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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? $$\boldsymbol{U}\left(\boldsymbol{W}_{\boldsymbol{p}}, \boldsymbol{W}_{\boldsymbol{c}}\right)=\boldsymbol{W}^{2 / 3} \boldsymbol{W}_{\boldsymbol{C}^{\prime}}^{1,3}$$ 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
Answer: For part a, the optimal allocation is 8.18 units of Bordeaux and 47.72 units of California wine. For part b, the optimal allocation is 16.36 units of Bordeaux and 32.90 units of California wine.

Step by step solution

01

Write down the budget constraint

In this case, the budget is \(\$ 300\). Let \(x\) and \(y\) represent the quantity of Bordeaux (\(W_F\)) and California (\(W_C\)) wines, respectively. The budget constraint can be written as: $$20x + 4y = 300$$
02

Rewrite the utility function

As given, the utility function is: $$U(x, y) = W_{F}^{2/3} W_{C}^{1/3}$$. Replace \(W_F\) and \(W_C\) with their respective quantities, \(x\) and \(y\), giving: $$U(x, y) = x^{2/3} y^{1/3}$$
03

Maximize the utility function subject to the budget constraint

To maximize utility, we need to solve for the Lagrange multiplier, \(\lambda\). Set up the Lagrangian: $$L(x, y, \lambda) = x^{2/3} y^{1/3} + \lambda (300 - 20x - 4y)$$ Now, take the partial derivatives with respect to \(x\), \(y\), and \(\lambda\): $$\frac{\partial L}{\partial x} = \frac{2}{3} x^{-1/3} y^{1/3} -20\lambda = 0$$ $$\frac{\partial L}{\partial y} = \frac{1}{3} x^{2/3} y^{-2/3} - 4\lambda = 0$$ $$\frac{\partial L}{\partial \lambda} = 300 - 20x -4y = 0$$
04

Solve for the optimal allocation

Solving this system of equations, we obtain: $$x=8.18 \quad \text{and} \quad y=47.72$$ #Part b#
05

Write down the new budget constraint

With the new price of the Bordeaux wine being $10 per bottle, the budget constraint becomes: $$10x + 4y = 300$$
06

Maximize the utility function subject to the new budget constraint

The utility function is unchanged, so we only need to update the Lagrangian multiplier amount accordingly: $$L(x, y, \lambda) = x^{2/3} y^{1/3} + \lambda (300 - 10x - 4y)$$ Now, take the partial derivatives with respect to \(x\), \(y\), and \(\lambda\): $$\frac{\partial L}{\partial x} = \frac{2}{3} x^{-1/3} y^{1/3} -10\lambda = 0$$ $$\frac{\partial L}{\partial y} = \frac{1}{3} x^{2/3} y^{-2/3} - 4\lambda = 0$$ $$\frac{\partial L}{\partial \lambda} = 300 - 10x -4y = 0$$
07

Solve for the optimal allocation

Solving this new system of equations, we obtain: $$x=16.36 \quad \text{and} \quad y=32.90$$ So, if the price of the French Bordeaux drops to \(\$ 10\) per bottle, our friend should purchase 16.36 units of French Bordeaux and 32.90 units of California wine to maximize their utility.

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Most popular questions from this chapter

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