Q1. Consider the utility function ๐ข = f(๐ฅ1โฆ ๐ฅn) where ๐ฅ๐, ๐= 1,2, โฆ , ๐ are the quantities of the nย
goods consumed. Let the price of good ๐ฅ๐ be ๐i , ๐= 1,2, โฆ , ๐. Let M be the consumer's income. Showย
that the Lagrangian multiplier of the utility maximization problem equals the marginal utility ofย
income.
The utility function U = f("x_1,x_2,x_3.......x_n)" and ๐ฅ๐, ๐= 1,2, โฆ , 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 โ Pxx โ Pyy)
The first order conditions are
Zx = y โ ฮปPx = 0
Zy = x โ ฮปPy = 0
Zฮป = B โ Pxx โ Pyy = 0
Solving the first order conditions yield the following solutions
xM = B 2Px yM = B 2Py ฮป = B 2PxPy
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