Microeconomics 2

Answers to the first 17 questions on the First Specimen Examination Paper

(the remaining 10 answers are elsewhere)

Rules of examination

• Please use the answer sheets attached to the examination paper. Students should not bring their own electronic calculators; standard university electronic calculators will be provided at each desk.

• There are 27 questions in sets of various sizes. Each set of questions is preceded by a preamble, which remains in force until the next preamble. Four marks are awarded for each correct answer and one mark will be deducted for each wrong answer. The resulting mark, denoted by x will be between -27 and 108. It will then be converted into a final mark for this module y using the formula y = (x+27)/1.35, which ensures

that the final mark will lie between 0 and 100.

Preamble to and Questions 1 to 4• Consider a market for a hypothetical good in which there are a number of

buyers and sellers, each of which wants to buy or sell one unit of the good. There are 6 buyers and their reservation prices are 10, 5, 11, 10, 7 and 11. There are 5 sellers and their reservation prices are 10, 5, 2, 6 and 2.

• Question 1: At what price or prices is demand equal to supply? (If the demand or supply at a price consists of a set of values because some buyers are indifferent about buying at that price or some sellers are indifferent about selling at that price, then interpret this condition as being satisfied if there is some possible value of demand at that price which is equal to some possible value of supply at that price.)

• Question 2:What is the maximum quantity exchanged when the price is such that supply equals demand?

• Question 3: What is the maximum surplus realisable in the market?• Question 4: If the buyers grouped together and chose a single price which

maximised their total buyer surplus, what price would they choose (assuming that any seller indifferent between selling and not selling actually sells and assuming that if there are two such

prices they choose the highest)?• (Could ask for other things.)

Answers to Questions 1 to 4found by drawing demand and supply ‘curves’

Question 1: Any price greater than or equal to 7 and less than or equal to 10 Question 2: 4Question 3: 27 Question 4: 6

Preamble to and Questions 5 and 6• Consider an individual with QUASI-LINEAR preferences between

money, denoted by m, and the quantity, denoted by q, of a DISCRETE good. Suppose there is a market in which this DISCRETE good can be sold or bought at a fixed price. Suppose the price at the moment is 38.4. The individual is endowed with 5 units of the discrete good and 48 units of money. The individual's indifference curve through this endowment point is given by the equation m = 240/q. (Note no figure this year.)

• Question 5: State whether the individual will be a buyer or a seller and how many units he or she will buy or sell at this price.

• Question 6: What will the individual's surplus be at this price?

Answers to Questions 5 and 6• Points on indifference curve m=240/q are [1,240], [2,120], [3,80],

[4,60], [5,48], [6,40], [7, 34.2857..], [8,30], [9,26⅔], [10,24].• Starts at [5,48] (good, money). Price is 38.4.• Cannot buy 2 – not enough money• If buys 1 – moves to [6,9.6] surplus -30.4 (note the minus sign)• If does nothing surplus nothing• If sells 1 – moves to [4,86.4] surplus 26.4• If sells 2 – moves to [3,124.8] surplus 44.8• If sells 3 – moves to [2,163.2] surplus 43.2• If sells 4 – moves to [1,201.6] surplus -38.4 (note the minus sign) • Hence sells 2 and has surplus 44.8.

Preamble to and Question 7

• Consider an individual with quasi-linear preferences between money and the quantity of a CONTINUOUS good. Suppose there is a market in which the CONTINUOUS good can be sold or bought at a fixed price. Suppose the price is such that the individual's optimal decision is to buy 5 units. The individual is endowed with 5 units of the good and 48 units of money. The individual's indifference curve through this endowment point is given by the equation m = 240/q. (Note no figure this year.)

• Question 7: What approximately is the price in the market and what approximately is the individual's surplus at this price?

Answers to Question 7

• The budget line must be tangential to an indifference curve at q=10 (‘optimal to buy 5 units’).

• The indifference curve m=240/q has slope -240/q2 in general and -2.4 at q=10.

• So the slope of the budget line must be -2.4.• So the price is 2.4.• The budget line passes through [5,48] and having slope 2.4

also passes through [10,36]. • The indifference curve m=240/q passes through [10,24].• So the surplus is 36-24 = 12.• The answers are 2.4 and 12.

Preamble to and Questions 8 and 9

• Consider an individual whose preferences are either Perfect Substitutes, Perfect Complements or Cobb-Douglas with parameter a, allocating a given endowment between two goods whose prices are p and 1 respectively. The individual's endowments of the two goods are 10 and 9 respectively. In the first situation the price p of Good 1 was 0.5 and the individual chose to consume 0 of Good 1 and 14 of Good 2. In the second situation the price p of Good 1 was 1/3 and the individual chose to consume 31 of Good 1 and 2 of Good 2. In a third situation the price p of Good 1 was 0.75.

• Question 8: What are the individual's preferences and the value of the parameter a? (This was two questions last year.)

• Question 9: What are the individual's gross demands for the two goods at the third price above? (New this year.)

Key results about demands(useful for questions 8 to 11)

• With Perfect Substitute preferences (with parameter a)...• ...the Individual always spends all his or her income on one of

the two goods, unless...• ... the price equals the parameter a. (If p>a then q1=0; if p<a

then q2=0)

• With Perfect Complement preferences (with parameter a)...• ...the ratio of the quantities (q2/q1) purchased is always equal

to the parameter a.• With Cobb-Douglas preferences (with parameter a)...• ...the individual always spends a fraction a of his or her

income on Good 1.

Answers to Questions 8 and 9

• Are prices different in the two situations?• Is the quantity purchased of either good zero?• Yes – so must be Perfect Substitutes.• Is there a situation with a non-zero demand for both goods? • Yes – this reveals that a=1/3. (because p=1/3 in this case).• Yes - hence Perfect Substitutes with a=1/3.• So one unit of Good 1 and 1/3 of a unit of Good 2 are the same

to the individual.• In situation 3, p=0.75 so the individual will just buy Good 2.• His/her endowment is worth 10*0.75+9=16.5.• Buys 0 of Good 1 and 16.5 of Good 2.

Preamble to and Questions 10 and 11

• Consider an individual whose preferences are either Perfect Substitutes, Perfect Complements or Cobb-Douglas with parameter a, allocating a given monetary income between two goods whose prices are p and 1 respectively. The individual's endowment of money is 98. In the first situation the price p of Good 1 was 1/3 and the individual chose to consume 73.5 of Good 1 and 73.5 of Good 2. In the second situation the price p of Good 1 was 1 and the individual chose to consume 49 of Good 1 and 49 of Good 2. In a third situation the price p of Good 1 was 1.25.

• Question 10: What are the individual's preferences and the value of the parameter a? (This was two questions last year.)

• Question 11: What are the individual's demands for the two goods at the third price above? (New this year.)

Answers to Questions 10 and 11

• Are prices different in the two situations?• Is the quantity purchased of either good zero?• No – so cannot be Perfect Substitutes.• Is the ratio of quantities (q2/q1) the same in the two

situations: 73.5/73.5 = 1 in situation 1 and 49/49 = 1 in situation 2.

• Yes – so it must be Perfect Complements with parameter a=1.

• In the third situation the quantity q of each good (which are equal for the two goods) must be such that 1.25q+q=98.

• That is q = 98/2.25 = 43.55556.

Preamble to and Questions 12 to 15

• Consider competitive exchange of two goods, Good 1 and Good 2, between two Individuals A and B. A starts with an endowment of 12 units of Good 1 and none of Good 2. B starts with an endowment of 12 units of Good 2 and none of Good 1. Individual A has Perfect Complement Preferences with a parameter 2. Individual B has Cobb-Douglas Preferences with a parameter 1/3.(In answering this question you should note a convention that we use here: in order for a situation to be termed a competitive equilibrium we require that both individuals are STRICTLY better off than with they were with their initial endowments.)

• Question 12: Determine whether a competitive equilibrium exists, and if so, determine the competitive equilibrium exchange rate.

• Question 13: If a competitive equilibrium exists, how many units of good 1 are exchanged?

• Question 14: If a competitive equilibrium exists, how many units of good 2 are exchanged?

• Question 15: Would dividing EQUALLY the initial endowments of the two goods be an efficient way of finally allocating the two goods to the two individuals?

The simplest way is with an Edgeworth Box

• Draw in the price-offer curves.• For PC it is easy. At the moment no PS (tricky).• For CD make use of the fact that A has all of Good

1 and B all of Good 2.• Here B wants to keep 2/3 of his/her Good 2.

(a=1/3).• If the price-offer curves intersect inside the box at

a point where both are better off then that is the competitive equilibrium. If not, not.

No need to draw indifference curves: Price-Offer curves are sufficient; if they intersect that gives the competitive equilibrium.

Answers to Questions 12 to 15

• The competive exchange is at [2,4].• The price line from the endowment point [12,12] to [4,8] has

slope (minus) 4/10 = 2/5.• The competitive exchange rate is therefore 5 of Good 1 for 2

of Good 2.• A gives up 10 of Good 1 for 4 of Good 2• B gives up 4 of Good 2 for 10 of Good 1• Diving equally at point [6,6] is not efficient. • So answers are: Yes: exchange rate of 5 of good 1 for 2 of

Good 2; 10; 4; No – moving from C west and north a little makes both of them happier.

Preamble to and Question 16

• Consider a perfectly competitive firm with a quadratic cost function C(q) = a + bq + cq2 where the parameters a, b and c are given below (note that the firm has to incur its fixed cost a whether it produces any output or not). Suppose that the given price for its output is 18. The value of a is 12, the value of b is 1, and the value of c is 2.

• Question 16: What is its profit maximising (loss minimising) output?

Answer to Question 16

• The cost function is C(q) = 12 + q + 2q2

• Hence Marginal Cost MC = 1 + 4q• The profit-maximising condition for a

competitive (price-taking) firm is price = MC• The price = 18.• The condition implies 18 = 1 + 4q• So q = 4.25.

Preamble to and Question 17• Consider a simple economy with two individuals, A and B, each of whom can

produce both of two goods, 1 and 2. If A works full-time on Good 1, he or she can produce 10 units of Good 1; if A works full-time on Good 2, he or she can produce 8 units of Good 2; more generally, if he or she works a fraction f of his or her time on Good 1 and a fraction (1-f) of his or her time on Good 2 then he or she can produce a quantity 10f of Good 1 and a quantity 8(1-f) of Good 2. If B works full-time on Good 1, he or she can produce 8 units of Good 1; if B works full-time on Good 2, he or she can produce 8 units of Good 2; ; more generally, if he or she works a fraction f of his or her time on Good 1 and a fraction (1-f) of his or her time on Good 2 then he or she can produce a quantity 8f of Good 1 and a quantity 8(1-f) of Good 2. Suppose that they agree that they jointly want to produce 6 units of Good 1.

• Question 17: Given their decision (above) on how much of Good 1 that they want to jointly produce, what is the maximum amount of Good 2 that the two individuals can produce?

Answer to Question 17 • For every 1 unit less of Good 1

produced A(B) can produce 8/10 (1) more of Good 2.

• Thus A (B) is relatively better at producing Good 1 (2) than B (A).

• So A (B) should specialise in Good 1 (2).

• Society’s production possibility frontier goes from (18,0) to (10,8) to (0,16).

• If 6 of Good 1 is produced then 11.2 is the most of Good 2 that they can produce. (Found by linear interpolation.)

End of the answers to the first 17 questions on the first specimen examination paper

• Go slow – there is plenty of time.• The questions are easy if you understand

microeconomics.• They are difficult if you do not.• You cannot memorise answers.• You may want to memorise methods - but that is

exactly what I have been trying to teach you. • Obviously this is a specimen – do look at the

document detailed description on the site.