Introductory Microeconomics: Problem Set Week 4

Utility maximisation and applications.

1. Gordon is an employee of a company that allows him to choose the number of hours he works per day. His preferences for consumption of goods and leisure can be represented as follows: U = C²F, where C stands for consumption (measured in expenditure) and F stands for free time or leisure. Gordon always sleeps for 8 hours each night and this is not included in F . The company pays Gordon a wage of £10 per hour and Gordon also has income from a trust fund that pays him £40 per day. Gordon spends all of his income on consumption goods.
   
   
1. How many hours a day does Gordon work and how much does he spend on consumption goods?
2. The government imposes a 50% tax on labour income. How do Gordon's work hours and consumption level change?
3. Explain the changes in part (b) in terms of income and substitution effects. Use a diagram in your answer.
4. Now the government decides to impose a lump-sum tax on each individual equal to the tax revenue collected under the previous income tax scheme. Now how many hours does Gordon work and how much does he consume?
5. Compare Gordon's utility in the two scenarios and comment on the difference.
2. Alice consumes only cheese and dates. Her utility function is U = 2c^0.5 + d, where c is the quantity of cheese she consumes and d is the quantity of dates. Her income is fixed at m > 0. The price of cheese is p > 0, and the price of dates is 1.
   
   
   1. What is Alice's budget constraint? Will she spend all her income on cheese and dates?
   2. What is Alice's demand function for cheese? (You may assume that Alice's income is sufficiently large that she buys positive quantities of each good.) Is there anything interesting or unusual about this demand function? Explain.
   3. Find an expression for Alice's demand for dates, and show that her income elasticity of demand is greater than 1.
   4. Bob obtains twice as much utility from consuming cheese and dates as Alice; his utility function is U = 4c^0.5 + 2d . Bob's income is twice that of Alice. Compare their demands for cheese and dates.