Introductory Microeconomics: Problem Set Week 34

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.


3. This question from Perloff relates obesity to innovations that make it easier to consume tasty but unhealthy food such as frozen chips. Jeff gets pleasure from the tastiness of his potatoes (T) and leisure (N). For each additional unit of tastiness, he must spend p hours preparing the food. His time constraint is thus N + pT = 24 (we ignore sleep and work here). His utility function is U = TN ^0.5.
   
   
   
   1. What is Jeff's marginal rate of substitution MU_T / MU_N?
   2. What is the marginal rate of transformation P_T / P_N?
   3. What is Jeff's optimal choice (T^*, N^*)?
   4. If innovation lowers the price of producing tastiness (which we are interpreting as unhealthy and obesity-inducing) will Jeff consume more tastiness or more leisure?


4. Consider a household which works today and retires tomorrow, and wishes to consume in both periods. Consumption today is C₁, consumption tomorrow C₂. Income (Y₁) saved today earns interest rate r, generating total savings tomorrow equal to (1 + r) x (Y₁ - C₁).
   
   
   
   1. Use a diagram with indifference curves and a budget constraint to show how the household responds to an increase in the interest rate r. Does it save more, or less?
   2. Explain your result in terms of income and substitution effects. Would your result be valid for any reasonable model of household preferences?


5. The overall rate of inflation masks widely-varying price changes for particular goods and services. From April 2022 to March 2023, CPI inflation averaged 8.8%. For natural gas, the rate was 115%, for child care services 4.1%, and for personal computers -6.5%.
   
   Imagine (counterfactually) that my employment contract had a cost of living adjustment built in: my pay would automatically rise each year by a percentage equal to the inflation rate that had prevailed over the previous year. Imagine too that the adjustment were based on a Brian-specific consumer price index, which rose or fell by just enough to allow me to buy again exactly the bundle of goods and services that I had bought a year earlier.
   
   Simplifying and imagining that I spend all of my income on just two goods, the prices of which change at different rates over the year, use a diagram with indifference curves and budget constraints to show whether the cost of living adjustment is too mean, too generous, or just right to keep me as happy as I was last year.