Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by utility \(=B \cdot T \cdot P\) 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 \(\left(P_{T} / P_{B}\right)\) 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 \(G\) and \(P\) ? d. Once Sarah decides how much to spend on \(G\), how will she allocate those expenditures between \(B\) and \(T ?\)

To define a composite commodity for ground transportation, we will combine bus \((B)\) and train \((T)\) travel into a single good \((G)\). Since the price ratio of train travel to bus travel \( \left( \frac{P_{T}}{P_{B}} \right) \) never changes, we can represent this composite commodity in terms of one of the individual grounds of transportation, say bus \((B)\). Then, the composite commodity for ground transportation can be defined as \(G = kB\) where \(k\) is a constant representing the fixed price ratio between train and bus travel.

Sarah's optimization problem can be rephrased as one of choosing between the ground (\(G\)) and air (\(P\)) transportation. Given her utility function, \(U = B \cdot T \cdot P\), we can now rewrite it using the composite commodity as \(U = \frac{G}{k} \cdot kT \cdot P\). The optimization problem is to maximize her utility function given her budget constraint, which can be written as \(P_{G} G + P_{P} P \leq I\) where \(P_{G}\) is the price of ground transportation, \(P_{P}\) is the price of air transportation, and \(I\) is her income.

To find Sarah's demand functions for \(G\) and \(P\), we need to solve her optimization problem. We can use the Lagrange multiplier method to find the optimal values of \(G\) and \(P\). The Lagrangian function is $$\mathcal{L}(G, P, \lambda) = \frac{G}{k} \cdot kT \cdot P + \lambda (I - P_{G} G - P_{P} P).$$ Taking the partial derivatives with respect to \(G\), \(P\), and \(\lambda\), we have: $$\frac{\partial \mathcal{L}}{\partial G} = \frac{1}{k} \cdot kT \cdot P - \lambda P_{G} = 0$$ $$\frac{\partial \mathcal{L}}{\partial P} = \frac{G}{k} \cdot kT - \lambda P_{P} = 0$$ $$\frac{\partial \mathcal{L}}{\partial \lambda} = I - P_{G} G - P_{P} P = 0$$ Solve this system of equations to obtain the optimal values of \(G\) and \(P\), which will represent Sarah's demand functions for ground \((G)\) and air \((P)\) transportation.

Once Sarah decides how much to spend on ground transportation (\(G\)), she will allocate her expenditures between bus \((B)\) and train \((T)\) travel based on the fixed price ratio, as given. Since the composite commodity is defined as \(G = kB\), where \(k = \frac{P_{T}}{P_{B}}\), she will spend \(k\) times more on train travel than on bus travel. That is: $$Expenditure_{T} = k \cdot Expenditure_{B}$$ So, the proportion of her ground transportation expenditures allocated to bus travel will be \(\frac{1}{(1+k)}\) of the total expenditure on \(G\), and the proportion allocated to train travel will be \(\frac{k}{(1+k)}\) of the total expenditure on \(G\).