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Variable returns to scale, urban unemployment and welfare: comment.

I. Introduction

Recently, Beladi [1] explored the question of gains from trade in the presence of urban unemployment and variable returns to scale (hereafter VRS).(1) That is, using a sector-specific minimum wage model, he examined the welfare implications of free trade with those of export-promoting policies and import-substituting policies in the context of VRS. His main results are that "for a small open economy, the import-substituting policies are suboptimal when compared to free trade and the export-promoting policies are superior to free trade if elasticity of returns to scale of the agricultural sector (exportable good) is larger than or equal to that of the manufacturing sector (importable good). However, if the elasticity of returns to scale of the manufacturing sector is larger than that of the agricultural sector, then the import-substituting policies seem preferable to the export-promoting policies [1, 42]."

However, he treated the variable which captures scale economies ([g.sub.j])as exogenous in the analysis of the positive aspects of the model presented in his appendix. Therefore, the existence of VRS has not been taken into account in the essential manner at all in the derivation of the effects of various policies on the outputs, the urban unemployment ratio and so on. It is easily seen that solutions of the matrix of his appendix are nothing but those corresponding to the case of constant returns to scale.

Then, in this note, we pursue the analysis of gains from trade in the presence of urban unemployment and VRS as Beladi exactly intended to do. We focus our attention to the case of wage subsidy for manufacturing (as an import-substituting policy) and wage subsidy for agriculture (as an export-promoting policy) for simplicity. The effect of the other policies will be discussed similarly.

II. The Model and Assumptions

Following Panagariya and Succar [6] and Beladi [1], we deploy the neoclassical version of the Harris-Todaro model by Corden and Findlay [2] incorporating VRS such as: There are two commodities, [X.sub.a] (agricultural output) and [X.sub.m] (manufacturing output), respectively produced in rural and urban sectors, using two factors of production, labor (L) and capital (K). Capital is fully utilized, but labor is fully employed only in the rural sector, where the real wage rate ([w.sub.a]) is flexible. In the urban sector where the real wage rate ([w.sub.m]) is rigid, there may be unemployment. Factors are perfectly mobile between sectors within an economy but perfectly immobile among economies. Each commodity has a different factor intensity which is non-reversible among commodities. Factor endowments are also assumed to be inelastically supplied.

The production function may be written as (1) [Mathematical Expression Omitted] where [g.sub.j] describes the role of output-generated externality and is a positive function defined on (O, [infinity]), [L.sub.j] [K.sub.j] are labor and capital employed in sector j. [F.sup.j] is homogeneous of degree one in the factors.

In capital market equilibrium, the marginal value product of capital is the same in both sectors, so that (2) [Mathematical Expression Omitted] where p is the relative price of the manufacturing good in terms of the agricultural good, r is the rental of capital, [Mathematical Expression Omitted] and so on.

With regard to the labor market, Harris and Todaro [3] argue that in the labor market equilibrium the wage rate in the agricultural sector ([w.sub.a]) equals the expected wage rate in the manufacturing sector, which equals the wage rate in the manufacturing sector ([w.sub.m]) times the probability of employment in the manufacturing sector. Let us denote [Lambda] as the ratio of the unemployed ([L.sub.u]) to the employed ([L.sub.m]) in the urban sector, and [s.sub.a] and [s.sub.m] as wage subsidies in the agricultural and manufacturing sectors, respectively. In the labor market equilibrium, therefore, (3) [Mathematical Expression Omitted] (4) [Mathematical Expression Omitted] and (5) [Mathematical Expression Omitted] Denoting the total endowments of capital and labor K and L, respectively, we can write (6) [Mathematical Expression Omitted] and (7) [Mathematical Expression Omitted] where 960 implies the full employment of capital, and (7) allows for the existence of urban unemployment.

Let U denote the social utility which a community derives from the consumption of the two commodities, with their demand denoted by [D.sub.a] and [D.sub.m]. Then, (8) [Mathematical Expression Omitted] The balance of trade equilibrium requires that (9) [Mathematical Expression Omitted] Differentiating (1), (8) and (9), we have (10) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the output elasticity of returns to scale of the jth sector. If we assume [s.sub.j] and O initially, it is necessary to examine the change in [X.sub.m] and [Lambda] in order to know the effect of the respective policy on welfare.

III. The Analysis in Terms of Rates of Changes

In this section, we will present the model in terms of rates of change. This method is useful in the consideration of stability and comparative statics presented in the following part of this note.

Differentiating (1) totally and using a circumflex to denote the rate of change in a variable, e.g., [Mathematical Expression Omitted], we have (11) [Mathematical Expression Omitted] where [Mathematical Expression Omitted], is factor i's distributive share in the jth industry. Note that due to external nature of economies of scale, (2), (3) and (4) imply average-cost-pricing in each industry. Therefore, we have (12) [Mathematical Expression Omitted] (13) [Mathematical Expression Omitted] Differentiating these conditions and making use of (11), we get (14) [Mathematical Expression Omitted] (15) [Mathematical Expression Omitted] where [Mathematical Expression Omitted]. Let us define the elasticity of factor substitution as (16) [Mathematical Expression Omitted] where [Mathematical Expression Omitted]. Using this definition, (11) yields (17) [Mathematical Expression Omitted] (18) [Mathematical Expression Omitted] Differentiating (6) and (7), respectively, we have (19) [Mathematical Expression Omitted] (20) [Mathematical Expression Omitted] where [[Lambda].sub.ij] (i = L, K; j = a, m) is factor i's allocative share in sector j. Substituting (17) and (18) into (19) and (20), and setting K [caret] = L [caret] = 0, we have (21) [Mathematical Expression Omitted] (22) [Mathematical Expression Omitted].

From (14) and (15), we obtain (23) [Mathematical Expression Omitted] (24) [Mathematical Expression Omitted], since w [caret] [.sub.m]] = 0. On the other hand, we have (25) [Mathematical Expression Omitted] (26) [Mathematical Expression Omitted]. Substituting (23), (24), (25) and (26) into (21) and (22), therefore, we have (27) [Mathematical Expression Omitted], where [Mathematical Expression Omitted]. This is the key equation in the positive aspects of the model. Finally, note that from (5) and (23) (28) [Mathematical Expression Omitted].

IV. Stability, Wage Subsidy and Welfare

Following Mayer [5], the dynamic adjustment mechanism of the system should be specified as follows: [Mathematical Expression Omitted] where "." denotes differentiation with respects to time and [d.sub.j] is the positive coefficient measuring the speed of adjustment. According to the Routh-Hurwitz theorem a necessary condition for local stability of the system is that the determinant of the Jacobian (J) is negative. Now we put forward the following assumption.

ASSUMPTION 1. Our equilibrium is stable which implies that /J/ < 0.

We can show that /J/ < 0 is equivalent to [Mathematical Expression Omitted], since [Mathematical Expression Omitted] where C = [[Lambda].sub.La] [[Theta].sub.Ka] [[Sigma].sub.a] + (1 + [Lambda]) [[Lambda].sub.Lm] [[Theta].sub.Km] [[Sigma].sub.m] and D = ([[Lambda].sub.Ka] [[Theta].sub.La] [[Sigma].sub.m].(2) Under constant to scale, i.e., [[Epsilon].sub.a] = [[Epsilon].sub.m] = 0, the condition ~[Lambda]~ > 0 is reduced to [[Lambda].sub.La] [[Lambda].sub.Km] - (1 + [Lambda]) [[Lambda].sub.Lm] [[Lambda].sub.Ka] > 0, which is Neary's condition of stability of the neoclassical Harris-Todaro model [6]. That is, the Neary condition is a special one for the case of constant returns to scale. For our present case of VRS, the stability of the system requires that ~[Lambda]~ > 0.

Wage Subsidy in Agricultural Sector

Assuming that initially [s.sub.a] = ([s.sub.m]) = 0, (10) shows that the welfare effect depends on the changes in [X.sub.m] and [Lambda]. Then, from (27) we have (29) [dX.sub.a] / [ds.sub.a] = [X.sub.a] (1 + [Lambda]) [[Lambda].sub.Lm] [[Lambda].sub.Km] / ~[Lambda]~ (30) [dX.sub.m] / [ds.sub.a] = [-X.sub.m] (1 + [Lambda]) [[Lambda].sub.Lm] [[Lambda].sub.Ka] / ~[Lambda]~. Therefore, the effect of wage subsidy for agriculture on outputs is not generally certain even if equilibrium is stable. We employ the following assumption which is originally made by Jones [4] for the analysis of VRS without urban unemployment.

Assumption 2. At constant commodity prices the expansion of any industry results in an increased demand for each factor of production. This implies that each [[Lambda].sub.ij] is positive.

Since ~[Lambda]~ and each [[Lambda].sub.ij] are positive by Assumption 1 and Assumption 2, respectively, it is clear from (29) and (30) that [dX.sub.a] / [ds.sub.a] > 0 and [dX.sub.m] / [ds.sub. a] < 0. Furthermore, these results and (28) show that d [Lambda] / [ds.sub.a] < 0. Therefore, Beladi's Proposition VII must be modifies as follows:

Proposition 1. Suppose that Assumption 1 and Assumption 2 are satisfied. In the presence of urban unemployment, free trade is inferior to a (small) wage subsidy to agricultural sector if [Mathematical Expression Omitted]. Furthermore, a (small) wage subsidy to agricultural sector causes a decline in urban unemployment rate. However, if [[Epsilon].sub.a] < [[Epsilon].sub.m], then free trade may be superior to restricted trade.

Wage Subsidy in Manufacturing Sector

The effects of wage subsidy for manufacture on [X.sub.a] and [X.sub.m] are obtained from (27) as (31) [dX.sub.a] / [ds.sub.m] = [-X.sub.a] [[Theta].sub.Lm] (A [Lambda].sub.Km] + B [[Lambda].sub.Lm] / ~[Lambda]~ < 0 (32) [dX.sub.m] / [ds.sub.m] = [X.sub.m] [[Theta].sub.Lm] (A [[Lambda].sub.Ka] + B [[Lambda].sub.La])/ ~[Lambda]~ > 0, since ~[Lambda~] and each [[Lambda].sub.ij] are positive by Assumption 1 and Assumption 2, respectively. Substituting (31) and (32) into (28) and setting [s.sub.m] = ([s.sub.a]) = 0 initially yield that d [Lambda] / [ds.sub.m] > 0. Therefore, we get the following proposition corresponding to Proposition IV by Beladi.

Proposition 2. Suppose that Assumption 1 and Assumption 2 are satisfied. A (small) wage subsidy to manufacturing sector will lead to a loss in welfare and a rise in the urban unemployment rate if [[Epsilon].sub.m]. However, if [[Epsilon].sub.a] < [[Epsilon].sub.m], the wage subsidy may not necessarily lead to a loss in welfare.

V. Concluding Remarks

Beladi is not derived the effects of wage subsidies on [X.sub.a], [X.sub.m] and [Lambda] under VRS. He derived them in fact under constant returns to scale by considering [g.sub.j] ([X.sub.j]) as exogenous in his appendix. Under VRS his results on [X.sub.a], [X.sub.m] and so on do not generally hold. Then, we have established some conditions under which the required results hold. We can obtain formally more elaborate condition under which the welfare effects of the subsidies are confirmed by substituting [dX.sub.m] / [ds.sub.j] and d [Lambda] / [ds.sub.j] into (10). However, it is not easy to derive any fruitful economic interpretations.

We have examined here only the effect of wage subsidies. The welfare implications of the other policies such as tariff, production subsidies and export subsidies will be discussed similarly. Shigemi Yabuuchi Aichi University Aichi, Japan (1)Panagariya and Succar [6] have discussed some positive aspects of the Harris-Todaro model in VRS. (2)Detailed calculation of this result as well as another part of this note is available from the author on request.


[1]Beladi, Hamid, "Variable Returns to Scale, Urban Unemployment and Welfare." Southern Economic Journal, October 1988, 412-23. [2]Corden, Max W. and Ronald Findlay, "Urban Unemployment, Intersectoral Capital Mobility and Development Policy." Economica, February 1975, 59-78. [3]Harris, John R. and Michael P. Todaro, "Migration, Unemployment and Development: A Two-Sector Analysis." American Economic Review, March 1970, 126-42. [4]Jones, Ronald W., "Variable Returns to Scale in General Equilibrium Theory." International Economic Review, October 1968-261-72. [5]Mayer, Wolfgang, "Variable Returns to Scale in General Equilibrium Theory: A Comment." International Economic Review, February 1974, 225-35. [6]Neary, Peter J., "On the Harris-Todaro Model with Intersectoral Capital Mobility." Economica, August 1981, 219-34. [7]Panagariya, Arvind and Patricia Succar, "The Harris-Todaro Model and Economies of Scale." Southern Economic Journal, April 1986, 986-98.
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Title Annotation:comment on Hamid Beladi's article in Southern Economic Journal, p. 412, October 1988
Author:Yabuuchi, Shigemi
Publication:Southern Economic Journal
Date:Apr 1, 1992
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