# American Institute of Mathematical Sciences

January 2018, 5(1): 41-59. doi: 10.3934/jdg.2018005

## Transitional dynamics, externalities, optimal subsidy, and growth

 Departamento de Economía, Universidad Autónoma Metropolitana-Azcapotzalco, Av. San Pablo 180, Col. Reynosa Tamaulipas, Delegación Azcapotzalco, 02200, Ciudad de México, México

* Corresponding author: Enrique R. Casares

Received  February 2017 Revised  August 2017 Published  January 2018

We develop an endogenous growth model with two sectors, manufacturing (learning) and non-manufacturing (non-learning). Domestic technological knowledge is only produced in the manufacturing sector through learning by doing. The knowledge produced in the manufacturing sector is available to the non-manufacturing sector. We obtain policy functions for the market economy and the social planner's economy. Thus, with the Pareto-optimal solution, we obtain the path of the optimal investment subsidy rate to the manufacturing sector for the market economy. The optimal investment subsidy rate increases as the market economy moves to the Pareto-optimal steady state.

Citation: Enrique R. Casares, Lucia A. Ruiz-Galindo, María Guadalupe García-Salazar. Transitional dynamics, externalities, optimal subsidy, and growth. Journal of Dynamics & Games, 2018, 5 (1) : 41-59. doi: 10.3934/jdg.2018005
##### References:
 [1] J. Aizenman and J. Lee, Real exchange rate, mercantilism and the learning by doing externality, Nber Working Paper Series, (2008), 1-17. doi: 10.3386/w13853. [2] K. J. Arrow, The economic implication of learning by doing, Readings in the Theory of Growth, 29 (1962), 131-149. doi: 10.1007/978-1-349-15430-2_11. [3] R. J. Barro and X. Sala-i-Martin, Economic Growth, 2$^{nd}$ edition, Cambridge University Press, 2004. [4] M. Ben-Gad, The two sector endogenous growth model: An atlas, Journal of Macroeconomics, 34 (2012), 706-722. doi: 10.1016/j.jmacro.2012.03.005. [5] E. W. Bond, P. Wang and C. K. Yip, A general two-sector model of endogenous growth with human and physical capital: Balanced growth and transitional dynamics, Journal of Economic Theory, 68 (1996), 149-173. doi: 10.1006/jeth.1996.0008. [6] B. Brou and M. Ruta, A Commitment Theory of Subsidy Agreements, The B. E. Journal of Economic Analysis & Policy, 2013. Available from: http://works.bepress.com/daniel_brou/10/. [7] J. Caballe and M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, 101 (1993), 1042-1067. doi: 10.1086/261914. [8] E. R. Casares and H. Sobarzo, Externalities, Optimal Subsidy and Growth, in Trends in Mathematical Economics. Dialogues Between Southern Europe and Latin America (eds. A. A. Pinto, E. Accinelli G., A. N. Yannacopulos and C. Hervés-Belo), Springer, (2016), 53-71. [9] S. Clemhout and H. Y. Wan, Learning-by-doing and infant industry protection, Review of Economic Studies, 37 (1970), 33-56. doi: 10.2307/2296497. [10] M. Dotsey and M. Duarte, Non traded goods, market segmentation, and exchange rates, Journal of Monetary Economics, 55 (2008), 1129-1142. [11] F. A. R. Gomes and L. S. Paz, Estimating the elasticity of intertemporal substitution: Is the aggregate financial return free from the weak instrument problem?, Journal of Macroeconomics, 36 (2013), 63-75. doi: 10.1016/j.jmacro.2013.01.005. [12] J. Gruber, A tax-based estimate of the elasticity of intertemporal substitution, Nber Working Paper Series, 11945 (2006), 1-31. doi: 10.3386/w11945. [13] R. E. Hall, Intertemporal substitution in consumption, Journal of Political Economy, 96 (1988), 339-357. [14] C. Hsieh and P. J. Klenow, Relative prices and relative prosperity, American Economic Review, 97 (2007), 562-585. [15] C. Jones, Economic growth and the relative price of capital, Journal of Monetary Economics, 34 (1994), 359-382. doi: 10.1016/0304-3932(94)90024-8. [16] A. Korinek and L. Serven, Undervaluation through foreign reserve accumulation: Static losses, dynamic gains, Journal of International Money and Finance, 64 (2016), 104-136. doi: 10.1596/1813-9450-5250. [17] R. E. Lucas, On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 3-42. [18] K. Matsuyama, Agricultural productivity, comparative advantage, and economic growth, Nber Working Paper Series, 3606 (1991), 1-27. doi: 10.3386/w3606. [19] K. Mino, Growth and Business Cycles with Equilibrium Indeterminacy, 1$^{st}$ edition, Springer, 2017. doi: 10.1007/978-4-431-55609-1. [20] A. K. M. M. Morshed and S. J. Turnovsky, Sectoral adjustment costs and real exchange rate dynamics in a two-sector dependent economy, Journal of International Economics, 63 (2004), 147-177. doi: 10.1016/S0022-1996(03)00038-2. [21] C. B. Mulligan and X. Sala-i-Martin, A note on the time-elimination method for solving recursive dynamic economic models, NBER, Technical Working Paper, 116 (1991), 1-28. doi: 10.3386/t0116. [22] C. B. Mulligan and X. Sala-I-Martin, Transitional dynamics in two-sector models of endogenous growth, Nber Working Papers Series, 3986 (1992), 1-73. doi: 10.3386/w3986. [23] D. Restuccia and C. Urrutia, Relative prices and investment rates, Journal of Monetary Economics, 47 (2001), 93-121. doi: 10.1016/S0304-3932(00)00049-0. [24] T. L. Roe, R. B. W. Smith and D. S. Saracoǧlu, Multisector Growth Models, Springer, 2010. doi: 10.1007/978-0-387-77358-2. [25] P. M. Romer, Increasing returns and long-run growth, Journal of Political Economy, 94 (1986), 1002-1037. doi: 10.1086/261420. [26] P. M. Romer, Capital Accumulation in the Theory of Long Run Growth, Modern Business Cycle Theory (Ed. R. Barro), Basil Blackwell, 1989. [27] P. Succar, The need for industrial policy in LDC's: A restatement of the infant industry argument, International Economic Review, 28 (1987), 521-534. [28] S. J. Turnovsky, International Macroeconomic Dynamics, MIT Press, 1997. [29] S. J. Turnovsky, Capital Accumulation and Economic Growth in a Small Open Economy, Cambridge University Press, 2009. [30] A. Valentinyi and B. Herrendorf, Measuring factor income shares at the sectoral level, Review of Economic Dynamics, 11 (2008), 820-835. doi: 10.1016/j.red.2008.02.003. [31] D. Xie, Divergence in economic performance: Transitional dynamics with multiple equilibria, Journal of Economic Theory, 63 (1994), 97-112. doi: 10.1006/jeth.1994.1034. [32] M. Yogo, Estimating the elasticity of intertemporal substitution when instruments are weak, Review of Economics and Statistics, 86 (2004), 797-810. doi: 10.1162/0034653041811770. [33] A. Young, Learning by doing and the dynamic effects of international trade, Nber Working Papers Series, 3677 (1991), 1-49. doi: 10.3386/w3577.

show all references

##### References:
 [1] J. Aizenman and J. Lee, Real exchange rate, mercantilism and the learning by doing externality, Nber Working Paper Series, (2008), 1-17. doi: 10.3386/w13853. [2] K. J. Arrow, The economic implication of learning by doing, Readings in the Theory of Growth, 29 (1962), 131-149. doi: 10.1007/978-1-349-15430-2_11. [3] R. J. Barro and X. Sala-i-Martin, Economic Growth, 2$^{nd}$ edition, Cambridge University Press, 2004. [4] M. Ben-Gad, The two sector endogenous growth model: An atlas, Journal of Macroeconomics, 34 (2012), 706-722. doi: 10.1016/j.jmacro.2012.03.005. [5] E. W. Bond, P. Wang and C. K. Yip, A general two-sector model of endogenous growth with human and physical capital: Balanced growth and transitional dynamics, Journal of Economic Theory, 68 (1996), 149-173. doi: 10.1006/jeth.1996.0008. [6] B. Brou and M. Ruta, A Commitment Theory of Subsidy Agreements, The B. E. Journal of Economic Analysis & Policy, 2013. Available from: http://works.bepress.com/daniel_brou/10/. [7] J. Caballe and M. S. Santos, On endogenous growth with physical and human capital, Journal of Political Economy, 101 (1993), 1042-1067. doi: 10.1086/261914. [8] E. R. Casares and H. Sobarzo, Externalities, Optimal Subsidy and Growth, in Trends in Mathematical Economics. Dialogues Between Southern Europe and Latin America (eds. A. A. Pinto, E. Accinelli G., A. N. Yannacopulos and C. Hervés-Belo), Springer, (2016), 53-71. [9] S. Clemhout and H. Y. Wan, Learning-by-doing and infant industry protection, Review of Economic Studies, 37 (1970), 33-56. doi: 10.2307/2296497. [10] M. Dotsey and M. Duarte, Non traded goods, market segmentation, and exchange rates, Journal of Monetary Economics, 55 (2008), 1129-1142. [11] F. A. R. Gomes and L. S. Paz, Estimating the elasticity of intertemporal substitution: Is the aggregate financial return free from the weak instrument problem?, Journal of Macroeconomics, 36 (2013), 63-75. doi: 10.1016/j.jmacro.2013.01.005. [12] J. Gruber, A tax-based estimate of the elasticity of intertemporal substitution, Nber Working Paper Series, 11945 (2006), 1-31. doi: 10.3386/w11945. [13] R. E. Hall, Intertemporal substitution in consumption, Journal of Political Economy, 96 (1988), 339-357. [14] C. Hsieh and P. J. Klenow, Relative prices and relative prosperity, American Economic Review, 97 (2007), 562-585. [15] C. Jones, Economic growth and the relative price of capital, Journal of Monetary Economics, 34 (1994), 359-382. doi: 10.1016/0304-3932(94)90024-8. [16] A. Korinek and L. Serven, Undervaluation through foreign reserve accumulation: Static losses, dynamic gains, Journal of International Money and Finance, 64 (2016), 104-136. doi: 10.1596/1813-9450-5250. [17] R. E. Lucas, On the mechanics of economic development, Journal of Monetary Economics, 22 (1988), 3-42. [18] K. Matsuyama, Agricultural productivity, comparative advantage, and economic growth, Nber Working Paper Series, 3606 (1991), 1-27. doi: 10.3386/w3606. [19] K. Mino, Growth and Business Cycles with Equilibrium Indeterminacy, 1$^{st}$ edition, Springer, 2017. doi: 10.1007/978-4-431-55609-1. [20] A. K. M. M. Morshed and S. J. Turnovsky, Sectoral adjustment costs and real exchange rate dynamics in a two-sector dependent economy, Journal of International Economics, 63 (2004), 147-177. doi: 10.1016/S0022-1996(03)00038-2. [21] C. B. Mulligan and X. Sala-i-Martin, A note on the time-elimination method for solving recursive dynamic economic models, NBER, Technical Working Paper, 116 (1991), 1-28. doi: 10.3386/t0116. [22] C. B. Mulligan and X. Sala-I-Martin, Transitional dynamics in two-sector models of endogenous growth, Nber Working Papers Series, 3986 (1992), 1-73. doi: 10.3386/w3986. [23] D. Restuccia and C. Urrutia, Relative prices and investment rates, Journal of Monetary Economics, 47 (2001), 93-121. doi: 10.1016/S0304-3932(00)00049-0. [24] T. L. Roe, R. B. W. Smith and D. S. Saracoǧlu, Multisector Growth Models, Springer, 2010. doi: 10.1007/978-0-387-77358-2. [25] P. M. Romer, Increasing returns and long-run growth, Journal of Political Economy, 94 (1986), 1002-1037. doi: 10.1086/261420. [26] P. M. Romer, Capital Accumulation in the Theory of Long Run Growth, Modern Business Cycle Theory (Ed. R. Barro), Basil Blackwell, 1989. [27] P. Succar, The need for industrial policy in LDC's: A restatement of the infant industry argument, International Economic Review, 28 (1987), 521-534. [28] S. J. Turnovsky, International Macroeconomic Dynamics, MIT Press, 1997. [29] S. J. Turnovsky, Capital Accumulation and Economic Growth in a Small Open Economy, Cambridge University Press, 2009. [30] A. Valentinyi and B. Herrendorf, Measuring factor income shares at the sectoral level, Review of Economic Dynamics, 11 (2008), 820-835. doi: 10.1016/j.red.2008.02.003. [31] D. Xie, Divergence in economic performance: Transitional dynamics with multiple equilibria, Journal of Economic Theory, 63 (1994), 97-112. doi: 10.1006/jeth.1994.1034. [32] M. Yogo, Estimating the elasticity of intertemporal substitution when instruments are weak, Review of Economics and Statistics, 86 (2004), 797-810. doi: 10.1162/0034653041811770. [33] A. Young, Learning by doing and the dynamic effects of international trade, Nber Working Papers Series, 3677 (1991), 1-49. doi: 10.3386/w3577.
The policy function $n=n(z)$, market economy with $\mu=0$.
The policy function $v=v(z)$, market economy with $\mu=0$.
The policy function $n=n(z)$, social planner's economy.
The policy function $v=v(z)$, social planner's economy.
The path of the optimal subsidy rate $\mu$.
Transitional dynamics of $n$ and $z$.
Transitional dynamics of $v$ and $z$.
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