Derivation and Discussion of Equilibria

1.1  Proposition 1: Autarky Equilibrium

Let's label one region's variables with a 1 at the end, and the other region's variables with a 2. To derive the autarky equilibrium for region 1, we can make the agricultural good the numeraire. 
However, the derivations as saved in the current notebook file have been calculated by not executing the following line.

Px1 = 1;

To keep this notebook file compatible with earlier versions of the paper, the following simultaneous equation system includes the possibility of capital accumulation under adjustment cost. The equilibrium as reported in proposition 1 of the most recent version of the paper results by setting Tobin's q to one (q1=1)  and the depreciation rate of capital to zero (so that δ=0). Then the equilibrium savings rate is zero (s1=0), the rate of change of Tobin's q is zero (gq1=0), and the capital stock is constant (so that gk1=0). 
So, to obtain the most recent reduced equilibrium we can set the variables below.

δ = 0 ;q1 = 
        1 ;

The remaining variables are defined just as in the most recent version of the paper. The equilbrium satisfies the following equation system.

FullSimplify[
      Solve[
{((θ*(1 - s1)*Y1)/(N1 *p1)) == (α/(1 - α))*A1*
              L0, 
(1 - θ)*(1 - s1)*
              (Y1/Px1) == 
            X1, 
w1 == ((γ * Px1 * X1)/((1 - λ1) * L1)), 
r1  == (1/q1) (((1 - γ) * Px1 * X1)/K1) + 
              gq1 + (1/(2  χ)) ((((q1 - 1))^2)/q1) - δ, 
p1  ==  
            (w1/(α * A1)), 
q1 == 
            1 + χ *(I1/K1), 
Y1 ==  
            w1 * L1  +  r1 * K1, 
p1*
              N1*(α/(1 - α))*A1*L0 ==  
            w1 *λ1*
              L1, 
I1 + (χ/2) ((I1^2)/K1) == 
            s1 * Y1, 
gk1 ==  ((I1 - δ K1)/K1), 
Px1*X1  +  Ω ==  
            w1 *(1 - λ1) * L1  +  
              r1 * K1}, 
{I1, s1, λ1, N1, p1, w1, 
          r1, Y1,  gk1, gq1,  Ω}]]

To clear the values set above, execute the following line.

Clear[Px1, δ, q1];

1.2  Proposition 2: Free Trade Equilibrium

After trade liberalization, the two regions face a new equilibrium. We could make the agricultural good the numeraire again, but now world-wide. 
So, to obtain the most recent reduced equilibrium we can execute the following line. However, the derivations as saved in the current notebook file have been calculated disregarding this line.

Px = 1;

The present version of the paper is based on a model of no capital accumulation, and the equation system below accounts for that. The equilibrium becomes

Solve[
{(p1/p2) ==  ((((A2/A1)))^(1 - α)), 
(p1*N1*A1  +  
                p2*N2*A2)*(α/(1 - α))*
            L0 ==  θ*(Y1 + 
                Y2), 
(1 - θ)*
              (Y1/Px) + (1 - θ)*(Y2/Px) == 
          X1 + X2, 

w1 == ((γ * Px * X1)/((1 - λ1) * L1)), 
r1  == (((1 - γ) * Px * X1)/K1), 
p1  ==  
          (w1/(α * A1)), 
Y1 ==  
          w1 * L1  +  r1 * K1, 
p1*
            N1*(α/(1 - α))*A1*L0 ==  
          w1*λ1*L1, 
Px*X1  ==  
          w1 *(1 - λ1) * L1  +  
            r1 * K1, 

w2 == ((γ * Px * X2)/((1 - λ2) * L2)), 
r2  == (((1 - γ) * Px * X2)/K2), 
p2  ==  
          (w2/(α*A2)), 
Y2 ==  
          w2*L2 +  r2* K2, 
p2*
            N2*(α/(1 - α))*A2*L0 ==  
          w2*λ2*L2, 
Px*X2  ==  
          w2*(1 - λ2)*L2  +  
            r2*K2}, 

{λ1, N1, 
        p1, w1, r1, Y1, λ2, N2, p2, w2, r2, Y2}]

To clear the numeraire value set above, execute the following line.

Clear[Px];

1.3  Corollary 2.3: Complete Specialization in Agriculture

One region may completely specialize in agriculture after trade liberalization. 
Again, we can make the agricultural good the numeraire but we need not. The specialization equilibrium below has been calculated ignoring the following line.

Px = 1;

By adjusting the first equation in the below system accordingly, the  complete specialization equilibrium becomes

Solve[
{((θ*(Y1 + Y2))/(N2 *p2)) == (α/(1 - α))*A2*
            L0, 
(1 - θ)*
              (Y1/Px) + (1 - θ)*(Y2/Px) == 
          X1 + X2, 

w1 == ((γ * Px * X1)/((1 - λ1) * L1)), 
r1  == (((1 - γ) * Px * X1)/K1), 
Y1 ==  
          w1 * L1  +  r1 * K1, 
Px*X1 ==  
          w1 *(1 - λ1) * L1  +  
            r1 * K1, 

w2 == ((γ * Px * X2)/((1 - λ2) * L2)), 
r2  == (((1 - γ) * Px * X2)/K2), 
p2  ==  
          (w2/(α * A2)), 
Y2 ==  
          w2 * L2 +  r2 * K2, 
p2*
            N2*(α/(1 - α))*A2*L0 ==  
          w2 *λ2*L2, 
Px*X2  ==  
          w2 *(1 - λ2) * L2  +  
            r2 * K2}, 

{λ1, 
        w1, r1, Y1, 
λ2, N2, p2, w2, r2, 
        Y2}]

To clear the numeraire value set above, execute the following line.

Clear[Px];

1.4  Table 2: Numerical Welfare Analysis and Simulations

To better understand the welfare implications of the model, we can simulate it for various levels of γ and θ. Implied utility levels for a liberalizing country that stops growing after to complete specialization in agriculture are reported in the following table

utilRatio[{γ_, θ_}] := 
      1 - 1/((1 - γ) (1 - θ) (((1 + (θ/(γ (1 - θ)))))^(θ + γ (1 - θ))) + (θ + γ (1 - θ)) (((1 + (θ/(γ (1 - θ)))))^(-(1 - γ) (1 - θ))))
MatrixForm[
      Table[utilRatio[{((n - 1)/5) +  .1, ((m - 1)/5) +  .1}], {n, 5}, {m, 
          5}]]

Given a hypothetical autarky growth rate of 5%, the lower bounds on ρ become

MatrixForm[
      Map[(#^(-1))* .05 &, 
        Table[utilRatio[{((n - 1)/5) +  .1, ((m - 1)/5) +  .1}], {n, 
            5}, {m, 5}]]]