Advanced Theory Field Exam–August 2008
Answer two of the following five parts; each part is worth 50% of the total grade. Use
separate blue books for the two parts, and clearly label which part of the exam you are
answering in each book.
1
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community
Ask the community for help and clear up your study doubts
University Rankings
Discover the best universities in your country according to Docsity users
Free resources
Our save-the-student-ebooks!
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Economic Theory Field Exam and Questions on Game Theory and Asset Pricing, Exams of Economics
A field exam for econ 206 with questions on principal-agent games, concave capacities, and axioms in decision theory. It also includes additional questions on game theory and asset pricing. The exam and questions cover topics such as mixed strategies, subgame-perfect equilibrium, present-biased preferences, and the stochastic discount factor.
Typology: Exams
2011/2012
1 / 8
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Economic Theory Field Exam and Questions on Game Theory and Asset Pricing and more Exams Economics in PDF only on Docsity!
Advanced Theory Field Exam–August 2008
Answer two of the following five parts; each part is worth 50% of the total grade. Use separate blue books for the two parts, and clearly label which part of the exam you are answering in each book.
Theory Field Exam - Econ 206 August 2008
Consider a standard hidden-action principal-agent game with the following tim- ing and assumptions. A principal makes an agent a take-it-or-leave-it (tioli) offer. If the agent leaves it, he gets utility 0. If he accepts, he expends effort a ∈ A and receives a payment contingent on an outcome, x ∈ X. Assume the agent’s utility is u(y) − c(a), where y is a payment from principal to agent, u(·) is an increasing, concave, and twice differentiable function, and c : A → R+. (a) Suppose that X = {xf , xs}, A = { 1 , 2 }, Pr{x = xs|a} = 1 − 1 /a, and c(1) < c(2). What contract would the principal offer in equilibrium if she wishes to induce a = 2? (b) Maintain all assumptions. Suppose that c(1) > 0 and u(0) = 0. Suppose, now, that, should outcome xf be realized, the principal goes bankrupt and can, consequently, pay the agent nothing. What contract would the principal offer in equilibrium if she wishes to induce a = 2? Consider the following changes in assumptions. Suppose that Pr{x = xs|a} = β − 1 /(a + 1), where β ∈ { 1 / 2 , 1 }. The parameter β is unknown by anyone. It is, however, common knowledge that Pr{β = 1} = 1/2. Let the agent’s utility be βˆ + u(y) − c(a), where βˆ is the estimate of β based on the realized outcome and the contract in place. Assume the principal has unlimited liability ( cannot default on a promised payment to the agent). (c) What contract would the principal offer in equilibrium if she wishes to induce a = 2? (d) Suppose that the prior on β were Pr{β = 1} = φ. From the principal’s perspective, what is the optimal value of φ if she wishes to induce a = 2? Interpret briefly. Return to the basic assumptions. Specifically, assume the agent’s utility is u(y) − c(a), and Pr{x = xs|a} = 1 − 1 /a. The principal has unlimited liability. Assume, however, that between the time the agent chooses his hidden action and the time that x is realized, that there is a period in which principal and agent can renegotiate the existing contract. Assume X ⊂ R+, the principal’s utility is x − y, and that xs is much larger than xf. (e) What contract would the principal offer in equilibrium assuming she gets to make a tioli offer during renegotiation? (f) What contract would the principal offer in equilibrium assuming the agent makes a tioli offer during renegotiation?
Theory Field Examination Game Theory (209A) Aug. 2008
Both questions carry approximately the same weight. Good luck!!!
Question 1 (static games)
[i] Consider a two-player simultaneous-move all pay (each player must pay his own bid) auction. Bids must be in nonnegative integers (no decimals). The higher bidder wins a nonnegative integer prize x. If the bids are equal, neither player receives the prize. Players are risk neutral so each player utility is simply her net earnings. Show that there is a symmetric mixed-strategy Nash equilibrium in which every bid less than x has a positive probability. [ii] Consider the following Matching Pennies game in which player 1 has an outside option x ∈ (0, 1)
H T Out x, 0 x, 0 H 1 , − 1 − 1 , 1 T − 1 , 1 1 , − 1
Find the set of set of mixed strategies for player 1 that survive iterated elimination of strictly dominated actions. Find the set of rationalizable strategies for player 1 and discuss the relationship between rationalizability and iterated elimination of strictly dominated actions.
Question 2 (dynamic games)
[i] Admati and Perry (1991) studied an infinite-horizon game in which two players i = 1, 2 make alternate contributions xi(t) to a public project. The cost of the public project is K, and the project is completed at the first date T at which P i=1, 2
P^ T t=
xi(t) ≥ K.
Each player i assigns the value V to the project so his payoff is equal to δTi V −
PT t=
δtxi(t)
if the project is completed at date T and − P∞ t=0 δ txi(t) if the project is never completed. To avoid trivialities, assume that the aggregate endowment is greater than the cost of the public project so that completion is feasible, and assume that the aggregate value of the good is greater than the cost so that completion is always efficient. Prove that the game has a unique subgame-perfect equilibrium. [ii] Consider a twice-repeated version of the Battle of the Sexes game
B S B 2 , 1 0 , 0 S 0 , 0 1 , 2
Can the history where players choose (B, S) in both periods be sup- ported by subgame-perfect equilibrium? Does it survive iterated elimination of weakly dominated actions?
rational, but she must maximize her utility at each continuation point). For what values of δ and η does there exist a personal equilibrium in which Iliyana for sure never does the task? Succinctly make clear how you reached this conclusion. For what values of δ and η does there exist a personal equilibrium in which Iliyana for sure does the task immediately? Succinctly make clear how you reached your conclusions. Compare your answer to part (a) above. Note: there may be more than one reasonable interpretation/variant of “personal equilibrium”; do whatever seems natural to you and be clear how you reached your conclusions.
234A, Macroeconomic Önance
Consider an endowment economy where log consumption follows
ct+1 = ct + + "t+1 + vt+
where is a constant, "t+1 N (0; ^2 ), vt+1 equals zero with probability 1 p and log [1 d] with probability p, 1 > d 0 , and p 0 is a small number. Here t+1 can be interpreted as an unlikely ìdisasterî event where consumption drops by d percent. We assume that " and are independent and i.i.d over time, and that the representative consume maximizes utility over lifetime consumption:
max Et
X^1
j=
j^
C t^1 + j 1
(a) Compute m = log [EtCt+1=Ct] the log growth rate of consumption. To do this, recall (i) that the expectation of the product of independent random variables is the product of their expectations; and (ii) the formula for the expected value of a lognormal random variable. To compute logEt exp [t+1] = log [1 p + p (1 d)], use the fact that for small x, log (1 x) x. How does the probability of disasters p a§ect expected consumption growth?
(b) Write down the stochastic discount factor in period t. Use the SDF to express the log riskfree rate rf. Write rf as a function of exogenous quantities using the approximation for the log function the same way as in (a). Does the presence of disasters (p > 0 ) increase or reduce the riskfree rate of return? Explain.
(c) Consider the risky asset which is a claim to next periodís total endowment Ct+1. Express the price of this asset, Pt, with the key asset pricing equation, using the SDF from (b). Compute log [Pt=Ct].
(d) Show that logE[1 + Rt+1] = logE[Ct+1=Ct] log [Pt=Ct] where Rt+1 is the return of the risky asset introduced in (c). Using this equation and your results from (a) and (c), compute the expected excess return ERt+1 Rf logE[1 + Rt+1] rf. Compare your result to the expected excess return in the absence of disasters (p = 0). Is the equity premium higher now? What asset pricing puzzles can this model help explain?
(e) Suppose that the probability of disaster p is time-varying. Based on your results in (b) and (d), what type of variation would you expect to see in the equity premium and the riskfree rate? Are these predictions consistent with facts?
(f) Qualitatively, what would the presence of disasters imply for the price of out-of-the- money put options on the stock market? Recall, the payo§ of a put option with strike X and expiration date T is max [X PT ; 0], and the option is out of the money if X is smaller than the current stock price (i.e., the payo§ from exercising the option today would be zero).