intro micro

Prelims Micro 2010

INTRODUCTION TO MICROECONOMICS

Trinity College, Oxford

Chris Wallace, Michaelmas Term 2010

Welcome (back) to Trinity!

This handout contains the ten topics with the readings, questions, and essays that make

up the Microeconomics part of the first-year Economics Course for PPE and E&M (or

second year for MEM and third year for EEM). The tutorials and classes will be held in

my office, (staircase 3, room 3) at Trinity College.

Tutorials and Classes

Each of the ten topics corresponds to one week of work: the course runs throughout

Michaelmas Term and for the first two weeks of Hilary Term. In the (odd) weeks that

correspond to a tutorial, the assigned work must be handed in by 5.00pm the day

before the tutorial to my pigeon hole at Trinity. I will mark your work and return it

during the tutorial. Each tutorial (of 2 students) will be one hour long.

There are five classes (of 4 students) which will be two hours long. In (even) weeks

corresponding to classes, you are not required to hand in the assigned work (you may if

you wish—and I will mark it): but you must attempt it! The work will be discussed

in class, and it will quickly become obvious if you have not done it.

Textbooks

The text for the course is Varian (2005) which you should buy if at all possible. Older

editions of the text (particularly the 5th and 6th) are virtually identical to this and may

be substantially cheaper. The chapter numbers may have changed however, so take care

to check you read the correct material (I refer to the 7th edition). In addition, the library

has multiple copies. We will be working steadily through most of this book.

For mathematics, the book I shall refer to most often is Anthony and Biggs (1996). A

more advanced text for those comfortable with mathematics already is Simon and Blume

(1994). Finally, if you have not done much mathematics before, and you would like a

companion text to the first book, try Renshaw (2005), although this book alone is not

sufficient. A very useful resource is the department’s maths workbook. Working through

the problems contained therein is certainly sufficient. It can be found at:

http://www.economics.ox.ac.uk/index.php/ . . .

. . . intranet/undergraduate maths workbook

Prelims Micro 2010

Lectures

Please go to the Introduction to Microeconomics Lectures; this is the best way to find

out what will be on the exam. Students who are less familiar with mathematics should

also attend the introductory mathematics lectures provided. The lecture lists contain

times and places; more details may be found at the department’s sites:

http://www.economics.ox.ac.uk/index.php/undergraduate/ . . .

. . . details/introductory microeconomics/

. . . details/prelims elementary mathematical methods/

Office Hours

I have an office hour on Wednesdays in Trinity at 11.30-12.30. Feel free to come along

if you have any problems or questions (academic or otherwise). If you cannot make this

time, get in touch, and we can sort out alternative times. My contact details are below.

Mailing address

Prof. C. C. Wallace

Trinity College

Oxford OX1 3BH

email address

[email protected]

Website

http://malroy.econ.ox.ac.uk/ccw/

Phone number

01865 2 71062

Prelims Micro 2010 MT Week 1

Consumers I: Preferences and Representation

One-Hour Tutorials

Suggested Reading

The main reading is Varian (2005, Ch. 1-7). If you have never done economics before, you

may find Begg, Fischer, and Dornbusch (2008) useful (or any earlier edition). Another

text to glance at as a complement is Morgan, Katz, and Rosen (2005). Take notes on:

• Preferences and Utility Representations.

• Indifference Curves and the Marginal Rate of Substitution.

• Budget Constraints and Choice.

• Consumer Demand, Substitutes, and Complements.

• Normal, Inferior, and Giffen Goods.

• Revealed Preferences.

Much of the mathematics you will need for this week may be found in Varian (2005).

Also try Anthony and Biggs (1996, Ch. 1-3, 7), or the more advanced Simon and Blume

(1994, Ch. 1-2). The workbook contains the appropriate material as well: see Ch. 1-4.

The same material (at a gentler pace) may also be found in Renshaw (2005).

Questions and Essay

You must hand in your work by 5pm the day before the tutorial to my pigeon hole at

Trinity. Please answer the following questions—keep your answers short!

(1) Suppose a consumer consumes a quantity x of good X and a quantity y of good

Y . Write the consumption bundle as (x, y). The consumer is equipped with a

strict preference relation � and with an indifference relation ∼. The consumer

has the following preferences:

(3, 3) � (3, 1) ∼ (1, 3) ∼ (2, 2) � (4, 0) ∼ (0, 4) ∼ (2, 1) ∼ (1, 2) � . . .

. . . � (3, 0) ∼ (0, 3) ∼ (1, 1) � (2, 0) ∼ (0, 2) � (1, 0) ∼ (0, 1) � (0, 0).

(a) Construct a utility representation for these preferences. With x = 0 what

are the values of the marginal utility of good Y for your representation?

(b) Construct a different utility representation for the same preferences. Again,

calculate the marginal utility values for good Y when x = 0. Comment.

(c) Plot indifference curves for these preferences on a graph.

(d) Suppose the consumer is currently consuming (2, 1). How much of good X

is the consumer prepared to give up in order to get an additional unit of Y ?

What is the associated marginal rate of substitution?

(e) Does the marginal rate of substitution depend upon the utility representation

you use to describe preferences? Does marginal utility? Comment.


(2) Now suppose the consumer has the following preferences:

(3, 3) ∼ (4, 0) � (3, 1) ∼ (1, 3) ∼ (2, 2) and

(2, 2) � (4, 0) ∼ (0, 4) ∼ (2, 1) ∼ (1, 2).

Is there a utility representation for these preferences? Why, or why not?

(3) Returning to the preferences of question (1), suppose that the consumer has an

income of 2 and that the price of X (pX) is equal to that of good Y (pY ) and both

have price 1. Draw the associated budget line in your graph. What is the optimal

choice of the consumer? Suppose the consumer’s income increases to 3. Now

what is the optimal choice? At this optimal choice, what is the market exchange

rate? What is the consumer’s marginal rate of substitution? Comment.

(4) Draw several indifference curves for each of the following utility functions: u(x, y) =

x1/2y1/2 ; u(x, y) = ln x+ ln y ; u(x, y) = x+ y ; u(x, y) = min{x, y}. What sorts

of preferences do these utility functions represent?

(5) Illustrate the difference between normal, inferior, ordinary, and Giffen goods

using income-offer curves, Engel curves, price-offer curves, and demand curves.

(6) Write a short essay (fewer than four written sides including diagrams) entitled:

“People do not spend their lives solving mathematical problems, so

the microeconomic theory of consumer choice has little to offer.”

Discuss.


Consumers II: Decompositions and Demand

Two-Hour Classes

Suggested Reading

The main reading is Varian (2005, Ch. 8-11, 14-16). Take notes on:

• Income, Substitution, and Endowment Effects.

• The Slutsky and Hicks Decompositions and Demand Curves.

• Labour Supply, Intertemporal Choice, and Asset Markets.

• Consumer Surplus, Compensating and Equivalent Variations.

• Market Demand and Elasticities.

• Market Equilibrium and Comparative Statics.

The mathematics you will need for this week may be found in Anthony and Biggs (1996,

Ch. 1-2, 6-7, 9), Simon and Blume (1994, Ch. 2-3), or the workbook Ch. 1-6.

Questions and Essay

If you would like to hand in work, please do so by 5pm the day before the tutorial (you

do not have to hand in work this week—but you must attempt the questions!)

(1) Illustrate the income and substitution effects on a graph. Hence distinguish be-

tween inferior, normal, and Giffen goods. Explain the endowment income effect.

(2) Graphically compare and contrast the Slutsky and Hicks decompositions of the

effect on demand of a price change.

(3) If leisure is an inferior good, what can be said about the slope of labour supply?

(4) Graphically compare and contrast compensating variation with equivalent varia-

tion, drawing on their relationship with substitution and income effects.

(5) Suppose that the quantity demanded of a commodity is given by qD = 10 − p

where p is price. Suppose that supply is given by qS = 23p.

(a) Calculate the equilibrium price and quantity for this market. Calculate the

consumer surplus, producer surplus, and revenue at this equilibrium.

(b) Now suppose supply shifts to qS = p. Perform the same calculations for the

new equilibrium. By how much have consumer surplus and revenue changed?

(c) Now suppose that supply shifts back to qS = 911p. Calculate the new equi-

librium. What is the price elasticity of demand at this equilibrium?

(d) What is the marginal revenue from increasing production from 4 to 5 units

in this market? How does this relate to the elasticity found in part (c)?

(6) Find the price elasticity of demand for the following demand functions: qD = 1−p;qD = p−1 ; qD = Ap−α, where A > 0 and α > 0.


(7) Suppose that the quantity demanded of a commodity is given by qD = 100 − pwhere p is price (in pounds). Suppose that supply is given by 25 + 1

2qS = p.

(a) Find the equilibrium price and quantity for this market. Calculate the con-

sumer surplus and the producer surplus.

(b) Suppose the government imposes a £15 tax per unit, collected from the

suppliers. Now how much do consumers pay? By how much has consumer

surplus fallen?

(c) How much do the suppliers receive (after the tax is collected) for each unit

they sell? What is the new producer surplus? Who actually pays the tax?

(d) What is the government’s revenue from the tax? What is the deadweight loss

associated with the introduction of the tax?

(e) Illustrate your answers in a graph.

(8) Write an essay plan (you do not need to write the essay) entitled:

How do the elasticities of demand and supply affect the impact of a

tax on (a) revenue, (b) consumer surplus, and (c) deadweight loss?


Producers I: Profits and Costs

One-Hour Tutorials

Suggested Reading

The main reading is Varian (2005, Ch. 18-21). Take notes on:

• Production Possibility Sets and Production Functions.

• Technical Rates of Substitution and Marginal Products.

• Increasing, Decreasing, and Constant Returns to Scale.

• Fixed and Variable Factors: the Short and Long Run.

• Profit Maximisation and Cost Minimisation.

• Average Cost, Variable Cost, and Marginal Cost Functions.

• Marginal Costs and Supply Functions.

The mathematics you need may be found in Anthony and Biggs (1996, Ch. 6-10), Simon

and Blume (1994, Ch. 2-5), or the workbook, Ch. 5-6. By the end of the week it is very

important that you fully understand calculus with one variable. In particular, you should

be able to differentiate functions of one variable, and find their maxima and minima.

Questions and Essay


Trinity. Please attempt all of the following questions.

(1) Under what conditions does the production function f(K,L) = AKαLβ, where

A > 0, exhibit constant, increasing, and decreasing returns to scale?

(2) Is it possible to have decreasing marginal products for every input and yet in-

creasing returns to scale? If so, give an example; if not, prove it.

(3) Can a profit-maximising competitive firm operating with constant returns to scale

make positive profits in equilibrium?

(4) Suppose a firm’s costs are given by c(q) = 100 + 15q−6q2 + q3, where q is output.

(a) What exactly is a cost function? How is it derived?

(b) Derive algebraic expressions for fixed, variable, average, marginal, average

fixed, and average variable costs. Sketch these functions for q = 0 to q = 10.

(c) At what output does minimum average cost occur? What about minimum

average variable cost? What is the value of marginal cost at these outputs?

(d) What is the supply curve of this (perfectly competitive) firm?

(5) Is it true that in the long-run a firm always operates at the minimum level of

average costs for the optimally sized plant to produce a given amount of output?



How does the profit-maximising output of a perfectly competitive

firm change in the short and long run following an increase in the

wage rate? How does its demand for labour change? Explain how

your answer might differ if it were the price of capital that increased.


Producers II: Perfect Competition and Monopoly

Two-Hour Classes

Suggested Reading


• Perfect (or Pure) Competition.

• Profits and Producer’s Surplus.

• Industry Supply in the Short and Long Run.

• Economic Rent.

• Monopoly Profit Maximisation and Supply.

• Monopoly Inefficiency and Deadweight Losses.

• Comparative Statics in the Short and Long Run.

The mathematics you need for this week may be found in Anthony and Biggs (1996,

Ch. 11-13), Simon and Blume (1994, Ch. 13-14, 17), or the workbook, Ch. 7-8. It is

important to be able to partially differentiate functions of more than one variable, and to

make a start on finding minima and maxima—a theme returned to in the coming weeks.

Questions and Essay



(1) A (perfectly competitive) firm has a production function f(K,L) = K1/3L1/3.

Suppose the price of capital is 1 and the price of labour is 1. Output sells for 9.

(a) Write down the firm’s profits in terms of K and L. How much of each factor

will the (profit-maximising) firm employ in production?

(b) How much output does the firm produce as a result?

(c) Write down costs in terms of K and L. Using the fact that output (labelled

q) is given by q = K1/3L1/3, substitute for L in terms of q and K.

(d) Hence show that the cost function is given by c(q) = 2q3/2.

(e) Write down profits in terms of q and find the optimal output. Comment.

(2) How does a perfectly competitive firm make its supply decision? Illustrate your

answer on a graph. Can the firm make positive profits?

(3) A perfectly competitive industry is composed of 60 firms. The industry demand

curve is Q = 1100 − 40P , where P is price and Q is the total market demand.

Firms have identical costs: C = 50 + 5q + 12q2, where q is the firm’s output.

(a) What is the market price and level of each firm’s output in the short run?

(b) How much profit does each firm make? Comment.

(c) How many firms operate in the long run? How much profit do they make?


(d) The government imposes a per-unit sales tax of 5. What happens to each

firm’s output in the short run? How many firms will survive in the long run?

(e) With this tax, how much does each firm produce in the long run? Comment.

(f) The government changes its mind, scraps the tax, and introduces a lump-sum

subsidy of 32 for each firm. What happens in the short and long run?

(4) How does a monopolist make its supply decision? Illustrate your answer on a

graph. Can the firm make positive profits?

(5) Using a diagram, illustrate the deadweight loss associated with monopoly pro-

duction. How is this affected by the elasticity of the demand curve?

(6) Suppose industry demand is Q = 100−P , where P is price and Q is the quantity

demanded. A firm has costs C = 10q, where q is the quantity it produces.

(a) What is the long run equilibrium quantity produced in a perfectly competi-

tive industry? How much would a monopolist produce? Comment.

(b) How much deadweight loss results in either case? Illustrate in a diagram.

(7) A monopolist has two plants, A and B, whose outputs of the same good are qAand qB respectively, and whose total costs are CA = 50 + 8qA + q2

A and CB =

50+22qB+q2B. Total demand for the good is P = 100−2q where P is price. What

is the profit-maximising price and division of output between the two plants?


Outline the main differences between perfect competition and mo-

nopoly. Under what conditions is each of these likely to occur?


Producers III: Monopolistic Competition and Factor Markets

One-Hour Tutorials

Suggested Reading

The main reading is Varian (2005, Ch. 19, 25-26). Take notes on:

• First, Second, and Third Degree Price Discrimination.

• Monopolistic Competition in the Short and Long Run.

• Comparative Statics in the Short and Long Run.

• Factor Demand Curve Derivation.

• Perfect Competition and Factor Demands.

• Monopoly and Factor Demands.

• Monopsony.

There is no “new” mathematics required for this week. You should continue to get to

grips with the material on partial differentiation and optimisation outlined last week.

Questions and Essay



(1) Explain why a first-degree price discriminating monopoly generates no deadweight

loss. What is second-degree price discrimination? Give examples of this practice.

(2) A monopolist has total costs given by C = 1000 + 10q + 16q2, where q is the

quantity produced. There are two markets for this good, A and B, with demand

qA = 110− pA and qB = 100− 2pB respectively. pA and pB are the prices charged

in markets A and B; qA and qB are the quantities sold in each of the markets.

(a) If the monopolist can third-degree price discriminate, how much will it sell

to market A? How much will it sell to market B? What are profits?

(b) If the monopolist must charge the same price in both markets, how much

will it produce? Now how much profit is made? Comment.

(c) What happens if fixed costs double? Briefly, do you think the government

ought to legislate against price discrimination?

(3) How are the factor demand curves of a perfectly competitive firm related to its

production function? Why are there no “Giffen” factors of production?

(4) How are the factor demand curves of a monopolist determined? How do these

compare to those derived in your answer to question (3) above?

(5) A firm sells a good in a perfectly competitive market, but is the only buyer of the

(single) factor used in its production. Illustrate the optimal factor-demand for

this monopsony in a graph. Show any associated efficiency losses in your diagram.



How does the imposition of a (i) lump-sum tax and (ii) quantity tax

affect the supply decision of a monopolistically competitive firm in

the short and long run? What happens to the number of firms?


Producers IV: Games and Oligopoly

Two-Hour Classes

Suggested Reading


• Games and Game Theory.

• Nash Equilibrium.

• Cournot Competition.

• Bertrand Competition.

• Comparative Statics.

• Stackelberg Leadership.

There is no new mathematics required for this week. You should now be comfortable

with optimisation and differentiation for multivariate functions. You may like to look

ahead to next week’s very important topic: constrained optimisation, see Anthony and

Biggs (1996, Ch. 21-22), Simon and Blume (1994, Ch. 18-19), or the workbook Ch. 9.

Questions and Essay



(1) What is a (strategic-form) game? How are games applied in economics?

(2) In each of the following representations of strategic-form games, player 1 chooses

between the rows, and player 2 chooses between the columns. Player 1’s payoff is

in the bottom-left corner of the cell, player 2’s payoff is in the top-right corner.

C D

C3

3

4

1

D1

4

2

2

S R

S5

5

4

0

R0

4

3

3

H T

H−1

1

1

−1

T1

−1

−1

1

Find the Nash equilibria of each. What do you think will happen in these games?

(3) There are two firms (A and B) in an industry. Firm A has costs C = F +cqA and

firm B has costs C = F + cqB, where qA and qB are the amounts produced by the

respective firms. Market demand is Q = a− P , and total supply is Q = qA + qB.

(a) Write down the profits for each firm, and hence find their reaction or best-

response curves. What would firm A produce if firm B set qB = 0?

(b) Identify the Nash (Cournot) equilibria of this game.

(c) What is the market price at equilibrium? What are profits?


(d) Suppose both firms chose to produce qA = qB = (a− c)/4. How much profit

would they make? Why is this not a Nash equilibrium?

(e) Calculate the output that would be sold under perfectly competitive condi-

tions in this market. What would a monopolist produce? How does Cournot

competition compare with perfect competition and monopoly?

(f) Illustrate your answers to (a)-(e) using a diagram.

(4) Two firms with identical marginal costs and no fixed costs simultaneously choose

prices. The firm with the lower price serves the entire market. If prices are equal,

the firms share the market equally. What is the Nash (Bertrand) equilibrium?

(5) Two firms compete in quantities. What happens if the marginal cost of one of

the firms rises? How does your answer change when the firms compete in prices?

(6) A man with a hand grenade comes up to you in the street and says, “Give me

one pound, or I will pull the pin.” What should you do? This game might be

represented by the following game tree.

..................................................

..................................................

..................................................

..................................................

.............................

.....................................................................................................................................................................................................................................

................................................

................................................

................................................

................................................

......................................

.....................................................................................................................................................................................................................................•

•

•

•

•

You

Man

Pull Pin

Don’t

Don’t

Give Cash −1, 1

0, 0

†R.I.P

, †R.I.P

The notation † indicates a very big (in absolute value) negative payoff. Interpret

the elements in the game tree above. How might you solve this sort of game?

Does the solution suggested by this method for the game above seem reasonable

to you? What should you do?

(7) Compare market prices, quantities produced, and profits generated when firms

choose quantities sequentially (Stackelberg) rather than simultaneously (Cournot).


Why do firms that set prices make positive profits?


Mathematical Interlude: Lagrangians

One-Hour Tutorials

Suggested Reading

There is no new reading from Varian (2005) specifically. The aim this week is to under-

stand constrained optimisation, and Lagrangian techniques in particular. Focus on:

• Constrained Optimisation Problems.

• Objective Functions.

• Constraints.

• The Lagrangian Technique.

• First Order Conditions.

• Lagrange Multipliers.

• Complementary Slackness.

A good starting place is Anthony and Biggs (1996, Ch. 21-22), or the workbook, Ch.

9. An excellent (if advanced) treatment can be found in Simon and Blume (1994, Ch.

18-22). Constrained optimisation is very important in economics, and is not something

you will have learnt elsewhere. You should be able to use the Lagrangian method (see the

theorem stated in the mathematical appendix to this handout for a correct, if technical,

statement of this method), although not necessarily understand why it works.

Questions and Essay


Trinity. Please have a go at as many of the following questions as you can.

(1) An individual consumes x1 of good 1 and x2 of good 2. Utility is given by

u = x1/21 x

1/22 . Suppose income is 40, the price of good 1 is 4 and the price of good

2 is 2. By writing down an appropriate Lagrangian, what quantities maximise

the consumer’s utility? What is the marginal utility of income?

(2) A firm’s production is Q = K1/3L2/3 where Q is output, and K and L are the two

inputs. The cost of a unit of K is r and the cost of a unit of L is w. What inputs

minimise the cost of producing an output of 32 units when w = 5 and r = 20?

Give an economic interpretation of the Lagrange multiplier in this example.

(3) A student’s utility is u = i2/3l1/3s1/2, where i, l, and s are the number of hours

spent idling, lazing about, and sleeping respectively. A day contains 24 hours.

(a) In a given day, how should the student allocate their time to maximise utility?

(b) Once a week the student is required to spend an hour at a tutorial (from

which utility is zero). How much less time is spent sleeping that day?


(4) A consumer has utility given by u = lnx + ln y where x and y are the amounts

consumed of two goods (X and Y respectively). The price of good X is p and the

price of good Y is 1, and the consumer has a money income of m. However, the

government, in its infinite wisdom, has forbade the consumption of more than x̄

of good X. Suppose that x̄ < m/p.

(a) What is the budget set faced by the consumer? Draw it in a graph.

(b) Write down all the constraints faced by the consumer, and hence construct

an appropriate Lagrangian.

(c) Suppose initially that x̄ > m/2p. Find the consumer’s optimal demands for

the two goods and illustrate them in your graph. What are the values of

the Lagrange multipliers in your solution? Interpret them in terms of the

consumer’s marginal utility.

(d) Now suppose x̄ < m/2p. What are the consumer’s optimal demands now?

What are the values of the multipliers? Illustrate your solution in the graph.

(e) What happens when x̄ = m/2p?

(5) A consumer obtains utility u(x, y) from consuming a quantity x of good X and a

quantity y of good Y . Suppose income is m and prices are pX and pY respectively.

Solve the consumer’s maximisation problem in each of the following cases:

(a) u(x, y) = xy;

(b) u(x, y) = x+ y;

(c) u(x, y) = ln(x+ y);

(d) u(x, y) = x2 + y2;

(e) u(x, y) = (x− γ)1/2(y − γ)1/2, where γ > 0;

(f) u(x, y) = (x+ γ)1/2(y − γ)1/2, where γ > 0.

In part (e) what happens if pX = pY = 1 and m = γ? How big does m need to

be? In part (f) what happens if pX = pY = 1 and m ≥ 2γ? What about m < 2γ?

(6) Write a very short essay (fewer than three sides including diagrams) entitled:

What is a constrained optimisation problem? Why are such prob-

lems important in economics and how may they be solved? What

economic concepts are captured by Lagrange multipliers?


Equilibrium I: Market Success and Efficiency

Two-Hour Classes

Suggested Reading


• Competitive General Equilibrium.

• Edgeworth Boxes and Exchange Economies.

• Robinson Crusoe Economies.

• Pareto Efficiency and Social Welfare.

• The First Welfare Theorem.

• The Second Welfare Theorem.

The mathematics this week continues to apply the Lagrangian technique introduced in

last week’s tutorial. Practice makes perfect.

Questions and Essay



(1) Illustrate a two-person exchange economy in an Edgeworth box. Identify the initial

endowment, prices, final allocation, and bundles consumed by the two agents.

(2) Carefully define a competitive (general or Walrasian) equilibrium. Identify an

equilibrium in the two-person economy illustrated in your answer to (1) above.

(3) What is a Pareto efficient allocation? Identify all such allocations in the two-

person economy illustrated in your answer to (1) above.

(4) What exactly does the first welfare theorem say about the relationship between

equilibrium and efficiency? Carefully state any assumptions required.

(5) What exactly does the second welfare theorem say about the relationship between

equilibrium and efficiency? Carefully state any assumptions required.

(6) Consider a two-person two-good exchange economy. Consumers (A and B) have

identical utility functions given by uA = ln xA + ln yA and uB = ln xB + ln yB,

where xA and yA are the quantities A consumes of the two goods (X and Y ), and

xB and yB are the quantities B consumes. Normalise the price of good X to 1,

and write p for the price of good Y . Suppose consumer A has an endowment of

10 units of good X whilst consumer B has an endowment of 10 units of good Y .

(a) Draw this economy in an Edgeworth box. Identify the initial endowment.

Sketch in a budget line and some indifference curves for each consumer.

(b) Write down the budget constraints for the two consumers.

(c) Carefully write down the consumers’ constrained optimisation problems.


(d) Solve each consumer’s problem using a Lagrangian. Evaluate the consumers’

demand curves for X and Y . What does the Lagrange multiplier represent?

(e) Write down two market clearing conditions.

(f) Hence find the general equilibrium. Sketch it in your diagram. Is it efficient?

(7) An economy contains a single turnip farmer who has one unit of (perfectly divis-

ible) time available for work and leisure, and a field. The farmer has preferences

represented by u(t, l) = ln t+ln(1− l), where t is the number of turnips consumed

and l is labour supply. Normalise the turnip price to 1; write w for the wage and

r for the rental price of the field.

(a) Argue that the farmer’s budget constraint may be written t ≤ wl + r.

(b) Hence (using a Lagrangian) find the farmer’s optimal demand for turnips

and supply of labour in terms of r and w. When is labour supply positive?

There is only one industry in this economy—turnips—which requires two inputs:

labour and fields. Turnips are competitively produced with a production function

f(L, F ) = L1/2F 1/2, where L is labour and F is the number of fields used.

(c) If the industry employs a total labour input of L and F fields, what are costs?

(d) Using a Lagrangian with multiplier λ, solve the cost-minimisation problem

and deduce the factor demand curves for labour and fields. Interpret λ.

(e) Remember that the supply of fields is FS = 1. Find turnip supply in terms

of r/w. Using three market-clearing conditions, find the general equilibrium.


What do the first and second welfare theorems say? Why are they

important, and why are they true?

Prelims Micro 2010 HT Week 1

Equilibrium II: General Equilibrium and Optimisation

One-Hour Tutorials

Suggested Reading

There is no new reading this week. Use the vacation to go over all the material that was

covered last term! Bring your questions to the tutorials.

Questions and Essay


Trinity. Please attempt both of the following questions.

(1) A corn producing economy consists of landowners and peasants. There is one

acre of land in total, which is equally divided amongst n landowners. There are

m peasants, who own no land. Both landowners and peasants have one unit

of (perfectly divisible) time available for work and leisure. They have identical

preferences over consumption and leisure:

u (c, l) = ln (c) + ln (1− l)

where c is an individual’s consumption of corn and l is labour supply.

(a) Normalise the price of corn to 1 (as numeraire) and let the wage rate be w.

Let q be the rental price of land (i.e. the amount per acre that an owner of

land earns each period). Write down an appropriate budget constraint for

the peasants. How does the budget constraint of the landowners differ? Find

the optimal consumption/leisure trade-off for the peasants and landowners

by maximising u subject to the relevant budget constraint.

(b) How low does the price of land need to be for landowners to choose to work?

(c) The technology for producing corn with a total labour input L and a total

stock of land S is Y = L1/2S1/2, where Y is the total output of corn. Corn

is produced competitively. Write q/w in terms of the total labour input L.

(d) Find the value of q/w which clears the labour market when both peasants and

landowners choose to work. How would you compute the general equilibrium?

(e) Find conditions on n and m such that the landowners choose not to work.

(2) This week’s essay should be a more substantial effort. Remember to use diagrams

in your essay, and feel free to make it as long as you like (within reason!)

“An economy in which people selfishly pursue their own objectives

can never achieve the objectives of a fair society.” “Competitive

markets are efficient, so we should not attempt to change them.”

Adjudicate.


Equilibrium III: Market Failure and Externalities

Two-Hour Classes

Suggested Reading


• Consumption and Production Externalities.

• Property Rights and Missing Markets.

• The Tragedy of the Commons.

• Correcting Externalities with Taxation.

• Public Goods: Non-Excludable Goods and Non-Rival Goods.

• Free-Riding and Efficient Provision.

• Asymmetric Information and the Market for Lemons.

There are no new mathematical techniques to be learnt this week. Continue getting to

grips with constrained optimisation and Lagrangians.

Questions and Essay



(1) What is the Tragedy of the Commons? Use a diagram to explain why it occurs

and how it might be resolved. Give some examples from the “real world”.

(2) What is the difference between marginal private cost and benefit and marginal

social cost and benefit? How can this difference result in welfare losses to society?

(3) How might a Pigouvian tax alleviate the problems arising in markets for goods

with negative externalities? Is there any other way to alleviate these problems?

(4) A student who lives in college has one hobby: listening to Mahler. The student’s

hi-fi system can produce noise levels of up to 100 decibels. His utility depends on

the loudness of Mahler, measured in decibels D, and the amount of money he has,

MS, so that preferences are represented by the utility function uS = 10D1/2 +MS.

(a) Term is over and the student has run out of money. What is his utility?

(b) A fellow lives on the same staircase. She is irritated by loudly played Mahler.

Her utility function is uF = 10(100−D)1/2 +MF , where MF is the amount

of money she has. What would be a Pareto-efficient noise level?

(c) The fellow has £100. What is the maximum bribe she is willing to pay the

student to turn down the music to a Pareto-efficient level? Is £50 enough?

(d) Draw an Edgeworth box to illustrate this problem. In your diagram, carefully

indicate the initial endowment, the contract curve, and the outcome if the

£50 bribe is accepted.


(5) What is a non-rival good? What is a non-excludable good? Give examples of

goods that have both, either, and neither of these properties.

(6) Why might a public good be under-provided by the market?

(7) What is the market of lemons? How does such asymmetric information result in

efficiency losses in such markets? Suggest some solutions to this problem.


How ought a government intervene in markets with externalities?

Prelims Micro 2010

A Mathematical Appendix: Kuhn-Tucker

For those of you who are a little uneasy with the presentation of constrained optimisation

in the lectures, workbook, and some textbooks, here is a more general statement of the

appropriate theorem (which, although you will use, you need not understand in any depth

at this stage). Let the problem to which you seek a solution be P1, so that

P1 : maxx1,...,xn

f(x1, . . . , xn) s.t. gi(x1, . . . , xn) ≤ ki, i = 1, . . . ,m.

The associated Lagrangian is L = f(x1, . . . , xn) +∑k

i=1λi {ki − gi(x1, . . . , xn)} .

P2 : 1. ∂L/∂xj = 0 for j = 1, . . . , n

2. λi ≥ 0 for i = 1, . . . ,m

3. ki − gi (x1, . . . , xn) ≥ 0 for i = 1, . . . ,m

4. λi {ki − gi (x1, . . . , xn)} = 0 for i = 1, . . . ,m

P2 are sometimes called the “Kuhn-Tucker conditions” (1. first-order conditions, 2. non-

negative multipliers, 3. constraints, and 4. complementary slackness). The following

theorem (cd means continuously differentiable) is often referred to as either Kuhn-Tucker

(which technically it is not) or better, as the concave programming theorem.

Theorem: If (i) the ‘objective’, f : X → R, is a concave cd function, (ii) for every

i = 1, . . . ,m the ‘constraint’, gi : X → R, is a convex cd function, (iii) there is some

(x1, . . . , xn) such that gi(x1, . . . , xn) < ki for all i = 1, . . . ,m and (iv) X is a convex

subset of Rn (R is the reals) then the solution, (x∗1, . . . , x∗n), to P1 and P2 coincides.

The last two assumptions for the theorem are technical and almost always satisfied. The

first two (concavity and convexity) are crucial—always check them. There are many

different versions of this theorem: this is the most general statement I think you will

need. It reduces to the various versions you have seen in lectures and textbooks.

Usefully, solutions to P1 are ‘very often’ solutions to P2: when the conditions in the

theorem fail, solve P2 anyway and check which candidate solution maximises f .

References

Anthony, M., and N. Biggs (1996): Mathematics for Economics and Finance. Cambridge UniversityPress, Cambridge.

Begg, D., S. Fischer, and R. Dornbusch (2008): Economics. McGraw-Hill, London, 9 edn.Morgan, W., M. L. Katz, and H. S. Rosen (2005): Microeconomics: European Edition. McGraw-

Hill, London.Renshaw, G. (2005): Maths for Economics. Oxford University Press, Oxford.Simon, C., and L. Blume (1994): Mathematics for Economists. Norton, New York.Varian, H. R. (2005): Intermediate Microeconomics: A Modern Approach. Norton, New York, 7 edn.

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