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Unformatted text preview: Economics 611 Game Theoretic Microeconomics Spring 2011
Final Exam All Syracuse University policies and procedures concerning academic honesty
apply to this course:
"Syracuse University students shall exhibit honesty in all academic endeavors.
Cheating in any form is not tolerated, nor is assisting another person to cheat. The
submission of any work by a student is taken as a guarantee that the thoughts and
expressions in it are the student's own except when properly credited to another.
Violations of this principle include: giving or receiving aid in an exam or where
otherwise prohibited, fraud, plagiarism, the falsification or forgery of any record,
or any other deceptive act in connection with academic work. Plagiarism is the
representation of another's words, ideas, programs, formulae, opinions, or
other products of work as one's own either overtly or by failing to attribute
them to their true source." (Section 1.0, University Rules and Regulations)
WARNING!!!
While homework problems may have been done cooperatively, exam s are
individual work. Do not communicate about this exam with anyone except the
instructor [x32345 or email to jskelly@maxwell.syr.edu]. Violation of this rule
will result in a grade of 0 for the exam. Any notices will be sent to you by email; check occasionally.
EXPLAIN your answers carefully.
*Keep a Xerox copy of your answers to the takehome portion of your exam*
Takehome portion DUE: Noon, Friday, April 7. Economics 611 Game Theoretic Microeconomics
Final Exam Spring 2011 EXPLAIN your answers carefully.
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 èÞi < 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 èÞi 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( èÞij – c.
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 èÞi 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. (GibbardSatterthwaite) [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 Takehome portion DUE: Noon, Thursday, April 6, in class.
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; x 2 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 principalagent 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? EXPLAIN your answers carefully. 4 ...
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 Fall '08
 Kelly,J
 Microeconomics

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