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 $$U(X, Y, Z)=X^{5} F-^{5}(1+Z)^{-5}$$ Suppose also that the prices for these goods are given by \(P_{x}=.25, P_{y}=1,\) and \(P_{x}=2\) and that this consumer's income is given by \(/=2\) a. Show that for \(Z=0\), maximization of utility results in the same optimal choices as in Ex ample \(4.1 .\) Show also that any choice that results in \(Z>0\) (even for a fractional \(Z\) ) re duces utility from this optimum. b. How do you explain the fact that \(Z=0\) is optimal here? (Hint: Think about the ratio \(M U J P_{z}\) c. How high would this individual's income have to be in order for any \(Z\) to be purchased?

To maximize the utility function with Z=0, we will have: $$U(X, Y) = X^{5}Y^{-5}$$ The consumer's budget constraint is: $$0.25X + Y + 0 \leq 2$$ Rearranging the budget constraint to isolate Y, we get: $$Y \leq 2 - 0.25X$$ Now, we will divide both sides of the constraint by their respective prices to obtain the quantities of X and Y: $$\frac{1}{4}X \leq \frac{3}{4}$$ So, the constraint for X is given by: $$X \leq 3$$ Now, we will compute the Marginal Rate of Substitution (MRS) for X and Y: $$MRS = \frac{U_{X}(x,y)}{U_{Y}(x,y)}$$ Taking the partial derivatives: $$U_{X}(x,y) = 5X^{4}Y^{-5}$$ $$U_{Y}(x,y) = -5X^{5}Y^{-6}$$ So MRS becomes: $$MRS = \frac{5X^{4}Y^{-5}}{-5X^{5}Y^{-6}} = -Y/X$$ We need to equate the MRS to the ratio of prices Px/Py: $$\frac{Y}{X}=-\frac{0.25}{1}$$ Solving for Y, we get: $$Y = -0.25X$$ Substituting this expression into the budget constraint and solving for X yields: $$0.25X + (-0.25X) \leq 2$$ $$X=3$$ and $$Y=2-0.25(3)=1.25$$ This is the same optimal choice of Example 4.1. Now to show that any Z>0 reduces utility, we can compare the utility levels at Z=0 and Z>0. At Z=0, the utility is given by: $$U(3,1.25)=3^{5}(1.25)^{-5}=27$$ At Z>0, let's assume Z=0.5 for example. The new budget constraint becomes: $$0.25X + Y + 2Z \leq 2$$ So $$0.25X+Y \leq 2-2(0.5) = 1$$ Solving for Y, we get: $$Y = 1 - 0.25X$$ The new MRS becomes: $$\frac{Y}{X}=-\frac{0.25}{1}$$ Solving for Y again, we get: $$Y = -0.25X$$ Substituting this into the budget constraint yields: $$0.25X + (-0.25X) \leq 1$$ $$X=1$$ and $$Y=1-0.25(1)=0.75$$ The new utility level is: $$U(1,0.75,0.5) = 1^{5}(0.75)^{-5}(1.5)^{-5} = 0.139$$ Comparing utility levels, we can see that any choice that results in Z>0 reduces utility (27 > 0.139).

To explain why Z=0 is optimal, we can look at the marginal utility per dollar spent on each good. $$MU_{X} = \frac{U_{X}(x,y)}{P_{x}} = \frac{5X^{4}Y^{-5}}{0.25}$$ $$MU_{Y} = \frac{U_{Y}(x,y)}{P_{y}} = \frac{-5X^{5}Y^{-6}}{1}$$ $$MU_{Z} = \frac{U_{Z}(x,y,z)}{P_{z}} = \frac{-25X^{5}Y^{-5}Z^{-6}}{2}$$ By comparing the marginal utility per dollar spent, if there is a good with a better marginal utility per dollar spent, the consumer will want to allocate resources towards that good in an optimal situation. In this case, we notice that both \(MU_{X}\) and \(MU_{Y}\) can both be positive and relatively large compared to \(MU_{Z}\). When Z=0, the marginal utility per dollar spent on Z is negative, meaning that the consumer derives negative utility from spending on Z, thus it would not be optimal to spend on Z.

To find the minimum income level for purchasing Z, we need to find a situation where the marginal utility per dollar spent on Z is at least equal to the marginal utility per dollar spent on X or Y. $$\frac{MU_{X}}{MU_{Z}} \leq 1$$ $$\frac{5X^{4}Y^{-5}}{0.25} \geq \frac{-25X^{5}Y^{-5}Z^{-6}}{2}$$ Simplifying, $$Y \geq -0.5XZ^{-6}$$ Since Z>0, Y must be positive. Therefore, $$-0.5XZ^{-6} > 0$$ We can consider the equality as a boundary: $$-0.5XZ^{-6} = 0$$ Now, we should find the value of Z when income is such that the consumer can consume a small quantity of Z. Let's assume Z=0.5. $$-0.5(3)(0.5)^{-6} = I$$ Solving for I, we get $$I = 48$$ So, the minimum income level for this individual to purchase any ice cream sundaes (Z) is $48.