Agent-Based Modeling of Rural-Urban Migration: An Emergent Harris-Todaro Equilibrium, Study notes of Economics

Download Agent-Based Modeling of Rural-Urban Migration: An Emergent Harris-Todaro Equilibrium and more Study notes Economics in PDF only on Docsity! Brazilian Journal of Physics, vol. 36, no. 3A, September, 2006 603 A Harris-Todaro Agent-Based Model to Rural-Urban Migration Aquino L. Espı́ndola, Instituto de Fı́sica, Universidade Federal Fluminense 24.210-340, Niterói, RJ, Brazil Jaylson J. Silveira, Depto de Economia, Universidade Estadual Paulista - UNESP 14.800-901, Araraquara, SP, Brazil and T. J. P. Penna Instituto de Fı́sica, Universidade Federal Fluminense 24.210-340, Niterói, RJ, Brazil Received on 6 September, 2005 The Harris-Todaro model of the rural-urban migration process is revisited under an agent-based approach. The migration of the workers is interpreted as a process of social learning by imitation, formalized by a com- putational model. By simulating this model, we observe a transitional dynamics with continuous growth of the urban fraction of overall population toward an equilibrium. Such an equilibrium is characterized by stabilization of rural-urban expected wages differential (generalized Harris-Todaro equilibrium condition), urban concentra- tion and urban unemployment. These classic results obtained originally by Harris and Todaro are emergent properties of our model. Keywords: Agent-based model; Harris-Todaro model; Econophysics; Complex system I. INTRODUCTION In this paper we turn upon the seminal Harris and Todaro [1] work, which together with Todaro [2] is considered one of the starting points of the classic rural-urban migration theory [3]. The hypothesis and predictions of Harris-Todaro model have been subjected to econometric evaluation and have been corroborated by several studies [4–7]. The key hypothesis of Harris and Todaro are that migrants react mainly to economic incentives, earnings differentials, and the probability of get- ting a job at the destination have influence on the migraton decision. In other words, these authors posit that rural-urban migration will occur while the urban expected wage [12] ex- ceed the rural wage. From this crucial assumption, as denom- inated by Harris-Todaro [1], is deduced that the migratory dy- namics leads the economic system toward an equilibrium with urban concentration and high urban unemployment. In our previous works [8, 9] we analyzed the rural-urban migration by means of an agent-based computational model taking into account the influence of the neighborhood in the migration decision. The inclusion of the influence of neigh- bors was done via an Ising like model. The economic anal- ogous to the external field in the Ising hamiltonian was the differential of expected wages between urban and rural sec- tors. Therefore, in theses works [8, 9] the crucial assumption of Harris and Todaro were taken for granted. Now, we are motivated by the following question: can the crucial assumption and equilibrium with urban concentration and urban unemployment obtained from the original Harris- Todaro model be generated as emergent properties from the interaction among adaptative agents? In order to answer this question we implemented an agent-based computational model in which workers grope for best sectorial location over time in terms of earnings. The economic system simulated is characterized by the assumption originally made by Harris and Todaro. The paper is arranged as follows. Section II describes the analytical Harris-Todaro model showing its basic equilibrium properties. In Section III we present the implementation of the computational model via an agent-based simulation and com- pare its aggregate regularities with the analytical equilibrium properties. Section IV shows concluding remarks. II. THE HARRIS-TODARO MODEL A. Assumptions Harris and Todaro [1] studied the migration of workers in a two-sector economic system, namely, rural sector and urban sector. The difference between these sectors are the type of goods produced, the technology of production and the process of wage determination. The rural sector is specialized in the production of agricultural goods. The productive process of this sector can be described by a Cobb-Douglas production function: Ya = AaNφ a , (1) where Ya is the production level of the agricultural good, Na is the amount of workers used in the agricultural production, Aa > 0 and 0 < φ < 1 are parametric constants. Similarly, the urban sector also has its productive process described as Cobb-Douglas production function:[13] Ym = AmNα m, (2) where Ym is the production level of the manufactured good, Nm is the quantity of workers employed in the production of manufactured goods, Am > 0 and 0 < α < 1 are parametric constants.[14] 604 Aquino L. Espı́ndola et al. Both goods and labor markets are perfectly competitive. Nevertheless, there is segmentation in the labor market due to a high minimum urban wage politically determined. In the rural sector, the real wage, perfectly flexible, is equal to the marginal productivity of labor in this sector:[15] wa = φAaNφ−1 a p, (3) where wa is the real wage and p is the price of the agricultural good, both expressed in units of manufactured good. In the urban sector, a minimum wage, wm, is assumed fixed institutionally at a level above equilibrium in this labor mar- ket. It can be formalized as [16] wm = αAmNα−1 m , such that Nm ≤ Nu, (4) where Nu is the amount of workers in the urban sector. The relative price of the agricultural good in terms of the manufactured good, p, varies according to the relative scarcity between agricultural and manufacturated goods. Then,[17] p = ρ ( Ym Ya )γ , (5) where ρ > 0 and γ > 0 are a parametric constants. γ is the elasticity of p with respect to the ratio Ym/Ya. The overall population of workers in the economy is N, which is kept constant during the whole period of analysis. By assumption there are only two sectors and rural prices are wholly flexible, which implies that there is full employment in the rural area, i.e., all workers living at the rural sector are employed at any period. Then at any period the following equality is verified: Na +Nu = N. (6) B. Temporary Equilibrium Given a parametric constant vector (Aa,Am,φ,α,ρ,γ), an initial urban population Nu, and a minimum wage wm one can calculate the temporary equilibrium of the economic system by using eqs. (1-6). From eq. (4) one can find the employment level at the man- ufacturing sector Nm = ( αAm wm ) 1 1−α . (7) Replacing eq. (7) in eq. (2) we get the production level of the manufacturing sector Ym = A 1 1−α m ( α wm ) α 1−α . (8) From eq. (6) one can obtain the relation Na = N−Nu, (9) which is used with eq. (1) to obtain the agricultural production Ya = Aa (N−Nu) φ . (10) By using eqs. (5), (8) and (10) the terms of trade are deter- mined p = ρ   A 1 1−α m ( α wm ) α 1−α Aa(N−Nu)φ   γ . (11) Finally, by using eqs. (3), (9) and (11), the rural wage in units of manufacturated good is obtained wa = φρA1−γ a A γ 1−α m ( α wm ) αγ 1−α 1 (N−Nu)1−φ+φγ . (12) In sum, the vector (Nm,Ym,Na,Ya, p,wa) configures a tem- porary equilibrium that might be altered whether occurs a migration of workers, induced by the differential of sector- ial wages, which changes the sectorial distribution of overall population. C. The Long Run Equilibrium Harris and Todaro, in determining the long run equilibrium, i.e., the absence of a net rural-urban migratory flow, argue that the rural workers, in their decision on migrating to the urban area, estimate the expected urban wage, we u, defined as: we u = Nm Nu wm. (13) The ratio Nm/Nu, which is the employment rate, is an estima- tive of the probability that a worker living at urban sector gets a job in this sector. As mentioned before, the key assumption of the model of Harris and Todaro is that there will be a migratory flow from the rural to the urban sector while the expected urban wage is higher than the rural wage. Thus, the long run equilibrium is attained when the urban worker population reaches a level such that the expected urban wage equates the rural wage: we u−wa = 0. (14) This equality is known in the economic literature as the Harris-Todaro condition. Harris and Todaro argue that the differential of expected wages in eq. (14) can be a constant Brazilian Journal of Physics, vol. 36, no. 3A, September, 2006 607 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 20 40 60 80 100 w u_ ex pe ct -w a simulation steps FIG. 5: Rural-urban expected wage differential (we u−wa) as function of simulation steps. From top to bottom the initial urban shares are 0.9, 0.8, 0.3, 0.2. FIG. 6: Equilibrium urban share nu as function of the technological parameter α and the minimum wage wm. White area is not a valid combination of parameters. wm, dramatically change the equilibrium values of the vari- able mentioned before. Higher values of γ generate a lower urban concentration, a higher gap in the expected wages and a higher unemployment rate in the equilibrium. Finally, in Fig. 12 is shown that the convergence of migra- tory dynamics for a urban share, compatible with historical data, is robust in relation to the variation of the key techno- logical parameters, α and φ. The impact of the variation of these parameters in the values of the equilibrium differential of expected wages, (we u−wa), and the equilibrium urban un- employment rate, (1−Nm/Nu) can be seen in Fig. 13 and 14, respectively. IV. CONCLUSION In this paper we developed and agent-based computational model which formalizes the rural-urban allocation of workers as a process of social learning by imitation. We analyze a two-sectorial economy composed by adaptative agents, i.e., FIG. 7: Equilibrium differential of expected wages as function of the technological parameter α and the minimum wage wm. White area is not a valid combination of parameters. FIG. 8: Equilibrium urban unemployment rate (1−Nm/Nu) as func- tion of the technological parameter α and the minimum wage wm. White area is not a valid combination of parameters. FIG. 9: Equilibrium urban share nu as function of the parameter γ and the minimum wage wm. White area is not a valid combination of parameters. 608 Aquino L. Espı́ndola et al. FIG. 10: Equilibrium differential of expected wages (we u −wa) as function of the parameter γ and the minimum wage wm. White area is not a valid combination of parameters. FIG. 11: Equilibrium urban unemployment rate (1− Nm/Nu) as function of the parameter γ and the minimum wage wm. White area is not a valid combination of parameters. FIG. 12: Equilibrium urban share nu as function of the technolog- ical parameters α and φ. White area is not a valid combination of parameters. FIG. 13: Equilibrium differential of expected wages (we u −wa) as function as function of the technological parameters α and φ. White area is not a valid combination of parameters. FIG. 14: Equilibrium urban unemployment rate (1− Nm/Nu) as function of the technological parameters α and φ. White area is not a valid combination of parameters. individuals that grope over time for best sectorial location in terms of earnings. This search is a process of imitation of successful neighbor agents. The dispersed and non-coordinated individual migration decisions, made based on local information, generate aggre- gate regularities. Firstly, the crucial assumption of Harris and Todaro, the principle that rural-urban migration will occur while the urban expected wage exceed the rural wage, comes out as spontaneous upshot of interaction among adaptative agents. Secondly, the migratory dynamics generated by agents that seek to adaptate to the economic environment that they co- create leads the economy toward a long run equilibrium char- acterized by urban concentration with urban unemployment. When this long run equilibrium is reached, the generalized Harris-Todaro condition is satisfied, i.e., there is a stabiliza- tion of the rural-urban expected wage differential. Thirdly, the impact of the minimum wage and elasticity of Brazilian Journal of Physics, vol. 36, no. 3A, September, 2006 609 terms of trade in a long run equilibrium obtained by simu- lations are in agreement with the predictions of the original Harris-Todaro model with Cobb-Douglas technology. Finally, the simulations showed an aggregated pattern not found in the original Harris-Todaro model. There is the possi- bility of small fluctuations of the urban share around an aver- age value. This phenomenon is known as reverse migration. Acknowledgments Aquino L. Espı́ndola thanks CAPES for the financial sup- port. Jaylson J. Silveira acknowledges research grants from CNPq. T. J. P. Penna thanks CNPq for the fellowship. [1] J. R. Harris and M. P. Todaro, American Economic Review 60, 126 (1970). [2] M. P. Todaro, American Economic Review 59, 138 (1969). [3] D. Ray, Development Economics (Princeton: Princeton Univer- sity Press, 1998). [4] L. Y. L. Yap, Journal of Development Economics 4, 239 (1977). [5] D. Mazumdar, Rural-urban Migration in Developing countries. In: Handbook of Regional and Urban Economics, Elsevier, Amsterdam, 1987. [6] L. Ghatak, P. Levine, and S. Price, Journal of Economics Sur- veys 10, 159 (1996). [7] J. G. Willianson, Migration and Urbanization. In: Chenery, H. and Srinivasan, T.N., Handbook of developments economics, Elsevier, Amsterdam, 1988. [8] J. J. Silveira, A. L. Espı́ndola, and T. J. P. Penna, physics/0506021, Physica A, to appear. [9] A. L. Espı́ndola, T. J. P. Penna, J. J. Silveira, forthcoming Inter- national Journal of Modern Physics C 16, issue 12 (2005). [10] R. H. Day et al, The Economic Journal 97, 940 (1987). [11] D. Stauffer and T. J. P. Penna, Physica A 256, 284 (1998). [12] The urban sector wage times the probability of getting a job in this sector. [13] Harris and Todaro set their model using sectorial production function with general functional form, i.e., in each sector i, Yi = fi(Ni) with f ′i (Ni) > 0 and f ′′i (Ni) < 0. Except where it is indicated, the results presented in this section are valid for this general case. The Cobb-Douglas form is a standard assumption about technology. [14] The endowment of land of the rural sector and the stocks of capital of both sectors are given for the period of analysis. [15] This marginal product is the derivative of the agricultural pro- duction function, eq. (1), with respect to Na multiplied by p. [16] The right-hand side of this equation is the marginal productivity of the manufacturing labor, i.e., the derivative of manufacturing production function, eq. (2), with respect to Nm. [17] Actually, Harris and Todaro worked with a general form, ρ de- notes a function in their paper not a constant value as used by us.