Question

Alice enjoys consuming goods x and y. Where (x,y) is the consumption bundle and 0 < α < 1, Alice’s utility is: u(x; y) = x^α*y^(1−α) (a) Originally, Alice has income m = 16, and faces prices px = py = 1, and consumes a bundle at which her utility (subject to her budget) is maximised. Then, py rises to 4. Alice is sadder with the new bundle she chooses, and tells you that her compensating variation is 16: that is, she would need an extra 16 dollars to be as happy at the new prices as she was at the old ones. Assuming Alice is telling you the truth, find α. (b) Find Alice’s equivalent variation.

Answer 1

Assume that Alice consumes two commodities X and Y and the utility function is given by U(X,Y) = X^?Y^(1-?), X?0 and Y?0. Generalized rule indicates that we have the constant budget share demand function X* = (?/? + 1 – ?) ×I/px or simply X* = ? (I/px). Similarly, Y* = (1 – ?/? + 1 – ?)×I/py or simply Y* = (1 – ?)*(I/py)

In our case, I (income) = 16 and px = py = 1. Hence initially the consumption bundle is x* = ? * (16/1) = 16? and y* = (1 – ?)*16/1 = 16 - 16?. Now we are given that compensating variation is $16.

Compensating variation is the amount of money necessary to be given to the consumer to keep her original bundle. There is a change in the price of py from 1 to 4, which means the new bundle of y is y* = (1 – ?)*16/4 or 4 - 4?.

The original bundle was 16 – 16? units of y so at the new price of $4 per unit, this bundle would have required $16 more than the current income and that is why the compensating variation is $6. Find the money necessary to buy the old bundle at new price which is 4* (16 – 16?). This is more than the current income by 16 which means we have

4*(16 – 16?) + 16? – 16= 16

64 – 64? + 16? = 32

32 = 48?

This gives ? = 2/3.

Equivalent variation measures the amount of money given up or needed by the consumer in order the buy the same bundle at the same old prices that entails a new utility. Find the demand function and place them into the utility function. Find the new utility level at the old income and new prices. Set this value equal to the utility function by using the unknown new income and old prices. Solving for the new income. Subtract this level of income from the old income. This is the Equivalent Variation

Demand function for x = (2/3)(I/Px) and demand function for y = (1/3)(I/Px)

Utility function U = x2/3y1/3

U = [(2/3)(I/Px)]^2/3*[(1/3)(I/Py)]^1/3

= (2/9)(I)*(1/Px)^2/3(1/Py)^1/3

U(old income and new price) = (2/9)(16)*(1/1)^2/3(1/4)^1/3 = 2.23986

2.23986 = (2/9)(Inew)*(1/1)^2/3(1/1)^1/3

Mnew = 10.07 or simply $10

Equivalent variation = old income – new income = 16 – 10 = $6.