Question

In: Economics

A young connoisseur has $600 to spend to build a small wine cellar. She enjoys two...

A young connoisseur has $600 to spend to build a small wine cellar. She enjoys two vintages in particular: a 2001 French Bordeaux (F) at $40 per bottle and a less expensive 2005 California varietal wine (C) priced at $8 per bottle. Her utility is ?(?, ?) = ? 2/3? 1/3 . a. What is her expenditure function? b. What are the required expenditures to reach the maximized utility level under the original prices?

Expert Solution

U(W_F, W_C) = W_F^2/3W_C^1/3

The Lagrangian condition is

L = W_F^2/3W_C^1/3 + l(I- p_FW_F -p_CW_C)

Taking first order conditions yields

W_F = 2/3(W_F^-1/3W_C^1/3) - l p_F = 0

W_C = 1/3(W_F^2/3W_C^-2/3) - l p_C = 0

W_l = I - p_FW_F - p_CW_C = 0

Taking the ratio of the first two expressions

2/3(W_F^-1/3W_C^1/3) = l p_F

1/3(W_F^2/3W_C^-2/3) l p_C

Simplifying

2 W_C/W_F = p_F/p_C

Thus, 2p_cW_C = p_FW_F .

Substituting into the budget constraint

Now, with our parameters,

600 = 3(8)W_C So Wc = 600/24 = 25

and

40W_F = 2(8)(25)

So W_F = 10.

(To check notice, that $40*10 + $8*25 = $600)

I = 2p_cW_C + p_cW_C = 3 p_cW_C

Rahul Sunny answered 2 years ago

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