Rural Poor Economies and Foreign Investors: An Opportunity or a Risk?

In the current age of commercial and financial openness, remote and poor local economies are becoming increasingly exposed to inflows of external capital. The new investors - enjoying lower credit constraints than local dwellers - might play a propulsive role in local development. At the same time, inflows of external capital can have negative impacts on local natural resource-dependent activities. We analyze a two-sector model where both sectors damage the environment, but only that of domestic producers relies on natural resources. We assess under which conditions the coexistence of the two sectors is compatible with sustainability, defined as convergence to a stationary state characterized by a positive stock of the natural resource. Moreover, we find that capital inflows can be stimulated by an increase in the pollution intensity of incoming activities, but also in the pollution intensity of the domestic sector; in both cases, capital inflows generate environmental degradation and a decrease in welfare for the local population. Finally, we show that a reduction in the cost of capital for external investors and the consequent capital inflows have the effect to increase wages, local investments and welfare of the local populations only if the environmental impact of the external sector is relatively low with respect to that of local activities. Otherwise, an unexpected scenario characterized by a reduction in domestic capital accumulation and the impoverishment of local agents can occur.


]
where λ is the co-state variable associated to K L . By applying the Maximum Principle, the dynamics of the economy are described by the equations: with the constraint: where C L and L satisfy the following conditions 1 : At the same time, the representative I-agent chooses the level of labor demand 1 − L and physical capital K I employed in the external production in order to maximize her profit function: This gives rise to the following first order conditions: The labor market is perfectly competitive and wages are flexible. I-and L-agents take w as given, but the wage rate and labor allocation between the two sectors continue to change until the labor demand is equal to labor supply. The labor market equilibrium condition is given by: By equation (S.8) we have: and substituting K I in (S.9) we obtain: where: α+β Function (S.11) identifies the labor market equilibrium value L * of L if the right side of (S.11) is lower than 1; otherwise, the equilibrium value of L is 1, that is: The economy is specialized in the production of the L-sector if L * = 1. Note that condition (S.5) excludes the specialization in the production of the external sector (i.e. L * > 0 always). Therefore two cases are distinguished, the case without specialization (in the local sector) and the case with specialization.
If Γ ( K α L E β ) 1 α+β < 1, then L-agents spend a positive fraction of their time endowment working in the external sector and condition (S.11) identifies the equilibrium value of L. The equilibrium wage rate is constant and is given by ; to prove such result, let us substitute (S.11) in (S.5) obtaining: In such a context, the dynamic system (S.1)-(S.3) can be written in the form (5).
α+β ≥ 1, then L-agents spend all their time endowment working in the L-sector, that is L * = 1, and the dynamic system (S.1)-(S.3) can be written in the form (6).
2) given ϵ, the function E 2 (ϵ) (see (18)) indicates the value of the parameter E such that the curves f (E) and g(E) have an intersection point along the horizontal line K L = K L (K L is defined in (9)) 2 ; proof: Remembering the values of Φ, Λ and K L , we obtain 3) given ϵ, the function E 3 (ϵ) (see (19)) indicates the value of the parameter E such that the curves f 1 (E) and g 1 (E) are tangent; proof: Let us rewrite the functions f 1 (E) and g 1 (E) as follows: . By solving of system: where f ′ 1 (E) and g ′ 1 (E) represent the derivatives with respect to E of f 1 (E) and g 1 (E), we get and where E T and E T indicate respectively, the values of E and E in the tangency point. proof: Substituting the equation (S.17) in (S.14) and solving for ϵ the following system: we obtain ϵ = ϵ T . From the system (S.19), we can see that ϵ is positively correlated to K L , then

5) the tangency point between the curves f (E) and g(E) lies below the horizontal line
only if the condition η < η T (see (21)) is satisfied ; proof: To prove this item we follow the steps: such that the curves f (E) and g(E) are tangent, see point 1) and c) By straightforward calculations, equation (S.21) can be written as: Substituting K L = K L in equation (S.22) we get η = η T , furthermore given the positive correlation between K L and η, we have that η < η T implies K L < K L ; (14)) if and only if η < η : proof: To prove the first part of the above item, note that being Now to prove that η > η T , we write this inequality as follows: then, substituting the values of ϵ T and η T (given by formulas (20) and (21) respectively) and noting that by (12), Ω = (K L Γ) α+β β holds, the inequality (S.23) becomes: , are two parallel straight lines; they coincide proof: Let us write equations (17) and (18) as follows: where it is easy to see that they have, in the plane (ϵ, E), the same slope Ωδ α .
To prove the second part of the above item, it is sufficient to show that the function: has the following properties: The propriety a) can be easily checked by noting that, by (21): To prove the propriety b), it is sufficient to check that the derivative of the function V (η) with respect to η, evaluated at η = η T , that is: is equal to zero, and that the second order derivative is strictly concave in ϵ, its graph lies below the straight line E 2 (ϵ) and is always tangent to it for ϵ = ϵ T .

Proof of Proposition 4:
This proof and the following one are built on [1]. The Jacobian matrix J 1 (P * ), evaluated at a stationary state with specialization P * = (E * , K * , λ * ), can be expressed as follows: The eingenvalues of J 1 (P * ) are the roots of the following characteristic polynomial: It easy to check that the determinant |J 1 | can be expressed as follows: where f ′ 1 and g ′ 1 are the derivatives of f 1 and g 1 evaluated at E * . Therefore, Since, at the stationary state A 1 , the condition f ′ 1 (E * ) < g ′ 1 (E * ) holds, A 1 is either a saddle with two eingenvalues with strictly positive real parts or a sink; however, A 1 cannot be a sink in that, by (S.29), At the stationary state B 1 , the condition f ′ 1 (E * ) > g ′ 1 (E * ) holds; therefore B 1 is either a saddle with two eingenvalues with strictly negative real parts or a source. [1] finds that M < 0, i.e. E * > 1 2 , is a sufficient condition for the saddle-point stability of B 1 . This completes the proof.

Proof of Proposition 5:
The Jacobian matrix J(P * ), evaluated at a stationary state without specialization P * = (E * , K * , λ * ), can be expressed as follows: The eingenvalues of J(P * ) are the roots of the following characteristic polynomial: where: It is easy to check that the determinant |J| can be expressed as follows: where f ′ and g ′ are the derivatives of f and g evaluated at E * . Therefore, At the stationary state B, |J| > 0 holds; therefore B is either a saddle with two eingenvalues with strictly negative real parts or a source. [1] finds that a positive determinant and a negative coefficient M are sufficient conditions for saddle-point stability. Notice that, if Λ > 0, the condition E − 2E * < 0 holds (see formula (11) of the paper and the definition of the stationary state of type B) and consequently M < 0. In case Λ < 0, from formula (7) of the paper and the equationK L = 0, we obtain: Substituting in (S.31) and remembering that K ⋆ L < K L = αδ −1 Γ −α−β , we can write: (S.33) Therefore, a sufficient condition for saddle-point stability is: This completes the proof.

Proofs of comparative statics
Below the straight line K L = K L , the stationary states (without specialization) are given by the intersections between the two following curves: where: Notice that Ω > 0 always holds while: Let us rewrite equations (S.36) and (S.37) as follows: Differentiating equations (S.41) and (S.42) with respect to the parameter y = E, ϵ, η, r we obtain: a) Proof of Proposition 8: Posing y = E, the system (S.43) becomes: > 0 (remember that Therefore:  (9)

∂Θ ∂r
where: The solution of such system is: 1−γ holds (see (S.41) and formula (9) of the paper) and solving the inequality: we obtain the sufficient condition for ∂E * ∂r > 0 given in the proposition.
Remember that, at the stationary state B, sign (Λ) = sign [g ′ (E * ) − f ′ (E * )] holds. Let us now consider the variations in K L , L and K I generated by an increase in r. Remember that, according to Proposition 6, K L is positively correlated with L and negatively correlated with K I .