# Help with checking work for preferences over consumption and leisure 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?

Work:  • How do you get to the equations you then solve? Do you do any utility maximisation?
– BrsG
Mar 1 at 22:12
• These equations make no sense. The first one interprets $C$ as income (or as quantity of the consumption good, if its price is normalized to \$1) if you work the full 4000 hours. The second one interprets$C$as hours worked. Moreover, "solving" these equations by substituting the first in the second gives you a negative$L\$. So where do you get these 3200 hours worked from? Mar 2 at 9:45

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:

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

1. $$w = 15, I = 10000$$

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.