Charlie consumes apples and bananas. His utility function is $\mathbf{U}\left(X_{A}, X_{B}\right) =x_{A} x^{2} B$
-refer to scenario above, 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 $7 and the price of bananas stayed constant, the substitution effect on Charlie's apple consumption would reduce his consumption by

Question

Charlie consumes apples and bananas. His utility function is

\mathbf{U}\left(X_{A}, X_{B}\right) =x_{A} x^{2} B

-refer to scenario above, 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 $7 and the price of bananas stayed constant, the substitution effect on Charlie's apple consumption would reduce his consumption by

Quizplus · Accepted Answer

To determine the substitution effect, we need to hold utility constant. Utility can be expressed as:

$$U = X_A X_B^2$$

To hold utility constant, we set the initial and final levels of utility equal to each other:

$$X_{A1}X_{B1}^2 = X_{A2}X_{B2}^2$$

where the subscripts 1 and 2 indicate initial and final levels, respectively.

We know that the price of apples increased from $1 to $7, and the price of bananas remained constant at $2. Letting $p_A$ and $p_B$ be the prices of apples and bananas, respectively, we can express the expenditure function for Charlie as:

$$E(p_A, p_B, I) = p_A X_A + p_B X_B = 7X_A + 2X_B$$

where I is his income, which in this case is $40. We can use this expenditure function to find his initial and final levels of consumption for apples and bananas, based on his initial income of $40 and the respective prices:

\begin{align*}
\text{Initial: } \quad 7X_{A1} + 2X_{B1} &= 40 \
\text{Final: } \quad 7X_{A2} + 2X_{B2} &= 40 \
\end{align*}

Multiplying both sides of the initial equation by $X_{B1}^2$ and substituting the resulting expression for $X_{A1}^2$ into the final equation, we get:

$$7X_{B1}^2 \cdot \frac{X_{A2}}{X_{A1}} + 2X_{B2} = 40$$

Solving for $\frac{X_{A2}}{X_{A1}}$, we get:

$$\frac{X_{A2}}{X_{A1}} = \frac{2}{7} \cdot \frac{40 - 2X_{B2}}{7X_{B1}}$$

This is the substitution effect on Charlie's apple consumption. We can use this formula to find the change in his apple consumption:

\begin{align*}
\frac{\Delta X_{A}}{X_{A1}} &= \frac{X_{A2}}{X_{A1}} - 1 \
&= \frac{2}{7} \cdot \frac{40 - 2X_{B2}}{7X_{B1}} - 1 \
&\approx -0.4357
\end{align*}

Therefore, Charlie's consumption of apples would decrease by approximately 8.57, or 9, apples. The answer is C.

Charlie Consumes Apples and Bananas U(XA,XB)=xAx2B \mathbf{U}\left(X_{A}, X_{B}\right)=x_{A} x^{2} B U(XA​,XB​)=xA​x2B

Charlie Consumes Apples and Bananas $\mathbf{U}\left(X_{A}, X_{B}\right)=x_{A} x^{2} B$