Question

7. Suppose Dennis has utility function over consumptionUD= 50√cD,While Emily has utility UE= 50√cE+3/4UD That is, Emily derives utility from her own consumption, as well as from Dennis’s consumption. Society has income of 2080 to divide between Dennis and Emily; currently, Dennis currently has income 520 while Emily has income 1560.(a) Are any pareto-improving redistribution possible here? (b) Find the income distribution that would maximize a simply util-itarian social welfare function.

Answer 1

Solution :

(a) Are any pareto-improving redistribution possible here?

Yes, Pareto improving redistribution is possible.

Let income of dennis rise to 600, then his utility will surely rise,

Now Emily will get = 2080 - 600 = 1480

Then UE = 50 * √1480 + 0.75 * 50 * √600

= 2842

Whereas utility of Emily before redistribution :

= 50 * √1560 + 0.75 * 50 * √520

= 2829

Thus both are better off with the redistribution.

(b) Find the income distribution that would maximize a simply util-itarian social welfare function.

Now utilitarian welfare function maximize joint welfare

So joint utility function U = UE + UD

= 50 * √CD + 50 * √CE + 0.75 * 50 * √CD

U = 50 * √CE + 87.5 * √CD

Now as CD + CE = 2080

put CD = 2080 - CE

So, U = 50*√CE + 87.5*√(2080-CE)

dU/dCE= 0

-87.5/2*√(2080-CE) + (50/2*√CE) = 0

By solving , we get

CE* = 512

CD* = 2080 - 512

CD = 1568