# Variable returns to scale, urban unemployment and welfare: reply.

I.

We wish to thank Professor Yabuuchi [2] for his comment regarding Beladi's [1] treatment of [g.sub.i] ([X.sub.i]) as exogenous. The term [g.sub.i] (X.sub.i]) captures interindustry scale economies, as can be seen by forming the elasticity of [g.sub.i] ([X.sub.i]) with respect to changes in industry output, [X.sub.i]: (1) [[Epsilon].sub.i] = ([dg.sub.i] / [dX.sub.i]) ([X.sub.i] / [g.sub.i]). [[Epsilon].sub.i] is defined on [- [infinity]], with [[Epsilon].sub.i] > 0 denoting increasing returns to scale (IRS), [[Epsilon].sub.i] = 0 denoting constant returns to scale (CRS) and [[Epsilon].sub.i] < 0 denoting decreasing returns to scale (DRS). Not taking the derivative of [g.sub.i] ([X.sub.i]) with respect to [X.sub.i] is equivalent to treating the industry as being characterized by CRS, as Yabuuchi notes.

In his comment, Yabuuchi derives two propositions concerning the relative merits of free trade versus a small wage subsidy in agriculture or manufacturing. These propositions are based on two assumptions. The first, that ~[Lambda]~ > 0, is an extension of Neary's stability condition; it is standard in the literature. The second, that [Mathematical Expression Omitted] = capital or L = labor; i [Mathematical Expression Omitted] a = agriculture as m = manufacturing), is the focus of this reply.

The assumption that [[Lambda].sub.ji] > 0 imposes limits on the degree of scale economies relative to the elasticity of substitution in agriculture [[Sigma].sub.a]), where [Sigma].sub.a] is convoluted by additional terms shown below. The terms [[Lambda].sub.ji] may be written as:(1) (2) [[Lambda].sub.Ka] = [[Lambda].sub.Ka] [1 - [[Epsilon].sub.a] + [[Sigma].sub.a + [[Sigma].sub.a] [[Epsilon].sub.a]] (3) [[Lambda].sub.La] = [[Lambda].sub.La] [1 - [[Epsilon].sub.a] [Phi]] (4) [[Lambda].sub.Km] = [[Lambda].sub.Km] [1 - [[Epsilon].sub.m] + [[Epsilon].sub.m] [[Psi].sub.K]] (5) [[Lambda].sub.Lm] = (+ [Lambda]) [[Lambda].sub.Lm] [1 - [[Epsilon].sub.m] + [[Epsilon].sub.m] [[Psi].sub.L]] where, [Mathematical Expression Omitted] it is apparent from (2) that [[Lambda].sub.Ka] > 0 iff: (6) [Mathematical Expression Omitted] which holds if [[Epsilon].sub.a] [is greater than or equal to] 0 or if [Sigma].sub.a] [is less than or equal to] 1 while 0 > [[Epsilon].sub.a]. Further, (7) [Mathematical Expression Omitted] Figure 1 provides a graphical representation of [[Lambda].sub.Ka]. It is obvious that [[Lambda].sub.Ka] [is less than or equal to] 0 for all [[Epsilon].sub.a < 1/(1 - [[Sigma].sub.a]).

It is also apparent from (3) that [[Lambda].sub.La] > 0 if [[Epsilon].sub.La] > 0 if [[Epsilon].sub.a] < (1/[Phi]) which clearly holds if [Phi] < 1. Since (d [[Lambda].sub.La]/d [[Epsilon].sub.a]) = -[Phi] < 0, we can drew Figure 2, showing that [[Lambda].sub.La] < 0 for [[Epsilon].sub.La] < 0 for all [[Epsilon].sub.a] > (/[Pi]) provided [Phi] > 1.

Thus far we have shown that [[Lambda].sub.La], [[Lambda].sub.Ka] > 0 is assured for: (8) [Mathematical Expression Omitted] with the larger inequality potential violated if [Phi] > 1 and the smaller inequality potentially violated if [[Sigma].sub.a] [is less than or equal to] 1.

Turning to [[Lambda].sub.jm] we se from (4) that [[Lambda].sub.Km] > 0 iff: (9) [Mathematical Expression Omitted] while from (5) [[Lambda].sub.Lm] > 0 iff: (10) [Mathematical Expression Omitted] Both relationships hold if [[Epsilon].sub.m] [is greater than or equal to] 0 or if [[Psi].sub.i] [is less than or equal to] 1 while 0 > [[Epsilon].sub.m]. Figures 3 and 4 graph [[Lambda].sub.Km] and [[Lambda].sub.Lm] respectively.

Thus, [[Lambda].sub.Lm], [[Lambda].sub.Km] > 0 is assured provided: (11) [Mathematical Expression Omitted] for [[Epsilon].subm] < 0, [[Psi].sub.i] > 1.

II. Summary

There have been two purposes of this reply. The first is to acknowledge that Yabuuchi correctly demands that analysis of variable returns to scale requires taking the derivative of [g.sub.i] (X.sub.i). The second had been to point out that his propositions 1 and 2 are based on somewhat more restrictive assumption concerning the magnitudes of [[Epsilon].sub.m] and [[Epsilon].sub.a] than is apparent at first reading. This is not a weakness of his model. Rather, it is a fact that the existence of variable returns to scale (VRS) precludes obtaining results as clean as those which occur with constant returns to scale. In a future paper we intend to provide a rigorous analysis of resource allocation across sectors under VRS in the context of the Harris-Todaro model of sector specific unemployment. Hamid Beladi University of Dayton Dayton, Ohio Charles A. Ingene University of Washington Seattle, Washington (1)Equations (2)-(5) correspond to the unnumbered definitions between equations (27) and (28) in Yabuuchi [2]. Our (4) and (5) differ slightly fron [[Lambda].sub.Km] and [[Lambda].sub.Lm in Yabuuchi, apparently due to typographical errors in the manuscript of the comment. However, even using Yabuuchi's definitions of these terms would yield similar results to those show below.

References

[1]Beladi, Hamid, "Variable Returns to Scale, Urban Unemployment and Welfare. Southern Economic Journal October 1988, 412,23. [2]Yabuuchi, Shigemi, "Variable Returns to Scale, Urban Unemployment and Welfare: Comment." Southern Economic Journal, April 1992.

We wish to thank Professor Yabuuchi [2] for his comment regarding Beladi's [1] treatment of [g.sub.i] ([X.sub.i]) as exogenous. The term [g.sub.i] (X.sub.i]) captures interindustry scale economies, as can be seen by forming the elasticity of [g.sub.i] ([X.sub.i]) with respect to changes in industry output, [X.sub.i]: (1) [[Epsilon].sub.i] = ([dg.sub.i] / [dX.sub.i]) ([X.sub.i] / [g.sub.i]). [[Epsilon].sub.i] is defined on [- [infinity]], with [[Epsilon].sub.i] > 0 denoting increasing returns to scale (IRS), [[Epsilon].sub.i] = 0 denoting constant returns to scale (CRS) and [[Epsilon].sub.i] < 0 denoting decreasing returns to scale (DRS). Not taking the derivative of [g.sub.i] ([X.sub.i]) with respect to [X.sub.i] is equivalent to treating the industry as being characterized by CRS, as Yabuuchi notes.

In his comment, Yabuuchi derives two propositions concerning the relative merits of free trade versus a small wage subsidy in agriculture or manufacturing. These propositions are based on two assumptions. The first, that ~[Lambda]~ > 0, is an extension of Neary's stability condition; it is standard in the literature. The second, that [Mathematical Expression Omitted] = capital or L = labor; i [Mathematical Expression Omitted] a = agriculture as m = manufacturing), is the focus of this reply.

The assumption that [[Lambda].sub.ji] > 0 imposes limits on the degree of scale economies relative to the elasticity of substitution in agriculture [[Sigma].sub.a]), where [Sigma].sub.a] is convoluted by additional terms shown below. The terms [[Lambda].sub.ji] may be written as:(1) (2) [[Lambda].sub.Ka] = [[Lambda].sub.Ka] [1 - [[Epsilon].sub.a] + [[Sigma].sub.a + [[Sigma].sub.a] [[Epsilon].sub.a]] (3) [[Lambda].sub.La] = [[Lambda].sub.La] [1 - [[Epsilon].sub.a] [Phi]] (4) [[Lambda].sub.Km] = [[Lambda].sub.Km] [1 - [[Epsilon].sub.m] + [[Epsilon].sub.m] [[Psi].sub.K]] (5) [[Lambda].sub.Lm] = (+ [Lambda]) [[Lambda].sub.Lm] [1 - [[Epsilon].sub.m] + [[Epsilon].sub.m] [[Psi].sub.L]] where, [Mathematical Expression Omitted] it is apparent from (2) that [[Lambda].sub.Ka] > 0 iff: (6) [Mathematical Expression Omitted] which holds if [[Epsilon].sub.a] [is greater than or equal to] 0 or if [Sigma].sub.a] [is less than or equal to] 1 while 0 > [[Epsilon].sub.a]. Further, (7) [Mathematical Expression Omitted] Figure 1 provides a graphical representation of [[Lambda].sub.Ka]. It is obvious that [[Lambda].sub.Ka] [is less than or equal to] 0 for all [[Epsilon].sub.a < 1/(1 - [[Sigma].sub.a]).

It is also apparent from (3) that [[Lambda].sub.La] > 0 if [[Epsilon].sub.La] > 0 if [[Epsilon].sub.a] < (1/[Phi]) which clearly holds if [Phi] < 1. Since (d [[Lambda].sub.La]/d [[Epsilon].sub.a]) = -[Phi] < 0, we can drew Figure 2, showing that [[Lambda].sub.La] < 0 for [[Epsilon].sub.La] < 0 for all [[Epsilon].sub.a] > (/[Pi]) provided [Phi] > 1.

Thus far we have shown that [[Lambda].sub.La], [[Lambda].sub.Ka] > 0 is assured for: (8) [Mathematical Expression Omitted] with the larger inequality potential violated if [Phi] > 1 and the smaller inequality potentially violated if [[Sigma].sub.a] [is less than or equal to] 1.

Turning to [[Lambda].sub.jm] we se from (4) that [[Lambda].sub.Km] > 0 iff: (9) [Mathematical Expression Omitted] while from (5) [[Lambda].sub.Lm] > 0 iff: (10) [Mathematical Expression Omitted] Both relationships hold if [[Epsilon].sub.m] [is greater than or equal to] 0 or if [[Psi].sub.i] [is less than or equal to] 1 while 0 > [[Epsilon].sub.m]. Figures 3 and 4 graph [[Lambda].sub.Km] and [[Lambda].sub.Lm] respectively.

Thus, [[Lambda].sub.Lm], [[Lambda].sub.Km] > 0 is assured provided: (11) [Mathematical Expression Omitted] for [[Epsilon].subm] < 0, [[Psi].sub.i] > 1.

II. Summary

There have been two purposes of this reply. The first is to acknowledge that Yabuuchi correctly demands that analysis of variable returns to scale requires taking the derivative of [g.sub.i] (X.sub.i). The second had been to point out that his propositions 1 and 2 are based on somewhat more restrictive assumption concerning the magnitudes of [[Epsilon].sub.m] and [[Epsilon].sub.a] than is apparent at first reading. This is not a weakness of his model. Rather, it is a fact that the existence of variable returns to scale (VRS) precludes obtaining results as clean as those which occur with constant returns to scale. In a future paper we intend to provide a rigorous analysis of resource allocation across sectors under VRS in the context of the Harris-Todaro model of sector specific unemployment. Hamid Beladi University of Dayton Dayton, Ohio Charles A. Ingene University of Washington Seattle, Washington (1)Equations (2)-(5) correspond to the unnumbered definitions between equations (27) and (28) in Yabuuchi [2]. Our (4) and (5) differ slightly fron [[Lambda].sub.Km] and [[Lambda].sub.Lm in Yabuuchi, apparently due to typographical errors in the manuscript of the comment. However, even using Yabuuchi's definitions of these terms would yield similar results to those show below.

References

[1]Beladi, Hamid, "Variable Returns to Scale, Urban Unemployment and Welfare. Southern Economic Journal October 1988, 412,23. [2]Yabuuchi, Shigemi, "Variable Returns to Scale, Urban Unemployment and Welfare: Comment." Southern Economic Journal, April 1992.

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Title Annotation: | response to Shigemi Yabuuchi, Southern Economic Journal, vol. 58, p. 1103, April 1992 |
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Author: | Ingene, Charles A. |

Publication: | Southern Economic Journal |

Date: | Apr 1, 1992 |

Words: | 880 |

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