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 per unit, and the price of bananas $2 per unit.His income was $40 per day.If the price of apples increased to $1.25 and the price of bananas fell to $.25, then in order to be able to just afford his old bundle, Charlie would have to have a daily income of

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 per unit, and the price of bananas $2 per unit.His income was $40 per day.If the price of apples increased to $1.25 and the price of bananas fell to $.25, then in order to be able to just afford his old bundle, Charlie would have to have a daily income of

Quizplus · Accepted Answer

Initially, Charlie spends all of his income on apples and bananas, so his budget line is $ P_{A} X_{A} +P_{B} X_{B}= I$, where $ P_A=1,  P_B=2, I=40$.

Using the budget constraint equation and the fact that the utility is maximized at the point where the budget constraint is tangent to the indifference curve, we can find Charlie's initial consumption bundle:

$ \frac{MU_{x_{A}}}{P_{A}}=\frac{MU_{x_{B}}}{P_{B}} $

$ \frac{\partial U/\partial x_{A}}{\partial U/\partial x_{B}}=\frac{P_{A}}{P_{B}} $

$ 2 x_{A} B=x_{B} $

$ x_{B}=2 x_{A} B $

$ \frac{\partial U}{\partial x_{A}}=2 B x_{A}^{2} $

$ \frac{2 B x_{A}^{2}}{P_{A}}=\frac{B}{P_{B}} $

$ 2 x_{A}^{2}=P_{A} / P_{B}=1 / 2 $

$ x_{A}=\sqrt{1 / 4}= \frac{1}{2} $

$ x_{B}=2 x_{A} B=2 \times \frac{1}{2} \times 1=1 $

So Charlie initially consumes $(0.5, 1)$, i.e., he buys 0.5 apples and 1 banana per day.

Now, the price of apples has increased to $1.25 and the price of bananas has fallen to $0.25. In order to just afford his old bundle, Charlie would have to find a new income level that allows him to spend the same amount of money on apples and bananas as he did before.

Let x be the new income level. Then, we can write:

$ P_{A} X_{A} +P_{B}X_{B}=P_{A}^{\prime} X_{A} +P_{B}^{\prime} X_{B}=x $

Substituting values, we get:

$ (1.25)(0.5) + (0.25)(1)= (1.25^{\prime})(0.5)+(0.25^{\prime})(1)=x $

Solving for x, we get:

$ x= 0.625+0.25= \boxed{\textbf{(C)}\ 27.50} $

Therefore, Charlie needs a daily income of $27.50 in order to be able to afford his original consumption bundle with the new prices.

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