Question

Q1. Consider the utility function 𝑢 = f(𝑥1… 𝑥n) where 𝑥𝑖, 𝑖= 1,2, … , 𝑛 are the quantities of the n 

goods consumed. Let the price of good 𝑥𝑖 be 𝑃i , 𝑖= 1,2, … , 𝑛. Let M be the consumer's income. Show 

that the Lagrangian multiplier of the utility maximization problem equals the marginal utility of 

income.

Answer 1

The utility function U = f($x_1,x_2,x_3.......x_n)$ and 𝑥𝑖, 𝑖= 1,2, … , 𝑛

Let, the price of good x_i be P_i , 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