Income gap in Ramsey model Ramsey model with income tax 

Question:

For this question you are required to setup a Ransey-Cass-Koopmans model economy with a Cobb-Douglas constant returns to scale production technology, with inputs of capital (K) and labour (L). Setup the model with a constant depreciation rate, constant rate of labour- augmenting technological progress and constant rate of population growth. Also incorporate a government that finances its spending each period via a proportional income tax system. Assume that both labour and capital income are taxed at the same rate – say τ. You can assume that government spending is completely wasteful; or you can make some other assumption about how the tax revenue is used, however do not allow government spending to generate endogenous growth.

I am asking only this past


  
    Use the model to explain the income gap that exists between wealthy and poor countries.
  


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What I did:

I solved this model with hamiltonian. And I got the following results (I posted picture of my results, because this is only extra information that helps to solve question.)



And the phase diagram



And at steady state, saving rate is

$$s^*={(delta +n + x)(1-τ)a\over p+delta+θx}$$

x is growth rath of technology. (Productivity growth rate) 

Delta depration rate. and production function $Y=K^a(AL)^{1-a}$

My interpretation: 

Economic growth is measured only by x. So only x and A affect the income. 

Assume that we have $k_i&lt;k^*$ and every parameters are constant for two countries except for x . Poor country has lower x. 

When x increases, saving decreases, MPK decreases, so capital rate of return decreases, capital stock increases. And eventually they have the same growth rate. (This is substitution effect) 

But I could not know how to explain this question better. Any help is appreciated. Really I need a help too much!
I think your solution is on the right direction. But allow me to make a few observations. 


The question talked about income gap and wanted you to use a Ramsey model, with exogenous technology growth, and no endogenous growth mechanism through public spending. Given these, I would think that the focus should not entirely be on the growth rate of technology. Using your example, if the two countries have the exact same set of parameters, then the country with higher x will have lower long run $k^*$. 
See the starting point of my (1) and your analysis: we assumed that the countries have identical parameters. For Rich(R) and Poor(P) countries, this is not a reasonable assumption. Furthermore, I am not sure if assuming that technological progress in R is faster than P is correct.


Might I suggest the following. Pick a parameter (say the tax rate). Make the tax rate higher in one of the countries. Which automatically divides the countries into R and P. Then even if you do not have different x, the income gap will remain. 

I assumed the following parameter values (L=Low, H=High):

$\delta = 0.1; n = 0.01; \rho = 0.05; \alpha = 0.3; \theta = 1/0.3$

$t_H = 0.2; t_L = 0.1; x_H = 0.5; x_L = 0.2;$  

My Utility function is $\frac{C^{1-\theta}}{1-\theta}$

The corresponding long run $k$ values are:

(i) $ k^*_{t_H x_H} = 0.583336 $

(ii) $k^*_{t_H x_L} = 0.943351  $

(iii) $k^*_{t_L x_H} = 0.69023 $

(iv) $k^*_{t_L x_L} = 1.11622 $

Now you can pick whichever pair among (i)-(iv) as your R and P. Suppose you say (ii) is P and (iv) is R. Then say both countries have the exact tech improvement and x rises from $x_L $ to $x_H$. (ii) moves to 0.58 and (iv) moves to 0.69. The gap (R-P) before the shock was 0.172869. After the shock income gap (R-P) is 0.106894. The gap went down. 

My code (Mathematica):  

kdot = (1 - t)*k^a - c - (n + d + x)*k;

Ham = c^(1 - \[Theta])/(
   1 - \[Theta]) + \[Lambda]*((1 - t)*k^a - c - (n + d + x)*k);

(*Costate1=\[Lambda]dot/\[Lambda]-(\[Rho]-n*(1-\[Theta])-x*(1-\[Theta]\
))==-D[Ham,k];*)

FOCc1 = D[Ham, c] == 0;

FOCc2 = FOCc1[[1]] + \[Lambda] == \[Lambda];

FOCc3 = FOCc2 /. {c -&gt; c[t], \[Lambda] -&gt; \[Lambda][t]};

FOCc4 = D[FOCc3, t][[1]]/c[t]^-\[Theta] == 
   D[FOCc3, t][[2]]/c[t]^-\[Theta];

FOCc5 = FOCc4 //. {c[t]^\[Theta] -&gt; \[Lambda][t]^-1} /. {\[Lambda]'[
      t]/\[Lambda][t] -&gt; \[Lambda]dot/\[Lambda]};

Euler = -((\[Theta] Derivative[1][c][t])/
     c[t]) - (\[Rho] - n*(1 - \[Theta]) - x*(1 - \[Theta])) == -D[Ham,
       k]/\[Lambda];

Euler2 = chat /. Flatten[Solve[Euler /. {c'[t]/c[t] -&gt; chat}, chat]];

kstar = k /. Flatten[Solve[Euler2 == 0, k]];

csol = c /. Flatten[Solve[kdot == 0, c]];

cstar = csol /. {k -&gt; kstar};

(*Parameters*)

d = 0.1; n = 0.01; \[Rho] = 0.05; a = 0.3; \[Theta] = 1/0.3;

Ht = 0.2; Lt = 0.1; Hx = 0.05; Lx = 0.02;

kstarHtauHx = kstar /. {t -&gt; Ht, x -&gt; Hx}
kstarHtauLx = kstar /. {t -&gt; Ht, x -&gt; Lx}
kstarLtauHx = kstar /. {t -&gt; Lt, x -&gt; Hx}
kstarLtauLx = kstar /. {t -&gt; Lt, x -&gt; Lx}