a. Suppose that a fast-food junkie derives utility from three goods: soft drinks \((X),\) hamburgers \((Y),\) and ice cream sundaes \((Z)\) according to the Cobb- Douglas utility function $$U(X, Y, Z)=X^{5} Y^{5}(1+Z)^{.5}$$ Suppose also that the prices for these goods are given by \(P_{x}=.25, P_{Y}=1,\) and \(P_{z}=2\) and that this consumer's income is given by \(I=2\) a. Show that for \(Z=0\), maximization of utility results in the same optimal choices as in Example \(4.1 .\) Show also that any choice that results in \(Z>0\) (even for a fractional \(Z\) ) reduces utility from this optimum. b. How do you explain the fact that \(Z=0\) is optimal here? (Hint: Think about the ratio \(\left.M U_{z} / P_{z^{*}}\right)\) c. How high would this individual's income have to be in order for any \(Z\) to be purchased?

We are given a Cobb-Douglas utility function: \(U(X, Y, Z)=X^{5}Y^{5}(1+Z)^{.5}\), and the prices for the goods are: \(P_X=0.25\), \(P_Y=1\), and \(P_Z=2\). The consumer's income is \(I=2\).

To check the utility for Z=0, we can plug this into the utility function: \(U(X, Y, 0) = X^{5}Y^{5}\), where X and Y can be any non-negative numbers. The consumer's budget constraint is \(0.25X + Y + 2(0) = 2\). The given information asks us to compare it with the result of Example 4.1. Assuming that Example 4.1 refers to the optimal solution for a Cobb-Douglas function with \(U(X, Y) = X^{5}Y^{5}\) and the same prices, the optimal choices for X and Y are the same in this case, because Z=0 does not affect the overall utility function.

We need to show that any choice resulting in Z>0 reduces utility from the optimum. Let's calculate the marginal utilities for each good: - \(MU_X = \frac{\partial U}{\partial X} = 5X^{4}Y^{5}(1+Z)^{.5}\) - \(MU_Y = \frac{\partial U}{\partial Y} = 5X^{5}Y^{4}(1+Z)^{.5}\) - \(MU_Z = \frac{\partial U}{\partial Z} = 0.5 X^{5}Y^{5}(1+Z)^{-.5}\) Now, observe the marginal utility to price ratios of each good: - \(\frac{MU_X}{P_X} = \frac{5X^{4}Y^{5}(1+Z)^{.5}}{0.25} = 20X^{4}Y^{5}(1+Z)^{.5}\) - \(\frac{MU_Y}{P_Y} = \frac{5X^{5}Y^{4}(1+Z)^{.5}}{1} = 5X^{5}Y^{4}(1+Z)^{.5}\) - \(\frac{MU_Z}{P_Z} = \frac{.5X^{5}Y^{5}(1+Z)^{-.5}}{2} = 0.25 X^{5}Y^{5}(1+Z)^{-.5}\) For any value of Z>0, the ratio \(\frac{MU_Z}{P_Z}\) will be smaller than the other two ratios due to the negative exponent in its term \((1+Z)^{-.5}\). This indicates that the optimal consumption bundle involves consuming only X and Y, and no Z, since the marginal utility gained per dollar spent on Z is lower than the other goods. Therefore, any choice of Z>0 will reduce utility from the optimum.

To find the minimum income required for the consumer to purchase any amount of Z, we will need to compare the marginal utility to price ratios again. The consumer will start buying Z when the ratio \(\frac{MU_Z}{P_Z}\) becomes greater than or equal to the ratios of the other goods. In this scenario: \(\frac{MU_Z}{P_Z} \geq \frac{MU_X}{P_X}\) and \(\frac{MU_Z}{P_Z} \geq \frac{MU_Y}{P_Y}\) Now, solve for I using the budget constraint: \(0.25X + Y + 2Z = I\) We want to maximize the ratio \(\frac{MU_Z}{P_Z}\) while decreasing the other ratios, so we can set them equal to each other: \(\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y} = \frac{MU_Z}{P_Z}\) Solving this equation for income, we get: \(I = 16X + 8Y\) Hence, the minimum income required for the consumer to purchase any amount of Z is when \(I = 16X + 8Y\). Without knowing the specific preferences of the consumer, we cannot determine an exact value for I, but this equation provides a general guideline for determining the minimum income necessary to incorporate Z into the consumer's consumption bundle.