INTERNATIONAL ECONOMIC REVIEW Vol. 42, No. 2, May 2001
OPTIMAL URBAN EMPLOYMENT POLICIES: NOTES ON CALVO AND QUIBRIA∗ By
Nancy H. Chau† and M. Ali Khan1 Cornell University, U.S.A. The Johns Hopkins University, U.S.A.
We show that Quibria’s insightful observation on the efficacy of urban policy in a model with an urban trade union and an urban informal sector holds, but with a revenue-neutral employment tax-subsidy combination levied on the urban employer. We also draw the relevance of the generalized Harris–Todaro model in providing intuition for the validity of this surprising result.
1.
introduction
2
The Harris–Todaro model is one articulation of why subsidization of urban employment does not succeed in checking rural–urban migration and in wiping out urban unemployment in LDCs. This result can be consolidated into the Bhagwati– Johnson general theory of distortions which prescribes that remedial policies should be directed to the source of the problems. Since there are two distortions in the model—a rigid urban wage and the requirement of expected wage equality—policies which are geared only to the former necessarily fail. This folk theorem has withstood generalizations to settings where the urban wage is no longer exogenously given. In a bold move which has not drawn as much attention as it deserves, Quibria (1988) has combined an urban informal sector, as introduced in Harberger’s ILO memorandum (1971), and an urban trade union, as introduced in Calvo (1978), and he has argued that a first-best competitive allocation of resources can indeed be obtained by focusing solely on the subsidization of formal urban employment. It is then natural to ask what is it in Quibria’s conception that is responsible for a result that does not hold in other formulations and, in particular, in a formulation that is especially close to his, namely Calvo’s, and one that seems to go against the grain of the conventional theory of distortions. An answer to this question is the primary purpose of these notes. We show that a first-best allocation of resources can be obtained by focusing solely on the formal urban sector, but urban subsidies by themselves will not work [see, for instance, Bhagwati (1971) and Johnson (1965)]. What is required is a somewhat subtler policy combination in which the urban employer is subsidized and taxed * Manuscript received September 1997; revised July 1999. † E-mail:
[email protected]. 1 The authors are grateful to Arnab Basu and Vandana Chandra for useful conversations on the subject of this paper. 2 See Harris and Todaro (1970), Harberger (1971), and references in Khan (1987).
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at the same time. Our answer can be seen as a particular instance of Tinbergen’s (1952) insistence on the equality of objectives and instruments, and the argumentation involves the generalized Harris–Todaro model as the relevant analytical backdrop and the use of geometry originally developed for it in Khan (1982). The analysis also gives a clearer understanding of why such a tax-subsidy combination does not work in Calvo’s model. Section 2 reproduces Quibria’s model, and Section 3 recalls and further develops the necessary geometry. Section 4 reduces Quibria’s model to a special case of the generalized Harris–Todaro model, one without an informal sector but with a rich specification of the determination of urban wages, and Section 5 revisits Calvo’s model from the point of view of urban wage policy. Section 6 concludes these notes.
2.
quibria’s model
Quibria (1988) works with a model of an economy with three primary factors of production, land, labor, and capital, and three internationally traded outputs. One of these is produced in the rural region and can be usefully thought of as a composite agricultural commodity, while the other two are produced in the urban region and constitute the manufacturing sector of the economy, disaggregated into formal and informal subsectors. The relevant variables of the rural region are subscripted by r and those of the urban region carry the subscript u or i depending on whether they pertain to the formal or the informal sector. The technologies are summarized by (1)
Xr = Fr Lr T
Xu = Fu Lu K
αXi = Li
α>0
where T and K represent the exogenously given amounts of land and capital; X, suitably subscripted, represents the three output levels; and L, again, suitably subscripted, represents the amount of labor input used in their production. There is a single technique of production in the informal sector, and 1/α represents both the average and marginal productivity of informal labor with this technique. It is assumed that Fu and Fr are continuously differentiable, exhibit constant returns to scale, and diminishing marginal productivity to each factor, this productivity being infinite at zero output levels. In addition to L and T , the aggregate resources of the economy include an exogenously given, homogeneous amount of labor . The full employment equation is given by (2) Lr + Lu + Li = Lr + 1 + λLu = where λ = Li /Lu is the proportion of informal to formal employment. Urban labor has two options for employment, and the Harris–Todaro equilibrium condition can be reformulated so that the guaranteed rural wage is equated to a weighted sum of the two urban wages, the weights being the employment rates in the two sectors, and representing proxies for the probability of finding employment in the two sectors. More formally, (3)
wr =
Lu Li 1 λ w + w ≡ w + w Lu + L i u Lu + L i i 1+λ u 1+λ i
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where w, suitably subscripted, represents wage rates. Quibria, however, modifies this equilibrium condition to require the equality of the expected utilities of urban and rural workers and thereby obtain (4)
Uwr =
1 λ Uwu + Uwi 1+λ 1+λ
where U· is a standard utility function with positive and nonincreasing marginal utility and normalized so that utility from zero income is zero. Quibria does not assume that wages in the urban formal sector are exogenously given, but follows Calvo (1978) in postulating an urban trade union which sets urban wages either through monopsonistic pricing or as a result of efficient bargaining. In either case, he derives the urban wage as a function of the wage in the informal sector: (5) ψsw < 0 ψs e < 0 ψw > 0 w u = ψ wi s w s e where sw and se are wage and employment subsidies, respectively, and ψj denotes derivative of ψ with respect to argument j.3 Finally, universal marginal production pricing is assumed. Given the constant returns to scale and the small country assumptions, we obtain the “price equals unit-cost” conditions as (6)
pr = Cr wr τ
pu = Cu 1 − sw wu − se R
pi = λwi
where τ and R denote the rentals to land and capital, and p, suitably subscripted, denotes the international prices.4 Since the international prices play no role in what follows, we shall, by a suitable choice of units, assume them all to be unity. The specification of Quibria’s model is now complete. We have to determine the allocation of labor Lu , Lr , and Li among the three sectors of the economy and thereby the probability, λ, of urban informal employment; the returns to land, capital, and rural and informal labor, τ, R, wr , and wi , and thereby the return to formal labor; and, finally, the three output levels, Xr , Xu , and Xi . In addition to the factor endowments and the international prices, the parameters are the technologies α, Fr , Fu , the utility function U, the formal sector wage function ψ, and the policy parameters sw and se .
3.
the generalized harris–todaro formulation
There is no informal sector in the generalized Harris–Todaro model, but the urban wage is seen as an abstract function of other variables, typically rates of return to domestic factors and of unemployment.5 We thus obtain a model in which α above is zero, Li represents urban unemployment, and ψ is replaced by a more general 3 See Equation (12) for one specific form of this function studied in Quibria (1988). For the specific form pertaining to Nash bargaining, see Equation (15c) in Quibria (1988). 4 Note that an exogenously given pi completely determines the informal wage, wi , as pi /α. 5 See Khan (1980, 1987).
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omega function wu = wr λ R τ sw se The rural–urban migration equilibrium condition is reformulated to be (7)
1 + λwr = wr wr λ R τ sw se
Khan (1982) reduces the analysis of this to two simultaneous equations in the two unknowns wr and λ and thereby to a diagrammatic analysis in the wr − λ plane. One of these equations is (7) above, while the other is the material balance, no longer with full employment, for which Equation (2) is to be rewritten as (8)
Lr wr + 1 + λLu 1 − sw − se =
The labor demand functions, Lr · and Lu ·, are obtained through the equalization of the relevant wage rates to the marginal revenue products.6 We now use this diagrammatic analysis to examine why urban wage subsidies do not generally lead to a first-best allocation of resources. We begin with the conventional Harris–Todaro model with a sector-specific urban wage. When the urban wage is exogenously given to be w¯ u , the omega function of the generalized Harris–Todaro model collapses to a constant, and we obtain (9)
1 + λwr = w¯ u
Lr wr + 1 + λLu 1 − sw w¯ u − se =
This equation system is depicted in Figure 1 with the first equation furnishing the W W curve and the second equation furnishing the LL curve.7 In the absence of wage subsidies, the optimum allocation of resources is given by the intersection of the two marginal revenue productivity schedules of Figure 2 and by the point wc 0 of Figure 1. For there to be a problem worth analyzing, w¯ u is assumed to be greater than wc . Note in particular that wr0 , as defined in Figures 1 and 2, is the rural wage at which aggregate labor demand is equal to , given the urban wage w¯ u , so that the LL curve intersects the wr -axis at the point wr0 0 in Figure 1. Meanwhile, for there to be no migration induced unemployment, the rural wage must be equal to w¯ u and, hence, the W W curve cuts the wr -axis at w¯ u 0. The intersection of the LL and W W curves, in turn, gives the equilibrium rate of urban unemployment and the rural wage. Upon introducing urban wage subsidies, the W W curve remains unchanged. On the other hand, since the effective wage facing the urban employer, 1 − sw w¯ u − se , is reduced, wr0 is bided upward and the LL curve shifts to the right. Thus, urban wage subsides increase the equilibrium rural wage and decrease the urban unemployment rate. The point, however, is that even though urban wage subsidies can shift the LL curve to the right so that it has one of its endpoints at wr0 = wc (curve L L ), they do note change the W W curve and, consequently, cannot eliminate urban unemployment 6 Since we have normalized international prices to be unity, the dependence of these demand functions on prices is not shown. 7 This figure follows Figure 2 in Khan (1982). We leave it to the reader to check through a straightforward argument, or by routine differentiation, that the slopes of the curves LL and W W have the signs as shown.
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Figure 1 the ww and ll curves
or depress the rural wage to the competitive one. As urban employment is subsidized further, and wr0 increases and the LL curve shifts further to the right beyond wc , we obtain a further increase in the rural wage and a further decrease in urban employment. When wr0 = w¯ u and the effective urban wage is decreased to wu0 , the latter is defined in Figure 2. This is an allocation of resources far from the competitive optimum. This argument extends to the generalized Harris–Todaro model without even appealing to any geometry. If the formal urban wage is responsive to employment subsidies and is not equal to the competitive wage when these policy parameters are zero, then we can choose their values which wipe out urban employment. This is simply to choose s¯w and s¯e such that the equation (10)
wr 0 Rc τc s¯w s¯e = wr
Figure 2 labor market equilibrium with urban minimum wage
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has the competitive wage wc as its solution, and where Rc τc are the competitive rentals to land and capital. Since this requires, by hypothesis, that at least one of the subsidies be nonzero, the urban employer will be facing an effective wage lower than the competitive one, leading to excess demand in the labor market. It is therefore clear that one policy instrument will not suffice for regulating two equilibrium conditions. However, we have two instruments, sw and se , at our disposal; it is thus natural to ask, following Tinbergen (1952), whether the two taken in conjunction can guide the allocation of resources to the first-best competitive one in a situation where, unlike the conventional Harris–Todaro model, the wage received by urban labor can also be manipulated by these instruments. We turn to this question.
4.
a reduction
We can now use the previous analysis as a background to record the basic observation that Quibria’s model is yet another special case of the generalized Harris–Todaro model. In order to see this, simply rewrite (4) as U wu + λU wi 1+λ U ψwi sw se + λU wi −1 = 1 + λU 1+λ = λ sw se
1 + λwr = 1 + λU −1
The geometry of Figures 1 and 2 continues to be relevant—we simply reinterpret w¯ u to be the root of the equation w¯ u = 0 0 0, the rural wage which induces a zero value of λ, in the absence of urban subsidies, sw = se = 0. If w¯ u = wc , then we have a competitive allocation of resources, and there is nothing more to be said or done.8 Thus, as in Figure 1, suppose w¯ u to be greater than the competitive wage wc , and equilibrium is again designated by the intersection of the LL and W W curves. Once urban wage subsidies are introduced, it is easy to check that the LL curve shifts to the right and the W W curve shifts below and to the left. Thus, in contrast to the Harris–Todaro model with an exogenous urban wage, both curves can be shifted, thereby raising expectation that they can be made to intersect each other at the point wc 0. The fact that this cannot be done is a nonexistence argument. Let us suppose that there exist wage subsidies ¯sw s¯e , at which wr0 = wc , and the LL curve begins at wc 0. In this case, the effective urban wage is 1 − s¯w ψwi s¯w s¯e − s¯e = wr = wc ,
8 We remind the reader that wi is essentially an exogenously given constant in our framework. In order to make the analysis meaningful, we choose its determining parameters, pi and α, to be such that it is less than wc .
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and we obtain
λ s¯w s¯e 1 + λ −1 U ψwi s¯w s¯e + λUwi 1 + λ = = U wr wr 1+λ ≥
1 + λ −1 U U ψwi sw se wr
≥
ψwi sw se w + s¯e >1 = c wr wc 1 − s¯w
Hence, the W W curve lies strictly above the point wc 0 Now suppose that there exist wage subsidies ˆsw sˆe , chosen in such a way that ψwi sˆw sˆe = wc , and thus the W W curve passes through the point wc 0. In this case, the effective urban wage wˆ u is given by 1 − sˆw ψwi sˆw sˆe − sˆe , which is less than wc We thus obtain (11)
1 + λ =
− Lr wc − Lr wc < =1 Lu wˆ u Lu w c
Hence the LL curve intersects the W W curve at a negative value of the unemployment rate, an absurdity. Thus there has to be something in the specific functional form of Quibria’s model— the Cobb–Douglas technology or the trade union utility function—that is responsible for his result. In the context of a monopsonistic trade union, Quibria derives the following equation for the determination of urban wages in the formal sector: 1 − a 1 − sw wu − se w (12) ln u = wi wu 1 − sw where a is a technological parameter with 1 > a > 0 If we denote both sides of (12) by y, we can plot it as two curves in the wu − y plane. This is done in Figure 3 with the curve beginning from wi being the left hand side of (12). When there is
Figure 3
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no employment subsidy, the right hand side of (12) is given by the horizontal line with height 1 − a; otherwise, the right hand side of (12) is represented by a hyperbola intersecting the wu -axis at the point s 0 in Figure 3, where s = se /1 − sw . It follows, therefore, that in the absence of wage subsidies, the urban wage is given by wi exp1 − a. It should also be clear from Figure 3 that with positive wage subsidies, we can steer the urban wage to be equal to wc . In the context of Figure 1, this is the value of the subsidies that have shifted the W W curve to intersect the wr -axis at wc . The problem, even with specific functional forms, is that the labor market is not in equilibrium, in precisely the same way as argued above for the general model. Since there are subsidies, the effective wage facing the formal urban employer is given by 1 − s¯w wc − s¯e , which is less than the competitive wage and hence implies an excess demand for labor or, tantamount to the same thing, a negative value of λ obtained by the LL curve intersecting the wr -axis at a point greater than wc . So far, however, we have not taken into account the fact that in (12) the wage subsidies enter in a particular multiplicative form. This is a particularity that can be exploited. If s¯ is the value of se /1 − sw that equates the formal urban wage, ψwi sw se , to wc , then by a suitable choice of se or of sw , we can allow formal urban employers to face an effective wage that is equal to the competitive one, wc , by solving either of the equations (13)
1 − sw wc − s¯1 − sw = wc
or
se w − s e = wc s¯ c
to obtain the optimal values of the policy parameters to be (14)
sw∗ = −
s¯ wc − s¯
and
se∗ =
s¯ wc − s¯
Equation (14) also implies that sw∗ wc = −se∗ . Hence, the financing of the employment tax–subsidy combination sw∗ se∗ necessitates a total expenditure on the part of the government that is equal to Lu wc sw∗ wc + se∗ = 0. In other words, the policy combination sw∗ se∗ not only reproduces the first-best outcome, it also requires no external means of finance due to its revenue neutrality.9 We can now summarize the discussion of this section in the following statement.10 in Quibria (1988). Theorem 1. In Quibria’s model there exist a jointly revenue-neutral employment subsidy and ad valorem wage tax that generate a first-best competitive allocation of resources. Such an allocation cannot be obtained through the use of only one of these instruments.
9 See, for instance, McCool (1982), which provides the normative ordering of various forms of employment subsidy finance in a Harris–Todaro economy. 10 We leave it to the reader to check that this discussion carries over without change to the bargaining setup; see (15c).
OPTIMAL URBAN EMPLOYMENT POLICIES
5.
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calvo’s model revisited
Calvo’s trade union model is obtained when the omega function of the generalized Harris–Todaro model collapses to11 (15)
wu = wr sw se
w > 0
s w < 0
s e < 0
to give us the following reduced form of the model: (16)
1 + λwr = wr sw se Lr wr + 1 + λLu 1 − sw wr sw se − se =
Quibria already emphasized the impossibility of attaining a first-best allocation of resources through wage subsidies in Calvo’s model, and we have also developed corresponding intuition regarding why we expect this to be true in a general setting. However, Theorem 1 suggests the possibility of exploiting specific features of Calvo’s model to obtain an analogous result. Furthermore, the geometric analysis of Khan (1982) is premised on w ≤ 1, an assumption not fulfilled in either version of Calvo’s analysis. Thus, additional scrutiny is warranted. We shall limit ourselves to the monopsonistic trade union case. Here (17)
wr sw se =
wr − 1 − ase /1 − sw a
When there is no employment subsidy, the urban wage is wr /a and migration equilibrium implies a constant rate of urban unemployment equal to 1 − a/a, which means that the W W curve is a horizontal line as shown in Figure 4. The LL curve designating equilibrium in the labor market remains as before, but with wr0 the solution to the equation (18)
Lr wr + Lu wr 0 0 = Lr wr + Lu wr /a ≡ Lr wr + Lau wr =
and with the equilibrium rural wage, wr∗ , the solution to the equation (19)
Lr wr + 1/aLu wr 0 0 = Lr wr + 1/aLu wr /a ≡ Lr wr + Lu wr /a =
Figure 5 justifies that these solutions are indeed as ordered in Figure 4. Equilibrium in Calvo’s trade union model is shown by the point wr∗ 1 − a/a in Figure 4. With the introduction of an employment subsidy, however small, the W W curve changes from a horizontal line to a hyperbola (curve W W ) which intersects the wr -axis at the point s = se /1 − sw , as shown in Figure 4. As either subsidy increases, 11
See Equations (13) and (29) in Calvo (1978) or Equation (17) below.
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Figure 4 urban wage and wage subsidies
s shifts toward the competitive wage, wc . The fact that the LL curve shifts to the right with an increase in either subsidy can be easily seen from (16); in particular, an increase in either subsidy decreases the urban wage and this requires an increase in wr0 to maintain equilibrium in the labor market. The question at hand, of course, is whether these two curves shift in such a way that they intersect at the point wc 0. The fact that this is impossible, which is to say that the gap between s and wr0 is maintained, is easy to see. From Equations (16) and (17), we obtain (20)
1−a se 1−a wr − = wr − s λ= awr 1 − sw awr
Hence, the value of s which generates first-best labor allocation, sc , is to be set so that sc = wr = wc . Meanwhile, the effective wage facing the urban employer is given
Figure 5 equilibrium rural wage and employment in the presence of a trade union
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by (21)
1 − sw wr sw se − se = 1 − sw wc sw se − se = 1 − sw wc − se = 0
and thus, given infinite marginal productivity at zero wage, there is no equilibrium in the labor market. A way of saying this in terms of the geometry of Figure 4 is that with the LL curve beginning at wc , the W W curve has shifted so much to the right that the intersection of the two curves, if it exists, is obtained at a negative value of λ. What is particularly interesting, and an important source of the difference between the Quibria and Calvo conceptions, is that the ability to maneuver the two instruments separately is of no consequence. This is in spite of the fact that the subsidies enter into Calvo’s omega function in a form identical to that of Quibria’s. Once s = se /1 − sw is set at sc , we cannot identify the two parameters separately. We have thus shown, Theorem 2. In Calvo’s model a competitive allocation of resources cannot be obtained with urban wage and employment policy instruments even if the possibility of taxation is admitted.
6.
concluding remarks
Note that Quibria’s model cannot be reduced to the generalized Harris–Todaro model if informal output is not internationally traded; a two-sector model can only go so far in accommodating a conception that is essentially based on three sectors. A special case of such a model, one in which the formal sector wage is exogenously given rather than being determined by a trade union, can be extracted from Chandra (1991) and Grinols (1991). However, Chandra, and work subsequent to hers, is directed to issues which require a disaggregated formal sector. What is clear in any case, if one is to be guided by this literature, is that a generalization to a setup with nontradeable informal output will not be easy.
references Bhagwati, J. N., “The General Theory of Distortions and Welfare,” in J. N. Bhagwati et al., eds., Trade, Balance of Payments and Growth (Amsterdam: North Holland, 1971). Calvo, G., “Urban Unemployment and Wage Determination in LDCs: Trade Unions in the Harris– Todaro Model,” International Economic Review 19 (1978), 65–81. Chandra, V., “The Informal Sector in Developing Countries: A Theoretical Analysis,” unpublished Ph.D. Dissertation, The Johns Hopkins University, 1991. Grinols, E. L., “Unemployment and Foreign Capital: The Relative Opportunity Costs of Domestic Labor and Welfare,” Economica 62 (1991), 59–78. Harberger, A. C., “On Measuring the Social Opportunity Cost of Labour,” International Labour Review 103 (1971), 559–79. Harris, J. R., and M. Todaro, “Migration, Unemployment and Development: A Two-Sector Analysis,” American Economic Review 60 (1970), 126–42. Johnson, H. G., “Optimal Trade Intervention in the Presence of Domestic Distortions,” in R. E. Caves et al., eds., Trade, Growth and the Balance of Payments (Amsterdam: North Holland, 1965).
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Khan, M. Ali, “Dynamic Stability, Wage Subsidies and the Generalized Harris–Todaro Model,” Pakistan Development Review 19 (1980), 1–24. , “Tariffs, Foreign Capital and Immiserizing Growth with Urban Unemployment and Specific Factors of Production,” Journal of Development Economics 10 (1982), 245–56. , “Harris–Todaro Model,” in J. Eatwell, P. Newman, and M. Milgate, eds., The New Palgrave (New York: Macmillan, 1987). McCool, T., “Wage Subsidies and Distortionary Taxes in a Mobile Capital Harris–Todaro Model,” Economica 49 (1982), 69–79. Quibria, M.G., “Migration, Trade Unions and the Informal Sector: A Note on Calvo,” International Economic Review 29 (1988), 557–63. Tinbergen, J., On the Theory of Economic Policy (Amsterdam: North-Holland, 1952).
