Chapter 4: Problem 2

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

Expert verified

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.