Economic analysis of e-waste market

Abstract

Despite international regulations that prohibit the trans-boundary movement of electronic and electric waste (e-waste), non-reusable e-waste is often illegally mixed with reusable e-waste and results in being sent to developing countries. As developing countries are not well prepared to properly manage e-waste, this illegal trade has important negative externalities and creates ‘environmental injustice’. The two main information problems on the e-waste market are imperfect monitoring and imperfect information on the so-called ‘degree of purity’ of the e-waste. In this paper, we use a simple bilateral North–South trade model and show that there exists an alternative e-waste market that is better than the standard e-waste market for developing countries. This alternative e-waste market is a joint trade in reusable and non-reusable e-waste. In both cases, we consider demand and supply sides, plus the equilibrium of the e-waste market to show that the alternative market that we propose is better for developing countries.

Appendix

Proof of proposition 6

The proof is divided into three cases. The first two cases are related to the alternative e-waste market, while the last case concerns the standard e-waste market.

Case 1 The reusable e-waste in the alternative e-waste market.

We use the supply X ₁ = S ₁(P ₁) and the demand X ₁ = D ₁(P ₁, P) of the reusable e-waste to get the following system.

$$\left\{ \begin{aligned} X_{1} - S_{1} (P_{1} ) = 0 \hfill \\ X_{1} - D_{1} (P_{1} ,P) = 0 \hfill \\ \end{aligned} \right.$$

• The effect of the disposal costs and the degree of monitoring.

$\frac{{{{d}}X_{1} }}{{{{d}}d_{N} }} = \frac{{{{d}}X_{1} }}{{{{d}}d_{S} }} = \frac{{{{d}}X_{1} }}{{{{d}}d_{N} - {{d}}d_{S} }} = \frac{{{{d}}X_{1} }}{{{{d}}\sigma }} = 0$, as X ₁ is not a function of d _N, d _S and σ. Likewise, $\frac{{dP_{1} }}{{dd_{N} }} = \frac{{dP_{1} }}{{dd_{S} }} = \frac{{dP_{1} }}{{dd_{N} - dd_{S} }} = \frac{{dP_{1} }}{d\sigma } = 0$, as P ₁ is not a function of d _N, d _S and σ.

• The effect of the resale price: (dX ₁/dP and dP ₁/dP).

By taking the derivative of the system with respect to P, we get:

$$\left\{ \begin{aligned} \frac{{dX_{1} }}{dP} - \frac{{\partial S_{1} }}{{\partial P_{1} }}*\frac{{dP_{1} }}{dP} = 0 \hfill \\ \frac{{dX_{1} }}{dP} - \frac{{\partial D_{1} }}{{\partial P_{1} }}*\frac{{dP_{1} }}{dP} - \frac{{dD_{1} }}{dP} = 0 \hfill \\ \end{aligned} \right.$$

and in matrix term:

$$\left( {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{1} }}{{\partial P_{1} }}} \\ 1 & { - \frac{{\partial D_{1} }}{{\partial P_{1} }}} \\ \end{array} } \right)*\left( \begin{aligned} \frac{{dX_{1} }}{dP} \hfill \\ \frac{{dP_{1} }}{dP} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} 0 \hfill \\ \frac{{dD_{1} }}{dP} \hfill \\ \end{aligned} \right)$$

We can then calculate the Jacobian determinant as:

$$\left| J \right| = \left| {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{1} }}{{\partial P_{1} }}} \\ 1 & { - \frac{{\partial D_{1} }}{{\partial P_{1} }}} \\ \end{array} } \right| = - \frac{{\partial D_{1} }}{{\partial P_{1} }} + \frac{{\partial S_{1} }}{{\partial P_{1} }} > 0$$

From the Jacobian determinant, we can deduce:

$$\frac{{dX_{1} }}{dP} = \frac{{\left| {\begin{array}{*{20}c} 0 & { - \frac{{\partial S_{1} }}{{\partial P_{1} }}} \\ {\frac{{dD_{1} }}{dP}} & { - \frac{{\partial D_{1} }}{{\partial P_{1} }}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{\frac{{\partial S_{1} }}{{\partial P_{1} }}*\frac{{dD_{1} }}{dP}}}{\left| J \right|} > 0{\text{ with }}\frac{{dD_{1} }}{dP} > 0{\text{ and }}\frac{{\partial S_{1} }}{{\partial P_{1} }} > 0$$

and

$$\frac{{dP_{1} }}{dP} = \frac{{\left| {\begin{array}{*{20}c} 1 & 0 \\ 1 & {\frac{{\partial D_{1} }}{\partial P}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{\frac{{dD_{1} }}{dP}}}{\left| J \right|} > 0 \, .$$

Case 2 The non-reusable e-waste in the alternative e-waste market.

We also use the supply X ₀ = S ₀(P₀) and the demand X ₀ = D ₀(P ₀,P) of the non-reusable e-waste to get the following system.

$$\left\{ \begin{aligned} X_{0} - S_{0} (P_{0} ,d_{N} ) = 0 \hfill \\ X_{0} - D_{0} (P_{0} ,d_{S} ) = 0 \hfill \\ \end{aligned} \right..$$

• The effect of the degree of monitoring and the resale price.

As neither the quantity of the non-reusable e-waste nor its price is a function of P and σ, we get no effect as follows.

$$\frac{{dX_{0} }}{dP} = \frac{{dX_{0} }}{d\sigma } = 0 \, \;{\text{and}}\;\frac{{dP_{0} }}{dP} = \frac{{dP_{0} }}{d\sigma } = 0.$$

• The effect of the disposal cost d_S in the South.

Taking the derivative of the system with respect to d_S, we get:

$$\left\{ \begin{aligned} \frac{{dX_{0} }}{{dd_{S} }} - \frac{{\partial S_{0} }}{{\partial P_{0} }}*\frac{{dP_{0} }}{{dd_{S} }} = 0 \hfill \\ \frac{{dX_{0} }}{{dd_{S} }} - \frac{{\partial D_{0} }}{{\partial P_{0} }}*\frac{{dP_{0} }}{{dd_{S} }} - \frac{{dD_{0} }}{{dd_{S} }} = 0 \hfill \\ \end{aligned} \right.$$

In matrix form we have:

$$\left( {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{0} }}{{\partial P_{0} }}} \\ 1 & { - \frac{{\partial D_{0} }}{{\partial P_{0} }}} \\ \end{array} } \right)*\left( \begin{aligned} \frac{{dX_{0} }}{{dd_{S} }} \hfill \\ \frac{{dP_{0} }}{{dd_{S} }} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} 0 \hfill \\ \frac{{dD_{0} }}{{dd_{S} }} \hfill \\ \end{aligned} \right)$$

$${\text{and}}\;\left| J \right| = \left| {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{0} }}{{\partial P_{0} }}} \\ 1 & { - \frac{{\partial D_{0} }}{{\partial P_{0} }}} \\ \end{array} } \right| = - \frac{{\partial D_{0} }}{{\partial P_{0} }} + \frac{{\partial S_{0} }}{{\partial P_{0} }} > 0$$

Then we deduce:

$$\frac{{dX_{0} }}{{dd_{S} }} = \frac{{\left| {\begin{array}{*{20}c} 0 & { - \frac{{\partial S_{0} }}{{\partial P_{0} }}} \\ {\frac{{dD_{0} }}{{dd_{S} }}} & { - \frac{{\partial D_{0} }}{{\partial P_{0} }}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{\frac{{\partial S_{0} }}{{\partial P_{0} }}*\frac{{dD_{0} }}{{dd_{S} }}}}{\left| J \right|} < 0{\text{ with }}\frac{{dD_{0} }}{{dd_{S} }} < 0{\text{ and }}\frac{{\partial S_{0} }}{{\partial P_{0} }} > 0$$

and

$$\frac{{dP_{0} }}{{dd_{S} }} = \frac{{\left| {\begin{array}{*{20}c} 1 & 0 \\ 1 & {\frac{{dD_{0} }}{{dd_{S} }}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{\frac{{dD_{0} }}{{dd_{S} }}}}{\left| J \right|} < 0 \,$$

• The effect of the disposal cost d_N in the North.

The derivative of the system with respect to d_N gives:

$$\left\{ \begin{aligned} \frac{{dX_{0} }}{{dd_{N} }} - \frac{{\partial S_{0} }}{{\partial P_{0} }}*\frac{{dP_{0} }}{{dd_{N} }} - \frac{{dS_{0} }}{{dd_{N} }} = 0 \hfill \\ \frac{{dX_{0} }}{{dd_{N} }} - \frac{{\partial D_{0} }}{{\partial P_{0} }}*\frac{{dP_{0} }}{{dd_{N} }} = 0 \hfill \\ \end{aligned} \right.$$

In matrix form we have:

$$\left( {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{0} }}{{\partial P_{0} }}} \\ 1 & { - \frac{{\partial D_{0} }}{{\partial P_{0} }}} \\ \end{array} } \right)*\left( \begin{aligned} \frac{{dX_{0} }}{{dd_{N} }} \hfill \\ \frac{{dP_{0} }}{{dd_{N} }} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} \frac{{dS_{0} }}{{dd_{N} }} \hfill \\ 0 \hfill \\ \end{aligned} \right)\;{\text{and}}\;\left| J \right| = \left| {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{0} }}{{\partial P_{0} }}} \\ 1 & { - \frac{{\partial D_{0} }}{{\partial P_{0} }}} \\ \end{array} } \right| = - \frac{{\partial D_{0} }}{{\partial P_{0} }} + \frac{{\partial S_{0} }}{{\partial P_{0} }} > 0$$

We deduce that:

$$\frac{{dX_{0} }}{{dd_{N} }} = \frac{{\left| {\begin{array}{*{20}c} {\frac{{dS_{0} }}{{dd_{N} }}} & { - \frac{{\partial S_{0} }}{{\partial P_{0} }}} \\ 0 & { - \frac{{\partial D_{0} }}{{\partial P_{0} }}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{ - \frac{{dS_{0} }}{{dd_{N} }}*\frac{{\partial D_{0} }}{{\partial P_{0} }}}}{\left| J \right|} > 0{\text{ with }}\frac{{dS_{0} }}{{dd_{N} }} > 0{\text{ and }}\frac{{\partial D_{0} }}{{\partial P_{0} }} < 0$$

and

$$\frac{{dP_{0} }}{{dd_{N} }} = \frac{{\left| {\begin{array}{*{20}c} 1 & {\frac{{dS_{0} }}{{dd_{N} }}} \\ 1 & 0 \\ \end{array} } \right|}}{\left| J \right|} = \frac{{ - \frac{{dS_{0} }}{{dd_{N} }}}}{\left| J \right|} < 0 \,$$

• The effect of the difference in the disposal costs between the North and the South (d _N-d _S = b).

The effect on the quantity of the non-reusable e-waste is calculated as follows.

$\frac{{dX_{0} }}{db} = \frac{{dX_{0} }}{{dd_{S} }}*\frac{{\partial d_{S} }}{\partial b} + \frac{{dX_{0} }}{{dd_{N} }}*\frac{{\partial d_{N} }}{\partial b}$ ${ = - }\frac{{dX_{0} }}{{dd_{S} }} + \frac{{dX_{0} }}{{dd_{N} }}$

$${ = - }\frac{{\frac{{\partial S_{0} }}{{\partial P_{0} }}*\frac{{dD_{0} }}{{dd_{S} }}}}{\left| J \right|} - \frac{{\frac{{dS_{0} }}{{dd_{N} }}*\frac{{\partial D_{0} }}{{\partial P_{0} }}}}{\left| J \right|} > 0$$

We also determine the effect on the price of the non-reusable e-waste.

$\begin{aligned} \frac{{dP_{0} }}{db} = \frac{{dP_{0} }}{{dd_{S} }}*\frac{{\partial d_{S} }}{\partial b} + \frac{{dP_{0} }}{{dd_{N} }}*\frac{{\partial d_{N} }}{\partial b} \hfill \\ \, \hfill \\ \end{aligned}$ $\begin{aligned} { = - }\frac{{dP_{0} }}{{dd_{S} }} + \frac{{dP_{0} }}{{dd_{N} }}\left\{ \begin{aligned} > 0{\text{ if }}\frac{{dP_{0} }}{{dd_{N} }} > \frac{{dP_{0} }}{{dd_{S} }} \hfill \\ < 0{\text{ otherwise}} \hfill \\ \end{aligned} \right. \hfill \\ \, \hfill \\ \end{aligned}$

Case 3 The standard e-waste market.

In the case of a standard e-waste market, we consider the supply X = S_e(P_e,dN,σ) and the demand X = D_e(P_e,P, d_S,σ) of e-waste that leads to the following system.

$$\left\{ \begin{aligned} X_{e} - S_{e} (P_{e} ,d_{N} ,\sigma ) = 0 \hfill \\ X_{e} - D_{e} (P_{e} ,P,\sigma ,d_{S} ) = 0 \hfill \\ \end{aligned} \right.$$

• The effect of the resale price P

$$\left\{ \begin{aligned} \frac{{dX_{e} }}{dP} - \frac{{\partial S_{e} }}{{\partial P_{e} }}*\frac{{{{d}}P_{e} }}{{{{d}}P}} = 0 \hfill \\ \frac{{dX_{e} }}{dP} - \frac{{\partial D_{e} }}{{\partial P_{e} }}*\frac{{{{d}}P_{e} }}{{{{d}}P}} - \frac{{dD_{e} }}{dP} = 0 \hfill \\ \end{aligned} \right.$$

In matrix form:

$$\left( {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ 1 & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right)*\left( \begin{aligned} \frac{{dX_{e} }}{dP} \hfill \\ \frac{{dP_{e} }}{dP} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} 0 \hfill \\ \frac{{dD_{e} }}{dP} \hfill \\ \end{aligned} \right)\;{\text{and}}\;\left| J \right| = \left| {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ 1 & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right| = - \frac{{\partial D_{e} }}{{\partial P_{e} }} + \frac{{\partial S_{e} }}{{\partial P_{e} }} > 0$$

$${\text{We}}\;{\text{deduce}}:\;\frac{{dX_{e} }}{dP} = \frac{{\left| {\begin{array}{*{20}c} 0 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ {\frac{{dD_{e} }}{dP}} & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{\frac{{\partial S_{e} }}{{\partial P_{e} }}*\frac{{dD_{e} }}{dP}}}{\left| J \right|} > 0{\text{ with }}\frac{{dD_{e} }}{dP} > 0{\text{ and }}\frac{{\partial S_{e} }}{{\partial P_{e} }} > 0$$

and

$$\frac{{dP_{e} }}{dP} = \frac{{\left| {\begin{array}{*{20}c} 1 & 0 \\ 1 & {\frac{{\partial D_{e} }}{\partial P}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{\frac{{dD_{e} }}{dP}}}{\left| J \right|} > 0 \, .$$

• The effect of the disposal cost d_N in the North.

$$\left\{ \begin{aligned} \frac{{dX_{e} }}{{dd_{N} }} - \frac{{\partial S_{e} }}{{\partial P_{e} }}*\frac{{dP_{e} }}{{dd_{N} }} - \frac{{dS_{e} }}{{dd_{N} }} = 0 \hfill \\ \frac{{dX_{e} }}{{dd_{N} }} - \frac{{\partial D_{e} }}{{\partial P_{e} }}*\frac{{dP_{e} }}{{dd_{N} }} = 0 \hfill \\ \end{aligned} \right.$$

In matrix form:

$$\left( {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ 1 & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right)*\left( \begin{aligned} \frac{{dX_{e} }}{{dd_{N} }} \hfill \\ \frac{{dP_{e} }}{{dd_{N} }} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} \frac{{dS_{e} }}{{dd_{N} }} \hfill \\ 0 \hfill \\ \end{aligned} \right)\;{\text{and}}\;\left| J \right| = \left| {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ 1 & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right| = - \frac{{\partial D_{e} }}{{\partial P_{e} }} + \frac{{\partial S_{e} }}{{\partial P_{e} }} > 0$$

We determine:

$$\frac{{dX_{e} }}{{dd_{N} }} = \frac{{\left| {\begin{array}{*{20}c} {\frac{{dS_{e} }}{{dd_{N} }}} & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ 0 & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{ - \frac{{dS_{e} }}{{dd_{N} }}*\frac{{\partial D_{e} }}{{\partial P_{e} }}}}{\left| J \right|} > 0{\text{ with }}\frac{{dS_{e} }}{{dd_{N} }} > 0{\text{ and }}\frac{{\partial D_{e} }}{{\partial P_{e} }} < 0$$

and

$$\frac{{dP_{e} }}{{dd_{N} }} = \frac{{\left| {\begin{array}{*{20}c} 1 & {\frac{{dS_{e} }}{{dd_{N} }}} \\ 1 & 0 \\ \end{array} } \right|}}{\left| J \right|} = \frac{{ - \frac{{dS_{e} }}{{dd_{N} }}}}{\left| J \right|} < 0 \,$$

• The effect of the disposal cost d_S in the South.

$$\left\{ \begin{aligned} \frac{{dX_{e} }}{{dd_{S} }} - \frac{{\partial S_{e} }}{{\partial P_{e} }}*\frac{{dP_{e} }}{{dd_{S} }} = 0 \hfill \\ \frac{{dX_{e} }}{{dd_{S} }} - \frac{{\partial D_{e} }}{{\partial P_{e} }}*\frac{{dP_{e} }}{{dd_{S} }} - \frac{{dD_{e} }}{{dd_{S} }} = 0 \hfill \\ \end{aligned} \right.$$

In the matrix form:

$$\left( {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ 1 & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right)*\left( \begin{aligned} \frac{{dX_{e} }}{{dd_{S} }} \hfill \\ \frac{{dP_{e} }}{{dd_{S} }} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} 0 \hfill \\ \frac{{dD_{e} }}{{dd_{S} }} \hfill \\ \end{aligned} \right)$$

and the Jacobian determinant is:

$$\left| J \right| = \left| {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ 1 & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right| = - \frac{{\partial D_{e} }}{{\partial P_{e} }} + \frac{{\partial S_{e} }}{{\partial P_{e} }} > 0$$

We deduce that:

$$\frac{{dX_{e} }}{{dd_{S} }} = \frac{{\left| {\begin{array}{*{20}c} 0 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ {\frac{{dD_{e} }}{{dd_{S} }}} & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{\frac{{\partial S_{e} }}{{\partial P_{e} }}*\frac{{dD_{e} }}{{dd_{S} }}}}{\left| J \right|} < 0{\text{ with }}\frac{{dD_{e} }}{{dd_{S} }} < 0{\text{ and }}\frac{{\partial S_{e} }}{{\partial P_{e} }} > 0$$

and

$$\frac{{dP_{e} }}{{dd_{S} }} = \frac{{\left| {\begin{array}{*{20}c} 1 & 0 \\ 1 & {\frac{{dD_{e} }}{{dd_{S} }}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{\frac{{dD_{e} }}{{dd_{S} }}}}{\left| J \right|} < 0 \,$$

• The effect of the difference in the disposal costs between the North and the South (d_N-d_S = b)

We determine the effect on the quantity of e-waste as:

$$\frac{{dX_{e} }}{db} = \frac{{dX_{e} }}{{dd_{S} }}*\frac{{\partial d_{S} }}{\partial b} + \frac{{dX_{e} }}{{dd_{N} }}*\frac{{\partial d_{N} }}{\partial b}{ = } - \frac{{dX_{e} }}{{dd_{S} }} + \frac{{dX_{e} }}{{dd_{N} }}{ = } - \frac{{\frac{{\partial S_{e} }}{{\partial P_{e} }}*\frac{{dD_{e} }}{{dd_{S} }}}}{\left| J \right|} - \frac{{\frac{{dS_{e} }}{{dd_{N} }}*\frac{{\partial D_{e} }}{{\partial P_{e} }}}}{\left| J \right|} > 0$$

The effect on the price of e-waste is:

$$\frac{{dP_{e} }}{db} = \frac{{dP_{e} }}{{dd_{S} }}*\frac{{\partial d_{S} }}{\partial b} + \frac{{dP_{e} }}{{dd_{N} }}*\frac{{\partial d_{N} }}{\partial b}{ = - }\frac{{dP_{e} }}{{dd_{S} }} + \frac{{dP_{e} }}{{dd_{N} }}\,\left\{ \begin{aligned} > \; 0 {\text{ if }}\frac{{dP_{e} }}{{dd_{N} }} > \frac{{dP_{e} }}{{dd_{S} }} \hfill \\ < 0\;{\text{otherwise}} \hfill \\ \end{aligned} \right. \,$$

• The effect of the degree of the monitoring σ

The derivative of the system gives:

$$\left\{ \begin{aligned} \frac{{dX_{e} }}{d\sigma } - \frac{{\partial S_{e} }}{{\partial P_{e} }}*\frac{{dP_{e} }}{d\sigma } - \frac{{dS_{e} }}{d\sigma } = 0 \hfill \\ \frac{{dX_{e} }}{d\sigma } - \frac{{\partial D_{e} }}{{\partial P_{e} }}*\frac{{dP_{e} }}{d\sigma } - \frac{{dD_{e} }}{d\sigma } = 0 \hfill \\ \end{aligned} \right.$$

In matrix form:

$$\left( {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ 1 & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right)*\left( \begin{aligned} \frac{{dX_{e} }}{d\sigma } \hfill \\ \frac{{dP_{e} }}{d\sigma } \hfill \\ \end{aligned} \right) = \left( \begin{aligned} \frac{{dS_{e} }}{d\sigma } \hfill \\ \frac{{dD_{e} }}{d\sigma } \hfill \\ \end{aligned} \right)\;\left| J \right| = \left| {\begin{array}{*{20}c} 1 & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ 1 & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right| = - \frac{{\partial D_{e} }}{{\partial P_{e} }} + \frac{{\partial S_{e} }}{{\partial P_{e} }} > 0$$

We deduce that:

$$\frac{{dX_{e} }}{d\sigma } = \frac{{\left| {\begin{array}{*{20}c} {\frac{{dS_{e} }}{d\sigma }} & { - \frac{{\partial S_{e} }}{{\partial P_{e} }}} \\ {\frac{{dD_{e} }}{d\sigma }} & { - \frac{{\partial D_{e} }}{{\partial P_{e} }}} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{ - \frac{{dS_{e} }}{d\sigma }*\frac{{\partial D_{e} }}{{\partial P_{e} }} + \frac{{\partial S_{e} }}{{\partial P_{e} }}*\frac{{dD_{e} }}{d\sigma }}}{\left| J \right|} \,$$

$${\text{with }}\frac{{dD_{e} }}{d\sigma } > 0{ , }\frac{{\partial S_{e} }}{{\partial P_{e} }} > 0,\;\frac{{\partial D_{e} }}{{\partial P_{e} }} < 0{\text{ and }}\frac{{dS_{e} }}{d\sigma } < 0{ (}\frac{dD}{d\sigma } < 0;\frac{dQ}{d\sigma } ) { ;}\,$$

$$\frac{{dX_{e} }}{d\sigma }\left\{ \begin{aligned} < 0 \hfill \\ > 0 \hfill \\ \end{aligned} \right.$$

and

$$\frac{{dP_{e} }}{d\sigma } = \frac{{\left| {\begin{array}{*{20}c} 1 & {\frac{{dS_{e} }}{d\sigma }} \\ 1 & {\frac{{dD_{e} }}{d\sigma }} \\ \end{array} } \right|}}{\left| J \right|} = \frac{{\frac{{dD_{e} }}{d\sigma } - \frac{{dS_{e} }}{d\sigma }}}{\left| J \right|} > 0 \, .$$