ECON 3152/4453/8053 Homework 4

1. (Tacit collusion) There are two firms. In each period t, the following events happen:

(1) Both firms simultaneously choose their outputs q_it ≥ 0.

(2) With probability µ ∈ (0, 1), the market price is p_t = H − q_1t − q_2t; with probability 1 − µ, the market price is p_t = L − q_1t − q_2t. (The market price is independent of events in earlier periods.)

(3) Firm i’s stage payoff is q_itp_t.

Here we assume that 0 < ² H < L < H. The game is repeated indefinitely and both firms discount

future payoffs by δ ∈ [0, 1).

(a) [5 marks] Show that the stage game has a unique Nash equilibrium in which both firms choose output 1 (µH + (1 − µ)L).

(b) [5 marks] Show that to maximize their total stage payoffs, the two firms must choose (q_1t, q_2t) so

that q_1t + q_2t = ¹ (µH + (1 − µ)L).

(c) [5 marks] First, consider the case where (q_1t, q_2t, p_t) are publicly observed at the end of each period t. Consider the following strategy profile: both firms choose output q^∗ = ¹ (µH + (1 − µ)L) until some firm chooses another output and they both choose output 1 (µH + (1 − µ)L) thereafter. Represent this strategy profile by an automaton and find the smallest δ for which it is a subgame perfect equilibrium. Denote the answer by δ.

(d) [5 marks] In what follows, assume that (q_1t, q_2t) is not publicly observed and only p_t is observed at the end of each period t.   Consider the following strategy profile:  both firms choose output    q^∗ = ¹ (µH + (1 − µ)L) until a price p_t outside the set {H − 2q^∗, L − 2q^∗} is observed; they both choose output 1 (µH + (1 − µ)L) thereafter. Represent this strategy profile by an automaton.

(e) [10 marks for ECON3152, 5 marks for ECON4453/8053] Is it possible that the above strategy profile is not a PPE even when δ > δ ?

(f) [5 marks] Now consider a new strategy profile: both firms choose output q^∗ = ¹ (µH +(1 − µ)L) until

a price p_t H − 2q^∗ is observed; they both choose output 1 (µH + (1 − µ)L) thereafter. Represent

this strategy profile by an automaton and find both players’ value functions. What are the limits of the value functions when δ → 1?

2. (Based on Aghion and Tirole (1997), “Formal and real authority in organizations”) There is a principal and agent deciding which projects to implement, if any. There are four projects to choose from. They are of three characters but it is not known which project has which character. The following table lists the possible characters and payoffs to both players when implemented. If no project is implemented, each

player receives zero payoff from “projects”. It is assumed that 0 < L < H and M is a big number (details below). The characters of the projects are represented by θ = (θ₁, θ₂, θ₃, θ₄) where θ_i ∈ {P, A, D} is the character of Project i. It is commonly known that there are two projects of the character D, one of the character P and one of the character A, and θ is equally likely to assume any one of its twelve possible values. (Therefore, choosing a project randomly yields payoff (H + L − 2M )/4 to both players; if one disastrous project has been excluded, choosing a project randomly among the remaining three yields payoff (H + L − M )/3 to both players.)

(1) Nature draws θ uniformly from Θ, as well as the two players lucks l_P and l_A independently and uniformly from [0, 1].

(2) Both players simultaneously choose efforts e_i ∈ [0, 1] to gather information about θ, for i =P, A.

(3) For each player i, nothing happens if e_i + l_i < 1 and Player i learns the value of θ (the characters of the projects) if e_i + l_i ≥ 1.

(4) If the principal learns θ, the game ends with the project of character P being implemented. Otherwise, the agent chooses to recommend one project or none of them.

(5) Observing the agent’s recommendation (or lack thereof), the principal chooses between implement- ing the recommended project (if any), implementing another project randomly and implementing nothing.

Each player i’s payoff is the difference between the project payoff (if any) and the cost 1 c_ie_i² of gathering

information. The following assumptions are maintained:

• 0 < L < H and H + L − M < 0.

• c_P > H and c_A > H.

The agent has three classes of information sets: I_A where he chooses effort, I_A0 where he chooses recommendation after failure to learn θ and I_A1 where he chooses recommendation knowing θ. The principal has three classes of information sets: I_P where she chooses effort, I_P 0 where she chooses which project to implement (if any) without knowing θ or receiving a recommendation, and I_P 1 where she chooses which project to implement (if any) based on the recommendation but not θ. We restrict attention to strategy profiles prescribing the same action for all information sets in the same class; in other words, we assume that decisions at I_A0, I_A1, I_P 0 and I_P 1 are independent of the players’ effort choices.

(a) [5 marks] Identify all the subgames of the game.

(b) [10 marks for ECON3152, 5 marks for ECON4453/8053] Consider the following strategy profile: the principal chooses e_P = c⁻¹(H − L), the agent chooses e_A = 1 and recommends the project of character A, the principal chooses a project randomly at I_P 0 and the recommended project at I_P 1. Are there parameter values for which this is a subgame perfect equilibrium?

(c) [5 marks] For the rest of the question, we will use the solution concept of sequential equilibrium. However, for simplicity, you are allowed to regard a strategy profile with pure effort choices strictly between zero and one (and fully mixed at all the other information sets) as “fully mixed”. No result will change due to this relaxation. We call a strategy profile a “part of sequential equilibrium” if we can find beliefs so that the beliefs and strategy profile together form a sequential equilibrium. Is the strategy profile in the previous part ever a part of sequential equilibrium? (In other words, are there beliefs consistent with the strategy profile that make it sequentially rational?)

(d) [5 marks] Consider the following strategy profile: the principal chooses e_P = c⁻¹H, the agent chooses

e_A = 0 and recommends no project at I_A0 and recommends a project of character D at I_A1, and the principal chooses not to implement any project at both I_P 0 and I_P 1.  Is this strategy profile part of a sequential equilibrium?

(e) [10 marks for ECON3152, 5 marks for ECON4453/8053] Now consider assessments where the agent recommends the project of character A at I_A1 and no project at I_A0, and the principal implements no project at I_P 0 and the recommended project at I_P 1. Find all sequential equilibria (if any) in this class of assessments. (Hint: first compute the two players’ efforts based on the assumption to figure out whether I_A1, I_A0, I_P 1 and I_P 0 happen with positive probabilities.)

3. Consider a two-period game in which two players play an investment game in both periods and discount future payoffs by δ ∈ (0, 1). In each period, both players choose their efforts e_i ∈ [0, 1] simultaneously and revenue R > 0 is generated with probability kq(e₁, e₂) and there is no revenue with probability   1 − kq(e₁, e₂), where k ∈ {0, 1}. The function q is twice continuously differentiable and we will denote its partial derivatives using q with appropriate subscripts; for example q₁₁ is the second partial derivative

with respect to its first variable. Each player i’s stage payoff is the difference between revenue (if any) and c(e_i). At the end of the first period, the revenue but not the efforts is publicly observed. Assume that for every R > 0, the stage game with k = 1 has a unique Nash equilibrium (e^∗(R), e^∗(R)) with e^∗(R) ∈ (0, 1), which yields expected payoff V (R) to both players. Throughout the question, assume that q(0, 0) > 0, q_i > 0, q_ii < 0 for i = 1, 2, q₁₂ > 0, c^′ > 0 and c^′′ > 0.

(a) [5 marks] (ECON4453/8053 only; however, undergraduates should read this part and assume its results for the rest of the question.) Show that e^∗ and V are both strictly increasing.

(b) [10 marks for ECON3152, 5 marks for ECON4453/8053] Assume that k = 1. Find all the PPEs

of the game. (Hint: this is a finite horizon game, so we use a backward-induction type of analysis instead of constructing automata.)

(c) [5 marks] For the rest of the question, assume that there are two types of environments, good (k = 1)

and bad (k = 0). Nature chooses the environment (once and for all) at the beginning of the game; Nature chooses k = 1 with probability µ ∈ (0, 1). Now the game no longer has public monitoring and we will use the solution concept of sequential equilibrium. Suppose that in a sequential equilibrium, both players choose the same effort e^∗0  in the first period.  Let µ_r (for r ∈ {0, R}) be the probability that both players assign to the environment being good when revenue r is generated in the first period and both players indeed choose e^∗0.  Show that both players choose effort e^∗(µ_rR) in this case.

(d) [5 marks] Show that µ_R = 1, µ₀ < µ, and µ₀ is strictly decreasing in e^∗0.

(e) [10 marks] Now consider a history where Player 1 chooses effort e^∗0,  Player 2 chooses effort e˜ ̸= e^∗0 and no revenue is generated in the first period. Show that Player 1 chooses effort e^∗(µ_rR) as in (c) in  the  second  period,  while  Player 2’s  effort  in  the  second  period  is  strictly  decreasing  in  e˜.  (This shows that in this game without public monitoring, Player 2’s best response to Player 1’s public strategy is a private strategy that depends on his private history e˜.)

4. (ECON4453/8053 only) Consider a stochastic model of growth of a closed economy with a representative consumer. The economy starts wit capital stock k₀ > 0. For simplicity, assume that there is an exogenous upper bound k¯ of capital stock, so that the state space [0, k¯] of our dynamic program is compact.  Each period t has an exogenous state s_t ∈ S, where S is a finite set. For every t ≥ 0, the probability that s_t+1  = j  is  p_ji  if  s_t  = i,  regardless  of  s_t′    for  t^′ < t and  other  past  events.   Denote  the  capital  stock in Period t as k_t. Then the output in Period t is f (k_t, s_t), where f (k, s) is continuously differentiable, strictly increasing and weakly concave in k, for every s ∈ S. The output can be consumed, invested or burned. Denote the consumption in Period t by c_t and the investment in Period t by i_t. Then the consumer’s stage payoff is u(c_t)

k_t+1 = βk_t + i_t,