Introductory Microeconomics: Problem Set 6
Utility maximisation and applications.
Alice consumes only cheese and dates. Her utility function is U = 2c0.5 + 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.
- What is Alice's budget constraint? Will she spend all her income on cheese and dates?
- 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.
- Find an expression for Alice's demand for dates, and show that her income elasticity of
demand is greater than 1.
- Bob obtains twice as much utility from consuming cheese and dates as Alice; his utility
function is U = 4c0.5 + 2d . Bob's income is twice that
of Alice. Compare their demands for cheese and dates.
- 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.
- How many hours a day does Gordon work and how much does he spend on consumption goods?
- In 1998, the government imposed a 50% tax on labour income. How did Gordon's work hours
and consumption levels change?
- Explain the changes in part (b) in terms of income and substitution effects. Use a diagram
in your answer.
- In 1999 the government decided to impose a lump-sum tax on each individual equal to the
tax revenue collected in 1998. Now how many hours does Gordon work and how much does he consume?
- Compare Gordon's utility in 1998 and 1999 and comment on the difference.
- A consumer's utility function is given by U = x0.2y0.8,
and her income is M = 1000. She initially faces a price vector p0 = (1,1),
which then changes to p1 = (2,1).
- Calculate the Compensating Variation and the Equivalent Variation of the price change.
- Illustrate your answer with an appropriate diagram.
- Think about the market for salt. Suppose your household only buys two goods, "salt" and "all
other goods". Suppose the price of salt trebles.
- Represent on a diagram the magnitude of the substitution effect. How big is this effect?
- What about the magnitude of the income effect?
- What can we conclude about the change in your optimal choice that is induced by this
enormous increase in the price of salt?
- What if your household were choosing between "housing" and "all other goods", and you were
analysing the impact on your optimal choice of an increase in, say, 50% of housing prices?
How would your optimal choice change?
- Consider the consumption and savings decision of a person who lives for two periods,
working in the first and enjoying retirement in the second. Explain with the aid of
indifference curve diagrams how her plans change in the following scenarios, carefully
stating any assumptions you are making.
- Wages (in the current period) rise.
- Interest rates (available in the current period) rise.
- Prices are expected to rise in Period 2.
- How does her utility vary in each case?
- Explain how a worker will vary the number of hours she works in response to a rise
in the wage rate, decomposing the change into income and substitution effects.
- Write down the Slutsky equation for this problem.
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