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 \(\$ 1,\) buns cost \(\$ .40,\) and mustard costs \(\$ .10\) 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 \(\$ 1.50 ?\)

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 \(U(h) = 15h = M'\).

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 \(p_h = \$1\), \(p_b = \$0.40\), and \(p_m = \$0.10\) respectively. Then, the total amount of money she spends is given by the expression \(M = p_hh + p_bb + p_mm\). 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: \(M = p_hh + p_bh + p_mh = (p_h + p_b + p_m)h\). 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 \(p_{h'} = \$1.50\), the expression for the total amount of money spent by Georgia on hot dogs, buns, and mustard becomes \(M' = p_{h'}h + p_bh + p_mh = (p_{h'} + p_b + p_m)h\). 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 \(U(h) = 15h = M'\).