Back Matter

Author(s):
Marco Pani, and Mohamed El Harrak
Published Date:
May 2010
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    1. Appendix The Model

    Production. Production in the rural and urban sectors of the economy employs labor and private and public capital. The value-added in each sector follows a Cobb-Douglas function:

    where the U and R superscripts denote the urban and rural sector respectively, h and k are, respectively, the stocks of public and private capital, At is a scale factor representing total factor productivity, and a, a’, β, and β’ are coefficients that represent the share of each factor in production.

    The public sector produces services free of charge, and its output is valued at factor costs and equal to the public wage bill:

    Labor demand. In each sector, the enterprises hire workers until the marginal productivity of labor (equal to (1aβ)yUt/LUt in the urban sector and to (1aβ)yRt/LRt the rural sector) is equal to the wage level. In the rural sector, the wage level is flexible and adjusts to equalize labor demand and labor supply, yielding:

    whereLR¯t¯is the rural labor supply (which in the short term is given). In the urban sector, wages are given, yielding the urban labor demand function

    Labor demand in the public sector is exogenously determined by the government.

    Labor supply. Total labor supply increases in line with population growth:

    where g is the natural growth rate of the population. Labor supply in each sector is fixed at any time. Labor supply for the public sector is not defined; for the purposes of the model, it is assumed that civil servants are hired randomly among the entire labor force (for instance, selecting the workers with the relevant skills which are assumed to be evenly distributed between rural and urban areas).

    Over time, labor supply in the rural and urban sector changes as a result of migration:

    where L¯U is the short-term urban labor supply and MIGt is the number of workers that migrate from the country to the city at the end of period t.

    Migration responds to the wage differential between the rural and urban sector according to the function

    where ξ is a coefficient that measures the speed of migration and τ is the income tax rate (it is assumed that income taxes are only paid in the urban and public sector).

    Unemployment is equal to the difference between urban labor supply and urban labor demand:

    In equilibrium, there is no migration and wages in the two sectors satisfy the equality:

    Capital and investment. The stock of capital (both private and public) increases over time as a result of (public and private) investment:

    where δ is the rate of depreciation, and IPR and IPB are private and public investment, respectively.

    Public investment in each sector is decided by the government, while private investment is equal to the sum of foreign direct investment (FDI) and domestic private investment. Domestic private investment, in turn, depends on disposable income and on the marginal rate of return on capital:

    Where IPR = IPRU+ IPRR is the sum of private investment in the urban and rural sector, ζis a coefficient, YD is disposable income (net of taxes), and rs the marginal rate of return on capital, which is equalized across sectors and equal to

    Private investment is, in turn, allocated across sectors to equalize the marginal rate of return.

    The government budget. Abstracting from other items, public expenditure is equal to the sum of the public wage bill and public investment:

    where PEt is public expenditure. Public expenditure is financed with income and consumption taxes and external grants, under a hard budget constraint. In this study the value of tax revenue and grants is taken as given.

    Income and taxes. The gross domestic product (GDP) is measured at market prices as the sum of the value-added of the three sectors and of consumption taxes:

    where TAXC is the total amount of consumption taxes paid by the citizens (consumption tax revenue for the government).

    Urban workers and investors, as well as civil servants, also pay income taxes on their wages; the net disposable income (that can be spent by the citizens for consumption and investment) is thus equal to

    where ITAX is the total amount of income taxes paid by the citizens (income tax revenue for the government). The average income tax rate is derived by dividing the income tax revenue by the income tax base, which is equal to the income of the urban and public sector (under the assumption that the rural sector pays no income taxes):

    Appendix 2.: The Labor Market

    The labor market in this study has been formalized on the basis of the framework proposed by Todaro (1969) and Harris and Todaro (1970) (see also Corden and Findlay, 1975), which has also been used in more recent macroeconomic models of developing countries (for instance, Bodart and Le Dem, 1996; Agénor et al, 2006).

    The labor market is articulated in two broad “private” sectors, the “rural” sector and the “urban” sector. In addition, the civil service can be considered as a third, “public” sector that plays a distinct role from the other two.

    At any moment in time, each worker that is not hired as a civil servant can only work in one of the two private sector depending on his place of residence: workers living in cities can only work in the urban sector, while workers living in the country can only be employed in rural activities. Workers can however change their place of residence (migrate) between one period and the next. Migration entails adaptation costs; hence, only a fraction of all the workers who would find migration convenient do actually migrate at any moment in time.

    The rural labor market is competitive; wages are set at the level that clears the market, and adjust in each period to the level that equalizes labor supply with labor demand. Labor supply is given in each period since workers only migrate between periods; labor demand, in turn, depends on the marginal productivity of labor, which is related to the stock of private and public capital that has been invested in that sector. Other things being equal, the productivity of labor declines as the number of workers increases, and the wage level therefore declines with the number of rural workers.

    The urban labor market is instead dominated by labor unions. Wages are set in bilateral negotiations between employers and union representatives, and do not normally clear the market. Labor demand, which equalizes the marginal productivity of labor with the negotiated wage, is typically lower than labor supply, which is given in each period; as a result, some urban workers remain involuntarily unemployed, or engage in precarious “informal” activities that yield a much lower income.1

    The public labor market is still different. Eligibility for public employment depends less on the place of residence than on other characteristics, such as the educational background. Labor demand is set by the government on the basis of staffing needs, which are less directly related to marginal productivity; civil servants are typically hired on long-term contracts, which limit new recruitment to levels close to the natural turnover.2 For the purposes of the model, it is assumed that new civil servants are randomly hired at any period from workers throughout the country, independently of their place of residence.

    At any period, workers decide whether they should migrate to the other sector in the next period, by comparing the wage they can expect to receive in either of the two private sectors. Since all rural workers are employed at any time, the expected wage in the rural sector is equal to the current wage offered to rural workers. In the urban sector, instead, some workers are unemployed and the expected wage is equal to the wage offered to the workers that are hired multiplied by the probability of being hired. Assuming that workers are hired randomly in each period among the entire pool of urban workers (that is, on short-term contracts), the probability of being hired is equal to the ratio between labor demand and labor supply. Workers thus migrate from the rural to the urban sector if and only if the expected value of urban wages exceeds the rural wages, that is, if and only if

    where wR is the level of rural wages, wU is the level of urban wages, LU is urban labor demand and U is (urban) unemployment (measured by number of workers without a job). IfwR>wU(LU/(LU+U)), workers migrate from the urban to the rural sector.

    As mentioned above, only a fraction of workers who are potentially willing to migrate does actually migrate at any time. Migration, in turn, reduces the wage differential and thereby plays an equilibrating role. If the expected urban wages are too high, workers migrate to the cities; since urban labor

    Figure A.1

    demand does not change migration results in an increase in unemployment, which reduces the expected value of urban wages; at the same time, migration reduces rural labor supply, producing an increase in rural wages. The opposite occurs when migrants move from the cities to the country. In equilibrium, the number of workers unemployed is just sufficient to equalize the expected value of urban wages with the rural wages, and migration stops. The wage differential between the two sectors results in persistent urban unemployment.

    The labor market equilibrium for different levels of wages is illustrated in Figure A.1. The segment OL measures the total private sector labor supply, defined as the number of workers not employed in the civil service; each point along this line represents a different division of labor supply between the two private sectors; labor supply in the rural sector is measured left to right, while labor supply in the urban sector is measured right to left. For instance, at point P the labor supply in the rural sector is equal to OP while the labor supply in the urban sector is equal to PL.

    The lines RR’ and UU’ represent the rural and urban labor demand, respectively, at different wage levels. The two lines interest at point Q, at a wage level equal to wQ; when this level is offered in the two sectors, total labor demand is equal to total labor supply and there is no unemployment.

    Assume now that, as a result of union demands or government regulation, wages in the urban sector are raised from wQ to w*; at this level, labor demand in the formal sector diminishes from PL to P1L along the line UU’; before migration occurs, a number P1P of urban workers remain unemployed, while the rural labor supply remains equal to OP and rural wages remain at wQ. Over time, migration restores equilibrium between the two sectors along the line SS’, which represents the long-term labor supply corresponding to the urban wage level w*. Along this line, the wage differential between the two sectors satisfies the equality WR = w* (LU / (LU + U)) and no workers are willing to migrate; if — by hypothesis — rural wages were equal to w*, all unemployed workers would migrate to the rural sector and the rural labor supply would be equal to OL1 ; when instead rural wages are below w*, migration stops before full employment has been reached; at this point, the risk of remaining unemployed in the city is exactly compensated by the prospect of earning a higher wage if employed, and no workers migrate either way. Equilibrium is reached at point E, where the rural labor demand and long-term labor supply curves intersect; at this point, rural wages are equal to wE and rural labor demand and supply coincide and are equal to OPE ; urban labor supply is equal to PEL, but urban labor demand is only equal to P1L, and PEP1 workers remain unemployed; the wage differential w* — WE is just sufficient to compensate for the risk of remaining unemployed in the urban sector, and no workers are therefore willing to migrate.

    It should be noted that the long-term labor supply curve shifts with the urban wage level (Figure A.2). If urban wages increase from w* to w**, the long-term labor supply curve shifts from SS’ to S*S*’ which satisfies the new equality wR = w**(LU / (LU + U)); in general, when migration stops, labor supply in each of the two sectors satisfies the equality wR = wU(LU /(LU +U)). If urban labor demand is sufficiently rigid, the new curve S2S2’ lies to the left of SS’ (Figure A.2); in the new equilibrium point E2, rural wages are higher (w2 > wE) but there is more unemployment, as less workers are now employed in the rural sector while urban labor demand is also lower; the larger wage differential has attracted workers to the urban sector despite the larger risk of remaining without a job.

    Consider, now, the impact of an exogenous shift in labor demand, caused, for instance, by an increase in productivity induced by technological change or by the accumulation of capital. If the rural labor demand curve shifts from RR’ to R3R3 (Figure A.3), at point E, the labor market is no longer in equilibrium; excess rural labor demand puts upward pressures on rural wages, attracting workers from the cities; this, in turn, reduces unemployment. The new equilibrium lies at point E3 where the new rural labor demand curve R3R3 intersects the long-term labor supply curve SS’; at this point, unemployment is lower while rural employment and wages are higher; urban employment and wages are unchanged.

    Figure A.2

    Figure A.3

    If the urban sector demand curve shifts to the left (as a result of an increase in productivity or capital accumulation), with an unchanged level of wages, labor demand increases from LP1 to LP4 (Figure A.4). The long-term labor supply curve also shifts to the left, from SS’ to S4S4, as the increased number of job opportunities attracts more workers to the cities. Point E is not an equilibrium as the expected wages in the city at that level of urban labor supply exceed the wages offered in the country. Workers therefore migrate until the economy reaches the new equilibrium at E4, where rural wages are higher, rural employment lower, the number of unemployed workers is larger but the proportion of unemployed to employed urban workers is lower, reflecting the narrower wage differential between the two sectors.

    Figure A.4

    References

      Agenor, Pierre-Richard; AlejandroIzquierdo; and HenningTarpJensen, (eds.), 2006, Adjustment Policies, Poverty, and Unemployment: The IMPPA Framework, Blackwell, pp. xii, 560

      Bodart, Vincent; and JeanLeDem, 1996, “Labor Market Representation in Quantitative Macroeconomic Models for Developing Countries: An Application to Côte d’lvoire,”IMF Staff Papers 43 (2), pp. 41951.

      Corden, W.M.; and Findlay, R.1975, “Urban Unemployment, Intersectoral Capital Mobility and Development Policies,”Economica, pp. 5978.

      Harris, R.John; and MichaelP.Todaro, 1970, “Migration, unemployment and development: A two-sector analysis,”American Economic Review60 (1), pp. 12641.

      Mongardini, Joannes; and IssoufSamake, 2009, “The Macroeconomics of Scaling Up Aid: The Gleneagles Initiative for Benin,”IMF Working Paper09/115, pp. 32.

      Todaro, MichaelP.,1969, “A Model of Labor Migration and Urban Unemployment in Less Developed Countries,”American Economic Review59 (1), pp.13848

    For simplicity, we shall assume that the unemployed earn no income at all; of course, all workers must be able to spend a minimum amount to ensure their subsistence during their period of unemployment, but these funds may be provided by sources other than labor income, such as accumulated savings or transfers from the government, NGOs, or employed and expatriated family members.

    The public sector was not included in the original Harris-Todaro (1970) model but is found in more recent models that have applied the Harris-Todaro framework, such as Bodart and Le Dem (1996) and Agenor et al. (2006).

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