Question

While visiting family in Mexico, your professor has a budget of $ 220 ($ = pesos) that he spends on tacos and tequila.

The price of tacos is $ 6 and a shot of tequila is priced at $ 7 .
During his visit the price of tacos changes to $ 4 .

Assume your professor has a Cobb-Douglas utility function with (tacos) parameter α = ( 23 / 100 ) .

How much utility does your professor's receive from his optimal consumption bundle before the price change ? after the price change?

Ans. 18.99 & 20.85

Answer 1

Utility function is U = TA^0.23 TQ^0.77

This gives the MRS = -(0.23TQ/0.77TA) and the slope of the budget constraint = -PTA/PTQ = -6/7

At the optimal bundle MRS = slope of the budget line so -(0.23TQ/0.77TA) = -6/7

This results in TQ = 2.8696TA

Use the budget line

220 = 6TA + 7*2.8696TA

Thus, TA = 220/26.08696 = 8.43 and TQ = 24.20

Utility before price change is 8.43^0.23 * 24.20^0.77 = 18.99

Now PTA = 4 so that slope of the budget constraint = -PTA/PTQ = -4/7

At the optimal bundle MRS = slope of the budget line so -(0.23TQ/0.77TA) = -4/7

This results in TQ = 1.913TA

Use the budget line

220 = 4TA + 7*1.913TA

Thus, TA = 220/17.3913 = 12.65 and TQ = 24.20

Utility after price change is 12.65^0.23 * 24.20^0.77 = 20.85