Chapter 6, Problem 6P

To determine

The consumer equilibrium in three commodity case and also draw the demand curve for good A.

Concept Introduction:

The Law of Demand: It states that, keeping other things constant, there is an inverse relation between price and quantity demanded.

Marginal Utility: It is the net addition to the total utility when an additional unit of good is being consumed.

  $\text{MU}={\text{TU}}_{\text{n}}-{\text{TU}}_{\text{n}-1}$

Consumer equilibrium: In order to maximize his utility, the consumer will spend his income in such a way so that the following condition is satisfied:

  $\frac{\text{MU from good} \text{'}\text{x}\text{'}}{\text{Price of good} \text{'}\text{x}\text{'}}=\frac{\text{MU from good} \text{'}\text{y}\text{'}}{\text{Price of good} \text{'}\text{y}\text{'}}$

Explanation of Solution

(a) With the help of given information, we can find out the consumers equilibrium in three commodity case. The following condition has to be satisfied in order to maximize the utility.

  $\frac{\text{MU from good} \text{'}\text{A}\text{'}}{\text{Price of good} \text{'}\text{A}\text{'}}=\frac{\text{MU from good} \text{'}\text{B}\text{'}}{\text{Price of good} \text{'}\text{B}\text{'}}=\frac{\text{MU from good} \text{'}\text{C}\text{'}}{\text{Price of good} \text{'}\text{C}\text{'}}$

The price of A = $2

The price of B = $3

The price of C = $1

Quantity	$M{U}_A$	$M{U}_B$	$M{U}_C$	$\frac{M{U}_A}{P_A}$	$\frac{M{U}_B}{P_B}$	$\frac{M{U}_C}{P_C}$
1	50	75	25	25	25	25
2	40	60	20	20	20	20
3	30	40	15	15	13.33	15
4	20	30	10	10	10	10
5	15	20	7.5	7.5	6.66	7.5

Daniel’s income is $24 per week.

The utility can be maximized when Daniel consumes 4 units of each good A, B and C because at this level, consumer’s equilibrium condition is satisfied and he is spending his whole income on purchase of these goods.

(b) If the price of A is $4 and other things are held constant.

Quantity	$M{U}_A$	$M{U}_B$	$M{U}_C$	$\frac{M{U}_A}{P_A}$	$\frac{M{U}_B}{P_B}$	$\frac{M{U}_C}{P_C}$
1	50	75	25	12.5	25	25
2	40	60	20	10	20	20
3	30	40	15	7.5	13.33	15
4	20	30	10	5	10	10
5	15	20	7.5	3.75	6.66	7.5

In this case, consumer will purchase 2 units of good A and 4 units of good B and C. In this way, he will be able to spend his entire income on the purchase of three goods and maximizes his utility.

(c) When the price of good A is $2, Daniel is consuming 4 units of good A.

When the price of good A rises to $4, Daniel is consuming 2 units of good A. We can plot these points on the graph and will get the demand curve.