<ul>
<li>Disposable Income $Y_D$ as a function of $Y_G$ and $Y_E$ is: 
$$Y_D = Y_G + (1-\alpha) Y_E$$</li>
<li>Given wage rate $w$, and $Y_E = wH$, the budget equation can be written as:
$$Y_D = Y_G + (1-\alpha)wH$$
Preferences of the individual over $Y_D$ and $H$ are represented by the utility function
$$U(Y_D, H) = \dfrac{Y_D}{(1+H)^2}$$
So the utility maximization problem is :
\begin{eqnarray*}\max_{Y_D, H} &amp;&amp; \ \ \dfrac{Y_D}{(1+H)^2} \\ \text{s.t} &amp;&amp;  \ \ Y_D = Y_G + (1-\alpha)wH \\ &amp;&amp;  \ \ H\geq 0 \end{eqnarray*}
Here is the picture to demonstrate optimal solution :</li>
</ul>

<p><a href="https://i.sstatic.net/2T94y.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2T94y.png" alt="enter image description here"></a></p>

<ul>
<li><p>Solving the above problem we get the optimal labor supply as
\begin{eqnarray*} H^*(w, \alpha, Y_G) = \begin{cases} 1-\dfrac{2Y_G}{(1-\alpha)w} &amp; \text{if } (1-\alpha)w - 2Y_G &gt; 0  \\ 0 &amp; \text{if } (1-\alpha)w - 2Y_G \leq 0 \end{cases}  \end{eqnarray*}</p></li>
<li><p>So the optimal utility as a function of $w, \alpha, Y_G$ is as follows:
\begin{eqnarray*} U^*(w, \alpha, Y_G) = \begin{cases}  \dfrac{((1-\alpha)w)^2}{4((1-\alpha)w - Y_G)} &amp; \text{if } (1-\alpha)w - 2Y_G &gt; 0  \\ Y_G &amp; \text{if } (1-\alpha)w - 2Y_G \leq 0 \end{cases}  \end{eqnarray*}</p>

<p>Now the Govt. chooses $(Y_G, \alpha)$ by solving :
\begin{eqnarray*}\max_{Y_G, \alpha} &amp;&amp; \ \ U^*(w, \alpha, Y_G) \\ \text{s.t} &amp;&amp;  \ \ Y_G = \alpha wH^*(w, \alpha, Y_G)  \end{eqnarray*}</p></li>
</ul>

<p>Solving it we get
$\alpha^* = 0$ and $Y_G^* = 0$</p>

CC BY-SA 4.0
2018-05-29T03:49:25.837Z
2018-05-29T03:49:25.837Z
Amit
<p>Consider a welfare system u dear which a single cash transfer(guaranteed income ) is given to every citizen. Then for each dollar the person earns the payment is reduced to by a where 00 you get a transfer. If t&lt;0, you pay taxes. If t=0, you don’t pay taxes or receive transfers. </p>

<p>Suppose that te wage rate is $w&gt;0$. And so a person who works H hours gets $y_e=wH$. A typical individuals utility function is $$u(y_d, H)=y_d/(1+H)^2$$</p>

<p>The question has multiple part. I tried to do most of them. But I’m not sure please check my solutions also I cannot do some parts. </p>

<p>First of all I write an expression for individual’s disposable income $y_d$ as a function of $y_g$ and $y_e$ and other parameters. </p>

<p>$$y_d=(1-a)y_e+y_g$$</p>

<p>secondly I try to draw indifference curves for that utility function and his disposable income line as follows </p>

<p><a href="https://i.sstatic.net/JABbO.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JABbO.jpg" alt="enter image description here"></a></p>

<p>I only put this image to show the graph. </p>

<p>Thirdly, I try to drive utility maximizing labor supply function $H^*$ by assuming that $(1-a)w-2y_g&gt;0$</p>

<p>$$u(y_d, H)=y_d/(1+H)^2$$</p>

<p>$$u(y_d, H)=((1-a)y_e+y_g)/(1+H)^2$$</p>

<p>Derivative with respect to $H$ </p>

<p>$${(1-a)w(1+H)^2-((1-a)wH+y_g)2(1+H)\over (1+H)^4}=0$$</p>

<p>$$\iff$$</p>

<p>$$(1+H)[(1-a)w(1+H)-2(1-a)wH-2y_g]=0$$</p>

<p>Since H>0, $(1+H)≠0$ </p>

<p>So,
$$[(1-a)w(1+H)-2(1-a)wH-2y_g]=0$$</p>

<p>$$H*={[(1-a)w-2y_g\over (1-a)w}&gt;0$$</p>

<p>Additionally,</p>

<p>$$\partial H^*/\partial y_g=-w/((1-a)w)&lt;0$$</p>

<p>By intuition, as the guaranteed income $yg$ increases labor supply will decrease. </p>

<p>$$\partial H^*/\partial a=-2y_g/. ((-a)^2w)&lt;0$$ </p>

<p>So for each dollar the person earns the reduction amount of a in the single cash transfer increases iff the labor supply decreases. </p>

<p><em>these intuitions are enough? What do you say about these results ?</em></p>

<p>Then, </p>

<p>Fourthly What is the utility maximizing labor supply when  $(1-a)w-2y_g\le 0$</p>

<p>In this case I said that </p>

<p>$$H*={[(1-a)w-2y_g\over (1-a)w}\le 0$$</p>

<p>Thus, the labor supply will be optimally zero. </p>

<p><strong>I am stuck at this point. Is what I said true?</strong></p>

<p>And then,</p>

<p>Assume that the government wants to choose its policy $(y_g, a)$ in such a way that a citizen utility is maximized subject to the constraint that the net transfer is zero   How can I derive the optimal policy? <strong>I tried to derive it. But I guess my solution is not true. How do you derive it? Is it correct? This point is really important.</strong></p>

<p>$$u(y_d, H)=Y_d/(1+H)^2=((1-a)wH+y_G)/(1+H)^2$$</p>

<p>Subject to $ T=Y_G-aY_E=0$ </p>

<p>So $y_g=aY_E=awH$</p>

<p>So 
$$u(y_d, H)=Y_d/(1+H)^2=((1-a)wH+awH)/(1+H)^2=wH/(1+H)^2$$</p>

<p>Derivative w.r.t. H; </p>

<p>$$(w(1+H)^2-wH2(1+H))/(1+H)^4=0$$</p>

<p>$$(w(1+H)^2-wH2(1+H))=0$$</p>

<p>Iff </p>

<p>$$w+wH-2wH=0$$</p>

<p>So $H^*=1/2$</p>

<p>So, $y_g^*=aw/2$ and $a^*=2y_g^*/w$</p>

<p>I am really stuck at this part.  </p>

<p><strong>And my last question is based on above assumption. But different citizens earn different wages. What do you think about the merit of this policy?</strong></p>

<p>Thank you in advance. </p>

<p><a href="https://i.sstatic.net/KcnYJ.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/KcnYJ.jpg" alt="enter image description here"></a></p>

CC BY-SA 4.0
2018-05-28T14:12:56.280Z
Can you post the exact question? I don't think i've understood what you are trying to say in the first few lines.
Amit
<microeconomics><utility><consumer-theory><self-study>
Maximization with disposable income
b11bb
# Maximization with disposable income

(Asked by: username_0 on 2018-05-21)

Consider a welfare system u dear which a single cash transfer(guaranteed income ) is given to every citizen. Then for each dollar the person earns the payment is reduced to by a where 00 you get a transfer. If t<0, you pay taxes. If t=0, you don’t pay taxes or receive transfers.

Suppose that te wage rate is $w>0$. And so a person who works H hours gets $y\_e=wH$. A typical individuals utility function is $$u(y\_d, H)=y\_d/(1+H)^2$$

The question has multiple part. I tried to do most of them. But I’m not sure please check my solutions also I cannot do some parts.

First of all I write an expression for individual’s disposable income $y\_d$ as a function of $y\_g$ and $y\_e$ and other parameters.

$$y\_d=(1-a)y\_e+y\_g$$

secondly I try to draw indifference curves for that utility function and his disposable income line as follows

[![enter image description here](https://i.sstatic.net/JABbO.jpg)](https://i.sstatic.net/JABbO.jpg)

I only put this image to show the graph.

Thirdly, I try to drive utility maximizing labor supply function $H^\*$ by assuming that $(1-a)w-2y\_g>0$

$$u(y\_d, H)=y\_d/(1+H)^2$$

$$u(y\_d, H)=((1-a)y\_e+y\_g)/(1+H)^2$$

Derivative with respect to $H$

$${(1-a)w(1+H)^2-((1-a)wH+y\_g)2(1+H)\over (1+H)^4}=0$$

$$\iff$$

$$(1+H)[(1-a)w(1+H)-2(1-a)wH-2y\_g]=0$$

Since H>0, $(1+H)≠0$

So,
$$[(1-a)w(1+H)-2(1-a)wH-2y\_g]=0$$

$$H\*={[(1-a)w-2y\_g\over (1-a)w}>0$$

Additionally,

$$\partial H^\*/\partial y\_g=-w/((1-a)w)<0$$

By intuition, as the guaranteed income $yg$ increases labor supply will decrease.

$$\partial H^\*/\partial a=-2y\_g/. ((-a)^2w)<0$$

So for each dollar the person earns the reduction amount of a in the single cash transfer increases iff the labor supply decreases.

*these intuitions are enough? What do you say about these results ?*

Then,

Fourthly What is the utility maximizing labor supply when $(1-a)w-2y\_g\le 0$

In this case I said that

$$H\*={[(1-a)w-2y\_g\over (1-a)w}\le 0$$

Thus, the labor supply will be optimally zero.

**I am stuck at this point. Is what I said true?**

And then,

Assume that the government wants to choose its policy $(y\_g, a)$ in such a way that a citizen utility is maximized subject to the constraint that the net transfer is zero How can I derive the optimal policy? **I tried to derive it. But I guess my solution is not true. How do you derive it? Is it correct? This point is really important.**

$$u(y\_d, H)=Y\_d/(1+H)^2=((1-a)wH+y\_G)/(1+H)^2$$

Subject to $ T=Y\_G-aY\_E=0$

So $y\_g=aY\_E=awH$

So
$$u(y\_d, H)=Y\_d/(1+H)^2=((1-a)wH+awH)/(1+H)^2=wH/(1+H)^2$$

Derivative w.r.t. H;

$$(w(1+H)^2-wH2(1+H))/(1+H)^4=0$$

$$(w(1+H)^2-wH2(1+H))=0$$

Iff

$$w+wH-2wH=0$$

So $H^\*=1/2$

So, $y\_g^\*=aw/2$ and $a^\*=2y\_g^\*/w$

I am really stuck at this part.

**And my last question is based on above assumption. But different citizens earn different wages. What do you think about the merit of this policy?**

Thank you in advance.

[![enter image description here](https://i.sstatic.net/KcnYJ.jpg)](https://i.sstatic.net/KcnYJ.jpg)

**Tags:** microeconomics, utility, consumer-theory, self-study

> **Comments:**
> **username_1**: Can you post the exact question? I don't think i've understood what you are trying to say in the first few lines.  

---

## 1 answers

### Answer by username_1 on 2018-05-29. Score: 1 | Accepted answer

* Disposable Income $Y\_D$ as a function of $Y\_G$ and $Y\_E$ is:
  $$Y\_D = Y\_G + (1-\alpha) Y\_E$$
* Given wage rate $w$, and $Y\_E = wH$, the budget equation can be written as:
  $$Y\_D = Y\_G + (1-\alpha)wH$$
  Preferences of the individual over $Y\_D$ and $H$ are represented by the utility function
  $$U(Y\_D, H) = \dfrac{Y\_D}{(1+H)^2}$$
  So the utility maximization problem is :
  \begin{eqnarray\*}\max\_{Y\_D, H} && \ \ \dfrac{Y\_D}{(1+H)^2} \\ \text{s.t} && \ \ Y\_D = Y\_G + (1-\alpha)wH \\ && \ \ H\geq 0 \end{eqnarray\*}
  Here is the picture to demonstrate optimal solution :

[![enter image description here](https://i.sstatic.net/2T94y.png)](https://i.sstatic.net/2T94y.png)

* Solving the above problem we get the optimal labor supply as
  \begin{eqnarray\*} H^\*(w, \alpha, Y\_G) = \begin{cases} 1-\dfrac{2Y\_G}{(1-\alpha)w} & \text{if } (1-\alpha)w - 2Y\_G > 0 \\ 0 & \text{if } (1-\alpha)w - 2Y\_G \leq 0 \end{cases} \end{eqnarray\*}
* So the optimal utility as a function of $w, \alpha, Y\_G$ is as follows:
  \begin{eqnarray\*} U^\*(w, \alpha, Y\_G) = \begin{cases} \dfrac{((1-\alpha)w)^2}{4((1-\alpha)w - Y\_G)} & \text{if } (1-\alpha)w - 2Y\_G > 0 \\ Y\_G & \text{if } (1-\alpha)w - 2Y\_G \leq 0 \end{cases} \end{eqnarray\*}

  Now the Govt. chooses $(Y\_G, \alpha)$ by solving :
  \begin{eqnarray\*}\max\_{Y\_G, \alpha} && \ \ U^\*(w, \alpha, Y\_G) \\ \text{s.t} && \ \ Y\_G = \alpha wH^\*(w, \alpha, Y\_G) \end{eqnarray\*}

Solving it we get
$\alpha^\* = 0$ and $Y\_G^\* = 0$


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