Intermediate macroeconomics: optimal bundle for quasilinear utility?

Question

How would I go about solving this question:

Assuming consumer's utility function is $U(C,L)=c+2l^{0.5}$, consumer earns a wage of 0.5/hour, $h=24$ and there is no real dividend and tax is $T=11$. find:

a. The maximum amount of leisure the house hold can have and still pay taxes
b. find the optimal bundle of consumption and leisure.

For part (a), I took the derivative of $U$ with respect to $L$ and set it to 0, giving me $l^{0.5}=0$ and $l=0$. Is this part correct? Intuitively, this feels very wrong to me.

I'm also stuck on part (b). I took the derivative of the utility with respect to $C$. From my understanding, MRS is the derivative of $U$ with respect to $L$ over the derivative of $U$ with respect to $C$. I ended up getting $1/(L^{0.5})$.

I know the optimal bundle is when the budget line is tangent to the indifference curve and I also know that the slope of the budget line would be $-W$. However, when I solve for $1/(L^{0.5})=0.5$, I end up getting $L=4$ and $C=-1$. It doesn't make sense to me that a consumer will be able to have negative consumption.

Would greatly appreciate it if someone could point out where I went wrong!

Answer 1

From the utility function it seems L is leisure.

a) I think you went about a) in a wrong way. For a, simply you want to work as little as possible but be able to pay 11 in taxes. So you will want to earn exactly 11USD and not more (earning more means working more, which you want to minimize). So at 0,50USD an hour you have to work 22 hours. Since h=24 the maximum leisure you can have is 24-22=2.

b) Where you are going wrong in your analytical approach for b) is not including the budget constraint in your problem (when taking derivatives), which you should as you explained for the graphic analysis.

For consumption you have expenditures - income: p*c=wage*labor - T (technically an inequality, but in this case you can assume it holds with equality). I would suggest using a langrangian with this constraint. You also have the constraint that Leisure=24-labor. This you can plug in directly and is advisable so that your problem only has two unknowns (c and labor). Solving this problem now should do the trick.

Answer 2

The first question has nothing to do with utility optimization, and so with the utility function. It just determines a bound, and it is the solution to the inequality

$$(24-l)\cdot \frac 12 \geq 11$$

since $(24-l)$ represents amount of work in hours.

The optimization task is

$$\max _{c,l}U(c,l)=c+2l^{1/2} \\$$

$$s.t. \;\;c = (24-l)\cdot \frac 12 -11,\;\; c\geq 0,\;\; l \geq 0$$

Due to the existence of the lump-sum tax, the non-negativity constraints matter. This is why the OP ended up with a negative consumption, because she did not consider the non-negativity constraints.

Now instead of going for a formal treatment with non-negative Karush-Kuhn-Tucker multipliers, the first question comes handy. Since there is a maximum amount of leisure at which point the consumer is just able to pay taxes (and so have zero consumption), it follows that the variable "leisure" cannot take a greater value than that, because it will lead to negative consumption.

Then one can reason by taking the partial derivatives of the utility function, and wonder "if a decrease leisure a little below its maximum permissible level, what do I lose in terms of utility from leisure lost, and what do I gain in terms of utility in terms of consumption gained?" It will also help to consider the two extremes: What is the utility when leisure is maximum and consumption is zero? What is the utility when leisure is zero and consumption is maximum? Don't forget to pay the (very heavy) taxes first.