Charlie’s utility function is xA ∗ xB . The price of apples used to be $1, the price of bananas used to be $2, and his income used to be $40. If the price of apples increased to $4 and the price of bananas stayed constant, how much the substitution effect on Charlie’s apple consumption would reduce his consumption? show all the calculations.
To calculate the substitution effect on Charlie's apple consumption, we need to compare his initial consumption of apples with his consumption after the price increase and income adjustment.
Initially, Charlie's income is $40, the price of apples is $1, and the price of bananas is $2. Let's assume he consumes xA apples and xB bananas.
His initial expenditure on apples is xA * $1 = xA.
His initial expenditure on bananas is xB * $2 = 2xB.
His initial utility is xA * xB.
Now, let's calculate Charlie's consumption after the price increase. The price of apples is now $4, and the price of bananas remains unchanged at $2.
Considering his income is still $40, his new expenditure on apples can be calculated as:
(xA - ΔxA) * $4, where ΔxA is the reduction in his apple consumption due to the substitution effect.
His expenditure on bananas remains unchanged at 2xB * $2 = 4xB.
Since his income remains the same, his new expenditure on apples and bananas should not exceed $40. Therefore, we can write the following equation:
(xA - ΔxA) * $4 + 4xB = $40
Simplifying the equation, we get:
4xA - 4ΔxA + 4xB = $40
Now, let's determine ΔxA, the reduction in Charlie's apple consumption due to the substitution effect.
Since Charlie's utility function is xA * xB, we can assume that the marginal utility of apples is equal to the marginal utility of bananas, given that we are only considering the substitution effect.
The marginal utility of apples is the derivative of his utility function with respect to xA:
∂U/∂xA = xB
The marginal utility of bananas is the derivative of his utility function with respect to xB:
∂U/∂xB = xA
Since we're assuming the marginal utilities are equal, we can equate these two derivatives:
xB = xA
Now, let's substitute this value into the earlier equation:
4xA - 4ΔxA + 4xA = $40
Simplifying this equation, we get:
8xA - 4ΔxA = $40
We know that xA * xB is Charlie's utility function, and we have assumed xB = xA. Therefore, we can substitute xB = xA into the utility function to express it solely in terms of xA:
U = xA * xA = xA^2
Now, we can differentiate this utility function with respect to xA to find the marginal utility of apples:
∂U/∂xA = 2xA
Since we are only considering the substitution effect, the marginal utility of apples remains constant. Therefore, we can equate the marginal utility of apples before the price increase (xB) to the marginal utility after the price increase (2xA):
xB = 2xA
Now, let's substitute this value into the equation we derived earlier:
8xA - 4ΔxA = $40
Since xB = 2xA, we can substitute this into the equation:
8xA - 4ΔxA = $40
8xA - 4(2xA) = $40
8xA - 8xA = $40
-4ΔxA = $40
Simplifying this equation, we find:
ΔxA = -$10
Therefore, the substitution effect on Charlie's apple consumption would reduce his consumption of apples by $10.