Introductory Microeconomics: Problem Set 3

Utility maximisation and applications.

1. 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.


2. 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.


3. Consider the market for salt (yes, it matters that it is salt and not, say, health care or foreign travel). Imagine your household's consumption as being divided between salt and a composite of all other goods (the 'quantity' being defined by your expenditure). Suppose the price of salt trebles.
   
   
   1. Use a diagram to represent the magnitude of the substitution effect. How big is this effect?
   2. What about the magnitude of the income effect?
   3. What can we conclude about the change in your optimal choice that is induced by this enormous increase in the price of salt?
   4. 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?


4. Two workers work together as a team. If each worker i puts in effort e_i, their total output is y = A ln(1 + e₁ + e₂ ). Their pay depends on their output: if they produce y, each receives w = w₀ + ky. A, w₀ and k are positive constants. Each of them chooses her own effort level to maximise her own utility, which is given by u_i = w - e_i.
   
   1. Write worker 1's utility as a function of her own and worker 2's effort.
   2. If worker 1 expects worker 2 to exert effort e₂, what is her optimal choice of effort?
   3. How much effort does each exert in a symmetric equilibrium?
   4. Is this a game? Is it a prisoners' dilemma?
   5. How could the two workers obtain higher utility?
   
   From the Maths Workbook.


5. Lucas is a bright young man who lives for two periods. He must decide how much to consume today (c₁) and in the next period (c₂). In the first period, his income is y₁, earned making rollerblades in a factory. Lucas has also received an offer to study at Nobel University. University studies, should he undertake them, occur in an instant at the beginning of period 2, when Lucas must also pay tuition fees p. Immediately thereafter he can take a job in the City and receive y₂ > y₁. If he does not study, he continues to earn y₁ at the factory in period 2. Lucas has access to a bank account that pays interest r on his savings in period 1. That is, whatever he does not consume stays in the bank, and (at the beginning of) the next period he receives back his principal plus interest. He can borrow from the bank at the same rate to fund consumption in period 1 (if he borrows x then at the beginning of period 2 he has to pay back x(1 + r)). He cannot borrow and repay within period 2 to fund his university fees.


1. The period 1 budget constraint is given by c₁ + s₁ = y₁, where s₁ denotes first period savings, money deposited in the bank. The period 2 budget constraint is given by
   c₂ = y₁ + s₁( 1 + r )    if he doesn't go to university, or
   c₂ = y₂ - p + s₁( 1 + r )    if he does.
   Show that his intertemporal budget constraint is the following:
   
2. What are the second period terms divided by 1 + r?
3. Why does it matter if c₁ is greater, equal to, or smaller than
4. Draw a diagram of the intertemporal budget constraint, with c₁ and c₂ on the axes, assuming the following parameter values: y₁ = 1, y₂ = 3, p = 1, r= 0.
5. Draw, in this diagram, indifference curves that show Lucas (i) going to university and (ii) choosing not to.
6. Now redraw the IBC assuming the following parameter values: y₁ = 1, y₂ = 3, p = 1, r= 2. (r = 2 means 200% interest; it's not a mistake. So do use the value 2 in your diagram.) With the preferences, i.e. indifference curve families, you drew before, will Lucas change his decision in either case? Why?

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