DSE 2014 Question 56: 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 now: each household maximises $U=w_t l_t^h - D$ ($D>0$) for any $l_t^h>0$, and $U=0$ for $l_t^h=0$ (fixed disutility of working). The new aggregate labour supply schedule by the households is