Question

Elaine is a typical student and she spends her $200 allowance on beer (X) and books (Y). Her utility Q1. function is given by U 20 XY and Px $2 and Py $5 - (a) Solve for Elaine's optimal purchases of X and Y (b) By how much would Elaine's utility change if her allowance were reduced by $0.50? (c) In an attempt to discourage drinking on campus, a hefty tax is imposed on beer and the price increases to $5.00. Solve for Elaine's new optimal purchases of X and Y (d) Calculate Elaine's elasticity of demand for beer and her cross price elasticity of demand for books with respect to the price of beer. Sketch Elaine's demand curve for beer using at least four price-quantity demanded combinations. (e) (g) up (say in the form of a lump-sum income tax) to prevent the increase in the price of beer, what would her answer be, assuming of course, she does not lie? If the government had asked Elaine how much of her income, at most she would be willing to give C

Please only do question G - Please show steps!

Answer 1

Solution:

The scenario described in part (g), where we are asked to find the loss in income a person is willing to undergo, to prevent price increase is called equivalent variation.

Finding the equivalent variation:

First we need to find the original/initial optimal bundle (before the price change). For such cobb-douglas function, we know demand functions are as follows:

With income, M = $200, Px = $2, Py = $5, and weights on each good from utility function can be seen to be 1 or a = b =1 (the power of both X and Y), so

X* = (a/(a+b))*(M/Px)

X* = (1/(1+1))*(200/2) = 200/4 = 50 beers

Y* = (b/(a+b))*(M/Py)

Y* = (1/(1+1))*(200/5) = 200/10 = 20 books

So, initial utility level is U = 20*50*20 = 20000

Now, we find the utility change due to such price increase

Utility at new optimaal bundele, found by new prices and initial income:

Using the same formula as above: X'* = (1/2)*(200/5) = 20 beers

Y'* = (1/2)*(200/5) = 20 books

So, new utility level (with price change) = 20*20*20 = 8000

At old prices, this new utility level or new optimal bundle would have been achieved at income of:

M' = Px*X'* + Py*Y'*

M' = 2*20 + 5*20 = 40 + 100 = $140

So, equivalent variation (or income given up such that the utility achieved is same under income given up or price increase) = M' - M

EV = 140 - 200 = -$60

With minus sign indicating the income given up.