Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by $$\text { utility }=\boldsymbol{B}-\boldsymbol{T}^{\prime} \boldsymbol{P}$$ where each letter stands for miles traveled by a specific mode. Suppose that the ratio of the price of train travel to that of bus travel \(\left(P, / P_{B}\right)\) never changes. a. How might one define a composite commodity for ground transportation? b. Phrase Sarah's optimization problem as one of choosing between ground (G) and air (P) transportation. c. What are Sarah's demand functions for Gand P? d. Once Sarah decides how much to spend on \(G\), how will she allocate those expenditures between \(B\) and \(7 ?\)

A composite commodity for ground transportation can be defined by combining the Bus and Train modes of travel, since they both belong to the same category of ground transportation. Let G represent the composite commodity for ground transportation, then G = B + T.

Sarah's optimization problem can be rephrased as choosing between ground (G) and air (P) transportation. Thus, her utility function can be represented as: $$\text{utility} = G - P^{'} P$$

To derive the demand functions for G and P, we need to find the marginal utility of each mode of transportation with respect to the price of the transportation. The marginal utility of G is: $$\frac{\partial (\text{utility})}{\partial G} = 1$$ The marginal utility of P is: $$\frac{\partial (\text{utility})}{\partial P} = -P^{'}$$ Now, we can find the demand functions for G and P by setting the marginal utility equal to their respective prices (denoted as \(P_G\) and \(P_P\)) and then solve for G and P. For G: $$1 = P_G$$ So, $$G = P_G K_G = K_G$$, where \(K_G\) is the constant of proportionality. For P: $$-P^{'} = P_P$$ So, $$P^{'} = \frac{1}{P_P} K_P = K_P$$, where \(K_P\) is the constant of proportionality.

Once Sarah decides how much to spend on G, she will allocate her expenditures between B and T based on the given price ratio, \(P(P_T / P_B)\). Since the price ratio never changes, Sarah will allocate her expenditures between B and T according to the same proportion. Let \(x\) be the proportion spent on Bus (B) and \(1-x\) be the proportion spent on Train (T). Then, the allocation of expenditures can be represented as: $$B = x(P_G K_G)$$ $$T = (1 - x)(P_G K_G)$$ In conclusion, by analyzing the given utility function, we defined a composite commodity for ground transportation, rephrased Sarah's optimization problem, derived the demand functions for ground and air transportation, and determined how she will allocate her expenditures between Bus and Train travel.