I want the FULL ANSWER please.

Question $4$

Consider $an$ economy where the population $of$ country $i{N}_{i,t}$ grows $at$ the constant rate $n_{i}0$ :

$N_{i,t+1}=(1+n_i)N_{i,t}.$

Output $is$ a linear function $in$ aggregate productivity $A_{i,t}$ and population $N_{i,t}$ :

$Y_t=A_{i,t}{N}_{i,t},$

The productivity $of$ country i evolves according $to$:

$A_{i,t+1}-A_{i,t}=A_{i,t}^{}{N}_{i,t}^{},$

with $>0,>0,$ which reflects the notions that a larger population generates more produc$-$

tive ideas $(hence$ productivity growth $is$ increasing $in$ population$)$ and that existing ideas beget new

ideas $(hence$ productivity growth $is$ increasing $in$ current productivity$).$

$(a)$ Characterize the long$-$run growth rate $of$ productivity $A_{i,t}of$ a country i first for the case $of$

$==1$ and then for the case

$(b)$ Consider two countries iin$\{1,2\}in$ the model under the parameter restrictions $=0\mathrm{.}5,$

$=1.In$ period $0,$ country $1$ has twice $as$ many people $as$ country $2,N_{1,0}=2N_{2,0},$ but only

a quarter $of$ the productivity, $A_{1,0}=\frac{1}{4}{A}_{2,0}.$ The population growth rates $of$ the countries are

identical, $n_1=n_2.$ How $do$ the growth rates $of$ productivity $of$ the two countries compare $in$

the initial period $(from$ period $0to1)?$ How $do$ the growth rates compare $in$ the long run $(i.e.,$

the balanced growth path$)?$ How $do$ the levels $of$ productivity compare $in$ the long run?