policy, Specialization, and Trade in the Two-sector Model: The Return of Ricardo? J Polit Econ 100:713–744

    Article  Google Scholar 

  • Bianconi M (1995) On Dynamic Real Trade Models. Econ Lett 47:47–52

  • Brecher RA, Chen Z, Choudhri EU (2002) Absolute and Comparative Advantage, Reconsidered: The Pattern of International Trade with Optimal Saving. Rev Int Econ 10:645–656

    Article  Google Scholar 

  • Brock PL, Turnovsky SJ (1993) The Growth and Welfare Consequences of Differential Tariffs. Int Econ Rev 34:765–794

    Article  Google Scholar 

  • Chatterjee P, Shukayev M (2012) A Stochastic Dynamic of Trade and Growth: Convergence and Diversification. J Econ Dyn Control 36:416–432

    Article  Google Scholar 

  • Chen Z (1992) Long-run Equilibria in a Dynamic Heckscher-Ohlin Model. Can J Econ 23:923–943

    Article  Google Scholar 

  • Davis, Donald R. and David E. Weinstein (1998) An Account of Global Trade NBER Working Paper 6785

  • Dixit, Avinash and Victor Norman (1980), Theory of International Trade: A Dual, General Equilibrium Approach, Cambridge: Cambridge University Press.

  • Eaton J (1987) A Dynamic Specific-Factors Model of International Trade. Rev Econ Stud 54:325–338

    Article  Google Scholar 

  • Eaton J (1988) Foreign-Owned Land. Am Econ Rev 78:76–88

  • Fedotenkov I, van Groezen B, Meijdam L (2014) Demographic Change, International Trade and Capital Flows. Open Econ Rev 25:865–883

    Article  Google Scholar 

  • Francois, J and Clinton R. Shiells (2008) Dynamic Factor Price Equalization and International Convergence. Johannes Kepler University of Linz, Working Paper No. 0820

  • Harrigan J (1997) Technology, Factor Supplies and International Specialization: Estimating the Neoclassical Model. Am Econ Rev 87:475–494

    Google Scholar 

  • Hu Y, Nishimura K, Shimomura K (2006) Dynamic Three Factor Models of International Trade. Asia-Pacific J Account Econ 13:73–85

    Article  Google Scholar 

  • Hu Y, Nishimura K, Shimomura K (2008) Specific Factor Models and Dynamics in International Trade. In: Marjit S, Yu ESH (eds) Contemporary and Emerging Issues in Trade Theory and Policy. Emerald Publishers, Bingley, pp. 191–207

  • Jones RW (1971) A Three-Sector Model in Theory, Trade, and History. In: Bhagwati JN, Jones RW, Vanek J (eds) Trade, Balance of Payments and Growth. North-Holland Publishing Company, Amsterdam

    Google Scholar 

  • Kaneko A (2006) Specialization in a dynamic trade model. Int Econ J 20:357–368

    Article  Google Scholar 

  • Komiya R (1967) Non-Traded Goods and the Pure Theory of International Trade. Int Econ Rev 8:132–152

    Article  Google Scholar 

  • Laitner J (2000) Structural Change and Economic Growth. Rev Econ Stud 67:545–561

    Article  Google Scholar 

  • Neary P (1978) Short-Run Capital Specificity and the Pure Theory of International Trade. Econ J 88:488–512

    Article  Google Scholar 

  • Oniki H, Uzawa H (1965) Patterns of Trade and Investment in a Dynamic Model of International Trade. Rev Econ Stud 32:15–38

    Article  Google Scholar 

  • Sen P (2013) Capital Accumulation and Convergence in a Small Open Economy. Rev Int Econ 24:690–704

    Article  Google Scholar 

  • Sen P (2015) Uncertain Lifetimes and Convergence in a Two-country Heckscher-Ohlin Model. Math Soc Sci 78:14–20

  • Stiglitz JE (1970) Factor Price Equalization in a Dynamic Economy. J Polit Econ 78:456–488

    Article  Google Scholar 

  • Tadesse B, White R (2010) Does Cultural Distance Hinder Trade in Goods? A Comparative Study of Nine OECD Member Nations. Open Econ Rev 21:237–261

    Article  Google Scholar 

  • Trefler D (1993) International Factor Prices: Leontief Was Right! J Polit Econ 111:961–987

    Article  Google Scholar 

  • Trefler D (1995) The Case of Missing Trade and Other Mysteries. Am Econ Rev 85:1029–1046

    Google Scholar 

  • Ventura J (1997) Growth and Interdependence. QJ Econ 112:57–84

  • Woodland, Alan (1982) International Trade and Resource Allocation, (Amsterdam) North Holland

Download references

Author information

Authors and Affiliations

  1. Centre for Development Economics, Delhi School of Economics, Delhi, India

    Partha Sen

  2. CESifo, Munich, Germany

    Partha Sen

  3. RIEB, Kobe University, Kobe, Japan

    Koji Shimomura

Authors
  1. Partha Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Koji Shimomura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Partha Sen.

APPENDIX A

APPENDIX A

Totally differentiating Equations (19), (20) and (21) we get Equations (A1), (A2) and (A3) below. Written in matrix form we get (A4). Equation (A5) is the positive because b32 (in the coefficient matrix B in Equation (A4) below) is the partial derivative of excess demands with respect to price.

$$ zdQ+\left(Q{z}_p-{R}_p\right)dp={R}_KdK $$
(A1)
$$ zd{Q}^{\ast }+\left({Q}^{\ast }{z}_p^{\ast }-{R}_p^{\ast}\right)dp={R}_{K^{\ast}}^{\ast }d{K}^{\ast } $$
(A2)
$$ {z}_p\left(dQ+d{Q}^{\ast}\right)-\left({R}_{pp}+{R}_{pp}^{\ast}\right)dp={R}_{pK}dK+{R}_{p{K}^{\ast}}^{\ast }d{K}^{\ast } $$
(A3)
$$ \left[\begin{array}{l}\\ {}\\ {}\\ {}\end{array}\right.\begin{array}{ccc}\hfill z\hfill & \hfill \left(Q{z}_p-{R}_p\right)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \left({R}_p-Q{z}_p\right)\hfill & \hfill z\hfill \\ {}\hfill {z}_p\hfill & \hfill {b}_{32}\hfill & \hfill {z}_p\hfill \end{array}\left.\begin{array}{l}\\ {}\\ {}\\ {}\end{array}\right]\kern1em \left[\begin{array}{l}\\ {}\\ {}\\ {}\end{array}\right.\begin{array}{c}\hfill dQ\hfill \\ {}\hfill dp\hfill \\ {}\hfill {dQ}^{*}\hfill \end{array}\left.\begin{array}{l}\\ {}\\ {}\\ {}\end{array}\right]=\kern0.5em \left[\begin{array}{l}\\ {}\\ {}\\ {}\end{array}\right.\begin{array}{c}\hfill {R}_KdK\hfill \\ {}\hfill {R}_K{dK}^{*}\hfill \\ {}\hfill 0\hfill \end{array}\left.\begin{array}{l}\\ {}\\ {}\\ {}\end{array}\right] $$
(A4)

Or compactly B.Z = S

$$ {b}_{32}=-\left({R}_{pp}+{R}_{pp}^{*}\right)<0 $$
$$ \varDelta =-{b}_{32}{z}^2>0 $$
(A5)
$$ dQ/dK=\left\{\left(Q{z}_p-{R}_p\right)\left(z{R}_{pK}-{R}_K{z}_p\right)-z{R}_K{b}_{32}\right\}/\varDelta ={z}^{-1}\left\{-{J}^y\left(dp/dK\right)+{R}_K\right\} $$
(A6a)
$$ dQ/d{K}^{\ast }=\left(Q{z}_p-{R}_p\right)\left(z{R}_{p{K}^{\ast}}^{\ast }-{R}_{K^{\ast}}^{\ast }{z}_p\right)/\varDelta =-\left({J}^y{z}^{-1}\right)\left(dp/d{K}^{\ast}\right) $$
(A6b)
$$ dp/d{K}^{\ast }=-z\left(z{R}_{p{K}^{\ast}}^{\ast }-{R}_{K^{\ast}}^{\ast }{z}_p\right)/\varDelta ={R}_K\left(1-p{z}_p{z}^{-1}\right)/\left(p{b}_{32}\right)={R}_K{\theta}_{\overset{\sim }{X}Q}/\left(p{b}_{32}\right)<0 $$
(A6c)
$$ dp/dK=-z\left(z{R}_{pK}-{R}_K{z}_p\right)/\varDelta ={R}_K\left(1-p{z}_p{z}^{-1}\right)/\left(p{b}_{32}\right)={R}_K{\theta}_{\tilde{X}Q}/\left(p{b}_{32}\right)<0 $$
(A6d)
$$ d{Q}^{\ast }/dK=\left({R}_p-Q{z}_p\right)\left(z{R}_{pK}-{R}_K{z}_p\right)/\varDelta ={z}^{-1}{J}^y\left(dp/dK\right) $$
(A6e)
$$ d{Q}^{\ast }/d{K}^{\ast }=\left\{-\left(Q{z}_p-{R}_p\right)\left(z{R}_{p{K}^{\ast}}^{\ast }-{R}_{K^{\ast}}^{\ast }{z}_p\right)-z{R}_{K^{\ast}}^{\ast }{b}_{32}\right\}/\varDelta ={z}^{-1}\left\{{J}^y\left(dp/d{K}^{\ast}\right)+{R}_{K^{\ast}}^{\ast}\right\} $$
(A6f)

1.1 APPENDIX B

In Appendix A, the values obtained in Equations (A6a) to (A6f) are modified to:

$$ dQ/dK=\left\{\left(Q{z}_p-{R}_p\right)\left(-{R}_K{z}_p\right)-z{R}_K{b}_{32}\right\}/\varDelta ={z}^{-1}\left\{-{J}^y\left(dp/dK\right)+{R}_K\right\} $$
(B1a)
$$ dQ/d{K}^{\ast }=\left(Q{z}_p-{R}_p\right)\left(-{R}_{K^{\ast}}^{\ast }{z}_p\right)/\varDelta =-\left({J}^y{z}^{-1}\right)\left(dp/d{K}^{\ast}\right) $$
(B1b)
$$ dp/d{K}^{\ast }=z\left({R}_{K^{\ast}}^{\ast }{z}_p\right)/\varDelta >0 $$
(B1c)
$$ dp/dK=z\left({R}_K{z}_p\right)/\varDelta >0 $$
(B1d)
$$ d{Q}^{\ast }/dK=\left({R}_p-Q{z}_p\right)\left(-{R}_K{z}_p\right)/\varDelta ={z}^{-1}{J}^y\left(dp/dK\right) $$
(B1e)
$$ d{Q}^{\ast }/d{K}^{\ast }=\left\{\left(Q{z}_p-{R}_p\right)\left({R}_{K^{\ast}}^{\ast }{z}_p\right)-z{R}_{K^{\ast}}^{\ast }{a}_{32}\right\}/\varDelta ={z}^{-1}\left\{{J}^y\left(dp/d{K}^{\ast}\right)+{R}_{K^{\ast}}^{\ast}\right\} $$
(B1f)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sen, P., Shimomura, K. Convergence and Overtaking in a Dynamic two Country Model. Open Econ Rev 28, 107–124 (2017). https://doi.org/10.1007/s11079-016-9413-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11079-016-9413-0

Keywords

Search

Navigation