Question

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?

Answer 1

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