Maximization with disposable income

Question

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

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

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.

Answer 1

• 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 :

• 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 $Y_G^* = 0$