Microeconomics 2

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Microeconomics 2

• Answers to the questions on the First Specimen
  Examination Paper for the new style examination

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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(xn)/n, which ensures that the final mark will
  lie between 0 and 100.

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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 1What is the maximum quantity demanded?
• Question 2What 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?

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Notes to Questions 1 to 4

• Aggregate demand equals aggregate supplyIf 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

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Answers to Questions 1 to 4found by drawing
demand and supply curves
Question 1 4 Question 2 3 Question 3 any price
between 7 and 9 Question 4 2
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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?

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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 (minus 10)
• If does nothing surplus nothing
• If sells 1 moves to (2,35) surplus 5
• If sells 2 moves to (1,50) surplus -10 (minus
  10)
• Hence sells 1 and has surplus 5.

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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?

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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.

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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?

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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 pgta
  then q10 if plta then q20)
• 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.

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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
  15.50.5/(150.58)0.5 in 1 and proportion
  15.22/(1528)0.796666 in 2.
• No so cannot be Cobb-Douglas.
• Is the ratio of quantities (q2/q1) the same in
  the two situations 7.75/15.51/2 and
  7.6/15.21/2.
• Yes - hence Perfect Complements with a1/2.

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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.

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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?

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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 60/203 in situation 1 and
  60/601 in situation 2.
• No cannot be Perfect Complements.
• Is the proportion of income spent on Good 1 in
  the two situations the same? proportion
  20/80¼ in situation 1 and proportion
  60(1/3)/80¼ in situation 2.
• Yes hence Cobb-Douglas with parameter ¼.

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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?

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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.

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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

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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
  profit-maximising (loss-minimising) output (a
  negative number if it makes a loss)?

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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 203.25 (15 73.2523.252)6.125

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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?

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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).
• Societys 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.

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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?

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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.

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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)?

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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.

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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 ?)
  where ? 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) vc).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?

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Answer to Question 21

• We have that U(c1, c2)u(c1) u(c2)/(1 ?)
• and that u(c) c1/2 vc
• 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 ?) 4 4/(1 ?) that is
• 2 3/(1 ?), that is, (1 ?) 3/2 1.5.
• Hence ? 0.5

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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)?

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Answer to Question 22

• The Certainty Equivalent is the amount of money
  received with certainty which is equivalent
  (given the individuals 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) 0.445 0.633 37.8.
• The CE must therefore satisfy u(CE) 37.8 and
  hence
• CE 37.8

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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).

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Answer to Question 23 (One way)

• Before the tax was imposed price and quantity
  were given by supplydemand that is, by 70-Q
  220.5Q 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.

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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.

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Answer to Question 23 (Other way)

• Before the tax was imposed price and quantity
  were given by supplydemand that is, by 70-Q
  220.5Q 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.

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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.

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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 ½3.6772.452 which is 5 to nearest integer.

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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?

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Answer to Question 24

• The monopolists 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 Q40 and so its price P 41.
• So the answers (price and output) are 41 and 40.

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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 27,213,39 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?

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Answers to Questions 25 and 26
Table for questions 25 and 26 Table for questions 25 and 26 Individual B Individual B Individual B
Table for questions 25 and 26 Table for questions 25 and 26 Column 1 Column 2 Column 3
Individual A Row 1 27,2 13,39 33,49
Individual A Row 2 8,12 38,50 40,23
Individual 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.)

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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?

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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 1s reaction curve.
• Doing the same for firm 2 gives
• 84 ? Q1 ? 2Q2 0 firm 2s reaction curve.
• Solving the two reaction curves simultaneously
  gives us Q128? and Q227?.

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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.

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First Specimen Paper

• Goodbye!