Question

a. A young connoisseur has \(\$ 300\) to spend to build a small wine cellar. She enjoys two vintages in particular: an expensive 1987 French Bordeaux \(\left(W_{F}\right)\) at \(\$ 20\) per bottle and a less expensive 1993 California varietal wine \(\left(W_{c}\right)\) priced at \(\$ 4 .\) How much of each wine should she purchase if her utility is characterized by the following function? $$u\left(w_{F}, w_{c}\right)-w$$ b. When she arrived at the wine store, our young oenologist discovered that the price of the 1987 French Bordeaux had fallen to \(\$ 10\) a bottle because of a decline in the value of the franc. If the price of the California wine remains stable at \(\$ 4\) per bottle, how much of each wine should our friend purchase to maximize utility under these altered conditions?

Short answer

Short Answer: When the price of French Bordeaux decreases, it will affect the budget constraint and change the optimal quantities of wine to be purchased. Using the method of Lagrangian multipliers, we can find the first-order conditions for the new budget constraint and solve for the optimal wine quantities under the altered conditions. Comparing the optimal wine quantities before and after the price change will provide insights into how the optimal combination of wines to maximize utility has been affected by the change in the price of the French Bordeaux.

Step-by-step solution

Part A: Setting up the utility function and budget constraint

: The utility function is given as: $$u\left(w_{F}, w_{c}\right)-w$$ The budget constraint is given as: \(20w_F + 4w_C = 300\)

Part A: Utility maximization using the Lagrangian multiplier method

: We will maximize the utility function subject to the budget constraint using the Lagrangian multiplier method. First, set up the Lagrangian function: $$\mathcal{L}\left(w_{F}, w_{c}, \lambda\right) = u\left(w_{F}, w_{c}\right) - w - \lambda \left(20w_{F} + 4w_{C} - 300\right)$$ Now, we need to find the first-order conditions by taking partial derivatives of the Lagrangian with respect to \(w_F\), \(w_C\), and \(\lambda\). This will give us three equations: 1) \(\frac{\partial \mathcal{L}}{\partial w_F} = 0\) 2) \(\frac{\partial \mathcal{L}}{\partial w_C} = 0\) 3) \(\frac{\partial \mathcal{L}}{\partial \lambda} = 0\)

Part A: Solving for the optimal wine quantities

: Solve the system of equations generated in the previous step to find the optimal values of \(w_F\) and \(w_C\) that maximize the utility under the budget constraint. After solving the equations and checking the non-negativity constraints, we will get the optimal wine quantities for part A. #tabpanel#

Part B: Changing the price of the French Bordeaux

: Now, the price of the French Bordeaux has fallen to \(10 per bottle. The budget constraint will now be: \)10w_F + 4w_C = 300$

Part B: Utility maximization using Lagrangian multiplier method in altered conditions

: Similar to part A, we will maximize the utility function subject to the new budget constraint using the Lagrangian multiplier method: $$\mathcal{L}\left(w_{F}, w_{c}, \lambda\right) = u\left(w_{F}, w_{c}\right) - w - \lambda \left(10w_{F} + 4w_{C} - 300\right)$$ We will find the first-order conditions by taking partial derivatives of the Lagrangian for the new budget constraint: 1) \(\frac{\partial \mathcal{L}}{\partial w_F} = 0\) 2) \(\frac{\partial \mathcal{L}}{\partial w_C} = 0\) 3) \(\frac{\partial \mathcal{L}}{\partial \lambda} = 0\)

Part B: Solving for the optimal wine quantities under the altered conditions

: Solve the system of equations generated in the previous step to find the optimal values of \(w_F\) and \(w_C\) that maximize the utility under the new budget constraint. After solving the equations and checking the non-negativity constraints, we will get the optimal wine quantities for part B. The student will be able to compare how the change in the price of the French Bordeaux affected the optimal wine quantities to maximize the utility. Please note that since the utility function is not provided in a specific format, it is not possible to derive explicit expressions for the optimal wine quantities. However, the solution process is still valid, and with a specific format of the utility function, the optimal quantities can be calculated.

Key concepts

Utility Maximization

Utility maximization is a fundamental concept in economics. It helps consumers make the best choices to maximize their satisfaction within their budget constraints. When we talk about maximizing utility, we consider how to get the most enjoyment from spending our limited resources.
Imagine you have a budget and two types of products to choose from, like the wines in our exercise. Your goal is to distribute your spending between these products in a way that gives you the highest satisfaction or happiness. You want to pick the combination that maximizes the utility function, which is a mathematical representation of your preferences.
In this context, utility is calculated as a function of the amount of each product purchased. The consumer will adjust their spending on each product to the point where the utility is maximized, i.e., they receive the greatest satisfaction possible from their chosen combination. The constraint? They cannot spend more than their budget allows.
In the wine example, the young connoisseur aims to allocate her $300 budget optimally between two wines to maximize her utility: the expensive French Bordeaux and the less expensive Californian varietal. Because the utility function itself isn't clearly defined here, she can only determine the best purchase decisions using specific mathematical methods, like the Lagrangian multiplier, to solve the constraint equations.

Lagrangian Multiplier Method

The Lagrangian multiplier method is a powerful tool in optimization problems, especially when dealing with constraints like budgets. It helps us determine how to allocate resources efficiently to achieve maximum utility without exceeding our limits.
To apply this technique, we set up a Lagrange function. This function combines the utility function and the budget constraint into a single equation. The form of the Lagrange function is:

• \(\mathcal{L}(w_{F}, w_{C}, \lambda) = u(w_{F}, w_{C}) - w - \lambda(20w_{F} + 4w_{C} - 300)\)

Here, \(\lambda\) is the Lagrangian multiplier, an additional variable introduced to account for the budget restriction. The beauty of this method is that it turns a constrained optimization problem into an unconstrained one, allowing us to use calculus to find where the function is maximized.
We first find the partial derivatives of the Lagrangian with respect to each wine (decision variables) and the multiplier \(\lambda\). Then, we set these derivatives to zero to find the points where the utility is maximized subject to the budget constraint:

• \(\frac{\partial \mathcal{L}}{\partial w_{F}} = 0\)
• \(\frac{\partial \mathcal{L}}{\partial w_{C}} = 0\)
• \(\frac{\partial \mathcal{L}}{\partial \lambda} = 0\)

Solving these equations gives us the optimal numbers of bottles for each type of wine to purchase. This method allows consumers to make well-informed decisions on their spending and maximize satisfaction.

Wine Purchase Decision

The wine purchase decision for our young connoisseur involves balancing her taste preferences with her budget. Initially, she wants to decide how many bottles of French Bordeaux and Californian varietal wine to buy with her \(300 budget.
The choice becomes interesting when the price of one of her preferred wines, the French Bordeaux, drops from \)20 to $10. With the change in price, her purchase decision needs to be reconsidered because her budget can now spread farther than before.
In the first scenario with the original prices, her budget constraint is given by the equation:

• \(20w_{F} + 4w_{C} = 300\)

The utility maximization process using the Lagrangian multiplier helps her distribute her budget effectively between the two wines.
When the price drops, the new budget constraint becomes:

• \(10w_{F} + 4w_{C} = 300\)

She can potentially purchase more bottles of her favorite wine for the same amount, leading her to revisit the optimization process. Her goal is still to maximize utility, but now she must consider how the price change increases affordability and potentially alters her taste balance.
Through recalculated optimal quantities using the revised budget constraint, she decides anew on how to balance her selections to derive the maximum satisfaction possible from her revamped buying potential. This demonstrates how changes in prices impact consumption decisions and utility.