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 $5 and the price of bananas stayed constant, the substitution effect on Charlie's apple consumption would reduce his consumption by A)4 apples. B)13 apples. C)8 apples. D)16 apples. E)None of the above.

Question

Quizplus · Accepted Answer

The substitution effect measures the change in consumption of a good due to a change in its relative price, holding utility constant.

To calculate the substitution effect for apples, we need to find the uncompensated price elasticity of apple demand. We can do this by taking the partial derivative of Charlie's utility function with respect to the price of apples, multiplied by the price of apples divided by the quantity of apples:

elasticity = (dU/dP) * (P/Xa)

= ((UxS1U1B1A - UxS1U1B0A)/(5-1)) * (1/Xa)

= ((1/5) - (0/1)) * (1/Xa)

= 1/5Xa

This means that a 1% increase in the price of apples will lead to a 1/5% decrease in the quantity of apples demanded. Now we can calculate the substitution effect:

ΔXa = - (ΔPa/Pa) * (Xa/Pa) * elasticity

= - ((5-1)/1) * (20/5) * (1/5Xa)

= - 4 * (4/Xa)

= -16/Xa

So the substitution effect on Charlie's apple consumption would reduce his consumption by 16/Xa apples. To find Xa, we need to solve for it using Charlie's budget constraint:

Px*Xa + Py*Xb = M

(5)(Xa) + (2)(Xb) = 40

Xa = (40-2Xb)/5

Now we can substitute this expression for Xa into the formula for the substitution effect:

ΔXa = -16/Xa

= -16/[(40-2Xb)/5]

= -8/(20-Xb)

So the substitution effect says that a price increase from $1 to $5 reduces Charlie's apple consumption by about 8/(20-Xb) apples. To find Xb, we need to use Charlie's utility-maximizing rule:

MUa/Pa = MUb/Pb

UxS1U1B1A / 5 = UxS1U1B1B / 2

Solving for Xb, we get:

Xb = (20/p) * (UxS1U1B1A / UxS1U1B1B)

So if we know the numerical value of UxS1U1B1B, we can solve for Xb and then use the formula for the substitution effect to get the numerical value of the reduction in Charlie's apple consumption. However, the problem does not provide us with UxS1U1B1B, so we cannot find the exact reduction in apple consumption. We can only conclude that it is approximately 8 apples, which corresponds to answer choice (C).

Charlie's Utility Function Is XAxB

Charlie's Utility Function Is X_Ax_B