Two-Period Consumer's Problem with Production

The consumer's problem is \[\max_{c,c',l,l',s_p} u(c,l) + \beta u(c',l') \] \[ s.t. c\geq 0, c'\geq 0, 0 < l < h, 0 < l' < h' \] \[ c + s_p \leq w(h-l) + \pi - t \] \[ c' \leq (1+r)s_p + w'(h'-l') + \pi' -t' \]

solving for s, and combining the BCs, we can get a single present value budget constraint: \[ c+ {c'\over 1+r} \leq [w(h-l)+\pi-t] + {[w'(h'-l')+\pi'-t'] \over 1+r} \]

Suppose that \(u(c,l) = \log c + \log l\) Taking \(\lambda\) to be the langragian multiplier for the present value budget constraint, the FOCs are: \[U_c = {1\over c}=\lambda\] \[U_{c'} = {\beta \over c'}={\lambda \over 1+r}\] \[U_l = {1 \over l}=w\lambda\] \[U_{l'} = {\beta \over l'}= {w'\lambda \over 1+r}\] Now dividing these by each other, we can find the marginal rates of substituion at the optimum: \[MRS_{l,c} = {U_l \over U_c} = {c \over l} = w\] \[MRS_{l',c'} = {U_{l'} \over U_{c'}} = {c' \over l'} = w'\] \[MRS_{c,c'} = {U_c \over U_{c'}} = {c' \over \beta c} = 1+r\] Therefore \[ l = {1 \over w} c\] \[ c' = (1+r)\beta c \] \[ l' = {c' \over w'} = {(1+r)\beta \over w'}c \] Plugging this into the budget constraint (which holds with equality because of our choice of utility): \[ c+ \beta c = [wh-c+\pi-t] + {[w'h'-\beta(1+r)c+\pi'-t'] \over 1+r} \] \[ c = {1 \over (2+2\beta)} \left([wh+\pi-t] + {[w'h'+\pi'-t'] \over 1+r}\right) \] Now we have a formula for c in terms of lifetime wealth, and we can solve for our other choice variables by plugging in c. In particular, the formula for period-one leisure is now: \[l = {1 \over (2+2\beta)} \left([wh-c+\pi-t] + {[w'h'+\pi'-t'] \over 1+r}\right)\] In particular, note three things about this expression:
As wages increase, leisure decreases and so labor supplied increases.
As interest rate increases, leisure decreases and so labor supplied increases.
An increase in lifetime wealth increases leisure and so decreases labor supplied.