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 the initial utility at the original prices to the hypothetical utility at the new prices, assuming his income remains the same.
Initial utility:
Let xA be the quantity of apples consumed and xB be the quantity of bananas consumed.
The original utility function is given by:
U = xA * xB
At the original prices, pA = $1 and pB = $2.
Total expenditure on apples:
EA = pA * xA = $1 * xA = xA
Total expenditure on bananas:
EB = pB * xB = $2 * xB = 2 * xB
Total expenditure on both apples and bananas:
EA + EB = xA + 2 * xB
Given Charlie's income used to be $40, we can set up the budget constraint:
EA + EB = 40
xA + 2 * xB = 40
Solving for xB:
xB = (40 - xA) / 2
Substituting this value of xB into the utility function:
U = xA * (40 - xA) / 2
= (40 * xA - xA^2) / 2
= 20 * xA - 0.5 * xA^2
Substitution effect:
Let xA' be the new quantity of apples consumed.
At the new price of apples, pA' = $4 and pB = $2.
Total expenditure on apples:
EA' = pA' * xA' = $4 * xA' = 4 * xA'
Total expenditure on both apples and bananas:
EA' + EB = 4 * xA' + 2 * xB
The budget constraint remains the same:
EA' + EB = 40
Substituting the expression for xB:
4 * xA' + 2 * ((40 - xA') / 2) = 40
Simplifying:
4 * xA' + (40 - xA') = 40
4 * xA' + 40 - xA' = 40
3 * xA' + 40 = 40
3 * xA' = 0
xA' = 0
The substitution effect on Charlie's apple consumption would reduce his consumption to zero.