Answer to Question #173897 in Microeconomics for RAJNISH SINGH

Question #173897

Q1. Consider the utility function ๐‘ข = f(๐‘ฅ1โ€ฆ ๐‘ฅn) where ๐‘ฅ๐‘–, ๐‘–= 1,2, โ€ฆ , ๐‘› are the quantities of the nย 

goods consumed. Let the price of good ๐‘ฅ๐‘– be ๐‘ƒi , ๐‘–= 1,2, โ€ฆ , ๐‘›. Let M be the consumer's income. Showย 

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

income.


1
Expert's answer
2021-03-23T08:40:54-0400

The utility function U = f("x_1,x_2,x_3.......x_n)" and ๐‘ฅ๐‘–, ๐‘–= 1,2, โ€ฆ , ๐‘›

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 + ฮป(B โˆ’ Pxx โˆ’ Pyy)

The first order conditions are

Zx = y โˆ’ ฮปPx = 0

Zy = x โˆ’ ฮปPy = 0

Zฮป = B โˆ’ Pxx โˆ’ Pyy = 0

Solving the first order conditions yield the following solutions

xM = B 2Px yM = B 2Py ฮป = B 2PxPy


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

suyash mishra
21.07.21, 09:36

Gracias Expert

Leave a comment