Question

Georgia always eats hot dogs in a bun together with 1 oz. of mustard. Each hot dog eaten in this way provides 15 units of utility, but any other combination of hot dogs, buns, and mustard is worthless to Georgia. a. Explain the nature of Georgia's utility function and indicate the form of her indifference curve map. b. Suppose hot dogs cost $\mathrm{\$}1,$ buns cost $\mathrm{\$}.40,$ and mustard costs $\mathrm{\$}.10$ per ounce. Show how Georgia's utility can be represented by the total amount of money she spends on these three items. c. How would your answer to part (b) change if the price of hot dogs rose to $\mathrm{\$}1.50?$

Short answer

Answer: When the price of hot dogs increases, Georgia's utility function remains the same, being represented by $U(h)=15h$. However, the total amount of money she spends on her preferred hot dog combination increases, which means she needs to spend more money to achieve the same level of utility. The new relationship between her utility and the total amount of money spent will be given by the equation $U(h)=15h=M^{\prime }$.

Step-by-step solution

Georgia only gets utility from consuming hot dogs in a bun together with 1 oz. of mustard. Each hot dog eaten in this way provides her with 15 units of utility. Any other combination of hot dogs, buns, and mustard does not provide her with any utility. Therefore, Georgia's utility function can be represented as $U(h)=15h$, where $h$ is the number of hot dogs consumed in the preferred combination. #Step 2: Identify the form of her indifference curve map#

Indifference curves represent various combinations of goods that give the consumer an equal amount of utility. In Georgia's case, only the combination of one hot dog, one bun, and 1 oz. mustard gives her any utility. As a result, her indifference curves will be L-shaped, with each curve indicating a different utility level corresponding to the number of hot dogs consumed in the preferred combination. #Step 3: Represent Georgia's utility by the total amount of money she spends on hot dogs, buns, and mustard#

Georgia's utility can be represented by the total amount of money she spends on hot dogs, buns, and mustard. Let's denote the prices of hot dogs, buns, and mustard as $p_h=\mathrm{\$}1$, $p_b=\mathrm{\$}0.40$, and $p_m=\mathrm{\$}0.10$ respectively. Then, the total amount of money she spends is given by the expression $M=p_{h}h+p_{b}b+p_{m}m$. Since Georgia consumes one bun and 1 oz. of mustard with each hot dog, we can write the relationship between these quantities as $b=h$ and $m=h$. Substituting these into the expression for $M$, we get: $M=p_{h}h+p_{b}h+p_{m}h=(p_h+p_b+p_m)h$. Therefore, Georgia's utility represented by the total amount of money spent on hot dogs, buns, and mustard is given by $U(h)=15h=M$. #Step 4: Analyze the change in Georgia's utility when the price of hot dogs increases#

If the price of hot dogs rose to $p_{h^{\prime }}=\mathrm{\$}1.50$, the expression for the total amount of money spent by Georgia on hot dogs, buns, and mustard becomes $M^{\prime }=p_{h^{\prime }}h+p_{b}h+p_{m}h=(p_{h^{\prime }}+p_b+p_m)h$. Since the new price is higher, Georgia needs to spend more money to consume the same number of her preferred hot dog combination. Her utility will still be represented by $U(h)=15h$, but now, each unit of utility will require spending more money. The new relationship between her utility and the total amount of money spent will be given by the equation $U(h)=15h=M^{\prime }$.

Key concepts

Indifference Curves

In the study of consumer behavior, indifference curves play a vital role by showing combinations of goods that provide the same level of satisfaction to a consumer. For Georgia, her utility is derived from a specific bundle of foods: one hot dog with a bun and 1 oz of mustard. This is the only combination that gives her utility, quantified as 15 units. As such, her indifference curves are unique and look quite different from typical ones.

Georgia's indifference map consists of L-shaped curves, because any deviation from her preferred combination doesn’t provide her any utility. Normally, indifference curves form a smooth curve on a graph, with each point representing the same utility level. But in Georgia's case, her indifference curves reflect that only the specific combination of one hot dog, bun, and mustard matters; hence the 'L' shape occurs at each preferred bundle, reflecting her strict consumption pattern.

Consumer Behavior

Understanding consumer behavior involves examining how consumers like Georgia make decisions to maximize their utility under budget constraints. Let's break down the components:

• Utility Maximization: Georgia aims to get the most utility or satisfaction from her spending. The utility is only obtained when she eats her preferred hot dog combination.
• Budget Constraint: Georgia’s budget determines how many hot dogs she can enjoy based on price. Initially, the total cost of each combination is given by the equation:

  $$M=(p_h+p_b+p_m)h$$

  where $p_h,p_b,$ and $p_m$ are prices of hot dogs, buns, and mustard respectively.

With the defined prices, Georgia has to decide how many preferred combinations she can purchase without exceeding her budget. Her behavior demonstrates "rational choice," trying to optimize satisfaction with the resources she has. Georgia always ensures she stays within her economic means while attempting to maximize utility, a fundamental principle of consumer choice theory.

Marginal Utility

Marginal utility represents the additional satisfaction a consumer receives from consuming one more unit of a good. For Georgia, the concept of marginal utility is simple but limited, as each time she consumes the full bundle of one hot dog, bun, and mustard, she gets exactly 15 units of utility.

This results in Georgia's marginal utility being the same for each additional hot dog unit that meets her criteria. It is a constant value, demonstrating that the utility she receives does not diminish or increase with each additional bundle. However, if any part of the combination is missing, the marginal utility becomes zero, as Georgia only values the complete set.

• With each preferred consumption, her total utility increases by the marginal utility:

  $$MU=\frac{\mathrm{\Delta }U}{\mathrm{\Delta }h}=15$$

  where $\mathrm{\Delta }U$ is the change in utility, and $\mathrm{\Delta }h$ is the change in quantity consumed.

Hence, understanding marginal utility helps clarify why consumers purchase specific quantities and the additional satisfaction those quantities bring, crucial components of consumer decision-making processes.