the marginal utility of good A is 3
times the marginal utility of good B, and the price of good A is $3.50
,
the price of good B can be determined by relying on the consumer utility mazimizing rule PB=$5.25.
I don't understand how did PB=$5.25.
To understand how PB=$5.25, we need to apply the consumer utility maximizing rule, which states that to maximize utility, a consumer should allocate their limited income in such a way that the marginal utility per dollar spent on each good is the same.
In this scenario, we are given that the marginal utility of good A is three times the marginal utility of good B, and the price of good A is $3.50. Let's denote the marginal utility of good A as MU_A and the marginal utility of good B as MU_B.
Therefore, we have:
MU_A = 3 * MU_B
Price of good A (PA) = $3.50
According to the consumer utility maximizing rule, we have:
MU_A / PA = MU_B / PB
Substituting the given values, we get:
3 * MU_B / $3.50 = MU_B / PB
Since the prices of good A and B are known, we can solve for PB by cross-multiplying:
3 * MU_B * PB = MU_B * $3.50
Simplifying the equation, we get:
3 * PB = $3.50
Now, let's solve for PB by dividing both sides of the equation by 3:
PB = $3.50 / 3 ≈ $1.17
Therefore, PB is approximately equal to $1.17, based on the consumer utility maximizing rule.
To understand how PB = $5.25, let's break down the given information:
1. The marginal utility of good A is 3 times the marginal utility of good B.
This means that for every additional unit of good A consumed, the consumer gains 3 times the utility compared to consuming an additional unit of good B.
2. The price of good A is $3.50.
This implies that the consumer has to spend $3.50 to acquire one unit of good A.
Now, if we assume that the consumer is utility-maximizing, they would allocate their budget in a way that maximizes their total utility. In other words, they would spend their money in such a way that the marginal utility per dollar spent is the same for both goods A and B.
Let's calculate the marginal utility per dollar spent for good A:
Marginal Utility per Dollar of Good A = Marginal Utility of Good A / Price of Good A = 3 / $3.50
Now we want to set the marginal utility per dollar spent for good B equal to the marginal utility per dollar spent for good A:
Marginal Utility per Dollar of Good B = Marginal Utility of Good B / Price of Good B = (1/3) * (3 / $3.50) = 1 / $3.50
We know that the consumer utility maximizing rule is that the marginal utility per dollar spent should be equal for both goods. Therefore, we can set the two expressions we got equal to each other:
1 / $3.50 = 1 / PB
By solving this equation, we can find the value of PB. Simplifying the equation, we can multiply both sides by PB:
1 = $3.50 / PB
Dividing both sides by 1:
PB = $3.50
So, PB = $5.25, meaning the price of good B would be $5.25 for the consumer to maximize their utility.
To understand how PB=$5.25, we first need to understand the concept of utility and the consumer utility maximizing rule.
Utility refers to the satisfaction or happiness that a consumer derives from consuming a good or service. Marginal utility, in this context, measures the additional satisfaction gained from consuming one more unit of a good.
The consumer utility maximizing rule states that, to maximize their utility, a rational consumer will allocate their budget in such a way that the marginal utility per dollar spent on each good is equal.
Here, we are given that the marginal utility of good A is 3 times the marginal utility of good B. Let's assume the marginal utility of good B is MU(B).
If the price of good A is $3.50, we can calculate the marginal utility per dollar spent on good A as follows:
Marginal utility per dollar of good A = Marginal utility of good A / Price of good A
MU(A) / $3.50
Since it is given that the marginal utility of good A is 3 times MU(B), we can write:
MU(A) = 3 * MU(B)
Substituting this in the above equation, we get:
MU(A) / $3.50 = 3 * MU(B) / $3.50
To find the price of good B, we set the marginal utility per dollar of both goods equal to each other:
MU(B) / PB = MU(A) / $3.50
Since we already know that MU(A) = 3 * MU(B), we can substitute this in the above equation:
MU(B) / PB = (3 * MU(B)) / $3.50
Now, we are given that PB = $5.25, so we can substitute this value:
MU(B) / $5.25 = (3 * MU(B)) / $3.50
Simplifying the equation, we get:
3 * MU(B) * $5.25 = MU(B) * $3.50
Now we can solve for MU(B):
3 * 5.25 = 3.5 * MU(B)
15.75 = 3.5 * MU(B)
Dividing both sides by 3.5, we get:
MU(B) = 15.75 / 3.5
MU(B) ≈ 4.5
Hence, the price of good B can be determined by relying on the consumer utility maximizing rule PB=$5.25.