QUESTION 5 The US customs know that a drug shipment is being smuggled into the country in the baggage of an unknown passenger on one of two flights, one from Bogota and one from Cartagena. Both flights arrive at the same time. There are 15 customs officers available, and each officer can inspect 10 passengers. The flight from Bogota is carrying 200 passengers and the flight from Cartagena 100 passengers. Nothing is known about the characteristics of the suspected smuggler, but it is known that, if the smuggler is coming from Bogota, he\\she is carrying a larger shipment than if he she is coming from Cartagena. All of this is common knowledge between the US customs and the drug kingpins. This situation can be seen as a simultaneous game, where the US customs must decide how many of the 15 officers should be assigned to the Cartagena flight (possibly none, possibly all; the remaining officers will be assigned to the Bogota flight), the drug kingpins must decide which flight to use to send the shipment and there are four possible outcomes: a large shipment is intercepted by the US customs (/ ), a small shipment is intercepted by the US customs ( ). a large shipment passes undetected through US customs (P,) and a small shipment passes undetected through US customs (P ). Both the US customs and the drug kingpins have von Neumann-Morgenstern preferences. For the US customs the utility of I, is twice the utility of 1 while the utility of P, is equal to the utility of P. For the drug kingpins the utility of P, is twice the utility of Po, while the utility of I, is equal to the utility of Is. (a) Write the normalized von Neumann-Morgenstern utility functions of the two players. (b) If the US customs office assigns 8 officers to the Bogota flight and 7 to the Cartagena flight, what is the probability that the drug shipment will be intercepted if it is on the Bogota flight? What is the probability that the drug shipment will be intercepted if it is on the Cartagena flight? (c) Does the US customs office have any strategies that are dominated? If your answer is No then prove it and if your answer is Yes then list the dominated strategies and state whether they are weakly or strictly dominated. (d) Prove that there are no pure strategy Nash equilibria. (e) Prove that there is no mixed-strategy Nash equilibrium where the US customs uses a pure strategy and the drug kingpins use a completely mixed strategy. f) Using the insight you got from the calculations for part (e), find a Nash equilibrium of the game and prove that it is a Nash equilibrium. [Note: remember that necessary conditions are not always sufficient.] (@) Now let us change the game as follows. It is a game of incomplete information where the size of the drug shipment was chosen earlier on and cannot be changed; the drug kingpins know the size of the shipment while the US customs do not know the shipment size and believe that it is large with probability p and small with probability 1- p ; these beliefs are common knowledge between the drug kingpins and the US customs. The drug kingpins choose on which flight to send the shipment and the US customs choose the number of officers to assign to the Cartagena flight (and the remaining are assigned to the Bogota flight); these choices are made "simultaneously" (that is, in ignorance of the other player's choice). The payoffs are the same as above. (g.1) Draw an extensive game that represents this situation and indicate what the payoffs are (you can do so as a function of n, where n is the choice of the US customs). (g-2) Suppose that P= = and the US customs use the mixed strategy found in part (f). Is there a pure strategy of the drug kingpins that is a best reply to that? (g-3) Does the drug kingpins' pure strategy of part (g.2) together with the US customs' mixed strategy of part (f) constitute a Bayesian Nash equilibrium?Question 1. Throughout this question we assume that all the optimization problems considered have unique solutions which are interior and characterized by the first-order equalities. We also assume that all decision makers take all prices as given, and that the Lagrange multipliers are nonzero. 1.1. This part covers standard price-taking production theory, and we minimally adapt the notation used in class and in Chapter 5 of Mas-Colell et al. We consider a price-taking firm that produces one output by using N inputs, according to a direct production function S: 14 - R : ( 3, .....2) =>/ ( =,.....> >) , where (z,,...,z,) = z denotes the input vector. We denote by q the amount of output, and in order to avoid notational confusion with Parts 1.2 below we use capital letters for prices, i. e., PER. denotes the price of the output, and W = (W,....W.)e R#, the vector of input prices. We consider the following optimization problems. Problem INPUTPROFITMAX[W, P]. Given (W, P), choose (z,,...,2,) in order to maximize Pf(2) - W . z. Denote by E(W, P) and by II(W , P) the solution and the value, respectively, of this problem. Problem COSTMIN[ W, q]. Given (W, q), choose (z,,...,z, ) in order to minimize W . z subject to f(2) 2q. Denote by 2(W.q) and by c(W ,q) the solution and the value, respectively, of this problem, and by B(W ,q) its Lagrange multiplier. Problem OUTPUTPROFITMAX[W, P]. Given (W, P), choose q order to maximize Pq-c(W ,q). Page 1 of 7 1.1(a). Prove that 2* solves INPUTPROFITMAX[W, P] if and only if, defining q* = /(z*): (i) z* solves COSTMIN[W, q*], and (ii) q* solves OUTPUTPROFITMAX[W, P]. Remark. It is here preferable that you write direct proofs by contradiction rather than manipulating first order conditions 1.1(b). Interpret B(W .q). 1.1(c). Argue that, if z* solves INPUTPROFITMAX[W, P], and q* =/(2*), then P=B(W ,q*). 1.2. We now consider a consumer with utility function u : R4 - R : (x,....x,) (x,....xx), which is in particular assumed to be locally nonsatiated, where (x,,..,x, )= x denotes her consumption vector. We denote by p= (p, .....P, )ER, the price vector for the consumption goods, and by w > 0 the wealth of the consumer. We consider the following optimization problems. Problem UMAX[p, w]. Given (p, w), choose x = (x,,..,x,) in order to maximize fi(x) subject to p. xS w.QUESTION 4 Consider an individual whose von Neumann-Morgenstern utility-of-wealth function is U(m) = IVm if she exerts no effort with c >0. Vm -c if she exerts effort The individual has an initial wealth of Wand faces a potential loss of f . The probability of her incurring a loss is p. if she exerts effort and p. if she chooses no effort, with 0 0 for k E {1. ..., L}. As usual we denote r' = (rj,...,;). Consider price vectors (p1, ..., pz) and wealth levels (w), ..., w) for which the solution to the utility maximization problem is interior for every consumer i c {1, ..., /}. a. ) Derive the Walrasian demand function for good j by consumer i. (Be careful with your calculations. Double-check! It is easy to make mistakes.) b.) What is the slope of consumer i's Engel curve for good j at (p, w')? c.) Find a condition as general as possible on parameters ox, B, i e {1, ...,/}, k E {1,..., L} guaranteeing the existence of a positive representative consumer. Do we need restrictions on parameters oz, ke {1,.... [}? d.) Consider now the special case with just a single consumer and two goods. The consumer's utility function is given by Derive the wealth-expansion path for a given price vector (p1, pz)- e.) In problem d.), when does the wealth-expansion path intersect the mpaxis and when does it intersect the ryaxis?Question 1 Consider an economy with / consumers and L goods. For each consumer i e {1, ..., /}, the consumption set is R4. Her utility function is given by K=1 where or, # > 0 for k E {1. ..., L}. As usual we denote r' = (rj,...,;). Consider price vectors (p1, ..., pz) and wealth levels (w), ..., w) for which the solution to the utility maximization problem is interior for every consumer i c {1, ..., /}. a. ) Derive the Walrasian demand function for good j by consumer i. (Be careful with your calculations. Double-check! It is easy to make mistakes.) b.) What is the slope of consumer i's Engel curve for good j at (p, w')? c.) Find a condition as general as possible on parameters ox, B, i e {1, ...,/}, k E {1,..., L} guaranteeing the existence of a positive representative consumer. Do we need restrictions on parameters oz, ke {1,.... [}? d.) Consider now the special case with just a single consumer and two goods. The consumer's utility function is given by Derive the wealth-expansion path for a given price vector (p1, pz)- e.) In problem d.), when does the wealth-expansion path intersect the mpaxis and when does it intersect the ryaxis?Question 2 It is intuitive to think that the presence of more agents in the economy "shrinks" its core, since there are more coalitions that can object a given allocation. You will understand in this question why this is indeed the case.' Fix a standard, two-person exchange economy & = ((u', w'), (u', w?)}. Define its replica as the four-person exchange economy & = (lu', w') (u, wa ). (u',w). (u',w')). where (u', w) = (u', w' ) and (u*, wi) = (u?, w?). 1. Argue that if (p, x', x') is a competitive equilibrium for &, then (p, x', x', x', x' ) with x3 = x' and x* = x', is an equilibrium for &?. 2. Argue that if both utility functions are strictly quasi-concave, and (p, x', x', x', x' ) is a competitive equilibrium for &', then, x' = x3 and x? = x*. 3. Argue that if both utility functions are strictly quasi-concave, and (x', x2, x3, x* ) is in the core of &', then, x' = x' and x' = x*. 4. Argue that if both utility functions are monotone and strictly quasi-concave, and (p, x', x?) is a competitive equilibrium for &, then (x], x2, x', x?) is in the core of 82. 5. Suppose that u' ( x ) = u'(x) = x'x'. w' = (1,0) and w = (0, 1). Argue that allocation ( (0,0), (1, 1)) is in the core of &, yet allocation ((0, 0), (1, 1), (0. 0), (1, 1) ) is not in the core of &2. 6. IIse these results to argue, informally, that the replication of agents does not affect the set of equilibrium allocations of the economy but shrinks its core.? 'We are using the term "shrink" loosely, since the presence of more agents changes the dimension of the allocation space, su courting the sizes of the cures will require some refinement of the arguincut. In the limit, one can show that replication ad infinitum reduces the core to just the set of equilibrium allocations