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
"u_0 = \u03bb_0Y_0 =(\\frac{10}{2},\\frac{10}{6}) =\\frac{25}{3}"
Calculate x.y Subject to 2x +3y = m'
"\u03b1 = x1 y1 + \u03bb(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"
Comments
Thanks a lot.
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