Income gap in Ramsey model
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}