Suppose Coke and Pepsi are perfect substitutes for me, and right and left shoes are perfect complements.
A. Suppose my income allocated to Coke/Pepsi consumption is $100 per month, and my income allocated to Right/Left shoe consumption is similarly $100 per month.
(a) Suppose Coke currently costs 50 cents per can and Pepsi costs 75 cents per can. Then the price of Coke goes up to $1 per can. Illustrate my original and my new optimal bundle with Coke on the horizontal and Pepsi on the vertical axis.
(b) Suppose right and left shoes are sold separately. If right and left shoes are originally both priced at $1, illustrate (on a graph with right shoes on the horizontal and left shoes on the vertical) my original and my new optimal bundle when the price of left shoes increases to $2.
(c) True or False: Perfect complements represent a unique special case of homothetic tastes in the following sense: Whether income goes up or whether the price of one of the goods falls, the optimal bundle will always lie on a the same ray emerging from the origin.
B. Continue with the assumptions about tastes from above.
(a)Write down two utility functions: one representing my tastes over Coke and Pepsi, another representing my tastes over right and left shoes.
(b) Using the appropriate equation derived above, label the two indifference curves you drew in A(a).
(c) Using the appropriate equation derived in B(a), label the two indifference curves you drew in A(b).
(d) Consider two different equations representing indifference curves for perfect complements: u1(x1,x2) =min{x1,x2} and u2(x1,x2) =min{x1,2x2}. By inspecting two of the indifference curves for each of these utility functions, determine the equation for the ray along which all optimal bundles will lie for individuals whose tastes can be represented by these equations.
(e) Explain why the Lagrange method does not seem to work for calculating the optimal consumption bundle when the goods are perfect substitutes.
(f) Explain why the Lagrange method cannot be applied to calculate the optimal bundle when the goods are perfect complements.