Help with checking work for preferences over consumption and leisure question

Question

I was wondering if anyone could help me check my work for the following question, and if I am wrong, help me correct my mistakes?

Question:

Work:

Answer 1

The optimization problem we want to solve is

$\max_{c,l} U(c,l) = c^{\frac{1}{3}} l^{\frac{2}{3}}$

subject to $c = w (4000-l) + I$

Here $w$ is the real wage (wage divided by the consumption good´s price) to make our lives easier, $I$ is the exogenous income (the stipend).

To save ourselves a few algebra steps with the Lagrangian, we know the condition for optimality is $MRS = \frac{p_c}{p_l}$ (relative prices):

$\frac{\frac{\partial U}{\partial c}}{\frac{\partial U}{\partial l}} = \frac{p_c}{p_l} \implies \frac{\frac{1}{3} c^{-\frac{2}{3}} l^{\frac{2}{3}}}{\frac{2}{3} c^{\frac{1}{3}} l^{-\frac{1}{3}}} = \frac{1}{w} \implies \frac{l}{2c} = \frac{1}{w} \implies \frac{2c}{l} = w \implies c = \frac{wl}{2}$

Substituting this expression for $c$ in our budget constraint,

$\frac{wl}{2} = w (4000-l) + I \implies l = 8000 -2l + 2I \implies 3l = 8000 + 2I \implies l^\star = \frac{8000 + 2I}{3} \implies l^\star = \frac{2}{3} (4000+I)$

Now we substitute back our marshiallian demand $l^\star$ back into our expression for $c$,

$c^\star = \frac{w}{3} (4000+I)$

Note the agent can't consume more than $4000$ hours of leisure, so we need to pay attention to the case where our expression for $l^\star \geq 4000$.

We have that

$l^\star \geq 4000 \iff \frac{2}{3} (4000 + I) \geq 4000 \iff 4000 + I \geq 6000 \iff I \geq 2000$.

So in reality,

$l^\star = \frac{2}{3}(4000 + I), I < 2000$

$l^\star = 4000, I \geq 2000$.

Note in the latter case consumption would equal the stipend $c^\star = I$.

Now we analyze both cases.

1. $w = 7.5, I = 20000$

Since $I = 20000 \geq 2000$, the agent wouldn't work and spend exactly their stipend on the consumption good:

$l^\star = 4000, c^\star = 20000$

2. $w = 15, I = 10000$

Since $I = 10000 \geq 2000$, the agent wouldn't work either:

$l^\star = 4000, c^\star = 10000$

Normally we would compute the values of $U(c^\star,l^\star)$ and compare, but in this case it is very clear that the agent would be better off under plan #1.

This happens because the agent wouldn't work, spend all their time on leisure, and consume with exactly their stipend in either case but in #1 the stipend is higher and the agent would be able to consume more.