econ11_08_ps5_sol

econ11_08_ps5_sol - Eco11 Fall 2008 Simon Board Economics...

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Eco11, Fall 2008 Simon Board Economics 11: Solutions to Practice Problems 5 (Week 6) November 4, 2008 1. Complements and Substitutes Suppose u ( x 1 , x 2 ) = x 1 x 2 . Are x 1 and x 2 gross complements or substitutes? Are they net complements or substitutes? Solution Marshallian demand is given by x * 1 = m/ 2 p 1 and x * 2 = m/ 2 p 2 . Since demand for good i is independent of p j these are neither gross complements or gross substitutes. Hicksian demand is given by h 1 = ( up 2 /p 1 ) 1 / 2 and h 2 = ( up 1 /p 2 ) 1 / 2 . Since h i is increasing in p j , these goods are net substitutes. 2. Choice with Partial Subsidies Suppose an agent chooses how much bread, x 1 , and wine, x 2 , to consume. The agent views the goods as perfect substitutes and has utility function: u ( x 1 , x 2 ) = x 1 + x 2 The price of bread is p 1 = 4, while the price of wine is p 2 = 2. The agent has income m = 4. (a) Solve for the agent’s optimal consumption. (b) In order to encourage the consumption of bread, the Government subsidises the first unit by $1. Hence the first unit costs p 1 = 3, while each unit thereafter costs p 1 = 4. (i) Draw the agent’s budget constraint. (ii) Solve for her optimal consumption. (iii) How much does the subsidy cost the Government? (c) The Government increases it’s subsidy on the first unit to $3. Hence the first unit costs p 1 = 1, while each unit thereafter costs p 1 = 4. (i) Draw the agent’s budget constraint. (ii) Solve for her optimal consumption. (iii) How much does the subsidy cost the Government?
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