u(c1)+v(l1)+u(c2) ,

where both u and v are assumed to be strictly increasing, strictly concave, and satisfy limc0 u(c) = liml0 v(l) = (so that we can ignore the NNCs on both c and l). To make things even simpler, let's way with labour in the second period, and make the pro- duction functions linear in both periods. That is, we assume output in the first period is given by y1 = z1(n1 + k1) and output in the second period by y2 = z2k2. Note that, as firms don't hire any labour in the second period, household labour income will be zero in that period. All other elements of the model are the same as in LN7.

(a) Write down the household's budget constraints for each period.