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?
Work:
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Sign up to join this communityI was wondering if anyone could help me check my work for the following question, and if I am wrong, help me correct my mistakes?
Work:
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.
Since $I = 20000 \geq 2000$, the agent wouldn't work and spend exactly their stipend on the consumption good:
$l^\star = 4000, c^\star = 20000$
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.