Question

Problem #1: Optimal labor supply Clark gains utility from consumption c and leisure l and his preferences for consumption and leisure can be expressed as U(c, l) = 2(√ c)(l). This utility function implies that Clark’s marginal utility of leisure is 2√ c and his marginal utility of consumption is l √ c . He has 16 hours per day to allocate between leisure (l) and work (h). His hourly wage is $12 after taxes. Clark also receives a daily check of $30 from the government no matter how much he works. Graph Clark’s budget constraint with leisure on the x-axis and consumption on the y-axis. What is Clark’s marginal rate of substitution (MRS) when l = 3 and he is on his budget line? At what wage rate would Clark be indifferent between working his first hour and being unemployed (his “reservation wage”)? Find Clark’s optimal amount of consumption and leisure (the values that maximizes his utility).

Answer 1

Clarks Budget Constraint can be written as C = wh + G Where C is the expenditure on consumption, w is the wage rate, h is the work hours such that wh is the labour earning G is the non labour income.
Clark has two alternative uses for her time work or leisure.
Symbolically, T = h + L Then, we can rewrite the budget constraint as
C = w ( T – L) +G
i.e, C = (wT +G) – wL -------------------------------(1)
The budget equation (1) is the equation of a straight line
Taking total differentiation of (1) with respect to L, we get
dC/dL= -w
This is the slope of the Budget line.
As w is fixed at $12 in the above problem, therefore the Budget Equation between leisure and consumption will be a negatively sloped straight line.
We can easily graph Clark’s Budget Constraint as the following

Point A said that how much Clark can consume if he does not work. I.e, he will get $30 from the government. Here OG =$30. He moves up towards the budget line as he trades off an hour of leisure and for additional consumption.

Here, Marginal Utility of consumption = l$\sqrt{c}$ and the Marginal Utility of Leisure = 2$\sqrt{c}$

Therefore the Marginal rate of Substitution (MRS) = MU_c/ MU_l = l$\sqrt{c}$ / 2$\sqrt{c}$ = l / 2. = 3/2

Clarks Optimal amount of consumption and leisure can be determined by maximising the utilioty function subject to the budget constraint.

For this we form the lagrange equation as

$u=2\sqrt(C)l + \lambda \left [C- (wT +G) + wL ]$