Writing down income

Question

Consider a welfare system under which a single cash transfer (a guaranteed income) is given to every citizen. Then for each dollar the person earns, this payment is reduced by $\alpha$ dollars, where $0<\alpha<1$. Let $Y_G\geq0$ be the guaranteed income. Then, if earned income is $Y_E$, the cash transfer is reduced by $\alpha Y_E$ dollars. So the net transfer is $T=Y_G-\alpha Y_E$. If $T>0$, you get a transfer. If $T<0$, you pay taxes. If $T=0$, you don't pay taxes or receive transfers.

• $(i)$ Find an expression for an individual's disposable income, $Y_D$ (i.e., his income after transfers) as a function of $Y_G$, $Y_E$, and other parameters.

Please help me to write disposable income.

Thanks a lot.

What I have thought

Disposable income = total income +transfers

So,

$$Y_D = Y +T$$

$$Y_D = Y +Y_G - \alpha Y_E$$

But I cannot decide what is Y ?

But possibly I think that Y is $ Y_E $

So

$$Y_D = Y_E + Y_G - \alpha Y_E$$

So,

$$Y_D = (1- \alpha )Y_E +Y_G $$

Answer 1

This is correct. Intuitively, disposable income is earned income $\pm$ transfer or taxes. You have already correctly identified that the transfer or taxes paid is $Y_T = Y_G - \alpha Y_E$, and it is given in the problem that earned income is $Y_E$.

So, $Y_D = Y_E + Y_G - \alpha Y_E=(1-\alpha )Y_E+Y_G$.