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Microeconomics 2 Answers to the questions on the First Specimen Examination Paper for the new style examination 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 (at the moment 27 there will be) n 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 -n and 4n. It will then be converted into a final mark for this module y using the formula y = 20(x+n)/n, which ensures that the final mark will lie between 0 and 100. Microeconomics 2 (ECO00001I) Answer Sheet Indicate your answers to the questions Instructions 1. Write your examination number in the boxes below, and then underneath it fill in the corresponding balls for each number. 2. Indicate your answers to the questions by filling in the appropriate ball in the right hand box. Fill in the ball neatly; do not go outside the ball; preferably use ink rather than pencil. Like this: Not like this: 3. If you cannot erase any wrong answer, put a big cross through the circle and indicate the correct answer. Like this: Write your examination number below and then fill in the corresponding balls (omitting the Y) Y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 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. The reservation prices for the buyers are 9, 7, 11 and 7. The reservation prices of the sellers are 6, 10 and 6. • Question 1:What is the maximum quantity demanded? • Question 2:What is the maximum quantity supplied? • Question 3: What is the price at which aggregate demand equals aggregate supply? • Question 4: What is the quantity exchanged when aggregate demand equals aggregate supply? Notes to Questions 1 to 4 • Aggregate demand equals aggregate 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 aggregate demand at that price which is equal to some possible value of aggregate supply at that price.” • Might also be questions on • Price-setting by sellers • Price-setting by buyers • Other forms of trade Answers to Questions 1 to 4 found by drawing demand and supply ‘curves’ Question 1: 4 Question 2: 3 Question 3: any price between 7 and 9 Question 4: 2 Preamble to and Questions 5 and 6 • Consider an individual with quasi-linear preferences whose indifference curve between money (on the vertical axis) and the quantity of a DISCRETE good (on the horizontal axis) is as given in Figure 1 attached to this script. Suppose the individual starts with an endowment at the point marked X in the figure. Suppose there is a market in which the DISCRETE good can be sold or bought at a fixed price. Suppose the price at the moment is 15. (You might like to know that the equation of the curve is m = 60/q where m and q are the variables on the vertical and horizontal axes respectively.) • 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 • • • • • • • Starts at (3,20) (good, money). Price is 15 Cannot buy 2 – not enough money If buys 1 – moves to (4,5) surplus -10 If does nothing surplus nothing If sells 1 – moves to (2,35) surplus 5 If sells 2 – moves to (1,50) surplus -10 Hence sells 1 and has surplus 5. (minus 10) (minus 10) Preamble to and Questions 7 and 8 • Consider an individual with quasi-linear preferences whose indifference curves between money (on the vertical axis) and the quantity of a CONTINUOUS good (on the horizontal axis) are as given in Figure 2 attached to this script. Suppose the individual starts with an endowment at the point marked X in the figure. Suppose there is a market in which the CONTINUOUS good can be sold or bought at a fixed price. Also inserted in the figure are the budget lines for 4 different prices. Suppose the price is such that the individual's optimal decision is to buy 2 units. (You might like to know that the equation of the curve is m = 60/q where m and q are the variables on the vertical and horizontal axes respectively. • Question 7: What approximately is the price in the market? • Question 8: What approximately is the individual's surplus at this price? Answers to Questions 7 and 8 • • • • • • If buys two units moves to flattest budget line This intersects the vertical axis at just over 27 Hence has a slope of –(27.2-20)/3 = - 2.4 Spends 4.8 So moves from (3,20) to (5,15.2) The original indifference curve m = 60/q passes through (5,12) an hence ends up 3.2 above it. • Hence the price is 2.4 and the surplus is 3.2. Preamble to and Questions 9 and 10 • 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 15 and 8 respectively. In the first situation the price p of Good 1 was 0.5 and the individual chose to consume 15.5 of Good 1 and 7.75 of Good 2. In the second situation the price p of Good 1 was 2 and the individual chose to consume 15.2 of Good 1 and 7.6 of Good 2. • Question 9: What are the individual's preferences? • Question 10: What is the value of the parameter a? Key results about demands (useful for questions 9 to 12) • 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 9 and 10 • • • • Are prices different in the two situations? Is the quantity purchased of either good zero? No – so cannot be Perfect Substitutes. Is the proportion of income spent on Good 1 in the two situations the same?: proportion 15.5*0.5/(15*0.5+8)=0.5 in 1; and proportion 15.2*2/(15*2+8)=0.796666 in 2. • No – so cannot be Cobb-Douglas. • Is the ratio of quantities (q2/q1) the same in the two situations: 7.75/15.5=1/2 and 7.6/15.2=1/2. • Yes - hence Perfect Complements with a=1/2. A little bit more on CD demands (useful for questions 13 to 16) • ...the individual always spends a fraction a of his or her income on Good 1. • Suppose the individual has an endowment only of Good 1: e1 • Suppose the prices are p1 and p2. The value of his endowment is p1e1. Spending a fraction a of this on Good 1 means that p1q1 = ap1e1 so that q1 = ae1. • Suppose the individual has an endowment only of Good 2: e2 • Suppose the prices are p1 and p2. The value of his endowment is p2e2. Spending a fraction a of this on Good 1 means that p1q1 = ap2e2 and spending the residual fraction on Good 2 means that p2q2 = (1-a)p2e2 so that q2 = (1-a)e2. Preamble to and Questions 11 and 12 • 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 80. In the first situation the price p of Good 1 was 1 and the individual chose to consume 20 of Good 1 and 60 of Good 2. In the second situation the price p of Good 1 was 1/3 and the individual chose to consume 60 of Good 1 and 60 of Good 2. • Question 11: What are the individual's preferences? • Question 12: What is the value of the parameter a? Answers to Questions 11 and 12 • • • • 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: 60/20=3 in situation 1 and 60/60=1 in situation 2. • No – cannot be Perfect Complements. • Is the proportion of income spent on Good 1 in the two situations the same?: proportion 20/80=¼ in situation 1; and proportion 60*(1/3)/80=¼ in situation 2. • Yes – hence Cobb-Douglas with parameter ¼. Preamble to and Questions 13 to 16 • • • • • 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 2/3. (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 they were with their initial endowments.) Question 13: Determine whether a competitive equilibrium exists, and if so, determine the competitive equilibrium exchange rate. Question 14: If a competitive equilibrium exists, how many units of good 1 are exchanged? Question 15: If a competitive equilibrium exists, how many units of good 2 are exchanged? Question 16: 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. For PS it would help to draw in some of the indifference curves. • 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 1/3 of his/her Good 2. • 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. Answers to Questions 13 to 16 • The competive exchange is at (4,8). • The price line from the endowment point (12,12) to (4,8) has slope 1. • The competitive exchange rate is therefore 1 for 1. • A gives up 8 of Good 1 for 8 of Good 2 • B gives up 8 of Good 2 for 8 of Good 1 • Diving equally at point (6,6) is not efficient. • So answers are: 1 for 1; 8; 8; No Preamble to and Question 17 • 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 20. The value of a is 15, the value of b is 7, and the value of c is 2. • Question 17: What profit does it make at its profitmaximising (loss-minimising) output (a negative number if it makes a loss)? Answer to Question 17 • The cost function is C(q) = 15 + 7q + 2q2 • Hence Marginal Cost MC = 7 + 4q • The profit-maximising condition for a competitive (price-taking) firm is price = MC • The price = 20. • The condition implies 20 = 7 + 4q • So q = 3.25 and hence • Profit =20*3.25 – (15 +7*3.25+2*3.252)=6.125 Preamble to and Question 18 • • 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 7 units of Good 1; if A works full-time on Good 2, he or she can produce 9 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 7f of Good 1 and a quantity 9(1-f) of Good 2. If B works full-time on Good 1, he or she can produce 14 units of Good 1; if B works full-time on Good 2, he or she can produce 11 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 14f of Good 1 and a quantity 11(1-f) of Good 2. Suppose that they agree that they jointly want to produce 14 units of Good 1. Question 18: 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 18 • For every 1 unit less of Good 1 produced A(B) can produce 9/7 (11/14) more of Good 2. • Thus A (B) is relatively better at producing Good 2 (1) than B (A). • So A (B) should specialise in Good 2 (1). • Society’s production possibility frontier goes from (21,0) to (14,9) to (0,20). • If 14 of Good 1 is produced then 9 is the most of Good 2 that they can produce. Preamble to and Question 19 • An individual is observed spending his or her monetary income on two goods, 1 and 2, in two situations, the price of Good 2 always being 1. In the first situation the individual's income was 14 and the price of Good 1 was 1. The individual was observed to purchase 2 units of Good 1 and 12 units of Good 2.In the second situation the individual's income was 10 and the price of Good 1 was 2/3. The individual was observed to purchase 10 units of Good 1 and 3 1/3 units of Good 2. • Question 19: Does this behaviour violate the weak axiom of revealed preference? Answer to Question 19 • In graph, the blue is the first situation (income 14 and prices both 1) and 1 the chosen point. • The red is the second situation (income 10 and prices 2/3 and 1) and 2 the chosen point. • The lines are the budget lines. • 1 revealed preferred to 2 (in the first situation) but we cannot infer anything about 1 and 2 from the second situation (because 1 was not available). • We can also draw convex indifference curves which would rationalise both choices. • So it is not a violation of the axiom. Preamble to and Question 20 • An individual has Perfect Substitute preferences over two goods, 1 and 2. The price of Good 2 is fixed at 1, and the individual's monetary income is 110. Initially the price of Good 1 is 1.0 but then it rises to 1.1.The value of the a parameter is 0.8. • Question 20: What is the Equivalent Variation of this price rise (that is, what reduction in his income would be equivalent to the effect of the price rise)? Answer to Question 20 • The blue (red) line is the budget line in the first (second) situation – income 110 and prices 1 and 1 (income 110 and prices 1.1 and 1). • The thin black line is the highest indifference curve reached in the first situation... • ... and the second. • So no loss of utility. • So EV is 0. Preamble to and Question 21 • Consider an inter-temporal choice problem with two periods where an individual's preferences over consumption, c1 and c2, in two periods 1 and 2 is given by U(c1, c2)=u(c1) + u(c2)/(1+ ρ) where ρ is the individual's discount rate (which should lie between 0 and 1) and u(c) is the square root of c (that is, u(c) = √c).The first consumption stream gives 36 in the first period and 1 in the second; the second consumption stream gives 16 in the first period and 16 in the second. • Question 21: What discount rate (if any between 0 and 1) would make the individual indifferent between these two streams of consumption? Answer to Question 21 • We have that U(c1, c2)=u(c1) + u(c2)/(1+ ρ) • and that u(c) = c1/2 = √c • We want the individual to be indifferent between (36,1) and (16,16). • So we need utilities to be the same. Hence • U(36, 1)= U(16, 16), that is: • 6 + 1/(1+ ρ) = 4 + 4/(1+ ρ) that is: • 2 = 3/(1+ ρ), that is, (1+ ρ) =3/2 = 1.5. • Hence ρ = 0.5 Preamble to and Question 22 • Consider an individual facing a risky choice (given below) who has preferences over risky choices given by the Expected Utility model. The individual's utility function is given by u(x) = xr where the parameter r (which indicates his risk attitude) is equal to 1. The risky choice gives outcome x equal to 45 with probability 0.4 and outcome x equal to 33 with probability 0.6. • Question 22: What is the individual's Certainty Equivalent for this risky choice (that is, what amount of money, received with certainty, would be equivalent for this individual to this risky choice)? Answer to Question 22 • The Certainty Equivalent is the amount of money received with certainty which is equivalent (given the individual’s preferences) to the risky choice. • So the individual is indifferent between the CE and the risky choice. The risky choice gives 45 with probability 0.4 and 33 with probability 0.6. • The utility function is u(x) = x. • So the Expected Utility of the risky choice is 0.4u(45) + 0.6u(33) = 0.4*45 + 0.6*33 = 37.8. • The CE must therefore satisfy u(CE) = 37.8 and hence • CE = 37.8 Preamble to and Question 23 • Consider a competitive market with linear demand and supply curves, as specified below. At the moment there is no tax imposed by the government.. The demand curve is given (in inverse form) by P = 70 - Q; the supply curve is given (in inverse form) by P = 22 + 0.5Q. • Question 23: Suppose that the government have decided to impose a tax at 10% of the price of the good. This will cause a higher price for the buyer, a lower price for the seller, and a deadweight loss of surplus in the market. Calculate this deadweight loss (to the nearest integer). Answer to Question 23 (One way) • Before the tax was imposed price and quantity were given by supply=demand; that is, by 70-Q = 22+0.5Q; that is, by P = 38 and Q = 32. • If we denote the price received by the seller as P, then the price paid by the buyer becomes 1.1P. • Hence the demand curve becomes 1.1P = 70-Q while the supply curve remains as before. Solving demand = supply we now get and Q = 29.548 and P = 36.774. So the tax is 3.677 and the quantity exchanged goes down by 2.452. Answer to Question 23 (one way) Hence the demand curve becomes 1.1P = 70-Q while the supply curve remains as before. Solving demand = supply we now get and Q = 29.548 and P = 36.774. Answer to Question 23 (Other way) • Before the tax was imposed price and quantity were given by supply=demand; that is, by 70-Q = 22+0.5Q; that is, by P = 38 and Q = 32. • If we denote the price paid by the buyer as P, then the price received by the seller becomes P/1.1 • Hence the demand curve remains as before while the supply curve becomes P/1.1 = 22 + 0.5Q. Solving demand = supply we now get and Q = 29.548 and P = 40.451.774. So the tax is 3.677 and the quantity exchanged goes down by 2.452. Answer to Question 23 (other way) Hence the demand curve remains as before while the supply curve becomes P/1.1 = 22 + 0.5Q. Solving demand = supply we now get and Q = 29.548 and P = 40.451.774. Answer to Question 23 (a simpler way) • 40.451 is price paid by buyers with tax. • 36.774 is price received by sellers with tax. • 29.548 is quantity exchanged with tax. • So the deadweight loss is the area of the triangle of height 3.667 and width 2.452; which is ½*3.677*2.452; which is 5 to nearest integer. Preamble to and Question 24 • Consider a monopolist with a linear demand function (as given below) and a linear cost function (as given below). The demand curve is given (in inverse form) by P = 81 - Q; the cost function is given by C(Q) = 39 + Q. • Question 24: What are the monopolist's optimal price and output? Answer to Question 24 • • • • • • • The monopolist’s Marginal Cost is MC =1. Its revenue is PQ = (81-Q)Q = 81Q – Q2. Hence its Marginal Revenue = MR = 81 – 2Q. For maximum profits MC = MR and hence 1 = 81 – 2Q and hence Q=40 and so its price P = 41. So the answers (price and output) are 41 and 40. Preamble to and Questions 25 and 26 • Consider a game between a row player (Individual A) and a column player (Individual B) where the payoff matrix is given in the 'table for the game theory questions' attached to this examination paper and is repeated here. The first row of the payoff matrix is [27,2][13,39] [33,49]. The second row of the payoff matrix is [8,12] [38,50] [40,23]. The third row of the payoff matrix is [23,25] [37,48] [15,5]. • Question 25: List all the Nash Equilibrium of this game when played simultaneously. • Question 26: Now suppose we change the rules of the game: with New Rules 1 we let A move first and then B responds; with New Rules 2 we let B move first and then A responds; What is A's best optimal row decision with New Rules 1; and what is B's best column decision with New Rules 2? Answers to Questions 25 and 26 Table for questions Individual B 25 and 26 Column 1 Column 2 Column 3 [27,2] [13,39] [33,49] Individual Row 1 Row 2 [8,12] [38,50] [40,23] A Row 3 [23,25] [37,48] [15,5] • For Question 25 you just try each cell and see it is a NE; you will find [2,2] as the only NE. • For Question 26 you just see what the starting player would end up with for each choice. (A would choose row 2; B column 2.) Preamble to and Question 27 • Consider a Cournot (quantity-setting) duopoly in which the aggregate demand curve (given below) and the cost functions (given below) are all linear. The demand curve is given (in inverse form) by P = 86 - Q, where Q is aggregate output; the cost function of firm 1 is given by C(Q1) = Q1, where Q1 is the output of Firm 1; the cost function of firm 2 is given by C(Q2) = 2 Q2, where Q2 is the output of Firm 2; of course Q= Q1 + Q2. • Question 27: What are the equilibrium (Cournot) optimal outputs for the two firms? Answer to Question 27 • • • • • • • Profits of firm 1 are PQ1 ‒ C(Q1) = [86-Q1-Q2] Q1 ‒ Q1 = 85Q1 ‒ Q12 ‒ Q1Q2 Maximising this wrt Q1 given Q2 gives 85 ‒ 2Q1 ‒ Q2 = 0: firm 1’s reaction curve. Doing the same for firm 2 gives 84 ‒ Q1 ‒ 2Q2 = 0: firm 2’s reaction curve. Solving the two reaction curves simultaneously gives us Q1=28⅔ and Q2=27⅔. That is the end of 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. First Specimen Paper • Goodbye!