Attempt the following please;

Univariate unconstrained maximization. (10 points) Consider the following maximization problem: max x f (x; x0) = exp((x x0)2) 1. Write down the first order conditions for this problem with respect to x (notice that x0 is a parameter, you should not maximize with respect to it). (1 point) 2. Solve explicitely for x that satisfies the first order conditions. (1 point) 3. Compute the second order conditions. Is the stationary point that you found in point 2 a maximum? Why (or why not)? (2 points) 4. As a comparative statics exercise, compute the change in x as x0 varies. In other words, compute dx/dx0. (2 points) 5. We are interested in how the value function f (x(x0); x0) varies as x0 varies. We do it two ways. First, plug in x(x0) from point 2 and then take the derivative with respect to x0. Second, use the envelope theorem. You should get the same result! (2 points) 6. Is the function f concave in x? (2 points) Problem 2. Multivariate unconstrained maximization. (13 points) Consider the following maximization problem: max x,y f (x, y; a, b) = ax2 x + by2 y 1. Write down the first order conditions for this problem with respect to x and y (notice that a and b are parameters, you do not need to maximize with respect to them). (1 point) 2. Solve explicitely for x and y that satisfy the first order conditions. (1 point) 3. Compute the second order conditions. Under what conditions for a and b is the stationary point that you found in point 2 a maximum? (2 points) 4. Asume that the conditions for a and b that you found in point 3 are met. As a comparative statics exercise, compute the change in y as a varies. In other words, compute dy/da. Compute it both directly using the solution that you obtained in point 2 and using the general method presented in class that makes use of the implicit function theorem. The two results should coincide! (3 points) 5. We are interested in how the value function f (x(a, b); y(a, b)) varies as a varies. We do it two ways. First, plug in x(a, b) and y(a, b) from point 2 into f and then take the derivative of f (x(a, b); y(a, b)) with respect to a. Second, use the envelope theorem. You should get the same result! Which method is faster? (3 points) 6. Under what conditions on a and b is the function f concave in x and y? When is it convex in x and y? (3 points) 1 Problem 3. Multivariate constrained maximization. (19 points) Consider the following maximization problem: max x,y u(x, y) = xy s.t. pxx + py y = M, with 0 < < 1, 0 < < 1. The problem above is a classical maximization of utility subject to a budget constraint. The utility function xy is also called a Cobb-Douglas utility function. You can interpret px as the price of good x and py as the price of good y. Finally, M is the total income. I provide these details to motivate this problem. In order to solve the problem, you only need to apply the theory of constrained maximization that we covered in class. But beware, you are going to see a lot more Cobb-Douglas functions in the next few months! 1. Write down the Lagrangean function (1 point) 2. Write down the first order conditions for this problem with respect to x, y, and . (1 point) 3. Solve explicitely for x and y as a function of px, py, M, , and . (3 points) 4. Notice that the utility function xy is defined only for x > 0, y > 0. Does your solution for x and y satisfies these constraints? What assumptions you need to make about px, py and M so that x > 0 and y > 0? (1 point) 5. Write down the bordered Hessian. Compute the determinant of this 3x3 matrix and check that it is positive (this is the condition that you need to check for a constrained maximum) (3 points) 6. As a comparative statics exercise, compute the change in x as px varies. In order to do so, use directly the expressions that you obtained in point 3, and differentiate x with respect to px. Does your result make sense? That is, what happens to the quantity of good x consumed as the price of good x increases? (2 points) 7. Similarly, compute the change in x as py varies. Does this result make sense? What happens to the quantity of good x consumed as the price of good y increases? (2 points) 8. Finally, compute the change in x as M varies. Does this result make sense? What happens to the quantity of good x consumed as the total income M increases? (2 points) 9. We have so far looked at the effect of changes in px, py , and M on the quantities of goods consumed. We now want to look at the effects on the utility of the consumer at the optimum. Use the envelope theorem to calculate du(x (px, py, M ) , y (px, py, M ))/dpx. What happens to utility at the optimum as the price of good x increases? Is this result surprising? (2 points) 10. Use the envelope theorem to calculate u(x (px, py, M ) , y (px, py, M ))/M. What happens to utility at the optimum as total income M increases? Is this result surprising? (2 points) Problem 4. Rationality of preferences (5 points) Prove the following statements: if is rational, then is transitive, that is, x y and y z implies x z (3 points) if is rational, then has the reflexive property, that is, x x for all x. (2 points) 2

1. Is it correct to state that 'if a firm has constant returns to scale, then the marginal products of labour and capital depend only on the ratio of labour and capital'? Explain. Can you state the same for an homothetic production function? Explain 2. Consider a Cobb-Douglas Production function: f(x) = x1 x 2 ; > 0; > 0: (i) Sketch the production possibility set Z in each of the three cases; 1. + < 1; 2. + > 1; 3. + = 1. For each case, sketch three typical 'slices' through Z perpendicular to each of the axes in turn. Also draw the marginal product, average product, and marginal rate of technical substitution, as functions of x1. (ii) Is the production function homothetic and/or homogeneous? If homogeneous, homogeneous in what, and to what degree? (iii) What returns to scale are there? 3. Consider the following four production plans obtained from the same technology (production possibility set). Each plan yields one unit of output 1 using two inputs. The plans are described by the following vectors: Plan 1: (1;1;6) Plan 2: (1;2;5) Plan 3: (1;3;3) Plan 4: (1;5;2) Is any of these production plans inefficient, in the sense that there exists no input price ratio at which a cost minimizing producer will choose the production plan in question? 4. Is it correct to state that for an homogeneous of degree one production function y = f(x1; x2), an increasing average product for one input, say x1, implies that the marginal product of the other input x2 is negative? 5. Consider a technology that produces a unique output and satisfied the following restrictive form of additivity: f(x + x0) = f(x) + f(x0): Further assume that the inputs are infinitely divisible, meaning that if x 2 V (y) then x 2 V (0y), for every 0 and 2 (0; 1). Show that this technology must be convex. 6. Does the input requirement set V (y) = f(x1; x2; x3) j x1 + minfx2; x3g 3y; xi 08i = 1; 2; 3g corresponds to a regular (closed and non-empty) input requirement set? Does the technology satisfies free disposal? Is the technology convex? 2

1. Consider two players that interact for two periods. In the first period the two players simultaneously and independently play the following version of the prisoners' dilemma game, labelled game A: C D C 4; 4 1; 5 D 5;1 0; 0 In the second and last period the two players simultaneously and independently play the following modification of the battle of sexes game, labelled game B: S B S 4; 4 0; 0 B 0; 0 4; 4 Assume that both players discount the future with a common discount factor such that 0 < < 1: (i) Start by identifying the mixed strategy Nash equilibria of the static normal form game A if this game is played once. (ii) Identify the mixed strategy Nash equilibria of the static normal form game B if this game is played once. 1 (iii) Identify the Subgame-Perfect-equilibrium strategies and the corresponding values of the discount factor of the dynamic game (in which game A is played in the first period and game B is played in the second period) such that both players get a payoff of 4 in both periods. 2. Consider two players playing the following two-period repeated game. In the first period the two players simultaneously and independently play the following game A: L R U 2; 4 7 3 ; 7 3 D 7 3 ; 7 3 4; 2 In the second and last period the two players simultaneously and independently play the following game B: S B S 2; 4 0; 0 B 0; 0 4; 2 Assume that both players discount the future with a common discount factor such that 0 < < 1: (i) Identify the mixed strategy Nash equilibria of both static normal form games A and B if each of these games is played once. (ii) Identify the Subgame-Perfect-equilibrium strategies and the corresponding values of the discount factor of the dynamic game (in which game A is played in the first period and game B is played in the second period) such that the two players' repeated game average discounted payoffs are (2; 4). 2 (iii) Identify the Subgame-Perfect-equilibrium strategies and the corresponding values of the discount factor of the dynamic game (in which game A is played in the first period and game B is played in the second period) such that the two players' repeated game average discounted payoffs are (4; 2). (iv) What is the lowest discount factor that supports the type of subgame perfect equilibrium strategies you identified in (b) and (c) above? Explain your answer. 3. Consider two players that interact for two periods. In each period, the two players play the following stage game, denoted game 1, (a version of the prisoners' dilemma game): C D C 6; 6 1; 7 D 7; 1 3; 3 Assume that both players discount the future with a common discount factor such that 2 3 < < 1: (i) What are the Nash equilibria of the stage game 1 if this game is played once? Explain your answer. (ii) What are the minmax payoffs of each player in the stage game 1? Explain your answer. (iii) What are the Subgame-Perfect-equilibrium strategies of the finite horizon dynamic game (in which game 1 is played in both periods)? Explain your answer. 3 4. Assume now that the game in Question 2 above is modified in the following way. Assume that there exists one column player, denoted C, and two identical row players, denotes R1 and R2. Assume further that these three players interact for two periods playing the following stage game, denoted 2. The stage game 2 is divided in two separate phases. In the initial phase of the stage game player C chooses one of the row players R1 or R2. The row player that is not selected receives a payoff equal to 0 while the column player and the selected row player move to the following phase. In the second phase of the stage game 2 the column player and the selected row player play the following prisoners' dilemma game: C D C 6; 6 1; 7 D 7; 1 3; 3 (i) What are the Subgame-perfect equilibria of the (dynamic) stage game 2 if this game is played once? Explain your answer. (ii) What are the minmax payoffs of each one of the two row players R1 and R2 in the two-phase stage game 2? Explain your answer. (iii) Can you construct a pair of Subgame-Perfect-equilibrium strategies of the finite horizon dynamic game (in which the stage game 2 is played in both periods) that achieve a payoff of 1 for the selected row player and a payoff of 7 for the column player in the first period? Explain your answer. 5. Compute the set of feasible payoffs in the infinitely repeated version of the following 'battle of the sexes' stage game B S S 0; 0 2; 1 B 1; 2 0; 0 4 What is the highest feasible symmetric payoff? Let = 0:9 and find a deterministic strategy profile for the repeated game G() with payoffs ( 3 2 ; 3 2 ). 6. Compute the set of feasible and individually rational payoffs of the infinitely repeated version of the following game: L C R T 1; 1 5; 0 0; 0 M 0; 5 4; 4 0; 1 B 0; 0 1; 0 1;1 5

2) Consider the following pricing game between Dell and Gateway. There are two types of demanders in the market, High and Low. High demanders value a computer at $4000. There are 100 of these people in the market. Low demanders value a computer at $1000. There are 200 of these people in the market. If Dell and Gateway set the same price, they split the market. If they set different prices, the lower price takes the entire market. Assume that the marginal cost of a computer is $500. a) Write the strategic form of the game (i.e. possible actions and payoffs). Assume that there are only two options for price $4,000 or $1000. b) What is each player's strategy? Explain the Nash equilibrium of the game. c) Suppose that Dell has the following beliefs about Gateway: Pr(P = $4,000) = 2/3 Pr(P = $1,000) = 1/3 Can this strategy be consistent with a nash equilibrium? d) Calculate the mixed strategy equilibrium for this game. e) What percentage of the time will computer prices be low? 3) Suppose that the elasticity of demand for tennis shoes is 4 (and is constant). Calculate the markup that would be charged is a monopoly controlled the market. How would your answer change if the market was oligopolistic with an HHI index of 5,000? 4) Suppose that the (inverse) demand curve for bananas is given by P = 400 5Q Where Q is total industry output. The market is occupied by two firms, each with constant marginal costs equal to $5. a) Calculate the equilibrium price and quantity assuming the two firms compete in quantities. Calculate the elasticity of demand facing each firm. How does this differ from industry elasticity? b) Repeat parts (a) assuming the competition is in prices rather than quantities. c) Suppose that each firm was capacity constrained. That is, each firm can only produce 100 units. How does this change your answers to (b)? 5) Explain the similarities/difference between Cournot competition and Bertrand competition. What are the key assumptions/results of each? 6) Explain the following statement: "If firms are competing in quantities, then it pays to be the first to the market. However, if firms arte competing in price, its worthwhile to wait for your opponent to make his move" 7) What is the chain store paradox? What is the major lesson we get from this game? 8) Suppose that the probability of getting in an accident is 3%. The average cost of an accident is $100,000. Suppose that the average car driver has preferences given by U(I ) = I a) Assuming that this individual earns $100,000 per year in income, calculate his expected utility if he buys no insurance. b) Calculate the cost of this policy for the insurance company. c) Suppose that half the population was made up of unsafe drivers (i.e. with a higher accident rate). How high would the unsafe driver's accident rate have to be for this market to break down? d) Explain how moral hazard and adverse selection are dealt with in the insurance industry. 9) Suppose that the market demand is described by P = 120 (Q + q) Where Q is the output of the incumbent firm, q is the output of the potential entrant and P is the market price. The incumbent's cost function is given by TC(Q) = 60Q While the cost function of the entrant is given by TC(Q) = 60q + 80 (80 is a sunk cost paid upon entering the market) a) If the entrant observes the incumbent producing Q units of output and expects this level to be maintained, what is the equation for the entrant's residual demand curve? b) If the entrant maximizes profits using the residual demand in (a), what output will the entrant produce? c) How much would the incumbent have to produce to keep the entrant out of the market? At what price will the incumbent sell this output

EconS 526 1. A monopolist has a linear cost function, 10y. It consulted with an economist to estimate the demand for its good. The economist derived the following demand equation, =100. a. How much should the monopolist produce and what price should it set to maximize profit? Show the monopolist's problem and solution. max(100)10 FOC: 100210=0. Therefore y=45 and p=100-45=55. b. If the government forced the monopolist to act like a competitive market, what price level would be set and how much would be produced? Derive the deadweight loss in the market given the monopolist operating in the market. In this case, price should equal marginal cost. 100=10. Therefore y=90 and p=10. The deadweight loss is a triangle bounded by the difference in output, difference in price and the demand curve. In this case it is (90-45)x(55-10)/2=1012.5 c. Another economist found that the correct demand equation should be =100/ if 20 and =0 if >20. Given this new information, what is the monopolist's profit maximizing price and quantity? A price higher than 20 leads to negative profit so price is less than or equal to 20. Note that revenue is fixed at 100. Therefore, the monopolist only needs to choose the lowest quantity to maximize profit. In this case, it is y=5 when p=20. d. If the government forced the monopolist to act like a competitive market, what price level would be set and how much would be produced? The government would equate price to marginal cost. So p=10. The resulting y=10. 2. A monopolist has a convex cost function () and is faced with an inverse demand (,) where y is output level and I is income. The good produced by the monopolist is a normal good. a. Set up and solve the monopolist's problem. Given your solution, derive an expression showing the effect of income on optimal output of the monopolist. What assumption(s) do you need to make to sign the comparative static? max(,)() FOC: (,)+(,)()=0. Thus, y*(I). SOC: +2<0 Get

The denominator is negative because that is the second order condition. The numerator is ambiguous. It will be positive if >0. Note >0 because the good is normal. So if that assumption holds, >0. b. How does an increase in income affect the price set by the monopolist? Show an expression that proves this relationship. =+ This is ambiguous. The direct effect is positive but the indirect effect is negative when >0 because <0. 3. There are two consumers with the following utility functions: 1=11 and 2=22 where 1>2. A monopolist supplies the good. It can produce the good at zero marginal cost but it can only produce 10 units of the good at most. The monopolist offers two price-quantity packages: (1,1) and (2,2) where is the cost of purchasing units of the ith good. a. Write the monopolist's profit maximization problem. Hint: It should have four constraints along with a capacity constraint. Identify which constraints are binding. max1,21+2 .. 11 1,222, 222211,111122,1+210 The binding constraints are 22=2 and 111=122 and 1+2=10. b. Substitute these constraints into the monopolist's profit maximization problem and derive the values of (1,1) and (2,2). So now, max1,21112+222 .. 1+2=10 or max 2110+2(21)2 So 1=10 and 2=0. This means that 2=0 and 1=101. 4. A monopolist sells the same product in two markets