Skilled-unskilled wage inequality and structural transformation in a dual economy

Abstract

In this paper, we develop a sectoral model of a less developed economy with a backward agricultural sector dependent on unskilled labour and with an advanced industrial sector dependent on capital and skilled labour and also with a private education sector whose role is to transform an unskilled labourer into a skilled one. Using a Ramsey framework of consumption saving allocation, we analyse how the economy grows over time through accumulation of capital and through transformation of unskilled labour into skilled labour. Such a model helps us to analyse how the structural shift from agriculture to industry takes place and what role do sector specific polices play in this context. We analyse short-run as well as long run effects of sector specific policies on the degree of wage income inequality as well as on the endogenous rate of economic growth; and find that sector specific policies produce opposite effects. Subsidization to agricultural sector leads to an improvement in the degree of wage income inequality but lowers the rate of economic growth in the long run. On the other hand, subsidization or protection to the manufacturing sector produces just the opposite result.

Appendix

1.1 Derivation of Eqs. (8), (9) and (20)

Differentiating both sides of Eq. (1), we obtain

$$ \widehat{{P_{\text{U}} }} = \widehat{{W_{\text{U}} }} $$

(A.1)

Differentiating both sides of Eq. (2), we obtain

$$ \widehat{{P_{\text{M}} }} = \theta_{\text{SM}} \widehat{{W_{\text{S}} }} + \theta_{\text{KM}} \widehat{r} $$

(A.2)

From Eqs. (3) and (4), we have

$$ W_{\text{S}} - W_{\text{U}} = a_{\text{SE}} W_{\text{S}} + a_{\text{KE}} r $$

(A.3)

Differentiating both sides of equation (A.3), we have

$$ \left( {\theta_{\text{SE}} - \frac{{W_{\text{S}} }}{{W_{\text{S}} - W_{\text{U}} }}} \right)\widehat{{W_{\text{S}} }} + \frac{{W_{\text{U}} }}{{W_{\text{S}} - W_{\text{U}} }}\widehat{{W_{\text{U}} }} + \theta_{\text{KE}} \widehat{r} = 0 $$

(A.4)

Using equations (A.1) and (A.4), we have

$$ \left( {\theta_{\text{SE}} - \frac{{W_{\text{S}} }}{{W_{\text{S}} - W_{\text{U}} }}} \right)\widehat{{W_{\text{S}} }} + \theta_{\text{KE}} \widehat{r} = - \frac{{W_{\text{U}} }}{{W_{\text{S}} - W_{\text{U}} }}\widehat{{P_{\text{U}} }} . $$

(A.5)

Solving equations (A.2) and (A.5), we have

$$ \widehat{{W_{\text{S}} }} = \frac{1}{\Delta }\left[ {\frac{{W_{\text{U}} }}{{W_{\text{S}} - W_{\text{U}} }}\theta_{\text{KM}} \widehat{{P_{\text{U}} }} + \theta_{\text{KE}} \widehat{{P_{\text{M}} }}} \right] $$

(A.6)

and

$$ \widehat{r} = \frac{1}{\Delta }\left[ { - \frac{{W_{\text{U}} }}{{W_{\text{S}} - W_{\text{U}} }}\theta_{\text{SM}} \widehat{{P_{\text{U}} }} - \left( {\theta_{\text{SE}} - \frac{{W_{\text{S}} }}{{W_{\text{S}} - W_{\text{U}} }}} \right)\widehat{{P_{\text{M}} }}} \right] $$

(A.7)

Equation (A.7) is same as Eq. (20).

From equations (A.1) and (A.6), we have

$$ \widehat{{W_{\text{S}} }} - \widehat{{W_{\text{U}} }} = \frac{{\theta_{\text{KE}} }}{\Delta }\widehat{{P_{\text{M}} }} + \left( {\frac{{\theta_{\text{KM}} \frac{{W_{\text{U}} }}{{W_{\text{S}} - W_{\text{U}} }}}}{\Delta } - 1} \right)\widehat{{P_{\text{U}} }} $$

(A.8)

where

$$ \Delta = \theta_{\text{KE}} + \theta_{\text{KM}} \frac{{W_{\text{U}} }}{{W_{\text{S}} - W_{\text{U}} }} $$

Equation (A.8) is same as Eq. (8).

From equations (A.6) and (A.7), we have

$$ \widehat{{W_{\text{S}} }} - \widehat{r} = \frac{{\frac{{W_{\text{U}} }}{{W_{\text{S}} - W_{\text{U}} }}}}{\Delta }\left[ { - \widehat{{P_{\text{M}} }} + \widehat{{P_{\text{U}} }}} \right] $$

(A.9)

Equation (A.9) is same as Eq. (9).

1.2 Derivation of Eq. (10)

Differentiating both sides of Eq. (6), we have

$$ \begin{aligned} \lambda_{\text{SM}} \widehat{{X_{\text{M}} }} + \lambda_{\text{SE}} \widehat{{X_{\text{E}} }} & = \widehat{S} - \lambda_{\text{SM}} \widehat{{a_{\text{SM}} }} - \lambda_{\text{SE}} \widehat{{a_{\text{SE}} }} \Rightarrow \lambda_{\text{SM}} \widehat{{X_{\text{M}} }} + \lambda_{\text{SE}} \widehat{{X_{\text{E}} }} \\& = \widehat{S} - \left( {\lambda_{\text{SM}} S_{\text{SS}}^{\text{M}} + \lambda_{\text{SE}} S_{\text{SS}}^{\text{E}} } \right)\left( {\widehat{{W_{\text{S}} }} - \widehat{r}} \right) \\ \end{aligned} $$

(A.10)

Similarly, differentiating both sides of Eq. (7), we have

$$ \lambda_{\text{KM}} \widehat{{X_{\text{M}} }} + \lambda_{\text{KE}} \widehat{{X_{\text{E}} }} = \widehat{K} - \left( {\lambda_{\text{KM}} S_{\text{KS}}^{\text{M}} + \lambda_{\text{KE}} S_{\text{KS}}^{\text{E}} } \right)\left( {\widehat{{W_{\text{S}} }} - \widehat{r}} \right) $$

(A.11)

Solving equations (A.10) and (A.11) we obtain

$$ \widehat{{X_{\text{E}} }} = \frac{1}{\left| \lambda \right|}\left[ {\lambda_{\text{SM}} \widehat{K} - \lambda_{\text{KM}} \widehat{S} + \left\{ {\lambda_{\text{KM}} \left( {\lambda_{\text{SM}} S_{\text{SS}}^{\text{M}} + \lambda_{\text{SE}} S_{\text{SS}}^{\text{E}} } \right) - \lambda_{\text{SM}} \left( {\lambda_{\text{KM}} S_{\text{KS}}^{\text{M}} + \lambda_{\text{KE}} S_{\text{KS}}^{\text{E}} } \right)} \right\}\left( {\widehat{{W_{\text{S}} }} - \widehat{r}} \right)} \right] $$

(A.12)

Here, $\left| \lambda \right| = \lambda_{\text{SM}} \lambda_{\text{KE}} - \lambda_{\text{SE}} \lambda_{\text{KM}}$.

Finally, using equations (A.8) and (A.12), we obtain

$$ \widehat{{X_{\text{E}} }} = \frac{1}{\left| \lambda \right|}\left[ {\lambda_{\text{SM}} \widehat{K} - \lambda_{\text{KM}} \widehat{S} + \frac{{\lambda_{\text{KM}} \left( {\lambda_{\text{SM}} S_{\text{SS}}^{\text{M}} + \lambda_{\text{SE}} S_{\text{SS}}^{\text{E}} } \right) - \lambda_{\text{SM}} \left( {\lambda_{\text{KM}} S_{\text{KS}}^{\text{M}} + \lambda_{\text{KE}} S_{\text{KS}}^{\text{E}} } \right)}}{\Delta }\frac{{W_{\text{U}} }}{{W_{\text{S}} \left( {1 - \eta } \right) - W_{\text{U}} }}\left\{ { - \widehat{{P_{\text{M}} }} + \widehat{{P_{\text{U}} }}} \right\}} \right] $$

(A.13)

Equation (A.13) is same as Eq. (10).

1.3 Derivation of Eq. (15)

The dynamic optimization problem is to maximize $\mathop \smallint \limits_{0}^{\infty } e^{ - \rho t} \frac{{C^{1 - \sigma } }}{1 - \sigma }{\text{d}}t$ with respect to control variable C subject to Eq. (12) and given $K(0)$ and satifying $0 \le C \le Y$.

The Hamiltonian to be maximized at each point of time is given by

$$ H = e^{ - \rho t} \frac{{c^{1 - \sigma } }}{1 - \sigma } + e^{ - \rho t} \lambda_{\text{K}} \left[ {Y - C} \right] $$

(A.14)

Where λ_K is the co-state variable. Assuming an interior solution, we obtain

$$ C^{ - \sigma } = \lambda_{\text{K}} $$

(A.15)

Also the optimum time path of $\lambda_{K}$ satisfies the following.

$$ \frac{{\dot{\lambda }_{K} }}{{\lambda_{K} }} = \rho - \frac{dY}{dK} $$

(A.16)

Using Eqs. (13) and (A.16) we have

$$ \frac{{\dot{\lambda }_{\text{K}} }}{{\lambda_{\text{K}} }} = \rho - r $$

(A.17)

Using equations (A.15) and (A.17), we have

$$ \frac{{\dot{C}}}{C} = \frac{1}{\sigma }\left( {r - \rho } \right) $$

(A.18)

Equation (A.18) is same as Eq. (15).

1.4 Derivation of Eq. (16)

Using Eqs. (6) and (7) we have

$$ X_{\text{S}} = \frac{{\left( {a_{\text{SM}} K - a_{\text{KM}} S} \right)}}{\left| \lambda \right|} $$

(A.19)

where, $\left| \lambda \right| = a_{\text{SM}} a_{\text{KE}} - a_{\text{SE}} a_{\text{KM}} = \lambda_{\text{SM}} \lambda_{\text{KE}} - \lambda_{\text{SE}} \lambda_{\text{KM}}$.

Using Eqs. (14) and (A.19) we obtain

$$ \frac{{\dot{S}}}{S} = \frac{{a_{\text{SM}} }}{\left| \lambda \right|}\frac{K}{S} - \frac{{a_{\text{KM}} }}{\left| \lambda \right|} - \eta $$

(A.20)

Equation (A.20) is same as Eq. (16).

1.5 Determinant and trace of Jacobian matrix

Here the Jacobian matrix corresponding to differential Eqs. (23) and (24) is given by

$$ J = \left[ {\begin{array}{*{20}c} {\frac{{\partial \dot{z}}}{\partial z}} & {\frac{{\partial \dot{z}}}{\partial y}} \\ {\frac{{\partial \dot{y}}}{\partial z}} & {\frac{{\partial \dot{y}}}{\partial y}} \\ \end{array} } \right] $$

where,

$$ \frac{{\partial \dot{z}}}{\partial z} = - \frac{{a_{\text{SM}} }}{\left| \lambda \right|}\frac{1}{{z^{2} }} - W_{\text{S}} $$

$$ \frac{{\partial \dot{z}}}{\partial y} = 1 $$

$$ \frac{{\partial \dot{y}}}{\partial z} = - W_{\text{S}} $$

and

$$ \frac{{\partial \dot{y}}}{\partial y} = 1 $$

So the determinant of the Jacobian matrix can be written as follows.

$$ \left| J \right| = - \frac{{a_{\text{SM}} }}{\left| \lambda \right|}\frac{1}{{z^{2} }} $$

This is same as that shown in Sect. 3.3.

The trace of the Jacobian matrix is given by

$$ {\text{Tr J}} = - \frac{{a_{\text{SM}} }}{\left| \lambda \right|}\frac{1}{{z^{2} }} - W_{\text{S}} + 1 $$

This is same as that shown in Sect. 3.3.