<p>This has to do with the form of the utility function. Assume instead that,say, we had</p>

<p>$$U(c,l) = c^{1/2} - \frac{1}{2}l^2$$</p>

<p>Does now $R$ affect the labor-supplied decision?</p>

<p>Solve it and explore. Check also the link offered in a comment to your question.</p>

CC BY-SA 3.0
2017-03-03T18:43:12.620Z
2017-03-03T18:43:12.620Z
Alecos Papadopoulos
<p>Given a Utility function $U(c,l) = c - \frac{1}{2}l^2$ , where $c$ is the consumption and $l$ is the number of hours of labour. Let $L$ be the maximum amount of labor, so the amount of leisure is $L - l$. The wage rate is $w$.</p>

<p>Assume a linear tax rate $r$, where $0 &lt; r &lt; 1$. Also, there is a fixed transfer of $R &gt; 0$ to every individual that is irrespective of labor supply choice. Solve for labor supply l that maximises utility.</p>

<p>My solution: </p>

<p>$c = (1-r)wl + R$ therefore we maximise $U(l) = (1-r)wl + R - \frac{1}{2}l^2$.</p>

<p>Differentiating $U$ w.r.t $l$ we get $(1-r)w - l = 0$; therefore $l = (1-r)w$. So this shows that $R$ has no relation to $l$. However, it should make a difference due to the income effect, so I'm not sure what I'm doing wrong :/</p>

<p>Thanks!</p>

CC BY-SA 3.0
2017-03-03T08:17:56.947Z
http://economics.stackexchange.com/q/5933/11590
Bayesian
CC BY-SA 3.0
2017-03-03T18:31:51.180Z
I don't understand why this question has been flagged for closing. The OP solved fully the exercise, and has difficulty _understanding and interpreting_ the result -which in my opinion is one of the most valid reasons why a forum like ours should exist, to help people interpret and understand.
Alecos Papadopoulos
<labor-economics><public-economics><consumption>
Effect of changing Non Labor Income on Consumption-Leisure problem
Abishek Puri
# Effect of changing Non Labor Income on Consumption-Leisure problem

(Asked by: username_0 on 2017-03-03)

Given a Utility function $U(c,l) = c - \frac{1}{2}l^2$ , where $c$ is the consumption and $l$ is the number of hours of labour. Let $L$ be the maximum amount of labor, so the amount of leisure is $L - l$. The wage rate is $w$.

Assume a linear tax rate $r$, where $0 < r < 1$. Also, there is a fixed transfer of $R > 0$ to every individual that is irrespective of labor supply choice. Solve for labor supply l that maximises utility.

My solution:

$c = (1-r)wl + R$ therefore we maximise $U(l) = (1-r)wl + R - \frac{1}{2}l^2$.

Differentiating $U$ w.r.t $l$ we get $(1-r)w - l = 0$; therefore $l = (1-r)w$. So this shows that $R$ has no relation to $l$. However, it should make a difference due to the income effect, so I'm not sure what I'm doing wrong :/

Thanks!

**Tags:** labor-economics, public-economics, consumption
> **Comments:**
> **username_1**: http://economics.stackexchange.com/q/5933/11590  
> **username_2**: I don't understand why this question has been flagged for closing. The OP solved fully the exercise, and has difficulty _understanding and interpreting_ the result -which in my opinion is one of the most valid reasons why a forum like ours should exist, to help people interpret and understand.  

---

## 1 answers

### Answer by username_2 on 2017-03-03. Score: 3 | Accepted answer

This has to do with the form of the utility function. Assume instead that,say, we had

$$U(c,l) = c^{1/2} - \frac{1}{2}l^2$$

Does now $R$ affect the labor-supplied decision?

Solve it and explore. Check also the link offered in a comment to your question.


---

