Question

The Phillie Phanatic (PP) always eats his ballpark franks in a special way; he uses a foot-long hot dog together with precisely half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. His utility is a function only of these four items, and any extra amount of a single item without the other constituents is worthless. a. What form does PP's utility function for these four goods have? b. How might we simplify matters by considering PP's utility to be a function of only one good? What is that good? c. Suppose foot-long hot dogs cost \(\$ 1.00\) each, buns cost \(\$ 0.50\) each, mustard costs \(\$ 0.05\) per ounce, and pickle relish costs S0.15 per ounce. How much does the good defined in part (b) cost? d. If the price of foot-long hot dogs increases by 50 percent (to \(\$ 1.50\) each), what is the percentage increase in the price of the good? How would a 50 percent increase in the price of a bun affect the price of the good? Why is your answer different from part (d)? f. If the government wanted to raise \(\$ 1.00\) by taxing the goods that \(\mathrm{PP}\) buys, how should it spread this tax over the four goods so as to minimize the utility cost to PP?

Short answer

Answer: To minimize the utility cost to PP, the government should implement taxes on each good proportional to the exponents in his utility function. Specifically, this involves distributing the tax revenue as follows: - Foot-long hot dogs: 20% of $1.00 tax, which is $0.20 - Buns: 10% of $1.00 tax, which is $0.10 - Mustard: 20% of $1.00 tax, which is $0.20 - Pickle relish: 40% of $1.00 tax, which is $0.40

Step-by-step solution

a. Identify the form of PP's utility function

Based on the given information, PP's utility function (U) depends on the consumption of four goods: foot-long hot dogs (H), buns (B), mustard (M), and pickle relish (R). However, he consumes these goods in fixed proportions: 1 hot dog, 1/2 bun, 1 ounce of mustard, and 2 ounces of pickle relish. Since the utility function is only affected by these items when consumed in these precise ratios, it means PP's utility function follows a Cobb-Douglas form: $$ U(H, B, M, R) = H^a \cdot B^b \cdot M^c \cdot R^d $$ Where \(a=1\), \(b=0.5\), \(c=1\), and \(d=2\), as given by the specific proportions.

b. Simplify the utility function

To simplify matters, we can consider PP's utility to be a function of only one good. Since all the goods are consumed in fixed proportions, we can define a composite good (G) as the combination of all four goods in the specific proportions mentioned: $$ G = H^{1} \cdot B^{0.5} \cdot M^{1} \cdot R^{2} $$ Now, the utility function becomes a function of that single composite good: $$ U(G) = G $$

c. Determine the cost of the composite good

To find the cost of the composite good (G), we first need to determine the cost of each component consumed in the specific proportions previously identified. The given prices are: \(\$ 1.00\) per foot-long hot dog, \(\$ 0.50\) per bun, \(\$ 0.05\) per ounce of mustard, and \(\$ 0.15\) per ounce of pickle relish. Using the fixed proportions, the cost of the composite good (G) is: $$ Cost(G) = 1 \cdot(\$1.00) + 0.5 \cdot(\$0.50) + 1 \cdot(\$0.05) + 2 \cdot(\$0.15) = \$ 1.00 + \$ 0.25 + \$ 0.05 + \$ 0.30 = \$ 1.60 $$

d. Effects of price changes on the composite good

If the price of foot-long hot dogs increases by 50 percent (to \(\$ 1.50\) each), we can calculate the percentage increase in the price of the composite good by comparing the new cost to the original cost: $$ New\ Cost(G) = 1 \cdot(\$1.50) + 0.5 \cdot(\$0.50) + 1 \cdot(\$0.05) + 2 \cdot(\$0.15) = \$ 1.50 + \$ 0.25 + \$ 0.05 + \$ 0.30 = \$ 2.10 $$ The percentage increase in the price of the composite good is: $$ \frac{(2.10 - 1.60)}{1.60} = 0.3125 = 31.25\% $$

e. Effects of a 50% increase in the price of buns

If the price of buns increases by 50 percent (to \(\$ 0.75\) each), the new cost of the composite good is: $$ New\ Cost(G) = 1 \cdot(\$1.00) + 0.5 \cdot(\$0.75) + 1 \cdot(\$0.05) + 2 \cdot(\$0.15) = \$ 1.00 + \$ 0.375 + \$ 0.05 + \$ 0.30 = \$ 1.725 $$ The percentage increase in the price of the composite good is: $$ \frac{(1.725 - 1.60)}{1.60} = 0.078125 = 7.81\% $$ This result is different from part (d) because buns are consumed in half proportion compared to the hot dogs, so an increase in their price has a smaller impact on the overall cost of the composite good.

f. Taxation to minimize utility cost to PP

To minimize the impact on PP's utility, the government should spread the tax in proportion to the elasticities of the goods. Since PP's utility function follows a Cobb-Douglas form and each good's exponent corresponds to its share in utility, the government should implement the tax proportional to the exponents: - Foot-long hot dogs: \(\frac{1}{1+0.5+1+2} = 20\%\) of \(\$ 1.00\) tax, which is \(\$ 0.20\) - Buns: \(\frac{0.5}{1+0.5+1+2} = 10\%\) of \(\$ 1.00\) tax, which is \(\$ 0.10\) - Mustard: \(\frac{1}{1+0.5+1+2} = 20\%\) of \(\$ 1.00\) tax, which is \(\$ 0.20\) - Pickle relish: \(\frac{2}{1+0.5+1+2} = 40\%\) of \(\$ 1.00\) tax, which is \(\$ 0.40\) By implementing the tax as stated above, the government would minimize the utility cost to PP.

Key concepts

Microeconomics

Microeconomics is a branch of economics that studies how individuals, households, and firms make decisions to allocate limited resources, typically in markets where goods or services are being bought and sold. It examines the mechanisms that influence the behavior of economic agents and how they interact within the market to understand their decision-making processes.

Specifically, it considers the concepts of supply and demand, price determination, and resource allocation. Consumer behavior, utility functions, and production choices are integral to microeconomic analysis. The study of how consumers maximize their utility, or satisfaction, given their budget constraints, is a key area within microeconomics. This involves making decisions on the consumption of various goods and services, and it often necessitates the formulation of a mathematical relationship known as the utility function. The Cobb-Douglas utility function, for instance, is a widely used form that describes a particular kind of consumer preference, as seen in the exercise involving the Phillie Phanatic's choices.

In microeconomic theory, the concepts like the Cobb-Douglas utility function help illustrate consumer behavior models and discuss how price changes can affect consumer choices and market demand curves. By exploring these relationships, microeconomics aims to better understand and predict the outcomes in a market-based economy.

Composite Good

A composite good in microeconomics is a theoretical construct that combines several related goods into a single aggregate good, which simplifies the analysis of consumer choice and utility maximization. The idea is that a combination of goods may provide utility to the consumer only when consumed together in certain proportions. This construct allows economists to reduce the complexity of the consumer's choice problem.

For example, consider the Phillie Phanatic's predetermined mix of a foot-long hot dog, half a bun, an ounce of mustard, and two ounces of pickle relish. Individually, each item has little value to the consumer, but together, they create a complete meal, which gives the Phanatic utility. By bundling these items into a single 'composite good,' it simplifies the economic modeling as well as the utility function, which now revolves around the composite good rather than each individual component.

The concept of a composite good is especially useful in cases where goods are perfect complements, as they must be used in fixed proportions. The simplification to a composite good not only makes for easier utility analysis but also helps in better understanding consumer purchasing patterns, budget constraints, and the effects of price changes on consumer behavior.

Utility Maximization

Utility maximization is the principle that consumers select goods and services to consume in combination to maximize their total satisfaction, or utility, subject to their budget constraints. In the context of microeconomics, utility maximization is the process by which individuals decide how to allocate their income across various goods and services to achieve the highest possible level of utility.

The decision-making process is often visualized through indifference curves and budget constraints on a graph. Indifference curves represent combinations of goods that provide the consumer with the same level of satisfaction, while the budget constraint represents the combinations of goods and services that can be afforded with the consumer's income. The point where the budget constraint is tangent to the highest indifference curve represents the utility maximization point.

When it comes to a Cobb-Douglas utility function, as seen with the Phillie Phanatic, the power exponents suggest the level of preference for each good in the utility function. As the prices of goods change, or income changes, the utility maximization point may shift, indicating a change in the quantity of goods to be consumed. For example, if the price of a hot dog rises by 50%, it affects the consumer's ability to maximize utility, as outlined in the textbook exercise. Understanding utility maximization enables consumers, businesses, and policymakers to make informed decisions that can facilitate better economic outcomes.