DSE 2014 Question 53: Harrod–Domar and Solow growth models

Consider an economy with $N$ identical competitive firms, each producing $Y^i=(K^i)^\alpha (L^i)^{1-\alpha}$, $0<\alpha<1$, with aggregate output $Y_t=(K_t)^\alpha (L_t)^{1-\alpha}$ (final good is numeraire; wage $w_t$, rental $r_t$). There are $H$ identical households, each with capital $k_t^h$ and 1 unit of labour, saving and investing half of total income ($dk^h/dt=\tfrac12 y^h$, no depreciation); capital is supplied inelastically. Labour supply is endogenous: each household supplies $l_t^h$ to maximise $U=w_t l_t^h - (l_t^h)^\delta$, $\delta>1$. The market-clearing wage rate in the short run (period $t$) is