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

• Author: Beladi, Hamid
• Position: Response to Shigemi Yabuuchi, Southern Economic Journal, vol. 58, p. 1103, April 1992

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...