Question

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?

Short answer

Answer: The minimum income level required for the consumer to purchase any ice cream sundaes is $48.

Step-by-step solution

a. Utility Maximization with Z=0

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).

b. Why Z=0 is optimal

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.

c. Minimum income level for purchasing 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.

Key concepts

Utility Maximization

Utility maximization refers to the process by which a consumer chooses a combination of goods that maximizes their overall satisfaction, given certain constraints like income and prices of goods. Suppose a fast-food enthusiast has a Cobb-Douglas utility function given by \(U(X, Y, Z) = X^5 Y^{-5}(1+Z)^{-5}\), where \(X, Y,\) and \(Z\) represent soft drinks, hamburgers, and ice cream sundaes, respectively.
When maximizing utility with \(Z = 0\), the consumer simplifies the function to \(U(X, Y) = X^5 Y^{-5}\). This scenario respects the budget constraint \(0.25X + Y \leq 2\). By solving this, the consumer finds that purchasing 3 soft drinks and 1.25 hamburgers maximizes their utility at 27 units. In this setup, any attempt to include even a small amount of \(Z\) reduces total utility because \(Z\) negatively affects the utility due to cost implications without proportional satisfaction increase.
Maximizing utility requires allocating resources efficiently to achieve the highest satisfaction, and for this particular case, it means ignoring the sundaes altogether.

Marginal Rate of Substitution

The Marginal Rate of Substitution (MRS) measures how much of one good a consumer is willing to forego to gain an extra unit of another good, while keeping their utility level constant.
For the utility function \(U(X, Y) = X^5 Y^{-5}\), the MRS between soft drinks and hamburgers is calculated as \(MRS = \frac{U_X}{U_Y} = -\frac{Y}{X}\). This is because derivatives of the utility function with respect to \(X\) and \(Y\) yield \(U_X = 5X^4 Y^{-5}\) and \(U_Y = -5X^5 Y^{-6}\), leading to an MRS of \(-Y/X\).
The MRS is equated to the price ratio \(\frac{P_x}{P_y}\), in this case \(-0.25/1\), to find the optimal allocation between the goods. This involves finding the negative slope of the indifference curve and aligning it with the consumer’s budget constraint.

Budget Constraint

A budget constraint represents all combinations of goods a consumer can afford with their income, given prices of the goods. In this case, the consumer's budget constraint is \(0.25X + Y \leq 2\).
This constraint ensures that the consumer’s spending on soft drinks and hamburgers does not exceed their total available budget. Rounding up resources wisely is crucial to not overspend.
Setting \(Z = 0\), rearranging and solving the constraint allows the consumer to afford up to 3 soft drinks and 1.25 hamburgers. However, introducing even minimal amounts of \(Z\) subtracts significantly from the budget, and thus from the possibility of maximizing utility, since \(Z\) is relatively expensive and offers less utility.

Marginal Utility per Dollar Spent

Marginal utility per dollar spent is an essential concept that helps consumers decide how best to allocate their limited resources across various goods.
It is calculated by \(MU_{X} = \frac{U_X}{P_x}\), \(MU_{Y} = \frac{U_Y}{P_y}\), and \(MU_{Z} = \frac{U_Z}{P_z}\). For instance, the calculation shows that \(MU_{X} > MU_{Z}\) and \(MU_{Y} > MU_{Z}\), making expenditures on \(X\) and \(Y\) more efficient in terms of utility gain per dollar than on \(Z\).
Since the marginal utility per dollar for the ice cream sundae \((Z)\) is negative, it implies that any monetary allocation towards \(Z\) results in reduced total satisfaction, thereby justifying why \(Z = 0\) is optimal. The consumer maximizes satisfaction by consuming more of the goods that provide higher utility relative to their prices.