Charlie's utility function is x_Ax_B.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 $6 and the price of bananas stayed constant, the substitution effect on Charlie's apple consumption would reduce his consumption by A)16.67 apples. B)5 apples. C)8.33 apples. D)13.33 apples. E)None of the above.

Question

Quizplus · Accepted Answer

The substitution effect measures the change in consumption due to the change in relative prices, holding the level of utility constant. In this case, the relative price of apples has increased (from 1 to 6), which makes apples relatively more expensive than bananas. As a result, Charlie will buy less apples and more bananas to maximize his utility level.

To calculate the substitution effect, we first need to find the optimal bundle of apples and bananas at the new prices, assuming that Charlie's utility level remains the same. The new budget constraint is 6x + 2y =40, and the utility function is xS1U1B1AS1U1B0xS1U1B1BS1U1B0.

Taking the partial derivative of the utility function with respect to x gives:
dU/dx = S1U1B1A / S1U1B1B

Setting this equal to the marginal rate of substitution (MRS), which is the ratio of the prices, we get:
S1U1B1A / S1U1B1B = 6/2
Simplifying, we get:
S1U1B1A / S1U1B1B = 3

Similarly, taking the partial derivative of the utility function with respect to y gives:
dU/dy = S1U1B0A / S1U1B0B

Setting this equal to the MRS (which is still 3), we get:
S1U1B0A / S1U1B0B = 3

Solving these two equations simultaneously gives:
x = 2S1U1B1A / S1U1B1B
y = 4S1U1B0A / S1U1B0B

Substituting in the values gives:
x = 2(1)/(1/2) = 4
y = 4(1)/(2/1) = 8

So, the optimal bundle at the new prices is (4,8).

To find the substitution effect on Charlie's apple consumption, we need to compare his original consumption of apples to his new consumption of apples, while keeping his utility level constant. Charlie's original budget constraint was x + 2y = 40, which means he was buying 20 apples (x=20/1) and 10 bananas (y=10/2).

At the new prices, he can no longer afford to buy 20 apples and 10 bananas. To see how many apples he will buy, we can use the following equation:

(6-p1)/(p2) = (dX/dP1)

Where p1 is the old price of apples, p2 is the old price of bananas, and dX/dP1 is the own-price elasticity of demand for apples.

Using the values given, we get:
(6-1)/(2) = (dX/dP1)
dX/dP1 = 2.5

This means that for every 1% increase in the price of apples, Charlie's apple consumption will decrease by 2.5%.

So, the percentage change in Charlie's apple consumption due to the price increase from $1 to $6 is:
(6-1)/1 * (-2.5) = -12.5

This means that Charlie's apple consumption will decrease by 12.5%. Since he was originally consuming 20 apples, this means he will reduce his consumption by 2.5 apples:

20 * 0.125 = 2.5, or approximately 8.33 apples.

Therefore, the answer is C, 8.33 apples.

Charlie's Utility Function Is XAxB

Charlie's Utility Function Is X_Ax_B