Practice Questions II
Economics 142
December 5, 2024
1. Choose the correct letter for each question.
(a) When one good is sold to one of the N buyers with value v1 > v2 > ... > vN > 0...
A. The buyer-optimal market clearing price is vN .
B. The buyer-optimal market-clearing price is v2.
C. The buyer-optimal market-clearing price is v1.
D. The buyer-optimal market-clearing price and the seller-optimal market-clearing price is the same.
(b) Consider an arbitrary bipartite network with N unit-sellers and N unit-buyers with heterogeneous values. They can trade only when they are connected. If we add one more buyer who is connected to every seller with value viN+1 > 0 for i = 1, ..., N, then....
A. All original N buyers must be weakly better off given the new buyer-optimal market clearing prices.
B. Some buyer must be strictly worse off given the new buyer-optimal market clearing prices.
C. The total values at the efficient assignment must increase weakly.
D. The total values at the efficient assignment must decrease weakly.
(c) Suppose that θ is either 1 and −1 with equal probability and let s ∈ {s1, ..., sm} be a signal about θ, which follows some conditional distribution Pr[s = si
|θ = z], z = 1, −1.
A. It is possible that the posterior belief Pr[θ = 1|s = si
] for θ = 1 is strictly larger than 0.5 for every i = 1, ..., m.
B. It is possible that the posterior belief Pr[θ = 1|s = si
] is strictly strictly smaller than 0.5 for every i = 1, ..., m.
C. If the posterior belief Pr[θ = 1|s = si
] is strictly larger than 0.5 for some si
, then it must be strictly smaller than 0.5 given some other sj ≠ si
.
D. The expected posterior belief E[Pr[θ = 1|s = si
]] = P m
i=1 Pr[θ = 1|s = si
] Pr[s = si
] can be be different from 0.5.
(d) Consider a star network with n + 1 nodes. Assume that each node i observes a noisy signal xi(0) = θ + ϵi of some hidden state θ ∈ R, where ϵi
is i.i.d. noise with mean 0. Let 0 be the center node. Assume that every node updates its x by taking the averages of x of its own and its neightbors (x0(t + 1) = n+1/Pni=0xi(t) for the center node and xi(t + 1) = 2/x0(t)+xi(t)
for other i = 0). Let xn(∞) be the consensus that is reached as t → ∞.
A. Given each θ, xn(∞) converges to some number as n → ∞ and may be biased (i.e. limn→∞ xn(∞) ≠ θ).
B. Given each θ, xn(∞) converges to θ.
C. Given each θ, xn(∞) is random even as n → ∞ and may be biased (i.e. E[limn→∞ xn(∞)] ≠ θ).
D. Given each θ, xn(∞) is random even as n → ∞ and is unbiased (i.e. E[limn→∞ xn(∞)] = θ).
2. Various types of memberships for some subscription service are on sale. Suppose that currently there is only one available slot of a special mem-bership with service level xs, then there are two slots for regular member-ships of service level xr ∈ (0, xs). Suppose that three people j = 1, 2, 3 are interested in these three memberships. Assume that j values service x by θjx, where θ1 > θ2 > θ3 > 0.
Answer the following questions.
(a) What is the most efficient assignment/allocation of the three mem-berships to these three people?
(b) Find the smallest (i.e. buyer-optimal) market-clearing prices.
(c) Find the largest (i.e. seller-optimal) market-clearing prices.
(d) Suppose that one regular membership is replaced with a upgraded membership“regular+”. The service xr+ for a regular+ member-ship is between xr and xs. Find the buyer-optimal market-clearing prices and the seller-optimal market clearing prices and compare them to your answer for (c). (We assume that one regular mem-bership is withdrawn so that there are three slots of memberships. As an exercise, do the same question assuming that two regular memberships are still available (so there are four membership slots available)).
3. Consider the model of observational learning where people decide whether to adopt a new technology or not sequentially. Each person decides to adopt a new strategy or stick to the old technology one by one after observing some information about the new technology and ob-serving the actions of the preceding people. There are two possible states: the new technology is pathbreaking (θ = P) with probability p or a failure (θ = F) with probability 1 − p. Each person’s pay-off from adopting a new technology is v + 80 if the state is P and v − 120 if the state is F, where v is the payoff from sticking to the old technology. Each person i observes a (independent and identically distributed) signal si ∈ {pb, f}, which is correct with probability 2/3 (i.e. Pr[si = p|θ = P] = Pr[si = f|θ = F] = 2/3).
(a) If the first person observes s1 = f, what is her posterior belief for θ = P and θ = F?
(b) For some range of initial belief p, everyone ignores her own signal and chooses the same action. Characterize such range of p.
(c) Suppose that p = 1/2. What is the probability that the tth person for every t ≥ 3 choose the same action independent of their own information?
(d) When p = 1/2, what is the probability that a sequence of people converge to choose the same bu a wrong choice?