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Study Questions 2014

Note: The following are a sample of questions drawn from assignments, tests, and

final exams. Please recognize that assignment questions are typically harder than exam

questions. ECON201 has been taught by a number of individuals in recent times and the

questions should not be attributed to any particular lecturer. Questions may also have been drawn from the recommended texts.

Please use the questions as a study guide. If you have difficulty with the question then, by all means, ask for assistance. But I will not provide answers; it is for you to attempt the question and check the answer out.

Consumer choice and demand1. Draw the following indifference curves:

(a)When making slurp I always combine 8 grams of lemonade (L) with 1 gram of jungle juice (J).

(b) BJ likes beer but hates hamburgers. She always prefers more beer no matter how many hamburgers she has.

(c)SS is indifferent between bundles of either three beers or two hamburgers. His preferences do not change as he consumes any more of either.

(d) AJ eats one hamburger and washes it down with one beer. He will not consume an additional unit of one item without an additional unit of the other.

(e)Suppose Bill views butter and margarine as perfectly substitutable for each other. Draw a set of indifference curves that describes Bill’s preferences for butter and margarine.

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(f) Following on from (e) above, Bill has $20 to spend each month. Butter costs $2 per package and margarine costs $1. What is Bill’s optimal consumption bundle? Illustrate your result using a suitably labeled diagram. Describe a representative indifference curve for each of the following situations.

(g) The two goods are right gloves (R) and left gloves (L) and the consumer has two hands.

(h) The two goods are wine (W) and lobster (L) tails. Eating more than a certain number of lobster tails makes the consumer sick.

(i) The two goods are Coke (C) and Pepsi (P) and the consumer perceives no difference between the two soft drinks.

(j) When optimizing his consumption over two periods Zac decides to save money in the first period. If the interest rate increases, what has happened to the price of consumption in period one? Will Zac remain a saver? Why or why not? Justify your answer using a suitably labeled diagram.

2. A consumer derives utility from consuming X and Y according to the function

.

(a) The consumer faces prices PX and PY and has income I. (b) Sketch the above utility function for U = 10, with X on the horizontal axis and

Y on the vertical axis. Show the income consumption curve in your graph.(c) Derive the Engel function for good Y. Illustrate the Engel function using a

suitably labeled diagram with Y on the horizontal axis.(d) Derive the demand functions for both X and Y. Using the demand function for

X, describe the impact of a rise in the price of Y on the demand for X. Briefly explain why this relationship exists.

3. Let X measure the amount of chocolate and Y measure the amount of a composite good SJ buys. We represent SJ’s preferences with the following utility function

. The price of the composite good is 1 per unit, let PX represent the price of X per unit. SJ has a budget of $10 per day.(a) Derive SJ’s demand function for chocolate. (b) Suppose the price of chocolate is initially $0.50 per unit. How many units of

chocolate and how many units of the composite good are in BJ’s optimal consumption basket?

(b) Suppose the price of chocolate drops to $0.20 per unit. How many units of chocolate and how many units of the composite good are in the optimal consumption basket?

(c) Illustrate the substitution and income effects that result from the fall in the price of chocolate.

4. A consumer has a utility function , income of $20, and faces PX

= 4 and PY = 5.(a) Find the consumer’s optimal consumption bundle (X*,Y*). Illustrate your

answer.(b) Let income = I. Derive an equation for the Engel curve. Illustrate your answer.

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5. Suzie purchases two goods, food (X) and clothing (Y). Her utility function is U(X,Y) = XY and she has income I. Let the price of food and clothing be given by PX and PY respectively.(a) Derive the demand functions for X and Y.(b) Is clothing a normal good? Draw her demand curve for clothing when the

level of income is I = 200. Label this demand curve D1. Draw the demand curve when I = 300 and label this demand curve D2. Note: your graph need only show the general shape and position of the demand curves.

(c) What can be said about the cross-price elasticity of demand of food with respect to the price of clothing?

(d) Now assume that there are n consumers. Each consumer i has the same utility function Ui (X,Y) = XY as defined above but different income Ii. Derive the market demand function.

6. A consumer with utility function, , has an initial endowment of and to allocate between consumption in period 1 and consumption in period 2

Her inter-temporal budget constraint is: , where is the

interest rate for borrowing and lending.

(a) Derive expressions for , and and hence derive a general expression for the first order condition for utility maximization for this consumer.

(b) Derive the demand functions for and .:(c) If , and :i. Find the optimal consumption bundle and determine the level of utility

associated with this consumption bundle. Is the consumer a borrower or a lender at this interest rate? How much does the consumer borrow or lend in equilibrium? Use a sketch to illustrate this consumption behaviour.

ii.If the interest rate increased to would you expect the consumer to be a borrower or a lender in equilibrium? Would you expect her utility to be higher or lower than the utility calculated in c.i.? Explain.

iii. Calculate the interest rate at which the consumer neither borrows nor lends in equilibrium.

7. DJ derives utility from always consuming 4 units of X to 1 unit of Y. No utility is gained from not following this rule. DJ has income I, and faces PX and PY.(a) Derive the demand function for X and Y. (b) Assume that I = 10 and PX = 1 and PY = 6. Calculate DJ’s optimal consumption bundle (X*,Y*). Now assume that the price of X falls PX = 0.5 and the price of Y is unchanged. Illustrate the income and substitution affects associated with the fall in the price of X?

8. Katy is a model whose diet consists solely of soup (S) and nuts (N) and whose preferences are represented by her utility function, u(S, N) = S2N. Her weekly

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food budget is I ($); the prices of soup and nuts are PS ($ per litre) and PN ($ per kilo).

(a) Write down Katy’s food budget constraint.(b) Find Katy’s demand functions for soup and nuts. Comment on how these

demand functions relate to the particular type of utility function that Katy has.

9. Zoë is a student and she drinks caffeine before ECON 201 (the lecturer isn’t all that great!). Her preferences are such that she always trades-off the drinks at the rate of: one glass of redcow (let this drink be X) for half a glass of boost (let this drink be Y). Zoe has a daily income of $20 and faces prices PX = 4 and PY = 5.(a) What is Zoë’s optimal choice? Illustrate your answer.(b) Let income = I. Derive an equation for the Engel curve. Illustrate your answer.

Risk & information

10. BJ is planning a skiing trip to Big Sky Montana. The utility from the trip is a function of how much she spends (y) and is given by U(y) = log y. BJ has $10,000 to spend on the trip. If she spends it all, her utility will be U(10,000) = log 10,000 = 4. ( we are using logarithms to base 10).(a) If there is a 25% probability that BJ will lose $1,000 of her cash on the trip,

what is the trip’s expected utility?(b) Suppose BJ can buy insurance against losing the $1,000 at an actuarially fair

premium of $250. Will BJ buy the insurance?(c) What is the maximum amount that BJ would be willing to pay to insure her

$1,000.(d) Suppose that people who buy insurance tend to become more careless with

their cash than those that don’t, and assume that the probability of their losing $1,000 is 30%. What will be the actuarially fair insurance premium? Will BJ buy insurance at the actuarially fair price?

11. TD’s utility depends on his wealth, w, according to the utility function, u(w) = √w. TD owns $50,000 worth of safe assets and he also owns a house that is located in an area where there are lots of forest fires. His house and land are currently valued at $200,000. However, if his house burns down, the remains of his house and the land it is built on would be worth only $40,000. The probability that his home will burn down is estimated to be 0.01.(a) Calculate TD’s expected utility if he doesn’t buy fire insurance.(b) Calculate the certainty equivalent of the gamble he takes if he doesn’t buy fire

insurance. Find the risk premium in this situation.(c) Suppose that TD can buy insurance at a price of $1 per $100 of insurance

coverage. Is this insurance actuarially fair or not? Explain.(d) If TD decides to fully insure against the risk of fire, what will his wealth be?

What will be his expected utility with full insurance?

12. You have just moved into a new apartment. Your assets are worth $10,000. If your apartment is burgled you will lose everything except the clothes you wear to university, reducing your wealth to $900. Based on historical records, the local

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police estimate the probability of a thief breaking into your apartment and stealing your assets is 0.1. Your utility function is where W is the value of your assets.

(a) Calculate and illustrate in a suitably labeled diagram: the expected value of risk; utility of expected value; expected utility

(b) If an insurance company offered you a fair contract to insure against burglary would you accept? Briefly justify your answer.

(c) What is the maximum price you would be willing to pay for insurance? Illustrate this price in the diagram presented in (a) above.

(d) Now assume that a new alarm system is available on the market. The new system does not alter the probability of burglary but it is might scare burglars away if they do break into your apartment. Insurance is still available at the fair price used in your answer to (b) above. What would the probability of scaring burglars away have to be in order for you to switch from the fair contract to buying the new alarm system? What is the maximum you would be willing to pay for the new alarm system?

13. You have been hired to advise the Ministry of Fisheries on its policy on illegal fishing. Thieves make money (W) out of illegal fishing. The existing probability of apprehending a fisheries thief is p < 0.5. If caught, they are fined F and their wealth is (W-F). The Ministry is proposing to deter illegal fishing by either doubling the fine (F) for illegal fishing (holding probability constant) or doubling the probability (p) of catching the fish thieves (holding the fine constant). As an economist, you are well aware of the relevance of an individual’s attitude to risk and suggest analysing the problem in terms of a risk averse fisher and a risk loving fisher

(a)If fish thieves are risk averse, what policy (doubling the fine or doubling the probability of getting a fine) will do better in deterring illegal fishing?

(b) How does your analysis change if fish thieves are risk loving? Which has the greater deterrence effect in this case, doubling the fine or doubling the probability of getting a fine?

Production & Costs

14. The production of wheat is represented by the following production function where F is fertilizer and W is water

(a)What returns to scale are evident?(b) Show one isoquant for wheat with water (W) on the

horizontal axis and fertilizer (F) on the vertical axis. Does the above function exhibit diminishing returns to scale? What is the marginal rate of technical substitution of water for fertilizer when W = 10 and F = 10.

(c) Now assume that genetic engineering has changed the production function to . How has genetic engineering impacted returns to scale? Using this new production function, what is the marginal rate of technical substitution of water for fertilizer when W = 10 and F = 10?

(d) Use two isoquants - one representing the old wheat variety and the other representing the genetically enhanced wheat variety – to illustrate the impact of genetic engineering. Briefly discuss your conclusions.

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15. A truck driver has an input bundle of 10 hours and 120 litres of diesel. Let q = kilometres traveled, h = hours, and d = litres of diesel. The travel production

function is .

(a) What is maximum number of kilometers her truck can be driven? What average speed is required to achieve this result? Given the same input bundle, how far can the truck be driven at 80 kph? At 160 kph?

(b) Find the total product function for d = 12, d = 27, d = 48. Illustrate the total production functions with h on the horizontal axis and q on the vertical axis.

(c) Suppose d = 12. Find the average product function.

16. The production function for an industry is given as where K is capital and L is labour.

(a) Graph an isoquant for this production function, with labour (L) on the horizontal axis and (K) on the vertical axis. Derive the rate of technical substitution and illustrate your result in the graph for a given quantity of labour and capital . What happens to output if labour and capital are now ? Does the rate of technical substitution change, justify your answer.

(b) An innovation in the industry has now changed the production relationship to where . Illustrate the impact of this innovation using a suitably labelled graph.

17. The production function of Lightning Direct Courier Service (LDCS) is given by: where q = kilometres of service, L = labour and G = litres of petrol. Let w = price of labour and v = the price of petrol.

(a)Suppose the price of labour is $6 and the price of petrol is $2. Find the cost-minimising combination of labour and petrol for 120 kilometres of service. Illustrate your result.

(b)Derive LDCS’s input demand functions for labour and petrol. Briefly comment on the relationship between input demand and the price of each input.

(c)Derive LDCS’s long run cost, average, and marginal, cost functions.

18. The production function for bikes requires capital (K) and labour (L) and is characterized by .(a) What is the average productivity of labour and capital for bike production? Graph the average productivity of labour curve for K = 100.(b) Sketch the q = 10 isoquant for this production function. What is the MRTS on the q = 10 isoquant at the points: K = L = 10; K = 25, L = 4; and K = 4, L =25. Does the function exhibit diminishing MRTS? Briefly justify your answer.

19. Suppose that a firm’s production function is given by and the

rental rates for capital and labour are v = 1 and w = 3 respectively.(a) Sketch the q = 100 isoquant for this production function. What is the relationship between capital and labour regardless of the level of production? (b) Calculate the firm’s long run total, average, and marginal cost functions.

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(c) Suppose that K is fixed at 10 in the short-run. Calculate the firm’s short-run total average, and marginal cost curves.

20. Suppose a firm’s production function is a Cobb-Douglas production function of the form Assume that the price of labour is given by w and the price of capital is given by r.

(a) Suppose that the firm’s capital is fixed at . What amount of labour will the firm hire to solve its short-run cost minimization problem?

(b) Now suppose that that the price of labour w is $5 per unit and the price of capital r is $20 per unit. What is the cost-minimising input combination if the firm wants to produce 1,000 units per year? Illustrate your result using a suitably labeled diagram.

(c) Derive the long-run total cost function. (Hint: find the factor demand functions first.) Illustrate the total cost curve when w = 25 and r = 100. What are the long-run average and marginal cost curves associated with the long-run total cost curve? Briefly explain the relationship between the long-run total cost curve and the long-run average and marginal cost curves.

(d) Using a suitably labeled diagram, explain why total costs are higher in the short run than in the long run.

21. Derive the production function for each of the following production relationships, construct an isoquant for the level of output indicated, and derive the cost function.

a. AJ uses a furnace and fuel to produce heat. The furnace can use either coal or wood for fuel. One tonne of coal produces 5 kilojoules of heat and one tonne of wood produces 2 kilojoules. Let q = kilojoules of heat, x1 = tonnes of coal and x2 = tonnes of wood. Construct, and illustrate using an appropriately labeled diagram, an isoquant for 20 kilojoules. Using w1 and w2 as the price of coal and wood respectively, derive the cost function.

b. TC works at a city nightclub. Her rum-and-Coke recipe calls for 10 millilitres of rum and 30 millilitres of Coke. Let q = number of drinks, x1 = millilitres of rum and x2 = millilitres of Coke. Construct, and illustrate using an appropriately labeled diagram, an isoquant for 2 drinks. Using w1 and w2 as the price of rum and Coke respectively, derive the cost function.

22. Suppose that a firm’s production function is given by and the rental rates for capital and labour are v = 1 and w = 3 respectively.(a) Sketch the q = 100 isoquant for this production function. What is the

relationship between capital and labour regardless of the level of production?(b) Calculate the firm’s long run total, average, and marginal cost functions.(c) Suppose that K is fixed at 10 in the short-run. Calculate the firm’s short-run

total average, and marginal cost curves.

23. The production function for bikes requires capital (K) and labour (L) and is characterized by .

(a) What is the average productivity of labour and capital for bike production? Graph the average productivity of labour curve for K = 100.

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(b) Sketch the q = 10 isoquant for this production function. What is the MRTS on the q = 10 isoquant at the points: K = L = 10; K = 25, L = 4; and K = 4, L =25. Does the function exhibit diminishing MRTS? Briefly justify your answer.

24. BJ’s is a small business that operates in the perfectly competitive residential window washing market. The short-run total cost function is

where q is the number of windows washed per day. The prevailing market price is $20 per window.

(a) How many windows should BJ wash to maximise profit? What is BJ’s maximum daily profit?

(b) Graph SMC, SAC, and show the profit-maximising daily profit. (5 marks)(c) What is BJ’s short-run supply function, assuming that all of the $40 per day fixed

costs are sunk?(d) What is BJ’s short-run supply function, assuming that if he produces zero output,

he can rent or sell his fixed assets and therefore avoid all his fixed costs?

Demand and Supply

25. The demand (Qd) and supply (Qs)functions for wine are and

(a) What are the equilibrium price and quantity? (b) At the equilibrium in part (a), calculate the consumer surplus and producer

surplus?(c) Suppose government imposes an excise tax of $6 per unit. What will the new

equilibrium quantity be? What price will consumers pay? What price will the sellers receive?

(d) At the equilibrium with the tax in part (c) above, calculate the consumer and producer surplus? Calculate the government revenue generated from the tax? Illustrate your results using a suitably labeled diagram.

26. The market supply functions for one of the key components of a drilling rig are: and where is measured in units and is measured in $ per unit.a. Determine the market equilibrium price and quantity.b. Determine the own price elasticity of demand and the own price elasticity of

supply at the market equilibrium.c. Suppose the government imposes a $45 per unit tax on this component. How

would this tax affect the market equilibrium? How would the tax burden be shared between buyers and sellers of the component?

27. Assume that the price per barrel of crude is about USD25 and world output was 70 million barrels/day. The short-run demand and supply functions are:

QD = 80 - 0.4P

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QS = 55 + 0.6P(a) Calculate the market equilibrium price & quantity; and the short-run supply

and demand price elasticities.(b) This demand curve is a ‘net’ demand curve -- net of the taxes paid by oil

consumers. These vary around the world, but are quite high compared with most other goods. Let us take $25 to be the average tax paid per barrel. Then we can construct the gross or retail demand curve which relates the demand for oil to the (tax-inclusive) actual price paid per barrel. Show that the formula for this demand function is:

QD = 90 - 0.4Pr

Where Pr is the price actually paid.(c)Calculate the retail market demand elasticity and explain in words why it is

larger in absolute value than the net demand elasticity of -0.15. )(d) Calculate the “ without the tax” equilibrium price and output .(e) Carry out a welfare analysis (changes in surpluses; deadweight loss) of the oil

tax (compared to no tax) from the perspective of an oil-importing economy such as New Zealand.

(f) The major player on the supply side of the world oil market is the OPEC (Organization of Petroleum Exporting Countries) cartel, which coordinates the supply of most of the world’s largest oil producers. OPEC is often accused of ‘monopolising’ the world market. Explain why OPEC cannot be maximising short-run profits.

(g) However, list some longer-run supply and demand considerations which may explain why OPEC would not want to set a price to maximise short-run profits

.

Competitive Equilibrium

28. Consider an industry in which there are 10 identical firms and 1,000 identical demanders. Each j demander has the following demand function .

Each of the i firms has the following short-run cost function .(a) Derive the short-run supply function for one of the firms.(b) Derive the short-run market supply function.(c) Calculate the short-run market equilibrium price and quantity.(d) Using an appropriately labelled graph: illustrate the firm’s supply function;

calculate and illustrate the firm’s profit.29. A perfectly competitive market has a demand function . If

every firm in the market has an average cost function .(a)Calculate the equilibrium price and quantity in the market.(b)Calculate the long-run equilibrium number of firms.(c)Derive the market supply function.(d)Calculate the total surplus for this competitive equilibrium.

30. All firms have identical average cost functions AC(q) = 40 − q + 0.01q2

and long run marginal cost functions MC(q) = 40 − 2q + 0.03q2. Market demand isD(p) = 25, 000 − 1, 000P.

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(a) Find the long run equilibrium quantity per firm, price and number of firms.

31. Kayaks are produced by a number (many) of identically sized firms. Total (long run) monthly costs for each firm are given by where q is the number of kayaks produced and demand is .

(a) Find the monthly output of kayaks by each firm and their market price.(b)What is the equilibrium number of firms?(c)What if demand increased to ? Assuming entry into the market is

free and does not alter costs, recalculate the equilibrium price, total number of kayaks and the number of new entrants into the market.

Monopoly

32. A firm is a monopolist with demand and total costs .(a)Derive the profit maximising level of output for the monopolist. At what price will

it sell this quantity? What are the profits of the firm?(b)Illustrate your results using a suitably labelled graph.(c)Calculate the optimal mark-up expressed as a percentage of price.(d)Suppose the government imposes a price ceiling of $65. What is the monopolist’s

profit maximising output? What is the monopolist’s profit?(e)Illustrate the results obtained in (d) above using a suitably labelled graph?

33. A monopolist can produce at a constant AC = MC = $5. It faces a market demand curve given by .

(a) Calculate the profit-maximising price and quantity for this monopolist. Also calculate its profits. (5 marks)

(b) Now suppose that a second firm enters the market. Market demand is now given by . Assume that their costs are identical and AC = MC = $5. Find the Cournot equilibrium. What are the resulting market price and profits for each firm. (10 marks)

(c) Suppose there are N firms in the industry, all with the same constant marginal cost, MC = $5. Find the Cournot equilibrium. How much will each firm produce, what will be the market price, and how much will each firm earn? Also, show that as N becomes “large”, the market price will approach the price that would prevail under perfect competition. (10 marks)

34. Consider a monopolist who faces a linear demand curve

.The monopoly’s total cost is a linear function of output

.(a) Calculate the monopolist’s profit maximizing output, market price, profit, and

consumer surplus. Illustrate your results in a suitably labelled graph.(b) Calculate the dead weight loss.

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(c) Now assume that the monopolist can perfectly discriminate so that each successive unit of q is sold for the maximum amount that consumers are willing to pay. Calculate the monopolist’s profit maximizing output, market price, profit, and consumer surplus. Illustrate your results in a suitably labelled graph.

(d) Now assume that aggregate demand and total cost remains unchanged but the monopolist can new separate consumers into two distinct markets, thus

and Calculate the monopolist’s profit maximizing output, market price, profit, and consumer surplus. Illustrate your results in a suitably labelled graph.

35. Suppose a textbook monopoly can produce any level of output it wishes at a constant marginal cost of $5 per unit. Assume that the monopolist sells its books in two different markets that are separated by some distance. Demand in the first market is given by

Demand in the second market is given by

(a) If the monopolist can maintain separation between the two markets, what level of output should be produced in each market and what price will prevail in each market? What are the total profits in this situation?(b) How would your answer change if it only cost demanders $5 to mail books between the two markets? What would be the monopolist’s new profit level in this situation?(c) How would your answer change if mailing costs were 0 and the firm was forced to follow a single-price policy?

Oligopoly and Game Theory

36. A homogeneous product industry has a market demand curve:P = 15 - 0.1QThe technology is represented by a cost function:TC = F + 3qfor any firm producing q units of output. Assume F = 0.

(a) Show that the reaction function of some firm A, giving its optimal output as a function of the output of other firms is: qA = 60 - 0.5QJ where QJ is the total output of the other firms in the industry.

(b)Suppose the industry is a duopoly. Show that the Cournot-Nash equilibrium has each firm producing 40 units, market price = $7, and profits/firm = $160.

(c) Use the reaction function to deduce that monopoly output would be 60.

(e) Suppose that the two firms were able to cooperate to set the joint-profit maximising price and split the market equally between them. Show that price = $9 and profits/firm = $180.

(f) Now show that the cooperative outcome is not a Nash Equilibrium, by calculating firm A’s optimal output if the other firm (call it firm B) produces only its half share of the joint-profit maximising industry output.

(g) Set out the above scenarios in the form of a payoff matrix, with each firm’s possible actions being ‘cooperate’ or ‘don’t cooperate’.

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(h) Suppose firm A advertises a ‘price protection promise’, under which it promises its customers that should they purchase the product from firm A, and then find it available from another supplier at a lower price, then firm A will refund the difference plus $1. Suppose that firm B makes a similar promise. Without being precise or technical about it, modify the payoff matrix to suggest how these promises may actually be used to convert the cooperative outcome into the Nash Equilibrium.

(i) Now assume that fixed costs, F = 200. Explain why the industry is now a ‘natural monopoly’ and explain why a regulator might wish to set a maximum price of $5.

37. Suppose Wai’s and H20’s demand functions are given by respectively. Wai’s marginal cost is

$5 per unit and H20’s marginal cost is $4 per unit.(a) What is Wai’s profit maximising price when H20’s price is $8?(b) What is the equation of Wai’s price reaction function?(c) What are Wai’s and H20’s profit maximising prices and quantities at the

Bertrand equilibrium?

38. Consider a duopoly. The demand for spring water is given by and the marginal cost of obtaining the water is zero.

(a) Assume a cartel combines the interests of both firms. What is the profit maximising price, output, and profit?

(b) Now assume the firms behave as Cournot duopolists. Derive the reaction function for each firm and derive the Cournot solution.

(c) On a graph, with price on the vertical axis and quantity on the horizontal axis, show the cartel (monopoly) solution, the Cournot solution, and the competitive solution

39. Suppose a market consists of N identical firms, that the market demand function is and the each firm’s marginal costs is c.

(a) What is the Cournot equilibrium quantity per firm?(b) What are the equilibrium market quantity and price?(c) What to the equilibrium quantity and price as the number of firms N gets

bigger?

40. The following table shows payoffs in millions of dollars associated with the release of new fashion designs between Kates Clothes (KC) and Ruby’s Fashions (RF).

KC

RFRelease Don’t release

Release 16,16 20,15Don’t release 15,20 18,18

(a)Assuming simultaneous moves, what is the Nash Equilibrium outcome for this game? Does the Nash Equilibrium maximise joint profits?

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(b)Now assume that both KC and RF include a third strategy and KC gets to move first. That is KC gets to decide what to do before RF. Construct a game tree for fashion game and identify the Nash Equilibrium.

KC

RFRelease Limited release Don’t release

Release 0,0 12,8 18,9Limited release 8,12 16,16 20,15Don’t release 9,18 15,20 18,18

41. “Tragedy of the Common” relates to overuse of a resource where players have open access. Assume two farmers A and B are deciding how many cows to graze on the village common. The village common is quite small and can easily be over-grazed. Let the payoff in milk per cow be given by

where YA and YB represents the number of cows brought to the commons by A and B respectively.

(a) Find the Nash equilibrium for this game and the return each famer makes.(b) Does the Nash Equilibrium maximise returns? If not how many cows does? Is

this equilibrium stable i.e. does the revenue maximising solution provide an incentive to cheat?

Pure Exchange Equilibrium

42. Person A’s utility function given by and person B’s is

. Their initial endowments are ; and

respectively.

(a) Using an Edgeworth Box, with person A’s origin at the “southwest corner” show their respective endowments of X and Y. Put X on the horizontal axis and Y on the vertical axis. Sketch in their respective indifference curves (note, a “rough” sketch is fine) at their initial endowments. Calculate their marginal rates of substitution at the initial endowment. Are they equal?

(b) Find an equilibrium set of prices using Px as the numeraire. Given your equilibrium prices find the quantity of X and Y each person consumes. Use the Edgeworth Box drawn in 1(a) above to illustrate the equilibrium consumption of X and Y. Calculate the MRS for each person at the equilibrium. Are they equal? Is the equilibrium efficient? Briefly justify your answer.

(c) Can you find another initial endowment, using the same set of prices you discovered in 1(b), that will result in the same equilibrium level of consumption? Illustrate your answer.

43. Ginger has a kilogram of sausages and no potatoes , and Fred has a kilogram of potatoes and no sausages . Assume Ginger has the utility

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function and Fred has the same utility function , and the parameter is the same for Ginger and Fred.

(a) Derive the contract curve and illustrate your result using the Edgeworth box.(b) Derive the competitive equilibrium.

44. Amanda has 4 bottles of wine and 2 loaves of bread and Sue has 2 bottles of wine and 8 loaves of bread . Their respective utility functions are .

(a) Illustrate the initial endowments using an Edgeworth box. Is the initial endowment efficient?

(b) Derive the competitive equilibrium and show that the equilibrium prices are Pareto efficient.

45. The total amount of food available on an island is 10 units and the total amount of energy available is 10 units. Robbie has 1 unit of food and 5 units of energy and Bonnie has 9 units of food and 5 units of energy. Their respective utility functions are .

(a) Is the initial endowment efficient?(b) Derive the contract curve.(c) Derive the competitive equilibrium using the price of energy as numeraire.

46. There are two goods in the world, milk (X) and honey (Y) and two consumers Ichi and Ni. Ichi got in first and cornered the total endowments of milk and honey, his endowment is and his utility function is . Ni’s utility function is . Use PX as the numeraire.

(a) Is the original endowment Pareto optimal? Explain.(b) Now suppose a benevolent dictator sets Ichi’s lump-sum tax at and

Ni’s lump-sum transfer at . Write down the new budget constraint for both Ichi and Ni. Solve for the competitive equilibrium .

Public Goods and Externality

47. There are three consumers of a public good. The demand for each consumer:P1 = 60 – QP2 = 100 – QP3 = 140 – QWhere Pi is the price (marginal value to person i) in dollars and Q measures the number of units of the good. The marginal cost of the public good is $180.

(a) What is the economically efficient level of production of Q? Illustrate your answer on a clearly labelled graph.

(b) What if the producer of Q privately contracted with person 2 and 3 over the supply of Q. What would they agree to? Is the contracted quantity of Q economically efficient? Justify your answer.

(c) How would your answer change if the marginal cost of producing the public good is $60? What if the marginal cost is $350?

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48. There are two groups of citizens on Waiheke Island. The first type is willing to pay for a marine reserve where Q is measured in hectares. The other group is willing to pay . The opportunity cost of setting aside the reserve is $20 per hectare.

(a) What is the optimal area of the reserve?(b) Assume that a private organisation is granted the rights to acquire the reserve

(at $20 per hectare) and charge an entry fee. Would this result in a different sized reserve? What pricing strategy would optimise the firm’s profit?

49. Suppose that a beekeeper is located next to a 20-hectare apple orchard. Each hive of bees can pollinate ¼ hectare of apple trees, thereby raising the value of apple output by $25.

a. Suppose the market value of the honey from one hive is $50 and that the bee-keeper’s marginal costs are given by

where Q is the number of hives employed. In the absence of any bargaining, how

many hives will the beekeeper have and what portion of the apple orchard will be pollinated?

b. What is the maximum amount per hive the orchard owner would pay as a subsidy to the beekeeper to prompt her to install extra hives? Will the owner have to pay this much to prompt the beekeeper to use enough hives to pollinate the entire orchard?

50. Two types of firms emit a nasty pollutant (we will call the pollutant z). There are ten firms of each type. The firms think of pollution emissions as a productive input much like any other input. The primary difference is that the price of pollution emissions is currently $0. Their input demand functions for pollution emissions are:w1 = 100 – z1

w2 = 150 – z2

(d)Given the current price of pollution emissions, how much does each firm emit, and what are total emissions?

(e)Suppose that we want to reduce pollution emissions to 1000 units in total and we must use an emission tax (T) per unit emited. What value of T will induce firms to reduce emissions to the target level?

References

Besanko, D. and R.R. Braeutigam. 2005. Microeconomics, John Wiley & Sons, Hoboken, NJ.

Nicholson, W. 1998. Microeconomic Theory, Harcourt Brace & Company, Orlando, Florida.

Perloff, J.M. Microeconomics with Calculus, Pearson, Essex, England.

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Serrano, R. and A.M. Feldman. 2013. A Short Course in Intermediate Microeconomics with Calculus, Cambridge University Press, NY.

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