# Maximization with disposable income

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

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, $$[(1-a)w(1+H)-2(1-a)wH-2y_g]=0$$

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

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

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?

• 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 May 28 '18 at 14:12

• 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 :
• 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 $Y_G^* = 0$