Bureaucratic efficiency, economic reform and informal sector

Abstract

In this paper, we develop a four sector general equilibrium model of a small open economy with three formal sectors and one informal sector. One formal sector’s output is used as an intermediate input in all other sectors. This intermediate input is defined here as bureaucracy. We have also incorporated an additional cost specific to the informal sector in addition to factor cost of production which is defined as the cost of corruption. In this context, this paper examines the impact of less protectionist policy and bureaucratic reform on the output levels and factor prices. It has been shown that the informal sector and manufacturing sector have contracted due to tariff reduction while informal wage goes up. Further, we have examined the effect of a decrease in bureaucratic (in)efficiency. This reduces the informal wage rate but informal sector expands. It is further examined that the effect of a decrease in the cost of corruption leads to an increase in the informal wage rate.

Appendices

Appendix 1: Tariff reform or a reduction in t

Differentiating Eqs. (1) (2), (3) and (4) using envelope condition, we get,

$$\hat{R}\theta_{TY} + \hat{r}\theta_{KY} + \hat{P}_{G} \theta_{GY} = 0$$

(9)

$$\hat{r}\theta_{KM} + \hat{P}_{G} \theta_{GM} = \alpha \hat{t}$$

(10)

$$\hat{r}\theta_{KG} = \hat{P}_{G}$$

(11)

$$\hat{W}\theta_{LI} + \hat{r}\theta_{KI} + \hat{P}_{G} \theta_{GI} = 0$$

(12)

where,

$$\alpha = \frac{t}{1 + t}$$

From Eqs. (10) and (11), we get

$$\hat{P}_{G} = \frac{{\alpha \theta_{KG} }}{{\left[ {\theta_{KG} \theta_{GM} + \theta_{KM} } \right]}}\hat{t}\quad {\text{and}}\quad \hat{r} = \frac{\alpha }{{\left[ {\theta_{KG} \theta_{GM} + \theta_{KM} } \right]}}\hat{t}$$

$\text{Let,} \quad \left[ {\theta_{KG} \theta_{GM} + \theta_{KM} } \right]$ = β

Therefore,

$$\hat{P}_{G} = \frac{{\alpha \theta_{KG} }}{\beta }\hat{t}$$

(13)

$$\hat{r} = \frac{\alpha }{\beta }\hat{t}$$

(14)

Using Eq. (13) in Eq. (9), it becomes

$$\hat{R} = - \frac{\alpha }{\beta }\frac{{\left( {\theta_{KY} + \theta_{KG} \theta_{GY} } \right)}}{{\theta_{TY} }}\hat{t}$$

(15)

Using the values of $\hat{r}$ and $\hat{P}_{G}$ in Eq. (12), we obtain

$$\hat{W} = - \frac{\alpha }{{\beta \theta_{LI} }}\left[ {\theta_{KI} + \theta_{KG} \theta_{GI} + \theta_{KG} } \right]\hat{t}$$

$\text{Let,} \quad \frac{1}{{\theta_{LI} }}\left[ {\theta_{KI} + \theta_{KG} \theta_{GI} + \theta_{KG} } \right] = \delta$

$${\text{Therefore}},\,\, \hat{W} = - \frac{\alpha }{\beta }\delta \hat{t}$$

(16)

Differentiating Eq. (7) and using elasticity of substitution

$$\hat{X}_{Y} = - \sigma_{Y} \theta_{LY} \left( {\hat{\bar{W}} - \hat{R}} \right)$$

Using the values of Eq. (15) in the above equation, it becomes

$$\hat{X}_{Y} = - A_{1} \hat{t} > 0\,\,{\text{as}}\,\,\hat{t} < 0$$

(17)

where $A_{1} = \left[ {\sigma_{Y} \theta_{LY} \frac{{\left( {\theta_{KY} + \theta_{KG} \theta_{GY} } \right)}}{{\theta_{TY} }}\frac{\alpha }{\beta }} \right] > 0$

Differentiating Eq. (5) and substituting the values from Eq. (17)

$$\lambda_{LM} \hat{X}_{M} + \lambda_{LG} \hat{X}_{G} + \lambda_{LI} \hat{X}_{I} = \lambda_{LY} A_{1} \hat{t}$$

(18)

From Eqs. (6) and (18), we get

$$\lambda_{KM} \hat{X}_{M} + \lambda_{KG} \hat{X}_{G} + \lambda_{KI} \hat{X}_{I} = \lambda_{KY} A_{1} \hat{t}$$

(19)

Equation (8) yields

$$\lambda_{GM} \hat{X}_{M} + \lambda_{GI} \hat{X}_{I} - \hat{X}_{G} = - \lambda_{GY} \hat{X}_{Y}$$

(20)

Using the values of (17), the above equation becomes

$$\lambda_{GM} \hat{X}_{M} + \lambda_{GI} \hat{X}_{I} - \hat{X}_{G} = \lambda_{GY} A_{1} \hat{t}$$

(21)

In matrix representation Eqs. (18), (19) and (21) can be shown as

$$\left[ {\begin{array}{*{20}c} {\lambda_{LM} } & {\lambda_{LI} } & {\lambda_{LG} } \\ {\lambda_{KM} } & {\lambda_{KI} } & {\lambda_{KG} } \\ {\lambda_{GM} } & {\lambda_{GI} } & { - 1} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\hat{X}_{M} } \\ {\hat{X}_{I} } \\ {\hat{X}_{G} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\lambda_{LY} A_{1} \hat{t}} \\ {\lambda_{KY} A_{1} \hat{t}} \\ {\lambda_{GY} A_{1} \hat{t}} \\ \end{array} } \right]$$

Using the Cramer’s rule we can solve

$$\hat{X}_{M} = \frac{1}{\left| \lambda \right|}A_{1} A_{2} \hat{t}$$

(22)

$$\hat{X}_{I} = \frac{1}{\left| \lambda \right|}A_{1} A_{3} \hat{t}$$

(23)

$$\hat{X}_{G} = \frac{1}{\left| \lambda \right|}\left[ {\lambda_{GY} \left( {\lambda_{LM} \lambda_{KI} - \lambda_{KM} \lambda_{LI} } \right) + \lambda_{GI} \left( {\lambda_{KM} \lambda_{LY} - \lambda_{KY} \lambda_{LM} } \right) + \lambda_{GM} \left( {\lambda_{KY} \lambda_{LI} - \lambda_{KI} \lambda_{LY} } \right)} \right]A_{1} \hat{t}$$

(24)

Therefore, the effect on G is ambiguous which depends on the relative changes of other sectors⁹

$$A_{2} = \left[ {\left( {\lambda_{LI} \lambda_{KY} - \lambda_{LY} \lambda_{KI} } \right) + \lambda_{KG} \left( {\lambda_{LI} \lambda_{GY} - \lambda_{LY} \lambda_{GI} } \right) + \lambda_{GY} \left( {\lambda_{LI} \lambda_{KG} - \lambda_{LG} \lambda_{KI} } \right)} \right] > 0$$

$$A_{3} = \left[ {\left( {\lambda_{KM} \lambda_{LY} - \lambda_{LM} \lambda_{KY} } \right) + \lambda_{GY} \left( {\lambda_{KM} \lambda_{LG} - \lambda_{LM} \lambda_{KG} } \right) + \lambda_{GM} \left( {\lambda_{LY} \lambda_{KG} - \lambda_{LG} \lambda_{KY} } \right)} \right] > 0$$

$$\left| \lambda \right| = \lambda_{GM} \left( {\lambda_{LI} \lambda_{KG} - \lambda_{LG} \lambda_{KI} } \right) - \lambda_{GI} \left( {\lambda_{LM} \lambda_{KG} - \lambda_{LG} \lambda_{KM} } \right) - \left( {\lambda_{LM} \lambda_{KI} - \lambda_{LI} \lambda_{KM} } \right)$$

If

• (i) I is more labor-intensive than G in comparison with capital, $\left( {\lambda_{LI} \lambda_{KG} - \lambda_{LG} \lambda_{KI} } \right)$ > 0;

• (ii) M is more capital-intensive than G in comparison with labor, $\left( {\lambda_{LM} \lambda_{KG} - \lambda_{LG} \lambda_{KM} } \right)$ < 0;

• (iii) I is more labor-intensive than M in comparison with capital. $\left( {\lambda_{LM} \lambda_{KI} - \lambda_{LI} \lambda_{KM} } \right)$ < 0.

Therefore, $\left| \lambda \right| > 0$.

Appendix 2: A decrease in bureaucratic (in)efficiency (bureaucratic reform) E

Equations (2) and (3) would be modified as

$$\hat{r}\theta_{KM} + \hat{P}_{G} \theta_{GM} = 0$$

(25)

$$\hat{r}\theta_{KG} - \hat{P}_{G} = \hat{E}$$

(26)

Using Cramer’s rule and solving the equations, we obtain

$$\hat{r} = \frac{{\theta_{GM} }}{{\left( {\theta_{KM} + \theta_{KG} \theta_{GM} } \right)}}\hat{E}$$

(27)

And

$$\hat{P}_{G} = - \frac{{\theta_{KM} }}{{\left( {\theta_{KM} + \theta_{KG} \theta_{GM} } \right)}}\hat{E}$$

(28)

Using Eq. (28) in Eq. (9)

$$\hat{R} = \frac{{\left( {\theta_{KM} \theta_{GY} - \theta_{KY} \theta_{GM} } \right)}}{{\theta_{TY} \left( {\theta_{KM} + \theta_{KG} \theta_{GM} } \right)}}\hat{E}$$

(29)

Using the values of (27) and (28) in Eq. (12), it becomes

$$\hat{W} = \frac{{\theta_{KM} \theta_{GI} - \theta_{KI} \theta_{GM} }}{{\theta_{LI} \left( {\theta_{KM} + \theta_{KG} \theta_{GM} } \right)}}\hat{E}$$

Let,

$$B_{1} = \left[ {\frac{{\theta_{KM} \theta_{GI} - \theta_{KI} \theta_{GM} }}{{\left( {\theta_{KM} + \theta_{KG} \theta_{GM} } \right)}}} \right]$$

Therefore,

$$\hat{W} = \frac{1}{{\theta_{LI} }}B_{1} \hat{E}$$

(30)

Differentiating Eq. (7) and using the elasticity of substitution

$$\hat{X}_{Y} = - \sigma_{Y} \theta_{LY} \left( {\hat{\bar{W}} - \hat{R}} \right)$$

Using the values of Eq. (29) in the above equation, it becomes

$$\hat{X}_{Y} = B_{2} \hat{E}$$

(31)

where $B_{2} = \left[ {\sigma_{Y} \theta_{LY} \frac{{\left( {\theta_{KM} \theta_{GY} - \theta_{KY} \theta_{GM} } \right)}}{{\theta_{TY} \left( {\theta_{KM} + \theta_{KG} \theta_{GM} } \right)}}} \right] > 0$

Differentiating Eqs. (5), (6) and (8) and arranging them in matrix form to yield the following set of equations

$$\hat{X}_{M} = \frac{1}{\left| \lambda \right|}B_{2} B_{3} \hat{E}$$

(32)

$$\hat{X}_{I} = \frac{1}{\left| \lambda \right|}B_{2} B_{4} \hat{E}$$

(33)

$$\hat{X}_{G} = \frac{1}{\left| \lambda \right|}\left[ {\lambda_{GY} \left( {\lambda_{KM} \lambda_{LI} - \lambda_{LM} \lambda_{KI} } \right) + \lambda_{GI} \left( {\lambda_{LM} \lambda_{KY} - \lambda_{LY} \lambda_{KM} } \right) + \lambda_{GM} \left( {\lambda_{KI} \lambda_{LY} - \lambda_{LI} \lambda_{KY} } \right)} \right]B_{2} \hat{E}$$

(34)

Again, the effect on G is also ambiguous¹⁰

$$B_{3} = \left[ {\left( {\lambda_{KI} \lambda_{LY} - \lambda_{LI} \lambda_{KY} } \right) + \lambda_{LG} \left( {\lambda_{KI} \lambda_{GY} - \lambda_{KY} \lambda_{GI} } \right) + \lambda_{GY} \left( {\lambda_{LG} \lambda_{KI} - \lambda_{KG} \lambda_{LI} } \right)} \right] < 0$$

$$B_{4} = \left[ {\left( {\lambda_{LM} \lambda_{KY} - \lambda_{KM} \lambda_{LY} } \right) + \lambda_{GY} \left( {\lambda_{LM} \lambda_{KG} - \lambda_{KM} \lambda_{LG} } \right) + \lambda_{GM} \left( {\lambda_{KY} \lambda_{LG} - \lambda_{LY} \lambda_{KG} } \right)} \right] < 0$$

$\left| \lambda \right| > 0$. The values are the same as before.

Appendix 3: A decrease in cost of corruption u

Differentiating Eq. (4)

$$\hat{W}\left( {\frac{{\theta_{LI} }}{{\theta_{GI} }}} \right) + \hat{r}\left( {\frac{{\theta_{KI} }}{{\theta_{GI} }}} \right) + \hat{P}_{G} = - \hat{u}$$

(35)

Solving Eqs. (9), (25) and (11), we can get

$$\hat{r} = \hat{R} = \hat{P}_{G} = 0$$

From Eq. (35)

$$\hat{W} = - \hat{u}\left( {\frac{{\theta_{u} }}{{\theta_{LI} }}} \right) < 0\,\,\,as\,\,\,\,\hat{u} < 0$$

(36)