Question

A consumer’s utility function is given by 𝑈(𝑥1

,𝑥2

) = 2𝑥1𝑥2 + 3𝑥1

. Where 𝑥1

and 𝑥2

denote the number of items of two goods 𝐺1

and 𝐺2

that are bought. Each item costs $1 for 

𝐺1

and $2 for 𝐺2

. Use Lagrange multipliers to find the maximum value of U if the 

consumer’s income is $83. Estimate the new optimal utility if the consumer’s income rises 

by $1

Answer 1

$u(X1X2)= 2X1X2+3X1$

where G₁=$1, G₂=$2 and B=$83

forming the budget constraint:

$B=G1X1+G2X2$

83=X₁+2X₂

$U=2X1X2+3X1+\lambdaλ(83-X1-2X2)$

$UX1=2X2+3-\lambda=0 ..................(i)$

$UX2=2X1-2\lambda=0 ..................(ii)$

$U\lambda=83- X1+2X2=0 ..............(iii)$

solving simultaneously:X₁=43 ; X₂=20 $\lambda =43$

substituting the values into the Lagrangian function

$U=2(43)(20)+3(43)+43(83-43-2(20))$

$U=2(860)+3(43)+43(0)$

$U=1720+129+0$

$U= 1849$

if the consumer's income increases by $1, the new budget becomes $84 and the new budget constraint becomes: $84=X$₁ +2$X$₂

$84-X$₁ $-2$ $X$₂

$U=2X1X2+3X1+\lambda(84-X1-2X2)$

$UX1=2X1X2+3-\lambda=0 ...........(i)$

$UX2=2X1-2\lambda=0 ..................(ii)$

$U\lambda=84 - X1+2X2=0 ............(iii)$

Solving simultaneously: X₁=43.5 ; X₂=20.25 and $\lambda$ = 43.5

substituting the values into the Lagrangian function gives the new utility as:

$U=2(43.5)(20.25)+3(43.5)+43.5(83-43.5-2(20.25))$

$U=2(880.75)+3(43.5)+43.5(0)$

$U=1761.75+130.5+0$

$U= 1892.25$

An increase of $1 in the income of the consumer will cause an approximate change of 43 as suggested by the lagrangian multiplier.