Long-Run Equilibrium

In the steady state, the motion of capital per head and money wages comes to a halt: 1$$ \dot{k}=0 $$2$$ \dot{w}=0 $$ Get rid of $$ \dot{w} $$ in $$ \dot{w}=\varepsilon w(N/\bar{N}-1) $$ by making use of (2) to attain: 3$$ N=\bar{N} $$ Therefore full employment prevails. Now divide $$ Y={{K}^{\alpha }}{{\bar{N}}^{\beta }} $$ through by $$ \bar{N} $$, which yields: 4$$ y={{k}^{\alpha }} $$ Then observe (1)in $$ \dot{k}=i-nk $$: 5$$ i=nk $$ Moreover substitute this together with c = (1 − s)y into y = c + i to acquire: 6$$ sy=nk $$ Further combine (4) as well as (6) and reshuffle terms: 7$$ k={{(s/n)}^{1/\beta }} $$8$$ y={{(s/n)}^{\alpha /\beta }} $$