Problem Set 1. Question 1.1. A farm produces yams using capital, labor, and land according to the production technology described by:

Transcription

1 Problem Set 1 Micro Analysis, S. Wang Question 1.1. A farm produces yams using capital, labor, and land according to the production technology described by: The firm faces prices for (a) Suppose that, in the short run, and are fixed. Derive the short-run supply and profit functions of the firm. (b) Suppose that, in the long run, and are marketable but is fixed. Derive the long-run supply and profit functions. If there were a market for land, how much would the firm be willing to pay for one more unit of land (the internal price of land)? (c) Suppose that, in the long run, all the factors and are marketable. Does this production function exhibit diminishing, constant, or increasing returns to scale? Suppose that competitive conditions ensure zero profits. Derive the long-run supply and demand functions. Question 1.2. Show that implies Question 1.3. Use a Lagrange function to solve for the following problem:, Question 1.4. Use a graph to solve the cost function for the following problem:, Page 1 of 10

2 Question 1.5. Find the cost function for the following problem:, Question 1.6. In the short run, assume is fixed: Find STC, FC, SVC, SAC, SAVC, SAFC, SMC, LC, LAC, and LMC for the following problem:, Question 1.7. Prove the first two properties of the cost function. Question 1.8. Prove the three properties of the demand and supply functions in Proposition Question 1.9. Consider the factor demand system: where are parameters. Find the condition(s) on the parameters so that this demand system is consistent with cost minimizing behavior. What is the cost function then corresponding to the above factor demand system? Question Show that if satisfies Assumptions 1.1 and 1.2 and then also satisfies Assumptions 1.1 and 1.2. Question A firm buys inputs at levels and on competitive markets and uses them to produce a level of output Its technology is such that the minimum cost of producing at input prices and is given by the cost function Page 2 of 10

3 where and are constant parameters. (a) What parameter condition does the homogeneity of this cost function imply? (b) Derive the conditional demand functions and Verify that the cross price effects are symmetric for these demand functions. (c) Show that the MC curve is upward sloping and that the AC curve is U-shaped (convex). Question The Ace Transformation Company can produce guns ( ), or butter ( ), or both; using labor ( ), as the sole input to the production process. Feasible production is represented by a production possibility set with a frontier (a) Write the production function on the implicit form Does satisfy Assumptions 1.1 and 1.2? (b) Suppose that the company faces the following union demands. In the next year it must purchase exactly units of labor at a wage rate or no labor will be supplied in the next year. If the company knows that it can sell unlimited quantities of guns and butter at prices and respectively, and chooses to maximize next year's profits, what is its optimal production plan? Page 3 of 10

4 Answer Set 1 Answer 1.1. (a) The short-run profit is implying implying implying implying (b) The Long-run profit is, The FOC's are: implying Substituting this solution into the first FOC, we can solve for implying implying The internal price of land will then be Page 4 of 10

5 (c) By the definition, the production exhibits CRS. The Long-run cost function is Take Then the FOC's are implying which imply that and Substituting these into the constraint, we can solve for and then and implying implying Competitive market ensures zero profit, which requires that in the long run. This means that no matter how much the firm produces the profit is always zero. Therefore, the output is indeterminate, meaning that the firm may produce any amount. Answer 1.2. For any and let and We then have Therefore, where the equality for is already given. Answer 1.3. See Varian (2nd ed.) p.31 33, or Varian (3rd ed.) p Page 5 of 10

6 Answer 1.4. From Figure 1.2, we see that the minimum point is on the ratio of Therefore, the cost is or That is, or depending x 2 ax 1 + bx2 = y Isoquant w x w x = c y/ a x 1 Figure 1.2. Cost Minimization with Linear Technology Answer 1.5. Since the production is not differentiable, we cannot use FOC to solve the problem. One way to do is to use a graph. x 2 ax = bx 1 2 y b y = f( x) y/ a x 1 Figure 1.3. Cost Minimization with Leontief Technology From Figure 1.3, we see that the minimum point is Therefore, the cost function is: Answer 1.6. See Varian, Example 2.16, p.55 and p.66. Page 6 of 10

7 Answer 1.7. The cost function and the expenditure function in consumer theory are mathematically the same. (,) Answer 1.8. (1) Since is linearly homogeneous and since is homogeneous of degree in Similarly for (2) By Hotelling s lemma, we have This immediately implies which gives the second property. (3) By the symmetry of the matrix we immediately have Answer 1.9. If the demand system is a solution of a cost minimization problem, then it must satisfy the properties listed in Proposition 1.6. Property (1) in the proposition is obviously satisfied. Property (2) requires symmetric cross-price effects, that is, or Therefore, With the substitution matrix is We have Page 7 of 10

8 and Thus, the substitution matrix is negative semi-definite. Finally, property (4) is implied by the fact that the substitution matrix is negative semi-definite. Therefore, to be consistent with cost minimization, we need and only need condition: Let Then the cost function is Answer Since Assumption 1.1 is satisfied. Since we have Multiply the first column of the right determinant by and then add what you have got to the jth column. This operation won t affect the value of the determinant. Thus, for any Therefore, also satisfies Assumption 1.2. Answer (a) By we immediately see that the linear homogeneity of cost function implies that (b) We have Then, Page 8 of 10

9 When the functions are differentiable, taking derivatives is often the easiest way to find monotonicity and convexity. Therefore, is upward sloping and is U-shaped. Answer (a) The production set is defined by which means that if the firm wants to produce it needs at least amount of labor. Since the labor is an input, it should be negative in the definition of implicit production function. This means that we can choose and define The production process is then defined by for We first have thus Assumption 1.1 is satisfied. The 2nd order conditions are / / / / and Page 9 of 10

10 Therefore, Assumptions 1.2 is satisfied. (b) The problem is The solution is: Therefore, the supplies are: Page 10 of 10

11 Problem Set 2 Micro Analysis, S. Wang Question 2.1. Show that strong monotonicity implies local nonsatiation but not vice versa. Question 2.2. A consumer has a utility function (a) Compute the ordinary demand functions. (b) Show that the indirect utility function is (c) Compute the expenditure function. (d) Compute the compensated demand functions. Question 2.3. Let demand for good is defined as and equal, they must all be one. be the consumer s demand for good The income elasticity of (,) Show that, if all income elasticities are constant Question 2.4. Show that the cross-price effects for ordinary demand are symmetric iff all goods have the same income elasticity: (,) (,) Question 2.5. A consumer has expenditure function / of? What is the value Question 2.6. Suppose the consumer s utility function is homogeneous of degree 1. Show that the consumer s demand functions have constant income elasticity equals 1. Page 1 of 10

12 Question 2.7. Use the envelope theorem to show that the Lagrange multiplier associated with the budget constraint is the marginal utility of income; that is, (,) Question 2.8. Suppose that the consumer's demand function for good has constant income elasticity Show that the demand function can be written as Question 2.9. Consider the substitution matrix (,) (a) Show that ( ) (,) of a utility-maximizing consumer. (b) Conclude that the substitution matrix is singular and that the price vector lies in its null space. (c) Show that this implies that there is some entry in each row and column of the substitution matrix that is nonnegative. Question An individual has a utility function for leisure and food of the form: Suppose that the individual has an income with wage rate and price of food (a) Derive the individual's compensated demand functions for food and leisure. (b) Verify Shephard's lemma and Roy's identity for this individual's demand functions. (c) Suppose that there is an increase in the price of food. Divide the total effect on the consumer demand for leisure into income and substitution effects. (d) Is there a price of food at which a further rise in the price will lead to a decrease in consumer demand for leisure? Question One popular functional form in empirical work for ordinary demand functions and is the double logarithmic demand system: where is the income and the price vector. The parameters are unknown and are to be estimated. Page 2 of 10

13 (a) Interpret and in terms of elasticity, where the price elasticity of demand for good is and the income elasticity of demand for good is (b) Show that in order that the above demand functions can be interpreted as having been derived from utility maximizing behavior, the following parameter restrictions must be imposed: If good 1 is a normal good and is not a Giffen good, are there additional parameter restrictions implied by this fact? If goods 1 and 2 are gross substitutes, are there additional parameter restrictions? Question A consumer has an intertemporal utility function of present consumption and future consumption He takes as given the spot prices He can borrow and lend freely at an interest rate He has an initial endowment of units of the commodity in the present and units of the commodity in the future. (a) Find the utility-maximizing consumption bundle of the consumer, and compute his marginal rate of substitution between present and future consumption. (b) What is the effect of a change in the interest rate on savings? (c) Suppose, in addition to his endowment, the consumer owns a firm with a production function where is the input in period 1 and is the output in period 2. (NOTE: and are in the units of the commodity in period 1; and are in the units of the commodity in period 2.) Determine the level at which the consumer will operate the firm and the utility-maximizing consumption bundle he attains. (d) Demonstrate that Fisher's Separation Theorem holds by showing that the problem can be decomposed into two separate problems: a maximization of profits; and a maximization of utility subject to a wealth constraint. Page 3 of 10

14 Answer Set 2 Answer 2.1. Since in any neighborhood of we can always find a point such that and strong monotonicity thus implies local nonsatiation. Suppose the preferences are defined by It is easy to see that the preferences satisfy local nonsatiation. But for two points and we have and but That is, the preferences don t satisfy strong monotonicity. Answer 2.2. (a) The consumer s problem is Let The FOC s imply Substituting this into the budget constraint will immediately give us By symmetry, we also have (b) Substituting the consumer s demands into the utility function will give us (c) Let i.e. which immediately gives us the expenditure function: (d) Substituting for in the consumer s demand functions we get Page 4 of 10

15 By symmetry, Answer 2.3. Using the adding-up condition we can take derivative w.r.t. on both sides of the equation to get: implying If then that is, Answer 2.4. By Shephard s lemma, By Slutsky equation, where is the income elasticity of demand for good Similarly, By (1) and the fact that then Page 5 of 10

16 Answer 2.5. Since is linearly homogeneous in Answer 2.6. We can easily show that given that fact that Then, is linearly homogeneous in and is homogeneous of degree in By Roy s identity, we then have Taking the derivative w.r.t., we then have Setting we then have Answer 2.7. The problem is The Lagrange function for this problem is We have Then by the Envelop Theorem,, Answer 2.8. Given for all we have Thus, Page 6 of 10

17 Therefore, Answer 2.9. (a) We have By taking derivative w.r.t. on both sides of above equation, we have (b) Part (a) implies (3) where is the substitution matrix, and By the assumption that (3) implies that must be singular. By the FOC we then have This means that where denotes the null space of 's. (d) For each by (2), since by assumption ( ) nonnegative. one of the must be Page 7 of 10

18 Answer (a) We have implying implying (b) Taking derivatives w.r.t. the prices, Therefore, the Shephard's Lemma is verified. From utility maximization, we can find the consumer demand functions: From the expenditure function, implying Therefore, the Roy's Identity is verified. (c) We have (e) We see that the two effects cancel out, and thus the total effect is zero. That is, changes in the price of food will not affect the demand for leisure. Page 8 of 10

19 Answer (a) We have (b) For any since is homogeneous of degree we have Therefore, Similarly, using the 2nd equation, we also have Normality implies that We hence have Since good 1 is not a Giffen good, If good 1 is a substitute for good 2, then If good 2 is a substitute for good 1, then We hence have We hence have We hence have Answer (a) The consumer's problem is The marginal rate of substitution between present and future consumption is This should be equal to the price ratio at the optimal consumption levels. That is, Thus, and hence from the budget constraint. (b) By (a), Page 9 of 10

20 Then, by the budget constraint, which implies that decreases as increases, and hence savings increases as increases. This is what we would expect in reality. (c) The consumer's problem is where This problem can be reduced to the following problem by eliminating and using the two restrictions:, We then have and Then, and (d) The profit maximization problem is which gives solution wealth constraint is The problem of utility maximization subject to which gives solution Separation Theorem is verified. Since the solutions in (d) and (c) are the same, Fisher's Page 10 of 10

21 Problem Set 3 Micro Analysis, S. Wang Question 3.1. Suppose that an expected utility function has constant absolute risk aversion () () What must the form of the utility function be? Question 3.2. Given any constant and a zero-mean random variable define by Denote Derive Question 3.3. For a quadratic utility function show that the expected utility of a random payoff is a function of the mean and variance of Question 3.4. A sports fan s preferences can be represented by an expected utility. He has subjective probability that the Lions will win their next football game and probability that they will not win. He chooses to bet on the Lions so that if the Lions win, he wins and if the Lions lose he loses The fan's initial wealth is (a) How can we determine his subjective odds by observing his optimal bet (b) Under what condition does an increase in lead to a higher bet Question 3.5. Suppose that a consumer has a differentiable expected utility function for money with The consumer is offered a bet with probability of winning and probability of losing Show that, if is small enough, the consumer will always take the bet. Page 1 of 8

22 Question 3.6. Let individual A have an expected utility function and let individual B have an expected utility function where is income. Let be a monotonic increasing, strictly concave function, and suppose that That is, is a concave monotonic transformation of (a) Show that individual A is more risk-averse than individual B in the sense of the absolute risk aversion. (b) Let be a random variable with Define risk premiums and by Here is initial wealth. If show that (c) Interpret the risk premium in words. Question 3.7. For Exercise 3.4, when the probability of winning goes up, do you expect the amount that a person is willing to gamble to go up? Prove your claim. Question 3.8. Suppose a farmer is deciding to use fertilizer or not. But there is uncertainty about the rain, which will also help the crops. Suppose that the farmer's choices consist of two lotteries: Suppose that the farmer is an expected utility maximizer and has monotonic preferences. What would the farmer choose if he were (i) risk loving? (ii) risk neutral? (iii) risk averse? Question 3.9. What axiom is violated by Question Show that the following two utility functions one is a monotonic transformation of the other imply the same preferences with certainty consumption bundles, but not with uncertainty consumption bundles: Page 2 of 8

23 Question For the insurance problem: where is the loss, is the probability of the bad event, is the price of insurance, is initial wealth, and (a) If the insurance market is not competitive and the insurance company makes a positive expected profit: will the consumer demand full-insurance underinsurance or over-insurance Show your answer. (b) Show the above solution on a diagram. Page 3 of 8

24 Answer Set 3 Answer 3.1. We have where is some constant. Then where and are some constants. Therefore, for are two constants and such that () () if and only if there Answer 3.2. By definition, (A) By Taylor's expansion, and Equalizing above two formulae immediately implies an approximated solution of Answer 3.3. We have Answer 3.4. (a) The individual problem is Page 4 of 8

25 The first-order condition implies that (1) implying By knowing and can then be determined using above equation. (b) By taking the derivative w.r.t on the FOC (1), we get (2) By (2), for a risk averse person with increasing utility function we have Answer 3.5. We need to show that (3) when is small for a differentiable utility function with ( may not be concave). By Taylor expansion, there are and such that Therefore, (3) is true if and only if Letting we have and and then Therefore, when is small, (3) is true. Answer 3.6. Since if we have Page 5 of 8

26 (b) We know that if is a convex function, then 1 By definition, Since is concave, is convex. Therefore, Assuming is strictly increasing, then (d) The risk premium is the maximum amount of money that an expect utility maximizer is willing to pay to avoid risk. Answer 3.7. For a risk averse person with increasing utility function, the answer is Yes. The first-order condition is By taking the derivative w.r.t. on above equation, we get Of course, for a risk loving person with increasing utility function, the opposite is true. Answer 3.8. If he is risk loving, then Since by monotonicity risk neutral, then he only cares about the expected income. Since this farmer will choose fertilizer. If he is this farmer will still choose fertilizer. If he is risk averse, then 1 For those who want to know, let 1 2 be a partition of the value space of the random variable and the probability of Then, by the continuity and convexity of we have Page 6 of 8

27 this farmer's choice will depend on his particular preferences. From the given information, we don't know what this farmer will choose. Note that by comparing the two distribution functions, the two lotteries don't dominate each other by FOSD or SOSD. Thus, stochastic dominance cannot help determine the preferences. Answer 3.9. If RCLA were not violated, then which would immediately imply a contradiction. Therefore, RCLA must has been violated. Answer Let Then Since is a strictly increasing function, and are equivalent over certainty consumption bundles. But for uncertainty consumption bundles: we have Hence, and are not equivalent over uncertainty consumption bundles. Answer (a) At the optimal point The expected profit is Then, Thus, Then, or have under-insurance. i.e., It implies that is, we (b) When in Example 1.12, we have shown that the solution must be on the line. When the budget line is flatter, and the tangent point must be below the line. That is, the individual is under-insured. Page 7 of 8

28 I 2 slope= 1-π π slope= 1-p p 45 o w-l... w I 1 Figure 5.1. Insurance in a non-competitive market Page 8 of 8

29 Problem Set 4 Micro Analysis, S. Wang Question 4.1. There are two consumers A and B with utility functions and endowments: Calculate the GE price(s) and allocation(s). Question 4.2 (PhD). We have is some initial bundle of goods agents with identical strictly concave utility functions. There Show that equal division is a Pareto efficient allocation. Question 4.3 (PhD). We have two agents with indirect utility functions and initial endowments Calculate the GE prices. Question 4.4 (PhD). Suppose that we have two consumers and functions with identical utility Suppose that the total available amount of good 1 is and the total available amount of good 2 is i.e., Draw an Edgeworth box to illustrate the strongly Pareto optimal and the (weakly) Pareto optimal sets. Question 4.5. Consider a two-consumer, two-good economy. Both consumers have the same Cobb-Douglas utility functions: 1/8

30 There is one unit of each good available. Calculate the set of Pareto efficient allocations and illustrate it in an Edgeworth box. Question 4.6. Consider an economy with two firms and two consumers. Denote as the number of guns, as the amount of butter, and as the amount of oil. The utility functions for consumers are.. Each consumer initially owns units of oil: Consumer 1 owns firm 1 which has production function consumer 2 owns firm 2 which has production function Find the general equilibrium. Question 4.7. Suppose that there are one consumer, one firm, and one good The firm is owned by the consumer. The consumer has an endowment of unit of time for working and enjoying leisure, and has utility function for good and leisure time The firm inputs amount of labor to produce amount of good. Find the GE. Question 4.8. Suppose that the economy is the same as in Question 4.7 except that the firm has production function Find the GE. Question 4.9. There are two goods and with prices and respectively, and two individuals and with and (a) Derive the contract curve. Suppose Draw it in an Edgeworth box. (b) Derive the GE price ratio(s) Question There are two goods and and two individuals with and (a) Find all the Pareto optimal allocations. Are they strongly Pareto optimal? (b) Find all the GE price ratio(s) 2/8

31 Answer Set 4 Answer 4.1. Individual A s utility function is equivalent to and Then the income is and the demands are: Let For individual B, by its utility function, we know that the demands must satisfy by budget constraint the demands are: Then In equilibrium, the total supply of good 1 must be equal to the total demand for good 1: Therefore, and the allocation is Answer 4.2. Denote another allocation such that If is not Pareto optimal, then there is (1) and By (1), Then, by concavity of By (2), Then above inequality implies This is a contradiction. Therefore, allocation must be Pareto optimal. Answer 4.3. Let and Then the incomes are By Roy's Identity, In equilibrium, the total supply of good 1 must be equal to the total demand of good 1: 3/8

32 Therefore, the equilibrium price ratio is: Answer 4.4. In the following charts, the left chart indicates the Edgeworth box and the indifference curves. The right chart indicates the Pareto optimal points. B F B u B E D A u A Figure 1. Pareto Optimal Points A C As indicated by the right chart, the set of weakly P.O. points consists of five intervals AC, CD, DE, EF, and FB: the set of strong P.O. points consists of only two points C and F: Answer 4.5. By Proposition 1.27, the following equation defines the set of P.O. points: Feasibility requires Let and Then above two equations imply Therefore, 4/8

33 This set is the diagonal line in the following diagram. y 2 P.O. y=x 1 x Figure 2. P.O. Allocations Answer 4.6. Denote price of guns price of butter price of oil (we can arbitrarily choose one of prices. We can do that because of the homogeneity of demand functions). The two consumers are: Firm 1 s problem:.... It implies Note that the only possible equilibrium is when here. Zero-profit argument is not accurate Firm 2 s problem: It implies Consumer 1 s problem:,.. Its solution is 5/8

34 Consumer 2 s problem:,.. The solution is Market clearing conditions: Because of Walras Law, we only need two of these three conditions to determine the equilibrium. They imply that and Therefore, the equilibrium is: Answer 4.7. Firm's problem:, The solution is The only possible equilibrium is when We thus only consider Consumer's problem:, gives solution Market clearing conditions: Because of Walras Law, we only need to use one of conditions to determine the equilibrium. The first condition implies that Then, and implies Therefore, the equilibrium is: 6/8

35 Answer 4.8. We can arbitrarily set Firm s problem:, gives Consumer s problem:, gives solution Market clearing conditions: Because of Walras Law, we only need to use one of conditions to determine the equilibrium. It implies that Therefore, the equilibrium is: Answer 4.9. (a) We have which gives the contract curve as goes from to We have 7/8

36 x y ω. B. u A u B Contract curve A x y Figure 3. Contract Curve and Equilibrium (b) We have Equilibrium condition implies which can be solved to get the equilibrium price ratio Answer (a) The contract curve is the diagonal line in the chart. The points on the contract curve are strongly P.O. y 2 W.. u 1 contract line u 2 1 x Figure 4. Contract Curve and Equilibria (b) The set of equilibria is That is, all the possible values of are equilibria. 8/8

37 Problem Set 5 Micro Analysis, S. Wang There are no exercises for Chapter 5. Page 1 of 1

38 Problem Set 6 Micro Analysis, S. Wang Question 6.1. You have just been asked to run a company that has two factories producing the same good and sells its output in a perfectly competitive market. The production function for each factory is: Initially, the capital stocks in the two factories are respectively and The wage rate for labor is and the rental rate for capital is In the short run, the capital stock for each factory is fixed, and only labor can be varied. In long run, both capital and labor can be varied. (a) Find the short-run total cost function for each factory. (b) Find the company s short-run supply function of output and demand functions for labor. (c) Find the long-run total cost function for each factory and the long-run supply curve of the company. (d) If all companies in the industry are identical to your company, what is the long-run industry equilibrium price? (e) Let Suppose the cost of labor services increases from to per unit. What is the new long-run industry equilibrium price? Can you determine whether the quantity of capital used in the long run will increase or decrease as a result of the increase in the wage rate from to? Question 6.2. Suppose that two identical firms are operating at the cooperative solution and that each firm believes that if it adjusts its output the other firm will adjust its output to keep its market share equal to What kind of industry structure does this imply? Question 6.3. Consider an industry with two firms, each having marginal costs equal to zero. The industry demand is where is total output. Page 1 of 8

39 (a) What is the competitive equilibrium output? (b) If each firm behaves as a Cournot competitor, what is firm 1 s optimal output given firm 2 s output? (c) Calculate the Cournot equilibrium output for each firm. (d) Calculate the cooperative output for the industry. (e) If firm 1 behaves as a follower and firm 2 behaves as a leader, calculate the Stackelberg equilibrium output of each firm. Question 6.4. Consider a Cournot industry in which the firms outputs are denoted by aggregate output is denoted by the industry demand curve is denoted by and the cost function of each firm is given by For simplicity, assume Suppose that each firm is required to pay a specific tax of on output. (a) Devise the first-order conditions for firm (b) Show that the industry output and price only depend on the sum of tax rates (c) Consider a change in each firm s tax rate that does not change the tax burden on the industry. Letting denote the change in firm s tax rate, we require that Assuming that no firm leaves the industry, calculate the change in firm s equilibrium output [Hint: use the equations from the derivations of (a) and (b)]. Question 6.5. (Entry Cost in a Bertrand Model). Consider an industry with an entry cost. Let where and are two constants. Find the equilibrium solution for the following twostage game. Stage 1. All potential firms simultaneously decide to be in or out. If a firm decides to be in, it pays a setup cost Stage 2. All firms that have entered play a Bertrand game. Question 6.6. Verify the socially optimal number of firms to be 6.9 of the book. () / / in Section Page 2 of 8

40 Answer Set 6 Answer 6.1. (a) For each factory with capital stock Therefore, the short-run cost functions are (b) The firm cares about the total profit from its two factories. The objective of firm is therefore to maximize the total profit: The FOCs give us the well-known equality:, We have is: and and Thus, and Then and imply that Therefore, the short-run supply function The labor demands for the factories are: Therefore, the labor demand is (c) The cost for each factory is, The Lagrange function is implying The total cost is then Page 3 of 8

41 From the profit function supply function: we immediately find the long-run That is, the long-run industry supply curve is horizontal. In this case, the equilibrium output is determined by demand (which is not given). (d) In a competitive market, with a horizontal industry supply curve, the long-run equilibrium price must be whatever the industry demand curve is. (e) The original long-run equilibrium price is and the new price is The total capital investment is With an increase in and output is reduced. With going down and going up, the change is in is ambiguous; it demands on the demand. p p.. s y D y Answer 6.2. Let be the market price of the good when the output is is the cost of firm when its output is The two firms have the same cost function. The cartel maximizes their total profit: The FOCs are, We look for a solution for which (the symmetric solution). Thus, the FOC becomes Page 4 of 8

42 We can rewrite (2) as where On the other hand, the Cournot output is determined by p Y c æ ö ç çè2 ø. B A.. C MR(Y ) D Y 1 MR( Y) - p ( Y) Y 2 Figure 6.1. A market-share Cournot equilibrium In the diagram, point is the competitive solution, for which each firm takes the market price as given; point is our solution, for which each firm acts upon a decreasing demand and assume equal market share as the other s reaction; point is the Cournot equilibrium. From the diagram, we can conclude that The equilibrium output at is lower than the output at the competitive solution and the output at the Cournot equilibrium. The equilibrium price at is higher than the price at the competitive solution and the price at the Cournot equilibrium. Answer 6.3. (a) For competitive output, firms take price as given in maximizing their own profits: which implies Page 5 of 8

43 That is, the firms supply curve is the horizontal line at So is the industry supply curve. The equilibrium industry supply is thus and the equilibrium price is (b) Firm 1 maximizes his own profit, given any which gives the FOC: Firm 1 s reaction function is thus (c) By symmetry, the outputs for the two firms should be the same in equilibrium. By the reaction function in (b), we hence have Cournot equilibrium is which gives Therefore, the (d) Suppose the two firms collude. They form a monopoly and maximizes their total profit: which gives the cartel output: (e) Firm 1 will behave as in (b), and reacts according to his reaction function Firm 2 will take this into consideration when maximizing his own profit: which implies Then, In summary, the competitive industry output is the highest, the Stackelberg industry output is the second, the Cournot industry output is the third, and cartel output is the lowest. Answer 6.4. (a) The profit maximization for firm is The FOC is (3) (b) By summarizing (3) from to we have (4) Page 6 of 8

44 This equation determines the industry output which obviously depends on rather than the individual tax rates s. (c) Since the total output depends only on and the latter has no change, doesn t change for a tax change. Then, by (3), i.e., where is determined by (4). Answer 6.5. This is from Example 12.E.2 on page 407 of MWG (1995). Once identical firms are in the industry, they play a Bertrand game. As we know, if the result is the competitive outcome, i.e., and the profit without including the entry cost is zero for all the firms. This means that each firm loses in the long run. Knowing this, once one firm has entered the industry, all other firms will stay out. Therefore, more intense competition in stage 2 results in a less competitive industry! This single firm will be the monopoly and produces at the monopolist output resulting the monopoly price The monopoly profit is As long as a firm will enter and that is the only firm in the industry. Answer 6.6. We have where Then, Page 7 of 8

45 implying / implying / / / / Page 8 of 8

46 Problem Set 7 Micro Analysis, S. Wang Try to do more problems in MWG (1995), Chapters 7 9. Question 7.1 (Mixed-Strategy Nash Equilibrium) (PhD). A principal hires an agent to perform some service at a price (which is supposed to equal the cost of the service). The principal and the agent have initial wealth and respectively. The principal can potentially lose If the agent offers low quality, the probability of losing is if the agent offers high quality, the probability of losing is The quality is unobservable to the principal. The price of a low quality product is (paid to the agent) is and the price of a high quality product is by the competitive market assumption, and are the costs of producing the products (the agent bears the costs). The agent is required by regulation to provide high-quality services, but he may cheat. After such a bad event happens, the principal can spend in an investigation; if the agent is found to have provided low-quality services, the agent will have to pay for the loss to the principal. This game can be written in the following normal form: low quality, high quality, investigate, not to investigate, where Find the mixed-strategy Nash equilibria. Question 7.2 (Pure-Strategy Nash Equilibrium) (PhD). Find the pure-strategy Nash equilibria in the above exercise. 1/12

47 Question 7.3. For the following game, find the pure-strategy NEs. Show whether or not they are trembling-hand perfect. Player 2 1, 6 0, 5 1, 1 1, 2 Question 7.4 (PhD). For the following game (Mas-Colell et al. 1995, p.271), find all the purestrategy Nash equilibria. P1 æ2ö 0 ç çè1 ø. P3 L1 R1 æ- 1ö 5 ç çè 6 ø P2. l r L2 R2 l P3. P3. r l r æ3ö 1 ç çè2 ø æ5ö 4 ç çè4 ø æ 0ö ç 1 ç- ç çè 7 ø æ- 2ö 2 ç çè 0 ø Question 7.5. In the following game, explain why there are mixed-strategy NEs in which P1 mixes and arbitrarily and P2 chooses P1 o æ0ö ç çè0 ø s11 L s M 1 21 s 1 31 R1. m1 H m2. L2 R L R P2 æ 1ö - ç ç- è 1 ø æ1ö ç çè2 ø æ 1ö - ç çè 0 ø æö 1 ç çè1 ø 2/12

48 Question 7.6. Consider the following game. L 1 P1 o R 1 x. P1 æ0ö ç çè0 ø ˆL 1 ˆR 1. m H m. 1 2 P2 L2 R2 L 2 R2 æ- 2ö ç ç- è 1 ø æ 1 ö ç ç- è 2 ø æ- 1ö ç çè 1 ø æ2ö ç çè3 ø (a) Find all pure-strategy NEs. (b) Find all SPNEs. (c) Find all BEs. (d) Are all the BEs subgame perfect? Question 7.7. Find all the mixed strategy SPNE in the following game. Firm E o Out 0 2 Small Niche Small Niche In x 1. Firm E. H I. Large Niche Small Niche Large Niche Firm I Large Niche Firm E' Firm I' s Payoff s Payoff /12

49 Question 7.8. For the following game, find all the pure-strategy NE, all the SPNEs and all the BEs. 0 2 σ 1E Out Firm E o m Fight σ I In Fight σ 2 E. 1 σ 1E z Firm E Accom 1 σ 2 E H I. Firm I. Accom σ I Fight σ 1-m Accom 1 σ 1 I I π E π I Question 7.9 (PhD). A revised version of Exercise 9.C.7 in Mas-Colell et al. (1995, p.304)]. (a) For the following game, find all the pure-strategy NEs. Which one is a SPNE? P1 o P2. δ1 δ 2 γ 1 γ 2 B T. P2 δ δ1 2 D U D U Figure 7.1. NEs and SPNEs (b) Now suppose that P2 cannot observe P1 s move. Draw the game tree, and find all the mixed-strategy NEs. 4/12

50 Question 7.10 (PhD). One problem with a BE is that it may not be trembling-hand perfect. Consider the following game.. P1 o L1 R1 μ 1 μ 2. P2 L2 R2 L2 R2 æ1 ö ç çè2 ø æ0ö ç çè2 ø æ0ö ç çè1 ø æö 3 ç çè3 ø (a) Show that we have the following BE: Figure 7.2. Trembling-Hand Perfect Equilibrium with payoff pair (b) Show that this BE is a SE. Note that we already know in Example 7.10 that this strategy profile is not trembling-hand perfect. 5/12

51 Answer Set 7 Answer 7.1. Assume that the principal can commit ex ante to investigate or not before a loss occurs. In other words, the principal can only make up his mind on investigation before she has suffered a loss. Before a loss occurs, the game box of surpluses is low quality, high quality, investigate, not to investigate, In each cell, the value on the left is the surplus of the principal and the value on the right is the surplus of the agent. The optimal choice of investigation: is to make the principal indifferent between investigation and no (1) implying implying implying The choice of is to make the agent indifferent between cheating and no cheating: (2) implying 6/12

52 Answer 7.2. By substituting the parameter values into the game box of surpluses, we have cheat, not to cheat, investigate, not to investigate, By Proposition 7.2 in the book, to find pure-strategy Nash equilibria, we can restrict to pure strategies only. Thus, simply by inspecting each cell one by one, we know that there is no purestrategy Nash equilibrium. Answer 7.3. This is a situation in which a player is indifferent from two alternative strategies, one of which is the equilibrium strategy. This player has no incentive to deviate if other players don t make any mistakes. However, the situation changes if possible mistakes by other players are taken into account. There two NEs: and In given player 1 is indifferent from and However, if player 2 may make some mistakes by taking with probability no matter how small is, player 1 will be strictly prefer to Thus, is not a trembling-hand NE, while is. Answer 7.4. The strategy sets for players 1 and 2 are simple: There are three information sets for player 3. Denote a typical strategy of player 3 as where is the action if the information set on the left is reached, is the action if the information set in the middle is reached, and is the action if the information set on the right is reached. Player 3 has eight strategies: The normal form is P1 plays P3 P2: 2,0,1-1,5,6 2,0,1-1,5,6 2,0,1-1,5,6 2,0,1-1,5,6 2,0,1-1,5,6 2,0,1 (-1,5,6) 2,0,1-1,5,6 2,0,1 (-1,5,6) 7/12

53 P1 plays P3 P2: 3,1,2 3,1,2 3,1,2 3,1,2 (5,4,4) (5,4,4) (5,4,4) (5,4,4) 0,-1,7 0,-1,7-2,2,0-2,2,0 0,-1,7 0,-1,7-2,2,0-2,2,0 All the pure strategy Nash equilibria are indicated in the boxes. To find all the Nash equilibria, we can check each cell one by one. A cell cannot be a Nash equilibrium if one of the players doesn t stick to it. In each cell, we can first check to see if player 3 will stick to his strategy, by which we can quickly eliminate many cells. A sequentially rational NE must be an outcome from backward induction. Example 7.14 in the book shows that backward induction only leads to one outcome: which is one of the Nash equilibria. Answer 7.5. Whatever P2 does, and are indifferent to P1. On the other hand, whatever P1 does, is always better to P2. Answer 7.6. (a) P2 has one information set containing two nodes. P1 has two information sets and where contains the initial node. Denote P1 s strategies as where is an action at and is an action at The normal form, where the payoff profile in each cell is (P2 s payoff, P1 s payoff), is: P2\P1 (0, 0) (0, 0) -1, -2 1, -1 0, 0 0, 0-2, 1 (3, 2) The pure-strategy NEs are indicated in the above table. (b) Since in the real subgame SG(x), there is only one NE in SG(x). Hence, there is one SPNE, which is 8/12

54 (c) Let us find BEs. For P2, iff or If so, P1 chooses at node Then, since choosing means a payoff of P1 chooses at the beginning. Since is not on the equilibrium path in this case, any belief is acceptable. Hence we have a BE: If then then P1 chooses at and then P1 choose at the initial node. In this case, consistency is required and it implies which can be satisfied. Hence, there is another BE: Further, if P2 is indifferent between and that is, P2 s strategy can be any mixed strategy with Then, at node P1 s preference would be iff However, this is completely impossible. We in fact always have Hence, P1 will always choose at Then, P1 s preference at the initial point is iff, i.e.,. Hence, if P1 s strategy is Since is not on the equilibrium path, any belief is acceptable. Thus, we have a BE: If is on the equilibrium path, by which consistency requires This is impossible. Hence, there is no BE in this case. If P1 is indifferent between and. In this case, if P1 takes with a positive probability, consistency is required and it cannot be satisfied. If P1 takes for sure, consistency on is not required and hence can be allowed. Hence, we have another BE: This BE4 can be combined with BE3. (d) Since the BE1, BE3 and BE4 (the strategies of these BEs) are not the SPNE, we conclude that BEs may not be SPNEs. Answer 7.7. In the proper subgame with the normal form: Firm I Small, Large, Firm E: Small, -6, -6 (-1, 1) Large, (1, -1) -3, -3 The equilibrium is to make firm E indifferent between his two strategies: 9/12

55 implying Since the game is symmetric, we also have We also have Out Firm E o In The game is reduced to: Then, the expected payoff is Then, firm E will choose out. Thus, the SPNE is Answer 7.8. Firm I has one information set containing two nodes. Based on this information, firm I has two strategies: Firm E has two information sets and where contains the initial node. Denote firm E s strategies as where is an action at and is an action at We can then find the normal form: Firm E <out, fight> <out, accom> <in, fight> <in, accom> Firm I: fight (2, 0) (2, 0) -1, -3-1, -2 accom 2, 0 2, 0-2, 1 (1, 3) We can easily find the pure-strategy Nash equilibria, as indicated in the above box: There is only one SPNE, which is NE3, i.e., One of BEs is This example indicates that BE and SPNE don t imply each other: BE eliminates two NEs, one of which is SPNE; SPNE also eliminates two NEs, one of which is BE. 10/12

56 There are three BEs: BE2 is the same as the SPNE. This example indicates that BE and SPNE don t imply each other: BE eliminates NE1; SPNE eliminates NE1 and NE2, one of which is BE. In other examples, we also know that BE sometimes eliminates SPNEs. Answer 7.9. (a) There are two information sets for P2. Let be a typical P2 s strategy, where is an action taken at the left information set and is an action taken at the right information set. The normal form of the game is P2 <D, D> <D, U> <U, D> <U, U> P1: B 4, 2 (4, 2) 1, 1 1, 1 T 5, 1 2, 2 5, 1 (2, 2) There are two pure-strategy NEs: and The first one is a SPNE. (b) The game tree is: P1 o. δ 1 δ 2 γ 1 2 B P2 H 2 T γ. δ δ 1 2 D U D U The normal form is P2 D U P1: B 4, 2 1, 1 T 5, 1 (2, 2) 11/12

57 There is a pure-strategy NE: Since playing is a strictly dominant strategy for P1, this NE is the NE. Answer It is simple. You do by yourself. 12/12

58 Problem Set 8 Micro Analysis, S. Wang Question 8.1 (Gibbons 1992, p.250, Exercise 4.10). It is a buyback solution to dissolve a partnership. Partners 1 and 2 own shares and of the partnership, respectively. Partner 1 is to name a price and then partner 2 chooses either to buy partner 1 s share for or to sell his share to partner 1 for Assume that partner 1 s value of the firm is if she owns the whole firm and zero otherwise; and partner 2 s value of the firm is if he owns the whole firm and zero otherwise. Suppose that each partner s valuation is private information and the other partner only knows the distribution only. Suppose and independently follow (the uniform distribution on ) What is the BNE? Question 8.2 (Gibbons 1992, p.250, Exercise 4.11). A buyer and a seller have valuations and respectively. The buyer s valuation is with known parameter The seller knows her own valuation (and hence but the buyer doesn t know The buyer knows that the seller s valuation follows (the uniform distribution on ). The buyer makes a single offer which the seller either accepts or rejects. Find the BNE. Question 8.3. (Gibbons 1992, p.253, the first part of Exercise 4.15) (PhD). Consider a legislative process in which a feasible policy is The status quo is and the ideal policy for the Congress is where The ideal policy for the president is which is private information of the president. The Congress only knows that follows The Congress proposes a policy and the president either signs or vetoes. Given a policy the payoffs of the Congress and the president are respectively and Find the BNE and verify in equilibrium. Question 8.4 (A cheap-talk game) (PhD). 1 The basic game setup is the same as in Question 8.3. Now, suppose that the president can engage in rhetoric (send a cheap-talk message) before the Congress proposes a policy. Consider a two-step PBE in which the president sends a message in the first period and the Congress proposes based on a belief which 1 This is from Gibbons (1992, p.253, the second part of Exercise 4.15). Ignore this exercise if I didn t cover cheap-talk games in class. Page 1 of 6

59 is the probability that the president has type when message is observed. The president may take a pure strategy or a mixed strategy with and (a) Define the PBE when the president takes a pure strategy. (b) Define the PBE when the president takes a mixed strategy. (c) Show that In equilibrium, there are only two possible proposals and. Derive the PBE and shows that Page 2 of 6

60 Answer Set 8 Answer 8.1. Partner 1 s problem is () () The FOC is implying This is the BNE. In equilibrium, who owns the firm? Partner 1 owns the firm if or otherwise partner 2 owns the firm. Note that, in the above, we assume that partner 1 decides the price and partner 2 decides whether to sell. If both partners have the right to decide whether to sell or buy, in order for partner 1 to have the firm, partner 1 should be willing to buy (when and partner 2 is willing to sell (when in order for partner 2 to have the firm, partner 1 should be willing to sell (when and partner 2 is willing to buy (when In this case, the firm goes to partner 1 iff and it goes to partner 2 iff This situation is complicated. Answer 8.2. If and only if problem is the seller will accept the price offer. Hence, the buyer s We have Hence, the optimal pricing is Page 3 of 6

61 Therefore, there is no trade if there is a trade if and there may or may not be a trade if Answer 8.3. If and only if the president will sign the proposal. Hence, the Congress s problem is () () () () We have () () ()() If then Then, the FOC for is or the right-hand side is negative. It is impos- The left-hand side is positive. But, since sible. Hence, we must have With we have The FOC for is implying implying implying implying Page 4 of 6

62 Hence, We have iff or which is always true. Hence, we indeed have. Answer 8.4. This problem is from Matthews (1989, QJE, ). When the Congress sees message it has the belief the probability of type is with message The Congress then responds with proposal The game is drawn below. In the figure, player P is the president and player C is the Congress. Let be type president s payoff under policy and be the payoff of the Congress under policy P. tl Nature t d 1- d H. P L R L R. C C. p C q C 1- p 1- q al ah al ah al ah al ah P P P P P P P P Figure 1. A Free-Talk Game (a) Following Gibbons (1992), we first consider a pure-strategy BE. Suppose that the president plays a pure strategy In the second step, when the Congress sees it guesses that the density of type is and its proposal is a solution of the following problem: (,)(,) (,)(,) (1) Page 5 of 6

63 In the first step, knowing the Congress s proposal message. His problem is the president considers how to send a (2) This implies Let be the true density function of the type. The equilibrium consistency condition requires that, if is a message that is sent in equilibrium, i.e., for some then () (3) Under three conditions (1) (3), we have a BE: (b) Following Crawford-Sobel (1982) and Matthews (1989), we now consider a mixedstrategy BE. Suppose that the president plays a pure strategy. In the second step, when the Congress sees it guesses that the density of type is and its proposal is a solution of the following problem: (,)(,) (,)(,) (4) In the first step, knowing the Congress s proposal the president considers how to send his message strategy. He applies a mixed strategy where, for a message if there is a such that then (5) Let be the true density function of the type. The equilibrium consistency condition requires that, if is a message that is sent in equilibrium, i.e., for some 2 then (6) Under three conditions (4) (6), we have a BE: (c) The mixed-strategy BE is the same as that in Matthews (1989). Hence, the solution can be found in Matthews (1989). 2 Following Matthews (1989), an alternative to this Page 6 of 6

64 Problem Set 9 Micro Analysis, S. Wang Question We have two agents with identical strictly convex preferences and equal endowments. Describe the core and illustrate it with an Edgeworth box. Question For a two-good two-agent economy, (a) Explain graphically that the core depends on the initial endowments. (b) Is it true that if the initial allocation is already in the core, then it is the only point in the core? Explain. (c) Try to suggest some mild conditions under which the statement in (b) is correct. Question In a two-agent two-good economy, suppose that the two agents are identical (with the same endowment and preferences) and they have strict monotonic and strict convex preferences. Show that the initial endowment point must be in the core. 1 1 Strict convexity of preferences means that: and for Page 1 of 3