Question

Andy loves buying raincoats (c) and oxford shoes (s) and he buys them in fixed proportions. He buys two coats with every pair of shoes he buys. A pair of shoes costs $30, and a raincoat costs 20. Andy´s total budget is $140.

a) Write a utility function for Andy´s preferences and his budget constraint. Graph Andy´s indifference curves and budget constraint.

b) How many pair of shoes and raincoats does Andy buy?

c) If the federal government taxes coats at $5 per coat. How many pair of shoes and coats does Andy consume after the tax is imposed?

d) One could interpret this problem as a one-good problem. Explain this interpretation. Graph Andy´s demand curve for this good.

e)Calculate the change in Andy´s consumer surplus after the imposition of the raincoats tax.

Answer 1

Let S denote consumption of pair of shoes & C for coats

And cost of pair of shoes by Ps  and cost of Raincoat by P_c

Since, Andy buys 2 coats along a pair of shoes. The Utility function for Andy looks like U = Min. [S,0.5C]

And the budget Constraint: P_s + 2P_c = M; Solving for the utility maximisation; Let z be the nos. of shoes the consumer buys, then we know that 2z is the number of coats he buys. Solving for budget constraint:  

P_s z + 2P_c z = 140;

30 z + 2 (20) z = 140

70 z = 140

z = 2

Hence, Andy will be buying 2 Coats & 1 Pair of Shoes.

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After the imposition of taxes, the price of Coat changes to $25. The new solution shall be:

P_s z + 2P_c z = 140;

30 z + 2 (25) z = 140

80 z = 140

z = 140/80 or 7/4

Now, since there is no way to consume Coat & shoes in fractions, we can approximate the consumption to the nearest integer (in this case again 2). The consumption basket for Andy remains same, 2 units of Coats & a pair of Shoes. Since the consumption was always in fixed proportion (Complementary goods) and taxing on one-good does discourage the consumption but not strong enough to off-set consumption of the other good.

There is no change in consumer surplus in this case as the consumption remains same even after the introduction of the taxes.