A rational consumer spends all of her income on two goods: Apple and Banana. Suppose the last dollar spent on Apple increased her total utility from 60 utils to 68 utils and the last dollar spent on Banana increased her total utility from 25 utils to 29 utils. If the price of a unit of Apple is 2 Birr, what is the price of a unit of Banana at equilibrium?
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To find the price of a unit of Banana at equilibrium, we need to determine the marginal utility per dollar spent on each good.
The marginal utility per dollar spent on Apple is the change in total utility divided by the change in the amount spent on Apple. In this case, the change in total utility is 68 - 60 = 8 utils and the change in the amount spent on Apple is 1 Birr (since the price of a unit of Apple is 2 Birr and the last dollar spent on Apple increased utility by 8 utils), so the marginal utility per dollar spent on Apple is 8/1 = 8 utils per Birr.
Similarly, the marginal utility per dollar spent on Banana is the change in total utility divided by the change in the amount spent on Banana. In this case, the change in total utility is 29 - 25 = 4 utils and the change in the amount spent on Banana is 2 Birr (since the last dollar spent on Banana increased utility by 4 utils and the price of a unit of Apple is 2 Birr), so the marginal utility per dollar spent on Banana is 4/2 = 2 utils per Birr.
At equilibrium, the marginal utility per dollar spent on each good should be equal. So we can set up the following equation:
8 utils per Birr (marginal utility per dollar spent on Apple) = 2 utils per Birr (marginal utility per dollar spent on Banana)
Simplifying the equation, we get:
8 = 2
This equation is not true, so there is no equilibrium price for Banana that satisfies the conditions given in the problem.
To find the price of a unit of Banana at equilibrium, we can use the concept of marginal utility and the principle of equi-marginal utility.
1. First, let's calculate the marginal utility of the last dollar spent on each good:
- On Apple: The increase in total utility from 60 utils to 68 utils implies that the additional utility from the last dollar spent on Apple is 68 - 60 = 8 utils.
- On Banana: The increase in total utility from 25 utils to 29 utils implies that the additional utility from the last dollar spent on Banana is 29 - 25 = 4 utils.
2. Next, we need to compare the marginal utility per dollar spent on each good. To do this, we divide the marginal utility by the price of each good:
- Marginal utility per dollar spent on Apple: 8 utils / 1 dollar = 8 utils/dollar.
- Marginal utility per dollar spent on Banana: 4 utils / (Price of Banana) dollars = 4 utils/(Price of Banana) utils/dollar.
3. At equilibrium, a rational consumer will allocate their budget in a way that the marginal utility per dollar spent is equal for both goods. Therefore, we can set the marginal utility per dollar spent on Apple equal to the marginal utility per dollar spent on Banana:
8 utils/dollar = 4 utils/(Price of Banana) utils/dollar.
4. Rearranging the equation and solving for the price of Banana:
8/(Price of Banana) = 4.
Price of Banana = 8/4 = 2 Birr per unit.
Therefore, the price of a unit of Banana at equilibrium is 2 Birr.
To determine the price of a unit of Banana at equilibrium, we need to use the concept of marginal utility and the principle of equimarginal utility.
First, let's calculate the marginal utility per dollar spent on Apple and Banana:
For Apple:
The increase in total utility from 60 utils to 68 utils when the last dollar is spent on Apple tells us that the marginal utility of the last dollar spent on Apple is 8 utils (68 utils - 60 utils = 8 utils).
For Banana:
The increase in total utility from 25 utils to 29 utils when the last dollar is spent on Banana tells us that the marginal utility of the last dollar spent on Banana is 4 utils (29 utils - 25 utils = 4 utils).
Now, let's calculate the price ratio between Apple and Banana:
The price of a unit of Apple is given as 2 Birr. Since the marginal utility per dollar spent on Apple is 8 utils, this means that each Birr spent on Apple generates 4 utils of utility (8 utils / 2 Birr = 4 utils/Birr).
Therefore, the price of a unit of Banana at equilibrium should be such that it also generates 4 utils of utility per Birr spent. In other words, the price of a unit of Banana should be equal to the utility generated by each Birr spent on Apple. So, the price of a unit of Banana at equilibrium is 2 Birr.
In summary, the price of a unit of Banana at equilibrium is 2 Birr.