Problem Set 5 Problem 1. Conflicts between me and myself Consider an economy in which there is only one possible investment. In period 0, nature chooses the value π of that investment. This value is selected from a distribution with continuous and differentiable c.d.f. F (π) in the support [π, π] and expected value E[π] = Rπ e π π dF (π) = π . In period 1, our risk-neutral agent is born. He knows the distribution from which π is drawn. He decides whether to learn or not the value of the investment drawn by nature. Learning has no cost but can be made in period 1 and only in period 1. In period 2, the agent decides whether to invest or not (note that he cannot commit in period 1 whether he will invest in period 2 or not). Investing requires to pay a cost c in period 2 and yields a benefit π in period 3. Not investing implies no cost and no benefit. Our agent discounts the future in a peculiar way. When he is in period 1, then period 2 is discounted at a rate βδ and period 3 is discounted at a rate βδ 2 where β ∈ (0, 1] and δ ∈ (0, 1). When he is in period 2, then period 3 is discounted at a rate βδ. The agent can never commit on his future actions. (1) Suppose that β = 1 (the standard “exponential” discounting). (1.a) If the agent learns at date 1 the value of the investment π, what is his optimal choice at date 2? Anticipating this choice at date 2, what is his expected payoff from his date 1 perspective if he decides to learn the value of the investment? (1.b) If the agent does not learn at date 1 the value of the investment π, what is his optimal choice at date 2? Anticipating this choice at date 2, what is his expected payoff from his date 1 perspective if he decides not to learn the value of the investment? (1.c) Combine (1.a) and (1.b) and determine under which conditions (if any) does the agent at date 1 prefer not to learn the value of the investment in period 1. (2) Suppose now that β < 1 (labelled in behavioral economics as “hyperbolic” discounting). Answer the same questions as before in this new case. (3) Interpret your results. Problem 2. More investment with more uncertainty Consider the investment under uncertainty game discussed in class. Suppose that instead of being drawn from a uniform distribution, θt is drawn from a general distribution. Denote F (θ) the c.d.f. and assume only that F (·) is continuous, twice differentiable and with support [θ, θ] with θ < 0 < θ. Also the density function f (θ) is positive for all θ ∈ [θ, θ]. Do not assume anything else about the distribution. Also, take any arbitrary number of periods T . 1 ∗ (1) Determine the recursive equation that links θt+1 (cutoff at t + 1) with θt∗ (cutoff at t). We call “option value of waiting at stage t” (denoted ovwt ) the difference between the minimum draw at which it is worth collecting the payoff even though there are still t periods left (θt∗ ) and the value of never collecting the payoff (0). (2) Show that ovwt+1 > ovwt for all t and independently of the distribution F (·). Interpret. (3) Show that ovwt+2 − ovwt+1 < ovwt+1 − ovwt for all t and also independently of the distribution F (·). Interpret. Suppose that another agent faces the same problem except that his θ is drawn from distribution G(θ) instead of F (θ). As before, G(·) is continuous, twice differentiable and with the same support [θ, θ]. Also the density function g(θ) is positive for all θ ∈ [θ, θ]. Suppose that F (θ) < G(θ) for all θ ∈ (θ, θ) (we say that F (·) first-order stochastically dominates G(·)). Denote θt∗∗ the analogue of θt∗ for the agent whose θ is drawn from distribution G(θ). As usual, θ0∗ = θ0∗∗ = 0 (where 0 is the last period). (4) Find if (i) θt∗ > θt∗∗ for all t > 0, or (ii) θt∗ < θt∗∗ for all t > 0, or (iii) the inequality can go in both directions. Interpret. Problem 3. Agreeing now to agree later Agent 1 wants agent 2 to give a speech next year. There are 3 possible topics λ ∈ {A, B, C}: social security (A), unemployment (B), and terrorism (C). Which one is better depends on what happens between now and next year. There are 2 possible states of the world: sA denotes social security becomes the hot topic and sC denotes terrorism becomes the hot topic. Each state occurs with probability 1/2. Both agents will observe the state in 10 months from now. There is no discounting. A hot topic attracts more audience. In state sA , the benefit for agent 1 of a speech on topic A is 20, on topic B is 17 and on topic C is 14. In state sC , the benefit for agent 1 of a speech on topic A is 14, on topic B is 17 and on topic C is 20. A hot topic is more difficult to prepare. In state sA , the cost for agent 2 of giving a speech on topic A is 10, on topic B is 9 and on topic C is 8. In state sC , the cost for agent 2 of giving a speech on topic A is 8, on topic B is 9 and on topic C is 10. Full contract. Agents 1 and 2 must sign a contract now that is implemented next year. 1. What is the “efficient” topic as a function of the state, that is, the topic that maximizes the sum of utilities of agents? 2. Suppose that agent 1 proposes the following contract: “I commit to pay you p and I will let you know in 11 months the topic of your speech”. What is the optimal price p for agent 1? Does it always implement the efficient topic (as defined in question 1)? 2 No contract. Suppose now that agents 1 and 2 cannot sign a contract. In 11 months, agents 1 and 2 bargain over the price that 1 pays to 2 and the topic of the speech. The price and topic selected in the bargaining process is the one that maximizes the product of the surplus of agents 1 and 2, where agent 1’s surplus is his net payoff (valuation minus price) and agent 2’s surplus is also his net payoff (price minus cost). 3. What is the outcome of the negotiation (price and topic) as a function of the state? Is it efficient (in the sense defined in question 1)? How much utility does agent 1 get? 4. Interpret. Partial contract. Suppose now that agent 1 can contract only on the price. That is, agent 1 proposes a price now and in 11 months agents 1 and 2 bargain over the topic at the price pre-specified in the contract. As before bargaining maximizes the product of the surplus of agents 1 and 2. 5. Suppose that the price proposed in the contract (and which cannot be renegotiated) ∗ is p ∈ [10, 20]. What is the outcome of the negotiation (topic) as a function of the state and p∗ ? Anticipating this future bargaining, what is the optimal price p∗ ? Does it implement the efficient topic? How much utility does agent 1 get? 6. Conclude. 3