Inflow of educational capital, trade liberalization, skill formation and informal sector

Abstract

This paper talks about a small open economy comprising of three sectors: two formal sectors and one informal sector. One of these formal sectors uses a specific factor which is specially trained labor. “Special training” can easily encompass the issues such as vocational training, technical training, computer literacy, software knowledge etc. Unskilled workers have the option of being trained to get a job in high wage skilled sector or to be employed in the informal sector. In such situation, an inflow of educational capital leads to a change in informal wage while formal wage and rental will remain unchanged. Under reasonable condition informal wage may even go up. Though we have mixed effect in formal sector, informal sector must shrink. An extended version of the model with the incorporation of the effect of trade liberalization policy predicts just the opposite results with regard to informal output and formal wage.

Appendices

Appendix A

Differentiating Eq. (2) and using zero-profit and envelope condition we get

$$\hat{r} = 0.$$

(12)

Similarly Eqs. (1) and (3) yield

$$\hat{{W_{S} }} = 0.$$

(13)

$$\hat{W} = \frac{{\hat{{P_{Z} }}}}{{\theta_{LZ} }}.$$

(14)

Differentiating Eq. (7) gives $\hat{{S_{1} }} = \hat{E}$ (15) as educational capital has no other alternative use except for training the informal workers.

Differentiating Eq. (4), (5) and (6) and then manipulating them we derive at the following sets of equations

$$\hat{X} = \hat{E}\lambda_{{S_{1} T}}$$

(15)

$$\lambda_{LY} \hat{Y} + \lambda_{LZ} \hat{Z} = - \hat{E}\lambda_{{S_{1} L}}$$

(16)

$$\lambda_{KY} \hat{Y} + \lambda_{KZ} \hat{Z} = - \lambda_{KX} \hat{E}\lambda_{{S_{1} T}}.$$

(17)

Rearranging Eqs. (16) and (17) in a matrix form we solve for

$$\hat{Y} = \frac{{\hat{E}}}{{\left| \lambda \right|}}\left( {\lambda_{{S_{1T} }} \lambda_{LZ} \lambda_{KX} - \lambda_{{S_{1} L}} \lambda_{KZ} } \right)$$

(18)

$$\hat{Z} = \frac{{\hat{E}}}{{\left| \lambda \right|}}\left( {\lambda_{KY} \lambda_{{S_{1} L}} - \lambda_{LY} \lambda_{KX} \lambda_{{S_{1} T}} } \right).$$

(19)

Substituting the values of $\hat{Y}$ and $\hat{Y}$ in Eq. (9) and manipulating it we derive the expression for

$$\hat{{P_{Z} }} = \hat{E}\left[ {\nu_{X} \lambda_{{S_{1} T}} + \frac{1}{\lambda}\left\{ {\lambda_{{S_{1} T}} \lambda_{KX} \left( {\nu_{Y} \lambda_{LZ} + \lambda_{LY} } \right) - \lambda_{{S_{1} L}} \left( {\nu_{Y} \lambda_{KZ} + \lambda_{KY} } \right)} \right\}} \right],$$

(20)

where $\nu_{X} = \frac{{\beta P_{X} X}}{{\left( {1 - \beta } \right)P_{z} Z}}$, $\nu_{Y} = \frac{{\beta P_{Y} Y}}{{\left( {1 - \beta } \right)P_{Z} Z}}.$

$P_{z}$ would increase if $\frac{{\lambda_{{S_{1} T}} }}{{\lambda_{{S_{1} L}} }}\lambda_{KX} < \frac{{\nu_{Y} \lambda_{KZ} + \lambda_{KY} }}{{\nu_{Y} \lambda_{LZ} + \lambda_{LY} }}$.

From (14) again $\hat{W} = \frac{{\hat{{P_{Z} }}}}{{\theta_{LZ} }}$. Thus $\hat{W} > 0$ if $\hat{{P_{Z} }} > 0$.

Differentiating Eq. (8) we have

$$\mu_{ES} \hat{R} = - \frac{{\hat{{P_{Z} }}}}{{\theta_{LZ} }} < 0\,{\text{as}}\,\,\hat{{P_{Z} }} > 0.$$

(21)

Note $S_{T} = S + S_{1}$

$$\lambda_{{S_{1} T}} = \frac{{S_{1} }}{{S_{T} }} < 1$$

$$\lambda_{{S_{1} L}} = \frac{{S_{1} }}{L} < 1$$

$$\lambda_{{S_{1} T}} > \lambda_{{S_{1} L}} \,\,{\text{as L}} > S_{T}.$$

$\left| \lambda \right| = \lambda_{LY} \lambda_{KZ} - \lambda_{LZ} \lambda_{KY}$ < 0 as Z is a labor intensive sector and Y is capital intensive sector.

Appendix B

Differentiating Eq. (10) and solving for $\hat{r}$ we get

$$\frac{{dP_{Y} (1 + t)}}{{P_{Y} (1 + t)}} + \frac{dt}{t}\frac{{tP_{Y} }}{{P_{Y} (1 + t)}} = \frac{{d\overline{W} }}{{\overline{W} }}\frac{{a_{LY} \overline{W} }}{{P_{Y} (1 + t)}} + \frac{{da_{LY} }}{{a_{LY} }}\frac{{a_{LY} \overline{w} }}{Y(1 + t)} + \frac{dr}{r}\frac{{a_{KY} r}}{{P_{Y} (1 + t)}} + \frac{{da_{KY} }}{{a_{KY} }}\frac{{ra_{KY} }}{{P_{Y} (1 + t)}}.$$

Since $\hat{W}$ and $P_{Y}$ do not change, and using the envelope condition [$a_{LY} \theta_{LY} + a_{KY} \theta_{KY} = 0$].

The above expression yields:

$\hat{r}\theta_{KY} = \alpha \hat{t}$, where $\alpha = \frac{t}{(1 + t)}$ and, the $\theta_{KY} = \frac{{a_{KY} r}}{{P_{Y} (1 + t)}}$e income share of physical capital in sector Y.

Thus

$$\hat{r} = \frac{{\alpha \hat{t}}}{{\theta_{KY} }} < 0\,\,{\text{as}}\,\,\hat{t} < 0$$

(22)

Now differentiating Eq. (1) and substituting the expression for $\hat{r}$ we get

$$\hat{{W_{S} }} = ( - )\frac{{\theta_{KX} }}{{\theta_{SX} }}\frac{{\alpha \hat{t}}}{{\theta_{KY} }} > 0{\text{ as }}\hat{t} < 0.$$

(23)

Again, differentiating Eq. (3) and substituting $\hat{r}$ we get

$$\hat{W} = \frac{{\hat{{P_{Z} }}}}{{\theta_{LZ} }} - \frac{{\theta_{KZ} }}{{\theta_{LZ} \theta_{KY} }}\alpha \hat{t}.$$

(24)

If E does not change and $S_{1}$, $L_{1}$ and K remains constant then no endowment effect (or Rybczzynski) is observed. However, there could be a possibility of factor substitution in X first and then in other commodities.

Equation (4) yields:

$$\hat{X} = ( - )\hat{{a_{SX} }}.$$

(25)

Using the expression for elasticity of substitution between S and K in X we have

$$\widehat{{a_{SX} }} = \widehat{{a_{KX} }} - \sigma_{X} (\widehat{{W_{S} }} - \hat{r})\,\,\,{\text{and}}$$

(26)

$$\widehat{{a_{KX} }} = \sigma_{X} \left( {\widehat{{W_{S} }} - \hat{r}} \right) + \widehat{{a_{SX} }}$$

(27)

Using the envelope condition Eq. (26) and (27) are modified as

$$\widehat{{a_{SX} }} = ( - )\widehat{{a_{KX} }}\frac{{\theta_{KX} }}{{\theta_{SX} }}\,\,{\text{and}}$$

(28)

$$\widehat{{a_{KX} }} = \left( - \right)\widehat{{a_{SX} }}\frac{{\theta_{SX} }}{{\theta_{KX} }}.$$

(29)

Again substituting these values in Eq. (26) we obtain

$$\widehat{{a_{KX} }} = \sigma_{X} \left( {\widehat{{W_{S} }} - \hat{r}} \right)\theta_{SX}$$

(30)

From Eq. (27) we have

$$\widehat{{a_{SX} }} = \left( - \right)\sigma_{X} (\widehat{{W_{S} }} - \hat{r})\theta_{KX}$$

(31)

Equation (22) and (23) yields

$$\widehat{{W_{S} }} - \hat{r} = ( - )\alpha \hat{t}\frac{1}{{\theta_{KY} \theta_{SX} }}$$

(32)

Substituting the above expression in Eq. (31) we have

$$\widehat{{a_{SX} }} = \sigma_{X} \alpha \hat{t}\frac{{\theta_{KX} }}{{\theta_{KY} \theta_{SX} }}$$

(33)

Using Eq. (25) we obtain

$$\hat{X} = ( - )\sigma_{X} \frac{{\alpha \hat{t}}}{{\theta_{KY} }}\frac{{\theta_{KX} }}{{\theta_{SX} }} > 0{\text{ as}}\,\,\,\hat{t} < 0$$

(34)

Differentiating Eqs. (5) and (6) and substituting the expression for elasticity of substitution we arrange them in matrix form to yield the following sets of equations

$$\hat{Y} = \frac{1}{\lambda} ( - \lambda_{LZ} \lambda_{KX} \sigma_{X} \frac{{\alpha \hat{t}}}{{\theta_{KY} }}\frac{{\theta_{KX} }}{{\theta_{SX} }}) < 0{\text{ if}}\,\,\lfloor {\lambda} \rfloor < 0$$

(35)

Similarly,

$$\hat{Z} = \frac{1}{{\left| \lambda \right|}}(\lambda_{LY} \lambda_{KX} \sigma_{X} \alpha \hat{t}\frac{{\theta_{KX} }}{{\theta_{KY} \theta_{SX} }}) > 0{\text{ if}}\left| \lambda \right| < 0$$

(36)

Substituting the values of $\hat{Y}$ and $\hat{Z}$ in Eq. (11) and manipulating it we derive the expression for $\widehat{{P_{Z} }} = \nu_{X} \hat{X} + \nu_{Y} \left( {\hat{Y} + t\hat{Y} + t\hat{t}} \right) - \hat{Z}$

Now,

$$\hat{{P_{Z} }} > 0 \,{\text{if}}\,\,\nu_{X} \hat{X} > \nu_{Y} \left( {\hat{Y} + t\hat{Y} + t\hat{t}} \right)$$

From Eq. (24), $\hat{W} > 0$.

The expression for the wage gap between the skilled and unskilled labor (or formal and informal labor) is given by

$$\left( {\hat{{W_{s} }} - \hat{W}} \right) = \frac{{\alpha \hat{t}}}{{\theta_{KY} }}\left( {\frac{{\theta_{KZ} - \theta_{KX} }}{{\theta_{LZ} \theta_{SX} }}} \right) - \frac{{\widehat{{P_{Z} }}}}{{\theta_{LZ} }} > 0{\text{ iff}}\,\theta_{KZ} > .\theta_{KX} .$$

(37)

Using this result and Eq. (8) we obtain $\hat{R}$ > 0.