Suppose that you have five consumption choices: good $x_{1}cdots{x}_5.$ An indifference surface is the set of consumption choices with a CONSTANT utility. For example if $(x_1,$cdots,$x_5)=(2,1,1,1,1)$ gives the same utility as $(x_1,$cdots,$x_5)=(1,1,1,1,2)$ than these are both points on the same indifference surface. An indifference map is the set of all indifference surface for EVERY given utility.

Consider the following utility map:

$U={\displaystyle \underset{i=1}{\overset{5}{}}}ln(x_i-a_i)$

Where $(a_1,$cdots,$a_5)=(4,7,5,7,5)$

The budget constraint gives the set of possible consumption choices with a given income. If you have an income of $$654$ and the price of good $x_i$ is given by $p_i.$ The equation for the budget line is given by: $654={\displaystyle \underset{i=1}{\overset{5}{}}}{p}_i{x}_i.$

A utility maximizing combination of goods $x_{1}cdots{x}_5$ occurs when the surface given by the budget constraint is tangent to an indifference surface.

Find $x_1$ as a function of $p_{1}cdots{p}_5$

$x_1=$

$$

$($Use p$1$ for $p_1$ and likewise for $p_2,p_3,p_4,p_5.$

The easiest way to solve this question is using Lagrange multiplier.

We define the Lagrange function to be:

$(x_1,$cdots,$x_5,)=U(x_1,$cdots,$x_5)-({\displaystyle \underset{i=1}{\overset{5}{}}}{p}_i{x}_i-654)$

Utility is maximized when all of the partial derivatives of the Lagrange function are equal to $0.$