Question

Consider a person who

consumes water and bread, deriving utility by xy if x is the amount of water consumed and

y is the amount of bread consumed. Suppose this person's income is Rs. 10, the unit

price of bread is Rs. 3 and the unit price of water facing this person is Re. 1. The price of

water incorporates a per unit subsidy of Re. 1, i.e., for every unit of water consumed by

this person, she pays Re. 1 to the water supplier and the government pays Re. 1 to the

water supplier. Suppose this person's demand is (x0, y0). ,If the government provides this person a lump-sum income subsidy that exactly offsets 

her utility loss on account of removal of the water subsidy, then the required lump-sum 

subsidy is

Answer 1

$u_0 = λ_0Y_0 =(\frac{10}{2},\frac{10}{6}) =\frac{25}{3}$

Calculate x.y Subject to 2x +3y = m'

$α = x1 y1 + λ(m' - 2x - 3y)$

$x_1 =\frac{ m'}{4}=\frac{\sqrt{200}}{4} ,y_1=\frac{ m'}{6}=\frac{\sqrt{200}}{6}$

$x_1y_1=\frac{ m'}{4}\times\frac{ m'}{6}=\frac{ m'^2}{24}$

but $\frac{ m'^2}{24}=u_0$

$\frac{ m'^2}{24}=\frac{25}{3}$

$m^2=200=\sqrt{200}$

This is the Income That spent at new Price Which Provide Same Utility

Subsidy = m' - m

$=\sqrt{200}-10$