Question

At a consumer optimum involving goods $\mathrm{A}$and $\mathrm{B}$, the marginal utility of good $A$is $2$utils, and the marginal utility of good $B$is $8$utils. How much greater or smaller is the price of good B compared with the price of good $\mathrm{A}$?

Short answer

The price of good B is $4$times of good $A$.

Step-by-step solution

Introduction

A consumer has many needs but only so much money to fulfil them. He wants to make the most of his limited resources in order to maximise his happiness. The utility theory explains how consumers pick between the various consumption options available to them.

Explanation

Consumer Optimum: When a consumer uses his limited income in the most efficient way possible, he has figured out what to buy and how much to buy so that his total utility is maximised.

Customer Optimum: If a consumer spends his money in such a way that the last dollar spent on various things offers him the same marginal value, he would maximise his pleasure.

$=\mathrm{MU}$from $good\mathrm{A}=\mathrm{MU}$from $good\mathrm{B}=\dots =\mathrm{MU}$from $goodZ$

Because different items are measured in different units (such as litres, metres, pounds, and so on) and have different prices, their marginal utilities can't be directly compared. To make them comparable, the ratios of their MU with their pricing should be removed.

$\frac{\text{MU of}goodA}{\text{Price of}goodA}=$$\frac{\mathrm{MU}\text{of}good\mathrm{B}}{\text{Price of}good\mathrm{B}}=\dots$$=\frac{\text{MU of}goodZ}{\text{Price of}goodZ}$

$=\frac{M{U}_A}{P_A}=\frac{M{U}_B}{P_B}=\dots \frac{M{U}_z}{P_z}$

Given his money income to spend on the two things, the customer considers two factors: the marginal utility of the two goods and their pricing.

Any change in the relative costs of one or more items will cause the consumer to re-evaluate his purchases, swapping one good for the other until the marginal utility of the last dollar spent on each good equalises.

Solution

Because there are only two products A and B involved:

$=\frac{M{U}_A}{P_A}=\frac{M{U}_B}{P_B}$

Where,

${\mathrm{MU}}_{\mathrm{A}}=2$Utils

${\mathrm{MU}}_{\mathrm{B}}=8$ Utils

Solution

Replace the following values in the equation:

$=\frac{M{U}_A}{P_A}=\frac{M{U}_B}{P_B}$

$=\frac{2}{P_A}=\frac{8}{P_B}$

$=\frac{P_B}{P_A}=\frac{8}{2}$

$=\frac{P_B}{P_A}=4$

$P_B=4\times P_A$

As a result, the cost of good B is four times that of good A.