Question

13. Pip consumes two goods, x and y. Pip’s utility function is given by u(x, y) = x 1/2y 1/2 The price of x is p and the price of y is 1. Pip has an income of M.

(a) Derive Pip’s demand functions for x and y. [5 marks]

(b) Suppose M = 72 and p falls from 9 to 4. Calculate the income and substitution effects of the price change. [5 marks]

(c) Calculate the compensating variation of the price change. [5 marks]

(d) Calculate the price elasticity of demand for x. [5 marks]

Answer 1

Part (A)

consumer maximizes utility where

substituting in budget constraint

Part (B)

bundle consumed is x=4 & y=36

price of good y remains the same and price of good x falls from 9 to 4

substitution effect measures the change in the consumption of good that occurs when the consumer moves along the same indifference curve due to a fall in the market price of good

to measure substitution effect we need to find the bundle which lies on the new budget constraint but on the existing indifference curve

change in consumption of good x from 4 units to 6 units is the substitution effect

the income effect measures the change in the consumption of good that occurs when the consumer moves to a higher or lower indifference curve (representing the change in real income)..

substituting the new utility maximizing condition in the new budget constraint

change in consumption of good x from 6 units to 9 units is the income effect

change in consumption of good x from 4 units to 9 units is the total price effect

fall in price of good x from 9 to 4 increases demand for good from 4 units to 9 units

Part (c)

compensating variation tells how much do we have to increase/decrease the consumer ’s income if we want her welfare to remain the same after a change in market prices

Part (D)

10 % fall in price increases demand by 22.5%