Insecure Property Rights and Growth: The Roles of Appropriation Costs, Wealth Effects, and Heterogeneity

yields q = (ρ — r)¹ /^{(1 ε}) and, hence, κ = u'(q) = [1 + (ρ — r) ^ε/^{(1 ε})](^{1 ε})/^ε.
σ = 1/(1 - ε) shows that

K = [1 + ( ρ — r )^{1 -σ} ]¹ /^{( σ-1}).

From (9), we have

dβ =     (ρ ^- r)γ^κ   < 0

dK     ( n — 1)(k — κ )2

and from (18) follows

dκ _ ln [1 + (ρ — r)^{1 σ}]      (ρ — r)^{1 σ} ln(ρ — r)

d σ ^K       ( σ — 1)2      ^*^( σ — 1)[1 + ( ρ — r )^{1 -σ} ]

The first term in the square brackets is always positive but the sign of the second term depends
on whether σ is greater or smaller than 1 and on whether ρ — r is greater or smaller than 1.
If σ>1 (which is equivalent to ε>0) and ρ — r ≥ 1, then it follows that the second term in
brackets is non-negative such that dκ/dσ < 0. This, in turn, implies together with (19) that
d//dσ > 0. We conclude that, under the stated assumptions, a higher elasticity of substitution
will lead to higher intensity of extraction.

4 Heterogeneous players

So far we have looked at the case of homogeneous players. The present section, on the other
hand, discusses the effects of differences between players. To simplify the analysis, we restrict
ourselves to case of only two types of players. More specifically, let us assume that there are
n1 ≥ 1 players described by the parameters (ρ1,γ1,κ1,r1) and the utility function U1 and n2 ≥ 1
players described by the parameters (ρ2 ,γ2 ,κ2,r2) and the utility function U2. The total number
of players is n = n1 + n2 . We assume that assumptions P and U hold for both types. For each
£ ∈ {1, 2}, we will denote by u_t and w_t the marginal utility of consumption u_t (q) = U_t(q, 1) and
the marginal utility of wealth w_t (q) = u_t (q) — u'_t (q) q.

In analogy to section 3, we define q_t as the unique positive number satisfying

( ρt — rt ) ut ( Kt ) = wt ( (it ),

and we set K_t = u'_t (q_t). Furthermore, we define

(     ( Pt — rt ) YtK t

At = —;---:---

к t — Kt

and B_t = R — ρ_t + A_t. Finally, we define

β^K1 =[B1 — (B1 — B2)n2]/(n — 1),

β^K₂ =[B₂ — (B₂ — B₁)n₁]/(n — 1).

12

<<   10   11   12   13   14   15   16   >>

More intriguing information