ECON 2001: Intermediate Microeconomics

Transcription

1 ECON 2001: Intermediate Microeconomics Coursework exercises Term Tutorial 1: Budget constraints and preferences (Not to be submitted) 1. Are the following statements true or false? Briefly justify your answers. It is not possible for all goods to be luxuries. If a good is a Giffen good then a rise in its price must lead to a fall in the quantity of some other good purchased. 2. A family of three can afford a package holiday to any one of three holiday destinations, Florida, Georgia or Hawaii. The father prefers Florida to Georgia to Hawaii. The mother prefers Georgia to Hawaii to Florida. The child prefers Hawaii to Florida to Georgia. They decide that the family weakly prefers one package X to another Y if and only if more family members prefer X to Y than prefer Y to X. Are the following statements true or false with regard to the suggested family preference relation over these three holiday packages? Briefly justify your answers. The family preference relation is transitive. The family preference relation is complete. 1. Internet access in a particular area is priced as follows.the charge is 1 per hour for the first 20 hours per month and 50p per hour after that. There is also a monthly subscription fee of 10 which is waived if the consumer spends more than 40. What does a typical consumer s budget constraint between hours of internet access x and other goods y look like? How is choice likely to be affected by the nonlinearities in the budget constraint? 2. A new university has 9m to spend on research x and other activities y, of which it currently plans to spend 3m on research. Its funding body considers the following plans in an attempt to encourage research. Plan A: Increase the funding unconditionally to 12m. Plan B: Increase the funding to an unconditional 9m plus an additional 3m if and only if the university spends at least 6m on research. Plan C: Increase the funding to an unconditional 9m plus half of whatever the university spends on research. Find expressions for the budget sets faced by the university under each plan and sketch them. The university s preferences over bundles (x, y) are such that it would choose to spend constant budget shares of 1 3 and 2 3 on x and y under any linear budget constraint. What bundle will the university choose for each of the plans? Which plan results in most being spent on research? Which plan results in the funding body providing lowest funding? 1

2 Tutorial 2: Choice and demand (To be submitted in week commencing 20 October) 1. A consumer has preferences over goods x and y represented by utility function u(x, y) = x 2 y 3. Are the following statements true or false? Briefly justify your answers. The same preferences are represented by utility function u(x, y) = (2x) 2 (3y) 3. The same preferences are represented by utility function u(x, y) = 0.4 ln x ln y. 2. A consumer has preferences over goods x and y. Are the following statements true or false? Briefly justify your answers. Preferences are quasilinear if and only if the marginal rate of substitution between the goods is independent of the amount of one of the goods consumed. If two goods are perfect substitutes then preferences between them are both quasilinear and homothetic. 1. A typical pensioner has an income of 50 of which 10 is spent on fuel. The government decides to instigate a reform cutting the pension to 45 but subsidising fuel for pensioners to the extent that the price is halved. Is the typical pensioner better or worse off? Will pensioner fuel consumption rise or fall? Will the cost of the subsidy exceed or fall short of the reduction in pension expenditure? Make as few assumptions as you need to answer these questions. 2. Consumers in a particular economy have preferences between clothes (c) and food (f) given by utility function u(c, f) = c + ln f. What will Engel curves for clothes and food look like? Explain why these show that consumers buy only food when income is low but buy a constant quantity of food once income is sufficiently high. Explain by drawing an indifference curve diagram why this arises. What sort of preferences are these and how realistic do you find preferences of this sort? How do changes in the prices of the two goods affect demand for each? 2

3 Tutorial 3: The Slutsky equation, indirect utility and expenditure functions (To be submitted in week commencing 10 November) 1. A consumer buys positive quantities of certain goods. Are the following statements true or false. Briefly justify your answer. If the relative prices of those goods change but the consumer can still buy what he bought before then he is no worse off. If the relative prices of those goods change and the consumer can not buy what he bought before then he can not be strictly better off. 2. Are the following statements true or false? Briefly justify your answers. If all prices increase by the same proportion then the true cost of living index is the same for rich and poor households. If prices of different goods increase by different proportions then the true cost of living index is the same for rich and poor households only if preferences are Cobb-Douglas. 1. You are studying demand for a low quality staple food in a low income country. You find that consumers demand for the good is well captured empirically by a double log equation ln q = A + α ln y + β ln p where q is quantity purchased, y is income, p is price and A, α and β are constants. However your estimates suggest that α is negative and β is positive. Use the Slutsky equation to show that this is compatible with optimising behaviour by consumers only if the budget share of the good is greater than β/α. Explain. 2. Preferences between fuel q 1 and other goods q 2 are given by u = q 1 (q 2 a) where a is a positive constant. Is fuel a necessity or a luxury? Establish the form of the indirect utility function and hence show that the expenditure function can be written as e(u, p) = 2 up 1 p 2 + p 2 a. Establish an expression for a cost of living index. Suppose the government places a tax on fuel which doubles p 1 while leaving p 2 unchanged. Would you say the cost of living had increased by more for richer or poorer households? Comment. 3

4 Tutorial 4: Demand with endowments, labour supply (Not to be submitted) 1. Are the following statements true or false? Briefly justify your answers. If a consumer is a net seller of a good then income and substitution effects of an increase in its price must be in opposing directions. If a consumer is a supplier of labour then an increase in the real wage will not lead them to reduce chosen hours of work to zero. 2. Suppose someone has an uncompensated labour supply schedule h(w, m) = α + βw where h is hours of work, w is wage rate and m is unearned income. Are the following statements true or false? Briefly explain. This is compatible with optimising behaviour only if β > 0. The consumers preferences are quasilinear. 1. (a) Sanjay has a fixed income from labour of y which is spent on wheat q 1 and rice q 2 at prices p 1 and p 2. Wheat and rice are perfect complements in his preferences ie his utility function is u = min[q 1, q 2 ]. His uncompensated demand for wheat is therefore q 1 = y/(p 1 + p 2 ). Is wheat a normal good? Is it a Giffen good? (b) Rajesh has similar preferences but in addition to his fixed labour income of y he has a smallholding producing ω units of wheat. What is his uncompensated demand for wheat? Suppose p 1 = p 2 = 1 and ω > y. What happens to his demand for wheat as p 1 rises? Is this at all puzzling given your earlier conclusion about normality of demand for wheat? Explain how the theory of consumer demand differs when agents have endowments of the goods in question and therefore how this can happen. (c) Show how the Slutsky condition is affected by the existence of initial endowments. The case of perfect complements is a special case in which substitution effects are zero. Explain therefore why the direction of the effect of changes in p 1 on his demand for wheat depends simply on whether Rajesh is a net seller or buyer of wheat. 2. John has 24 hours in his day (T = 24) which he can devote to working and earning income or to leisure (including sleep). His utility function is given by u(r, L) = RL R+L where John devotes L hours to leisure and h hours to work. The wage rate is ω so he earns R = ωh if he works h hours. (i) Calculate his labour supply curve (ii) Calculate the elasticity of his supply with respect to the wage. (iii) Suppose now that John has to decide how much to work on two separate days. On the first day the wage rate is equal to ω=1 and on the other day ω= 4. Assume that he has to consume his income on the day on which he earns it. How much will he work on each day? Explain. 4

5 Tutorial 5: Intertemporal choice, uncertainty (To be submitted in week commencing 24 November) 1. An individual lives for two periods, earning Y in the first and nothing in the second. If he consumes C 0 when young and C 1 when old then his lifetime utility is C 0 + β ln C 1 where β is a preference parameter reflecting impatience and you should assume Y > β. Savings invested in the first period earn interest at a real rate r. Are the following statements true or false? Briefly justify your answers. Chosen saving is insensitive to either income Y or interest rate r. A higher interest rate makes the consumer worse off. 2. Are the following statements true or false. Briefly justify your answer. A risk averse individual will always prefer the expected value of their wealth with certainty to the uncertain distribution of their wealth. A risk averse individual will always purchase insurance if it reduces the uncertainty associated with their wealth. 1. (a) An individual expects to live for two periods and has convex preferences captured by lifetime utility function U(C 0, C 1 ) where C 0 is consumption when young and C 1 is consumption when old. Income when young is Y and resources when old consist only of what is saved when young. Those savings can either be invested in bonds where they earn a rate of return r or be invested in a family farm. An amount X invested in the farm when young will yield a return F (X) when old. Assume that F (X) is a concave function with F (0) > 1 + r. The returns are certain, however invested. Write down the budget constraint and, by substituting into lifetime utility or otherwise, explain why intertemporal optimisation requires U/ C 0 U/ C 1 = F (X) = (1 + r). Illustrate the choice diagrammatically. income Y? Comment. Does the chosen value of X depend on 2. (a) Consider the choice whether and how much to bet on the the outcome of the Cup Final. Let π denote an individual s assessed probability of victory for United. The best odds offered by any bookmaker are ρ (which is to say that the bookmaker will return (1 + ρ) times the stake if United win). Show that if individuals are risk averse then the only individuals who will bet on United winning are those for whom ρ > (1 π)/π. Explain this condition and illustrate the choice diagrammatically. (b) Individuals maximise expected utility and have within-state utility functions ln W where W is wealth. Supposing that ρ > (1 π)/π what fraction of wealth will an individual be willing to stake? If ρ = 4/11 and π = 3/4 then calculate the size of the bet. 5

6 Tutorial 6: Exchange equilibrium (To be submitted in week commencing 1 December) 1. Are the following statements true or false? Briefly justify your answers. Walras Law for the consumer says that, at all prices, the value of what he wishes to buy is equal to the value of what he wishes to sell. Walras Law for the economy implies that if all but one market clear then the remaining market must also clear. 2. In a pure exchange economy with endowments ω the price vector p constitutes a competitive equilibrium. Are the following statements true or false? Briefly justify your answers. The price vector 2p is a competitive equilibrium in the same economy. The price vector 2p is a competitive equilibrium in an economy with twice these endowments 2ω. 1. The population of a country consists of two types of people in equal numbers. Northerners consume salt x 1 and sugar y 1 and have preferences summarised by the utility function u 1 (x 1, y 1 ) = α ln x 1 + (1 α) ln y 1. Southerners also consume salt x 2 and sugar y 2 and have preferences summarised by utility function u 2 (x 2, y 2 ) = β ln x 2 + (1 β) ln y 2. Each Northerner has an endowment ω 1 of salt and each Southerner has an endowment ω 2 of sugar. Let p be the relative price of salt. (a) Derive an expression for the value of p at which aggregate excess demand for sugar is zero. What is the aggregate excess demand for salt at that price? Comment on the reasons for this result. (b) What are the equilibrium allocations of salt and sugar? (c) Discovery of new salt resources in the North increases each Northerner s endowment of salt. What happens to the equilibrium price? Who is better off as a result of the discovery? Explain. (d) Suppose the salt discovery happens in the South instead so that each Southerner acquires a small endowment of salt. Who benefits? Explain the contrast. 2. The owl and the pussycat dine on mince x and slices of quince y. The owl has an endowment of X units of mince whereas the pussycat has an endowment of Y slices of quince. Suppose that the two trade mince and quince with each other at prices p x and p y. Their preferences are each described by utility function u = α ln(x A)+(1 α) ln y. If either has an endowment of value E then show that their demand for mince is A + α(e p x A)/p x. Hence show that the market for mince clears if the relative price is p x /p y = is increasing in A. αy (1 α)(x 2A). Explain why the equilibrium relative price of mince 6