Question

John has a utility function
, where A, and are constants, B is burritos, and Z is pizzas. If the price of burritos, Pb is 10 and the price of pizzas, Pz, is N$5, and Y is N$1790, what is John’s optimal bundle

Answer 1

Question #76823, Economics, Microeconomics

John has a utility function $U(B,Z) = AB^{\wedge}(1 / \alpha) Z^{\wedge}(1 / \beta)$, where A, and are constants, B is burritos, and Z is pizzas. If the price of burritos, P b is $10 and the price of pizzas, P z, is $5, and Y is $1790, what is John's optimal bundle?

$$U (B, Z) = A B ^ {\frac {1}{\alpha}} Z ^ {\frac {1}{\beta}}$$ $$I = p _ {B} * B + p _ {Z} * Z$$ $$1 7 9 0 = 1 0 * B + 5 * Z$$ $$M U _ {b} = A \frac {1}{\alpha} B ^ {\frac {1}{\alpha} - 1} Z ^ {\frac {1}{\beta}}$$ $$M U _ {z} = A \frac {1}{\beta} B ^ {\frac {1}{\alpha}} Z ^ {\frac {1}{\beta} - 1}$$ $$M R S _ {b, z} = - \frac {\frac {1}{\alpha} B ^ {\frac {1}{\alpha} - 1} Z ^ {\frac {1}{\beta}}}{\frac {1}{\beta} B ^ {\frac {1}{\alpha}} Z ^ {\frac {1}{\beta} - 1}} = - \frac {\beta Z}{\alpha B}$$ $$M R S _ {b, z} = - \frac {p _ {b}}{p _ {z}}$$ $$\frac {p _ {b}}{p _ {z}} = \frac {\beta Z}{\alpha B}$$

For $Z^{*}$

$$p _ {b} B = \frac {\beta}{\alpha} p _ {z} Z$$

From the budget constraint, we have the following:

$$\frac {\beta}{\alpha} p _ {z} Z + p _ {z} Z = I$$ $$\frac {\beta}{\alpha} * 5 Z + 5 Z = 1 7 9 0$$ $$5 Z \left(\frac {\beta}{\alpha} + 1\right) = 1 7 9 0$$ $$\frac {\beta + \alpha}{\alpha} * 5 Z = 1 7 9 0$$ $$Z = \frac {1 7 9 0}{5} * \frac {\alpha}{\alpha + \beta} = \frac {3 5 8 \alpha}{\alpha + \beta}$$

For $B^{*}$

$$p _ {b} B + p _ {z} \frac {3 5 8 \alpha}{\alpha + \beta} = 1 7 9 0$$ $$\begin{array}{l} 10B + 5 \frac{358\alpha}{\alpha + \beta} = 1790 \\ 10B + 1790 \frac{\alpha}{\alpha + \beta} = 1790 \\ 10B = 1790 \left(1 - \frac{\alpha}{\alpha + \beta}\right) \\ B = 179 \left(\frac{\alpha + \beta - \alpha}{\alpha + \beta}\right) = \frac{179\beta}{\alpha + \beta} \\ \end{array}$$

Optimal bundle is $B = \frac{179\beta}{\alpha + \beta}$ and $Z = \frac{358\alpha}{\alpha + \beta}$

Answer provided by https://www.AssignmentExpert.com