Question

Alt Alt Fn question 2 (22 pts.) Consider a representative agent with preferences over consumption c and leisure l ue l) = In c + In I. Her budget constraint is c wM, where w is the wage hours worked. The representative agent also chooses how to allocate her time between w activities given her time constraint l + N = h, where h is the total number of hours. represented by rate and N - the number of ork and leisure a) (1 pt.) Combine the budget constraint and the time constraint of single constraint. the representative agent intoa b) (2 pts.) List all endogenous and exogenous variables. c) (1 pt.) Set up the Lagrangean function.

Answer 1

(a) The two constraints are $\dpi{80} c = wN$ (supposing all income is exhausted in consumption) and $\dpi{80} l + N = h$ or $\dpi{80} N = h - l$ . Putting $\dpi{80} N = h - l$ in $\dpi{80} c = wN$ , we have $\dpi{80} c = w(h - l)$ or $\dpi{80} c + wl = wh$ , which is the representative budget constraint.

(b) The exogenous variables are w (wage) and h (total number of hours). These variables are to be having given values in the model. The endogenous variables are c (consumption), l (leisure) and N (hours worked).

(c) The Lagrangian function would be $\dpi{80} L = U + \lambda (wh - c - wl)$ or $\dpi{80} L = ln(c) + ln(l) + \lambda (wh - c - wl)$ .

(d) The FOCs are as below.

$\dpi{80} \frac{\partial L}{\partial c} = 0$ or $\dpi{80} \frac{\partial }{\partial c}(ln(c) + ln(l) + \lambda (wh - c - wl)) = 0$ or $\dpi{80} \frac{1}{c} + \lambda (- 1) = 0$ or $\dpi{80} \frac{1}{c} = \lambda$ .

$\dpi{80} \frac{\partial L}{\partial l} = 0$ or $\dpi{80} \frac{\partial }{\partial l}(ln(c) + ln(l) + \lambda (wh - c - wl)) = 0$ or $\dpi{80} \frac{1}{l} + \lambda (- w) = 0$ or $\dpi{80} \frac{1}{l} = \lambda w$ .

$\dpi{80} \frac{\partial L}{\partial \lambda} = 0$ or $\dpi{80} \frac{\partial }{\partial \lambda}(ln(c) + ln(l) + \lambda (wh - c - wl)) = 0$ or $\dpi{80} wh - c - wl = 0$ or $\dpi{80} c + wl = wh$ .

(e) Comparing the first two FOCs, we have $\dpi{80} \frac{1/c}{1/l}= \frac{\lambda}{\lambda w}$ or $\dpi{80} \frac{1/c}{1/l}= \frac{1}{w}$ , which is the utility maximizing combination. But this combination would be as $\dpi{80} MRS = 1/w$ , for -1/w is the slope of the constraint, considering leisure is on the x-axis. Hence, we have $\dpi{80} MRS = - \frac{1/c}{1/l}$ or $\dpi{80} MRS = - l/c$ . The MRS is basically the slope of the utility curve for a particular utility level. This means that to remain on the same utility level, the agent would have to sacrifice MRS amount of consumption to get a marginal unit of leisure.