Question

# A consumer has a demand function for good 2, ?2, that depends on the price of...

A consumer has a demand function for good 2, ?2, that depends on the price of good 1, ?1, the price of good 2, ?2, and income, ?, given by ?2 = 2 + 240/(??2) + 2?1. Initially, assume ? =40, ?2 = 1, and ?1 = 2. Then the price of good 2 increases to ?2′ = 3.

a) What is the total change in demand for good 2?

b) Calculate the amount of good 1 consumed at the optimal bundle before and after the price change.

c) What is the amount of income needed to make the initial optimal bundle just affordable at the new prices? [5 marks]

d) Find the optimal consumption of good 2 at the income level you found in part c) after the price change. [1 mark]

e) Calculate the amount of change in good 2 due to the Slutsky substitution effect of the price change. [3 marks]

f) Calculate the amount of change in good 1 due to the Slutsky substitution effect of the price change. [2 marks]

g) Calculate the amount of change in each good due to the income effect of the price change. [3 marks]

h) For good 2, check that the Slutsky identity does indeed hold in this example. [1 mark]

i) Assume well-behaved preferences and draw a diagram, with ?1 on the horizontal axis and ?2 on the vertical axis, showing the Slutsky substitution effect and income effect you calculated in parts e) through

g). Label the initial budget line, the final budget line, and the line that passes through the initial bundle and is parallel to the final budget line, and show the axis intercepts for each. Show the coordinates of the optimal bundles on each budget line and draw the indifference curves through these optimal bundles. Show the substitution and income effects on both goods with arrows. [6 marks]

j) Using your diagram, for each good, determine whether it is an inferior good or normal good. Justify your answer. If there is not sufficient information to determine the type of good, explain why. [3 marks]

k) Using your diagram, for each good, determine whether it is an ordinary good or Giffen good. Justify your answer. If there is not sufficient information to determine the type of good, explain why.

Solution :

A) total change in demand for x2= -4 (demand decreased by 4 units)

B) before price change optimal x1= 14

After price change, optimal x1=8

C) Amount of income needed to make the initial bundle just affordable at new prices is 64

D) optimal consumption of good 2 at income in part (c) and new prices is 7.25