Answer in Microeconomics for RAJNISH SINGH #173897
March 27th, 2023
Q1. Consider the utility function = f( 1… n) where , = 1,2, … , are the quantities of the n
goods consumed. Let the price of good be i , = 1,2, … , . Let M be the consumer’s income. Show
that the Lagrangian multiplier of the utility maximization problem equals the marginal utility of
income.
The utility function U = f(“x_1,x_2,x_3…….x_n)” and , = 1,2, … ,
Let, the price of good xi be Pi , i=1,2,3……n
Maximize: U= xy
Subject to constraint
B= “P_xx+P_yy”
The Lagrangian for this problem is
Z = xy + λ(B − Pxx − Pyy)
The first order conditions are
Zx = y − λPx = 0
Zy = x − λPy = 0
Zλ = B − Pxx − Pyy = 0
Solving the first order conditions yield the following solutions
xM = B 2Px yM = B 2Py λ = B 2PxPy
