Harris-Todaro Model: Urban-Rural Labor Migration and Endogenous Unemployment, Study notes of Dynamics

Download Harris-Todaro Model: Urban-Rural Labor Migration and Endogenous Unemployment and more Study notes Dynamics in PDF only on Docsity! 1 The Harris-Todaro Labor Allocation Mechanism as an Evolutionary Game By Jaylson Jair da Silveira and João Rogério Sanson* Abstract The Harris-Todaro model is reinterpreted under an evolutionary game approach, including the behavior of migrants under bounded rationality in a process of imitation or learning. A first feature is an improvement upon Harris-Todaro by replacing the sign-preserving function of the adjustment mechanism by a replicator dynamics, which is deduced from the assumptions about the interaction process among heterogeneous workers in a bounded rationality context. A second feature is an interpretation of the equilibrium condition as a mixed strategy equilibrium of an evolutionary game. The Harris-Todaro equilibrium condition then appears as a spontaneous result of the model. Key words : migration, labor market, evolutionary game, bounded rationality JEL codes: J6, O15, C7 Developing countries generally show some kind of dualism in their labor markets, be it with respect to production structure (traditional and modern), to geographical location (rural and urban), to legal nature of the activities (formal and underground), or to the composition of the labor force (skilled and non-skilled). As shown by Agénor and Montiel (1996, p. 63), duality in the labor market implies a segmentation, i.e., a situation in which identical workers earn different wages depending on where they are employed. Todaro (1969) has built up a seminal model to analyze rural-urban migration in a developing country, extending and formalizing ideas from various authors that followed Lewis (1954). As the main assumption of the model, the decision of the rural worker to migrate depends upon expected differential wage. In Michael Todaro's analysis, the decision to move is seen as an investment decision tied to expected net returns. These expected returns crucially depend upon the probability of getting a job in the modern, or formal, sector versus a job at the traditional urban sector, also called underground or informal. The model takes this * Silveira: Department of Economics, Paulista State University (UNESP), Araraquara, SP, Brazil (e-mail: jaylson@fclar.unesp.br); Sanson: Department of Economics, Federal University of Santa Catarina (UFSC), Florianopolis, SC, Brazil (e-mail: jrsanson@mbox1.ufsc.br). We benefitted from comments and discussions that resulted from presentations at the First Brazilian Workshop of the Game Theory Society in 2002, the 2003 southern regional meeting of the National Association of the Graduate Centers in Economics (ANPEC), the 2003 national meeting of ANPEC, the Pontificial Catholic University of Rio Grande do Sul (PUC-RS), the UNESP, and the UFSC. Silveira thanks the National Council for Scientific and Technological Research (CNPq) 2 probability as endogenous and influenced by the creation of urban employment and the number of unemployed urban workers. The objective of the model is to show how the employment rate tends to an equilibrium below full employment even in the long run. Harris and Todaro (1970) is a general equilibrium analysis in which the artificial upholding of the wage differential between the rural and urban sectors leads to an inefficient equilibrium. Although the informal sector is now excluded from the model, the concept of expected wage is kept. What determines the expected wage is employment or unemployment in the urban sector. Despite defending the principle that workers take into consideration the present value of the expected real income flow, Todaro (1969) as well as Harris and Todaro (1970) in fact work with the postulate of myopic expectations. Todaro (1969, sect. IV) formally obtains his main conclusions from the assumption that the time horizon of the workers is only one period. Thus, in their model, the differential in real income for time t appears in place of the differential of the present value of the flow of expected real income along two or more periods. This is done in the adjustment mechanism associated with the aggregate supply of workers to the urban sector. Todaro (1969, p. 143, n. 10) justifies this by saying that this "... assumption is made necessary by mathematical convenience but is in fact probably a more realistic formulation in terms of actual decision making in less developed nations ..." [Our italics.] We consider this implicit idea of bounded rationality of the workers, and propose a model based on the tools of the theory of evolutionary games.1 In our model, the migratory movement of workers is interpreted as a process of imitation/learning in an environment of bounded rationality. 2 With the help of an evolutionary game, we deduce a replicator dynamics that replaces the adjustment mechanism postulated both in Todaro (1969) and Harris and Todaro (1970). Starting with their same general equilibrium structure, but including it in our model of evolutionary game, we show that their classical results appear as a consequence of the interaction among economic agents that try to get higher wages in a context of bounded rationality. for a young professional research fellowship, developed at the Department of Economics of UFSC. Sanson thanks CNPq for a research grant that included an undergraduate fellowship for Rimenez Tuon. 1 On this theory, see Weibull (1995), Vega-Redondo (1996), Samuelson (1997), and Hofbauer and Sigmund (1998). 2 The model is based on one initially presented in Silveira (2001), where it was used to represent the competition among workers along business cycles. 5 (5). Finally, the real wage of the rural sector in terms of the manufactured good, aω , is obtained by (3), given an and p. In short, the vector ),,,,,( ** mamaa xxnnpω configures a temporary equilibrium or a short run, which is defined for a given vector of exogenous variables6 and a given urban population of workers un . This temporary equilibrium will be disturbed when a redistribution of the population of workers between sectors happens, i.e., when there is a migration of workers that changes the urban population. Harris and Todaro, for determining the long run equilibrium, argue that the rural workers reckon, in their decision on migrating to the urban area, the expected wage, e uω , defined as: (7) m u me u n n ωω * = . 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 real wage is higher than in the rural sector. Thus, the long run equilibrium, i.e., the absence of a rural-urban migratory flow is established when the urban worker population reaches a level such that the expected urban real wage equates the rural real wage, i.e.: (8) a e u ωω = . This equality is known in the literature as the Harris-Todaro condition. The level of the urban population that satisfies the Harris-Todaro condition, * un , is attained from the solution of the equation resulting from substitution of equations (3), (5), (6), and (7) in (8): (8-a) ( ) ( ) 0)1()1( ***** =−−′− umumum ngxngnn ρω . Figure 1 illustrates this solution. Note that the position of the curve labeled ( ))1()1( * umua ngxng −−′= ρω also depends on the minimum wage, since the relative price, p, depends upon the urban production, * mx . Combined with the diagram that represents the 6 The endowment of land, the stocks of capital and the minimum wage. 6 manufacturing demand curve for labor, we can also see urban unemployment, given by the difference between * un and * mn . [FIGURE 1] Given the properties assumed for the functions )(⋅g and )(⋅ρ , we can establish the following proposition concerning the Harris-Todaro equilibrium. PROPOSITION 1: Suppose that )(⋅′g and )(⋅ρ are continuously differentiable functions in the interval ++ℜ∈)1,[ * mn . The existence and uniqueness of the Harris-Todaro equilibrium with urban unemployment are guaranteed if (9) ( ) mmumu n nngxng u ωρ ** 1 ])1()1([lim >−−′ −→ and (10) ( ) mmmm ngxng ωρ <−−′ )1()1( *** . PROOF: Let ( ) ( ))1()1()( ** umumumu ngxngnnn −−′−≡Φ ρω . The function )( unΦ is continuous in the interval )1,[ * mn because it is the sum of continuous functions. By conditions (9) and (10), it follows that 0)(lim 1 <Φ −→ u n n u and 0)( >Φ ∗ mn . If 0)(lim 1 <=Φ −→ cnu nu , with c a real constant, then for every 0>ε there is some 0>δ such that, for all un , if 11 <<− unδ then ε<−Φ cnu )( . Specifically, let 0>−= cε . Thus there is some 0>δ such that, for all un , if 11 <<− unδ then ccnu −<−Φ )( and this last inequality implies 0)( <Φ un . If −∞=Φ −→ )(lim 1 u n n u , then for every 0<A there is some 0>δ such that, for all un , if 11 <<− unδ then Anu <Φ )( . In both cases, we can take 2 1 δ−=un such that 0 2 1 <     −Φ δ . Since 0)( >Φ ∗ mn , 0 2 1 <     −Φ δ and Φ is continuous in close interval )1,[ 2 1, ** mm nn ⊂    − δ , we can apply the intermediate 7 value theorem and conclude that there is some )1,[ 2 1, ** mmu nnn ⊂    −∈∗ δ such that 0)( =Φ ∗ un . Moreover, considering (1), (2) and (5) we have 0 )( )( )()()( 2 * 2 * <      ⋅ ⋅′ ⋅′−⋅⋅′′+−= ∂ Φ∂ g g xg n n n m u mm u ρρ ω for all )1,[ * mu nn ∈ . Hence, since un∂ Φ∂ is continuous in interval )1,[ * mn , there is only one )1,[ * mu nn ∈∗ such that 0)( =Φ ∗ un . † By condition (9), as the numbers of rural workers tend to zero, both the labor productivity in this sector and the terms of trade tend to extreme high values. Thus the rural real wage becomes greater than the expected urban real wage. On its part, condition (10) establishes that, at the value for which the minimum wage restriction is binding, the real rural wage must be smaller than the expected urban real wage, which in the present case equals the exogenously determined minimum wage. Harris and Todaro (1970, p. 129), in order to evaluate the long run equilibrium, postulate a mechanism of adjustment that is based on the following function of sign preservation: (11) )( a e uun ωωψ −=& , with 0)( >⋅′ψ and 0)0( =ψ . Thus, while a e u ωω > , 0>un& will be the result, and only if a e u ωω = , 0=un& will occur. The differential equation that governs the state transition in the model of Harris and Todaro is obtained by substituting the left-hand side of (8-a) in (11): (12) ( )))1(()1()( ** umumumu ngxngnnn −−′−= ρωψ& . It should be stressed that the state space for (11) is the real interval )1,[ * mn , which is positively invariant.7 7 Considering (10) in * mu nn = , this results in 0>un& . Given condition (9) and the fact that ( ) mmmum n nnn u ωω ** 1 lim = −→ , it follows that 0lim 1 < −→ u n n u & . 10 rural sector that will actually migrate is assumed to be directly proportional to the difference in income ru yy − .9 In Table 1, we summarize the income gaps that may appear in each possible comparison. The income of workers of the ith sector was separated in two components: one is the real wage iω (null when the worker is unemployed) and the other, the sum of incomes unconnected to the direct sale of his labor force, iε , assumed exogenous and hereafter named non-wage income.10 From Table 1, we can estimate the migratory flow of workers to the urban sector. When employed rural workers relate their income to the one of the employed urban workers (event 1) and 0)()( ≤−−+=− rruuru yy εωεω , then the quantity of immigrant rural workers will be zero. However, when this comparison yields 0)()( >−−+=− rruuru yy εωεω , the estimated inflow of employed workers in the urban sector is directly proportional to this differential in income (normalized). In a compact way, the subsequent expression gives both possibilities: (14) { }0),()(max)1( rruuuuru vnvn εωεω +−+− . When employed rural workers compare their income with unemployed urban workers (event 2), there will be immigration to the urban sector provided the non-wage income at the urban sector is high enough as weighed against the expected total income in the rural sector: (15) { }0),(max)1()1( rruuuru vnvn εωε +−−− . The same type of comparison is valid for events 3 and 4. Provided 0>−+ ruu εεω , the proportion of rural workers that effectively immigrate is directly proportional to this income difference: (16) { }0,max)1)(1( ruuuuru vnvn εεω −+−− . 9 In the theory of evolutionary games, this assumption may have several microeconomic foundations. See the appendix to this paper. 11 Likewise, an encounter of unemployed rural workers with unemployed urban workers, combined with appropriate differential in incomes will result in an inflow of workers to the urban sector given by (17) { }0,max)1()1)(1( ruuuru vnvn εε −−−− . By aggregating expressions (14), (15), 16) and (17), we get the expected migratory flow from the rural to the urban sector: (18) [ ]. }0,max{)1)(1(}0,max{)1( }0),(max{)1(}0),(max{)1( ruurruuur rruurrruuuruu vvvv vvvvnn εεεεω εωεεωεω −−−+−+−+ +−−++−+− B. Urban-Rural Migratory Flows A worker from the urban sector becomes a potential emigrant only if he compares his income with the income of a worker from the rural sector and discovers that uuurrr yy =+>+= εωεω . As before, four events are possible in this pairwise comparison of incomes, according to the situation of employment of each worker in the urban and the rural sectors. Take, for example, event 1. As employed urban workers contrast their income with the income of their equals as to the employment situation in the rural area, two results are possible. If 0)( ≥+−+=− rruuru yy εωεω , no employed urban worker leaves his area. However, if 0)( <+−+=− rruuru yy εωεω , the migratory outflow from the urban area will be directly proportional to this difference in income, or: (19) }0),(min{)1( rruuruuu vnvn εωεω +−+−− . 10 The non-wage income may be generalized to include net benefits of the public budget plus other usual sources of income that are studied in the literature on migration. Thus, even under unemployment in both areas, it may be worth the effort to migrate to the urban sector. 12 The other three events are interpretable likewise. Thus, from event 3, the result of the meeting of an employed urban worker with unemployed rural workers may be described by: (20) }0,min{)1)(1( ruuruuu vnvn εεω −+−−− Event 2, when unemployed urban workers come across employed workers from the rural sector, results in: (21) }0),(min{)1)(1( rruruuu vnvn εωε +−−−− . At last, in event 4, unemployed workers from both areas meet and the corresponding expected migration is given by: (22) }0,min{)1)(1)(1( ruruuu vnvn εε −−−−− By aggregating expressions (19) to (22), we attain the expected migratory outflow from the urban to the rural sector: (23) [ ].}0,min{)1)(1(}0),(min{)1( }0,min{)1(}0),(min{)1( rururruru ruururruuruuu vvvv vvvvnn εεεωε εεωεωεω −−−++−−+ −+−++−+−− C. Replicator Dynamics Given these possibilities of migration for the workers, the change in the proportion of workers located in the urban sector is approximated by the difference between the rural-urban flow and the urban-rural flow, i.e., by the difference between (18) and (23): (24) [ ]. }0,min{)1)(1(}0),(min{)1( }0,min{)1(}0),(min{ }0,max{)1)(1(}0,max{)1( }0),(max{)1(}0),(max{)1( ruurrruru ruururruuru ruurruuur rruurrruuuruuu vvvv vvvv vvvv vvvvnnn εεεωε εεωεωεω εεεεω εωεεωεω −−−++−−+ −+−++−++ −−−+−+−+ +−−++−+−=& 15 (26) 0 )( )( )()()( 2 * 2 * <      ⋅ ⋅′ ⋅′−⋅⋅′′+−= ∂ Φ∂ g g xg n n n m u mm u ρρ ω , for all )1,[ * mu nn ∈ . Thus in the interval )1,[ * mn , if ),[ * ∗∈ umu nnn , then 0)( >Φ un and if )1,( ∗∈ uu nn , then 0)( <Φ un . Hence we deduce that for any initial condition )1,[)( * 0 mu ntn ∈ the economic system converges to the Harris-Todaro (mixed strategy) equilibrium with urban unemployment, * un . „ If the urban population is below this equilibrium, i.e., ),[ * ∗∈ umu nnn , then the urban employment rate is too high, the marginal productivity of rural labor is too low and the terms of trade are biased towards manufactured goods. This situation gives rise to an average real wage in the urban sector greater than the real wage in the rural sector. Such a favorable income differential for the urban sector generates a positive net rural-urban migratory flow. This keeps going until its negative effect on the rate of urban employment and its positive effect on the marginal productivity of rural labor and on the terms of trade equalize average real wage between the two sectors. On the contrary, when the urban population is above the level of the Harris-Todaro mixed strategy equilibrium, )1,( ∗∈ uu nn , the urban employment rate is too low, the marginal productivity is too high and the terms of trade are biased towards agricultural goods. Now the average real wage in the urban sector is lower than the real wage in the rural sector. This leads to a reversal in rural-urban migration, constraining the migratory movement and taking the system back to the mixed strategy equilibrium. To put in a nutshell, the Harris-Todaro condition emerges as the outcome of the interaction among heterogeneous workers that are worried à la Keynes with their relative wages in conditions of bounded rationality. IV. Conclusions In this paper, we developed an evolutionary game model, more specifically a replicator dynamics, which formalizes the sector allocation of labor as a process of imitation/learning in a milieu of bounded rationality. The Harris-Todaro condition comes out as a spontaneous upshot of interaction among heterogeneous workers. 16 In the model we propose here, due to the separation of income in two parts – wage and non-wage income – there is room for studying the role of public policies that influence the non-wage component of income and therefore influence the migratory dynamics of Harris and Todaro. Analyzing this dynamics is directly relevant to a better understanding of the process of urban concentration. APPENDIX: AN ALTERNATIVE DERIVATION OF THE REPLICATOR DYNAMICS (24-A): The evolutionary game models based on an imitation process are formed by two essential pieces as highlighted by Weibull (1995, p. 152): "the time rate at which agents review their strategy choice" and "the choice probabilities of a reviewing agent." Let ix be the proportion of individuals from the population that opt for the strategy si (type i agents), among h possible strategies and ),,,,,( 21 hi xxxxx KK= be the distribution of strategies in the population. The (average) rate of revision of a type i agent refers to the average number of times this agent revises his strategy per time interval, and may be taken as function of the state of the population, )(xri . This rate of revision may be considered as an arrival rate of a Poisson process, i.e., a realization of Poisson process. Let us assume that the frequency with which an agent reconsiders his strategy in a given time interval does not affect the frequency of revisions of the other agents, which is the same as saying that the Poisson process of individual revisions are statistically independent. We then conclude that the aggregate rate of revision of the subpopulation given by type i agents is by itself a Poisson process with an average rate of revision )(xrx ii . If changes in individual strategies are statistically independent random variables, then the flow of agents from the type i subpopulation to the type j subpopulation is an aggregate Poisson process with the arrival rate )()( xpxrx j iii , where )(xp j i is the probability of a type i agent becoming a type j. The choice probabilities of a type i agent form a probability distribution, represented by ( ))(),...,(),...,()( 1 xpxpxpxp h i j iii = , naturally with 1)( 1 =∑ = h j j i xp . 17 In the case of only two possible strategies, 2=h , we have 121 =+ xx . Thus we need to consider only one state variable. Without loss of generality, we will take 1x as the reference state variable, so the vector of distribution of strategies will be perfectly determined once 1x is given, because )1,(),( 1121 xxxxx −== . The inflow to the subpopulation of type i agents can be approximated by (27) )()( 1 222 xpxrx and the outflow by (28) )()( 2 111 xpxrx . The rate of variation of subpopulation 1 will be determined by the difference between inflow and outflow: (29) )()()()( 2 111 1 2221 xpxrxxpxrxx −=& . As explained by Weibull (1995, sect. 4), there are several assumptions pertaining to the revision rate and to the probability of choice that may be adopted. In section 3 above, we implicitly assumed that all reviewing agents reassess their choice together and just once per time interval, or (30) 1)()( 21 == xrxr As in section 3, we assume a direct imitation process (Weibull, 1995, p. 155-158), i.e., each reviewing agent randomly meets another agent so that the probability of a reviewing agent encountering an agent that follows strategy 1s is 1x and one that follows 2s is 12 1 xx −= . As done by Weibull (1995, p. 155-158), we suppose that, after the reviewing agent meets another agent, he detects, with some noise, his own payoff, 11 ii x απ +)( , and the payoff linked to the strategy of the other agent, jj x απ +)( . Since iα and jα are stochastic 11 )(xiπ is the expected result for an agent that opts for a strategy si when the population state is )1,(),( 1121 xxxxx −== . 20 Equation (38) is the replicator dynamics (24-a), except for the scalar 1/a, which does not alter the qualitative structure of the solutions of the replicator dynamics under inquiry. As a result, (24-a) may be interpreted as a normalized version of (38). Normalizing 1=a , we can say that the income difference in (24-a) may now be expressed in units of the maximum differential in average income that is a priori possible. REFERENCES Agénor, P.-R. and Montiel, P. J. Development Macroeconomics. Princeton: Princeton University Press, 1996. Harris, J. R. and Todaro, M. P. “Migration, unemployment and development: two sector analysis.” American Economic Review, March 1970, 15(1), pp. 126-142. Hofbauer, J. and Sigmund, K. Evolutionary Games and Population Dynamics. Cambridge: Cambridge University Press, 1998. Lewis, W. A. “Economic Development with Unlimited Supplies of Labour.” Manchester School, May 1954, 22, pp. 139-191. Nachbar, J. H. “‘Evolutionary’ Selection Dynamics in Games: Convergence and Limit Properties.” International Journal of Game Theory, 1990, 19, pp. 59-89. Samuelson, L. Evolutionary Games and Equilibrium Selection. Cambridge: MIT Press, 1997. Silveira, J. J. Ciclos Goodwinianos e o Processo de Concorrência num Ambiente de Racionalidade Limitada: uma Análise a partir da Teoria dos Jogos Evolucionários. Thesis (Doctorate in Economics) – Instituto de Pesquisas Econômicas, Universidade de São Paulo, São Paulo, 2001. Todaro, M. P. “A Model of Labor Migration in Less Developed Countries.” American Economic Review, March 1969, 59(1), pp. 138-148. Vega-Redondo, F. Evolution, Games and Economic Behaviour. Oxford: Oxford University Press, 1996. Weibull, J. W. Evolutionary Game Theory. Cambridge: MIT Press, 1995. 21 Table 1 – Income differential and number of estimated comparisons between workers of urban and rural sectors, according to employment situation Event 1 2 3 4 Situation Employed Unemployed Employed Unemployed Income )( uy uu εω + uε uu εω + uε U rb an s ec to r w or ke r Sub- population uu vn )1( uu vn − uuvn )1( uu vn − Situation Employed Employed Unemployed Unemployed Income )y( r rr εω + rr εω + rε rε R ur al s ec to r w or ke r Sub- population ru vn )1( − ru vn )1( − )1)(1( ru vn −− )1)(1( ru vn −− Income diferential )yy( ru − )()( rruu εωεω −−+ )( rru εωε +− ruu εεω −+ ru εε − Number of estimated comparisons ruuu vnvn )1( − ruuu vnvn )1)(1( −− )1)(1( ruuu vnvn −− )1)(1)(1( ruuu vnvn −−− 22 Figure 1 – Harris-Todaro model with endogenous terms of trade