Domestic Income Transfer in an Open Dual Economy

Abstract

This chapter investigates the welfare effects of an income transfer from urban manufacturing workers to rural agricultural workers in an open dual economy where the urban manufacturing wage is fixed under the minimum wage legislation. We show that the utility of a rural worker may be reduced by the transfer if capital is specific, but such a transfer paradox never appears if capital is mobile between industries. We also derive the result that the transfer causes urban unemployment to decrease in the sector-specific capital case but possibly increase in the mobile capital case.

Appendix 12.1

In this appendix we prove that if T^AL = T^ML = 0, the equilibrium is globally stable under factor movement dynamic process (D). Total differentiation of (D) with respect to the endogenous variables yields

$$ d{\dot{L}}_A\equiv {dw}^A-\frac{\overline{w}\left(L-{L}_A\right){dL}_M+\overline{w}{L}_M{dL}_A}{{\left(L-{L}_A\right)}^2}, $$

$$ d{\dot{K}}_A={pG}_{KK}{dK}_A+{pG}_{KL}{dL}_A-{F}_{KK}{dK}_M-{F}_{KL}{dL}_M. $$

Therefore Jacobian matrix of (D) is given by

$$ J\equiv \left[\begin{array}{cc}{pG}_{LL}-\frac{\overline{w}{L}_M}{L-{L}_A}& {pG}_{LK}-\frac{\overline{w}{F}_{LK}}{\left(L-{L}_A\right){F}_{LL}}\\ {}{pG}_{KL}& {pG}_{KK}+{F}_{KK}-\frac{{F_{KL}}^2}{F_{LL}}\end{array}\right] $$

since pG _LL dL _A  + pG _KK dK _A  = dw^A, F _LL dL _M  + F _LK dK _M  = 0, and dK _M  + dK _L  = 0.

Now, every diagonal element of J is negative. The determinant of J is derived as

$$ {\displaystyle \begin{array}{ll}\mid J\mid & =\left({pG}_{LL}-\frac{\overline{w}{L}_M}{{\left(L-{L}_A\right)}^2}\right)\left({pG}_{KK}+{F}_{KK}-\frac{{F_{KL}}^2}{F_{LL}}\right)-{pG}_{KL}\left({pG}_{KL}-\frac{\overline{w}{F}_{LK}}{\left(L-{L}_A\right){F}_{LL}}\right)\\ {}& =\left(\frac{{F_{KL}}^2}{F_{LL}}-{pG}_{KK}-{F}_{KK}\right)\frac{\overline{w}{L}_M}{\left(L-{L}_A\right)}+{pG}_{KL}\frac{\overline{w}{F}_{LK}}{\left(L-{L}_A\right){F}_{LL}}\\ {}& =\frac{p\overline{w}{G}_{KL}}{L-{L}_A}\frac{L_A{L}_M}{K_A{K}_M}\left(\frac{K_M}{L-{L}_A}-\frac{K_A}{L_A}\right),\end{array}} $$

implying that ∣J∣ is positive in sign if the urban area is more capital intensive than the rural area.

Applying the stability theorem in Oleck (1963) to these results, we can assert that the equilibrium is globally stable in the case where T^AL = T^ML = 0.