Question

Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by \\[\text { 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 \(t\) ?

Short answer

Short answer: To allocate her expenditures between bus and train travel, Sarah will always allocate her expenses proportionally based on the price ratio of train to bus travel. By solving for her demand functions for ground (g*) and air (p*) transportation and using the budget constraint, she can determine the optimal allocation of her expenditures between bus and train travel based on the price ratio and her demand functions for ground and air transportation.

Step-by-step solution

a. Defining a composite commodity for ground transportation

The composite commodity for ground transportation can be defined as the combination of bus and train transportation. We can represent ground transportation \((g)\) as the sum of bus and train travel: \[g = b + t\]

b. Phrasing Sarah's optimization problem

We can rewrite Sarah's utility function in terms of ground \((g)\) and air \((p)\) transportation as: \[\text{utility} = (g - t) \cdot t \cdot p\] Now, let's phrase the optimization problem for Sarah. She needs to maximize her utility function while considering her income, prices for each transport mode, and her consumption of ground and air transportation services. \[\max U(g, p) = (g - t) \cdot t \cdot p\] subject to the budget constraint: \[I = p_b \cdot b + p_t \cdot t + p_p \cdot p\] \[I = p_b \cdot (g - t) + p_t \cdot t + p_p \cdot p\] where \(I\) represents Sarah's income, and \(p_b, p_t, p_p\) represent the prices of bus, train, and plane travel, respectively.

c. Finding the demand functions for g and p

To solve the optimization problem, we can use the method of Lagrange multipliers. First, let's denote the Lagrange multiplier as \(\lambda\). Then, the Lagrangian function is: \[L(g, p, \lambda) = (g - t) \cdot t \cdot p + \lambda(I - p_b \cdot (g - t) - p_t \cdot t - p_p \cdot p)\] Now, we will differentiate the Lagrangian function with respect to \(g\), \(p\), and \(\lambda\): \[\frac{\partial L}{\partial g} = tp - \lambda p_b = 0\] \[\frac{\partial L}{\partial p} = (g - t)t - \lambda p_p = 0\] \[\frac{\partial L}{\partial \lambda} = I - p_b \cdot (g - t) - p_t \cdot t - p_p \cdot p = 0\] Now, we need to solve these three equations simultaneously to find the demand functions for \(g\) and \(p\). Let's rearrange the first two equations: \[g \cdot p = \frac{\lambda p_b}{t}\] \[gt - t^2 = \frac{\lambda p_p}{p}\] We can then substitute \(gp = \frac{\lambda p_b}{t}\) into the second equation: \[t = \frac{p_p p}{p_p + p_b}\] Now, plug this into the equation for \(g\): \[g = (g - t) + t\] \[g = (g - \frac{p_p p}{p_p + p_b}) + \frac{p_p p}{p_p + p_b}\] From the budget constraint, we can solve for \(\lambda\): \[ \lambda = \frac{I - p_t(\frac{p_p p}{p_p + p_b}) - p_p \cdot p}{p_b(g - \frac{p_p p}{p_p + p_b})}\] Now, we can find the demand functions for \(g\) and \(p\) by solving for them from the previous equations: \[g^{*} = \frac{\lambda p_b t^{*}}{p}\] and \[p^{*} = \frac{\lambda p_p}{p(g^{*} - t^{*})}\]

d. Allocating expenditures between bus and train travel

Once Sarah decides how much to spend on ground transportation, she will allocate her expenditures between bus and train travel. Since the price ratio of train to bus travel never changes, Sarah will always allocate her expenditures proportionally between bus and train based on their price ratio \((\frac{p_t}{p_b})\). To find how much Sarah spends on bus and train travel, we can use the budget constraint and the demand functions for \(g^{*}\) and \(p^{*}\): \[I = p_b \cdot b + p_t \cdot t + p_p \cdot p\] Plugging in the demand functions for \(g^{*}\) and \(p^{*}\): \[I = p_b \cdot (g^{*} - t^{*}) + p_t \cdot t^{*} + p_p \cdot p^{*}\] Let \(x = \frac{p_t}{p_b}\) be the price ratio. Then, we can solve for the expenditures on bus and train travel: \[x = \frac{p_b \cdot b^{*}}{p_t \cdot t^{*}}\] From here, Sarah can determine the optimal allocation of her expenditures between bus and train travel based on the price ratio and her demand functions for ground and air transportation.

Key concepts

Composite Commodity

Understanding the concept of a composite commodity is essential when we delve into the realms of microeconomic optimization. In our example, we witness Ms. Sarah Traveler's predicament in choosing between various transportation modes. A composite commodity, in this context, can be explained as a homogenized aggregation of similar goods or services. In Sarah's case, the composite commodity for ground transportation (denoted as 'g') combines both bus (b) and train (t) travel into one entity.

To visualize, imagine we represented apples and oranges together as 'fruit' for simplicity in calculations when both have similar nutritive value. It simplifies analysis by reducing the number of variables we have to juggle with. When Sarah contemplates her travel options, considering ground transportation (bus plus train) as one composite good allows her to broker a decision between just two choices: ground (g) or air (p), streamlining her utility maximization problem.

Budget Constraint

Moving forward, let's break down the 'budget constraint', which is a cornerstone of any optimization query in microeconomics. A budget constraint represents all combinations of goods and services that a consumer can purchase, given their income levels and the prices of these goods and services. For Ms. Traveler, her budget constraint is mathematically illustrated as:
\[I = p_b \times b + p_t \times t + p_p \times p\]
where I symbolizes her income, while p_b, p_t, and p_p reflect the prices of bus, train, and plane travel respectively. Sarah's choices are restricted by this constraint because she cannot spend more than her income. It's akin to an individual managing a fixed monthly budget - they allocate funds to different needs without exceeding what they earn.

Demand Functions

Demand functions in the context of this exercise reveal how Ms. Traveler's consumption of ground (g) and air (p) transportation vary with changes in price, income, and other factors. They're derived from the optimization problem and indicate the relationship between the quantity demanded of a commodity and the factors affecting its purchase. These functions are the solutions to the optimization problem and can be considered Sarah's strategy for transportation choices given her budget and preferences. In mathematical terms, her demand functions for 'g' and 'p' will show us the precise quantities of ground and air miles she would choose to maximize her utility within her budget.

Lagrange Multipliers

When it comes to solving these optimization problems, the use of Lagrange multipliers offers a slick solution. They are introduced when the optimization must adhere to an external constraint, such as Ms. Traveler's budget. A Lagrange multiplier, represented by λ (lambda), helps in adjusting the level of utility Sarah could achieve by relaxing her budget constraint by one unit.

Mathematically, it's used to combine the utility function and the budget constraint into a single Lagrangian function. The settings where these multipliers are applied are where trade-offs come at a cost: for instance, how much additional utility Sarah would get if she had an extra dollar to spend, and conversely, how much utility she might lose for every dollar less.

Price Ratio of Transportation Modes

Finally, the price ratio of transportation modes significantly influences how Ms. Traveler allocates her resources between bus and train travel. This ratio remains unchanged in our scenario and thus simplifies the allocation of expenditures. For example, if the price of train travel is consistently double the price of bus travel, Sarah would allocate her budget for ground transportation between bus and train travel based on this immutable ratio.

The idea of a fixed price ratio is similar to the concept of relative prices in economics, where it is the price of one good or service in comparison to another. It implies that instead of only looking at the price tag, we also need to consider how the cost of one item relates to another in making lined-up purchase decisions.