ECN611 Final Exam S10

ECN611 Final Exam S10 - E co no m ics 6 11 G am e T h eo re...

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Unformatted text preview: E co no m ics 6 11 G am e T h eo re tic M icro ec on om ics S pring 2 0 10 Final E x am A ll S y ra c u s e U n iv e rs ity p o lic ie s a n d p ro c ed u re s c o n c e rn in g a c a d e m ic h o n e s ty apply to this cou rse: "S y racuse U n iv ersity s tu d en ts sh all e x h ib it h o n esty in a ll a cad em ic en d eav o rs. C he a ting in a ny form is n ot tole ra te d, n or is a s s is ting a nothe r p e rs on to c he a t. T he s ubm is s ion of any w ork by a s tude nt is take n a s a g ua ra nte e tha t the thoughts a nd ex p ressio n s in it a re th e stu d en t's o w n e x cep t w h en p rop erly c redited to a n o th er. V io latio n s o f th is p rin cip le in clu d e: g iv in g o r receiv in g a id in a n e x am o r w h ere othe rw is e p rohibite d, fra ud, p la gia ris m , the fa ls ific a tion or forge ry of a ny re c ord, o r a n y o th e r d e c e p tiv e a c t in c o n n e c tio n w ith a c a d e m ic w o rk . P lag iaris m is the r e pres e nt ation of a not her's w ords , ideas, p rogram s , for m ulae , o pinions, o r o th er p ro d u cts of w o rk a s o n e's o w n e ith er o v ertly o r b y f ailin g t o a ttrib u te th em to their true sou rce." (S ectio n 1 .0 , U n iv ersity R u les an d R eg u latio n s) W A R N IN G !!! W h ile h o m e w o rk p ro b le m s m a y h a v e b e e n d o n e c o o p e ra tiv ely , e x a m s a re in d ivid u a l w o rk . D o not c om m unic a te a bout this e xa m w ith a ny one e xc e pt the in s tr u c to r [x 3 -2 3 4 5 o r e -m a il to js k e lly @ m a x w e ll.s y r.e d u ]. V io la tio n o f t h is r u le w ill r e s u lt in a g r a d e o f 0 fo r th e e x a m. A ny n o tices w ill b e sent to y ou b y e m a il; c h e c k o c c a s io n a lly . E X P L A IN y o u r a n sw e r s c a r e fu lly . * K e e p a X e r o x c o p y o f y o u r a n s w e r s to th e ta k e -h o me p o r tio n o f y o u r e x a m* T a k e -h o m e p o r tio n D U E : N o o n , F r id a y , A p r il 7 . Ec on omic s 6 11 G a m e T he or et ic M ic r oe c on omic s S pr in g 2 0 1 0 S econd Exam E X P L A IN y o u r a n s w e r s c a r e fu lly . 1. (Public goods mechanism design) [20 points] Individuals have quasilinear preferences given by ui(x, èi) = kèi + (mi + ti). Where è = (è1, è2, ... , èI), suppose a social choice function f with that satisfies f(è) = (k(è), t1(è), t2(è), ... , tI(è)) (A) (B) Ó ti(è) = – ck(è). Revelation mechanism with income tax: Choose k(èÞ) = 1 if Ó èÞi ￿ c; 0 otherwise. ti(èÞ) = – (c mi / Ómi )k(èÞ) [You pay in proportion to your income. The mi values are common knowledge.] Assume Èi = {èi} for i not equal to 1; È1 = ￿+ A. Show, by an example, that for some values of m1, ..., mI, c, è1, ..., èI, individual 1 has an incentive to understate his preference for the public good, by choosing èÞ1 < è1. Extra points for an example where this causes Ó èiÞ < c even though Ó èi ￿ c. B. Show, by a different example, that for some values of m1, ..., mI, c, è1, ..., èI, individual 1 has an incentive to overstate his preference for the public good, by choosing èÞ1 > è1. Extra points for an example where this causes Ó èiÞ ￿ c even though Ó èi < c. 1 2. (Public goods mechanism design) [20 points] With the same utility functions as in question 1, now consider the revelation mechanism Choose k(èÞ) = 1 if Ó èÞi ￿ c; 0 otherwise. ti(èÞ) = ￿)i( èij – c. Þ Assume Èi = ￿+ for all i. In class I said that for this rule, no one individual has an incentive to choose a èiÞ different from èi. Show by an example that when everyone else is telling the truth, two individuals both choosing a èiÞ different from their èi can improve their outcome over what they get if they tell the truth. Give specific numerical values for è1, è2, ... , èI and c. Extra points for an example where this causes Ó èÞi < c even though Ó èi ￿ c, or where this causes Ó èÞi ￿ c even though Ó èi < c. 3. (Gibbard-Satterthwaite) [20 points] There are, say, 3 alternatives and 6 individuals, each with strong orderings over the alternatives. For each of the following two social choice rules present a preference profile at least someone has an incentive to misrepresent their preference ordering. Rule I. If one alternative is at the top of the preference ordering for more individuals than is true for any other alternatives, that alternative is chosen. If there is a tie between two or more alternatives for the largest number of top positions, the alphabetically earlier of those is selected by Rule I. Rule II. Look at the top two alternatives for individual #1. Have a majority vote by the remaining 5 individuals, 2, 3, ..., 6 and the rule selects the majority winner. For each rule, present a preference profile at least someone has an incentive to misrepresent their preference ordering. 2 T a k e -h o m e p o rtio n D U E : N o o n , T h u r s d a y , A p r il 6 , in c la s s . 4. (Weierstrass) [20 points] A fishery earns a profit of ð(x) in a year from catching and selling x fish in that year. The firm owns a pool which currently has y1 fish in it. (Note: y1 is a fixed parameter.) If x ￿ [0, y1] fish are caught this period, the remaining y1 – x fish will grow to f(y1 – x) by the beginning of the next period, where f: ￿+ ￿ ￿+ is the growth function for the fish population. The fishery wishes to set the volume of its catch in the next three period to maximize the sum of its (undiscounted) profits over this time. That is, it wishes to solve the problem of maximizing with respect to x1, x2, and x3, the sum ð(x1) + ð(x2) + ð(x3) subject to x1, x2, x3 ￿ 0, x1 ￿ y1; x2 ￿ y2 = f(y1 – x1); x3 ￿ y3 = f(y2 – x2). The terms y2 and y3 are just for exposition; they are not additional variables or parameters; we could have written the constraints as: x1 ￿ y1; x2 ￿ f(y1 – x1); x3 ￿ f(f(y1 – x1) – x2). Assume that the functions ð and f are continuous on ￿+ and show that there does exist a global maximum for the firm’s problem. Do NOT assume that ð and f are increasing functions of their arguments. 3 5. (Moral hazard with endogenous probability of detection) [20 points] Consider an extension of our basic principal-agent model of employment. This time, the firm first selects a high or low level of surveillance. The low level of surveillance, which is costless, results in a probability of workers getting caught shirking of ð(E) = 1 – E. The high level of surveillance, which entails fixed costs C, results in a probability of workers getting caught shirking of ð(E) = 1 – E2. Then the firm announces a (w, L) contract. The worker then chooses whether or not to accept the contract and, if it accepts, chooses a level of effort, E. The firm has revenue function: V(LE) = V(L ￿ E) = (L ￿ E)½ and labor costs of W ￿ L. Employees have utility function U = W ￿ (1 – E) and exogenous reservation utility: W. Assuming that employees can observe the surveillance level before they select their effort level, what are the subgame perfect equilibria? E X P L A IN y o u r a n s w e r s c a r e fu lly . 4 ...
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