What is phi as function of theta provide an analytical

Question

"Endogenous Ranking and Equilibrium Lorenz Curve Across Ex-ante Identical Countries", by Matsuyama, K. (ECMA 2013). Please try to provide a detailed explanation of each question you attempt to answer.

Consider a world economy consisting of J ≥ 2 exante identical countries trading a continuum of tradable goods indexed by s ∈ [0, 1].

Tradable goods are produced by using a single non-tradable, primary factor and a continuum of non-tradable local producer services.

Local producer services are produced by using a primary factor. The author shows that the S₀ = 0, S₁, S₂........S_J = 1 where Sj is the cumulative share of j poorest countries in the world, solve the second-order difference equation

S_j+1 - S_j/S_j - S_j-1 = (Γ(S_j, S_j+1)/Γ(S_j-1, S_j))^θγ(s) where Γ(S_j-1, S_j) = _Sj-1∫^S_j γ(s)ds/S_j-S_j-1

After taking a limit as J → ∞, the author establishes that the limit equilibrium Lorenz curve, Φ solves

Question 1: Suppose

γ(s) = 

What is Φ (as function of θ)? Provide an analytical solution of Φ when θ = 3?

Question 2: Gini index of income inequality is measured as twice the area between the diagonal and the Lorenz curve. What is Gini index of income inequality (as function of θ) if γ(s) is given by (3)? What happens to Gini index of income inequality as θ increases?

What is Gini index of income inequality when θ = 3.