ECON 3152/4453/8053 Homework 2 2022

1. An entrepreneur needs a fixed investment I to start a project but only have cash L ≤ I .  After this investment is made, the entrepreneur chooses an effort level e ∈ [0, ] to work on the project, where   < 1.  The project succeeds with probability e, in which case it generates revenue R > I; the project fails with probability 1 − e, in which case it generates no revenue. The bank may offer the entrepreneur a contract to lend the entrepreneur I − L on the condition that the entrepreneur is to repay D when the project succeeds.  (If the project fails, the entrepreneur has no cash left and is protected by limited liability, so he pays the bank nothing.)  The timing is as follows: the bank offers a loan contract with repayment D or offers nothing; observing D  (if a loan is offered), the entrepreneur decides whether to accept the loan; if a loan contract is accepted, the investment is made, the effort is chosen by the entrepreneur and the outcome is realized. The bank’s payoff is (1+ r)−1  times the repayment it receives minus the loan it lends out, where r  > 0 is a parameter known to both parties.  The entrepreneur’s payoff is L if the project is not implemented and the difference between (1+r)−1 times his wealth at the end and c(e) if the project is implemented, where c : [0, ] → R is the cost of effort function known to both parties. It is assumed that c is continuously differentiable and strictly convex, c(0) = 0, c\ (e) → 0 as e → 0 and c\ (e) → ∞ as e → . Let g be the inverse function of the strictly increasing function c\ . In many parts of this question, it may be convenient to express your answers in terms of g .

(a)  [5 marks] First, consider the efficiency benchmark where the entrepreneur has cash L = I to imple-

ment the project without help. Find the minimum revenue R for the entrepreneur to invest and the first order condition for his optimal effort when he does invest. This minimum revenue R is denoted by Rm(FB)   (with  “FB” standing for  “first best”).  A project with R  > Rm(FB)   is considered to have potential to generate a positive net present value (when invested by a self-funded entrepreneur).

(b)  [10 marks for ECOn3152, 5 marks for ECON4453/8053] In what follows, assume that L < I; we

will treat L as a fixed number and consider how R affects equilibrium outcomes.  Show that there exists an Rm(SB)   (with  “SB” standing for  “second best”) such that there exists a subgame perfect equilibrium in which the project is implemented if and only if R ≥ Rm(SB) . (Hint: the bank is making a take-it-or-leave-it offer, so it is safe to assume that the entrepreneur accepts the contract when indifferent.)

(c)  [20 marks bonus] (Optional) Compare the equilibrium outcomes (minimum R and effort e) with the efficiency benchmark in (a); show that Rm(SB)   > Rm(FB) .  Even if you do not attempt this part, you should assume its result in later parts of the question.

(d)  [5 marks] That the equilibrium effort is lower than the efficient effort is not surprising:  the en- trepreneur does not receive the full benefit of his effort when he has to payback a debt in the future. The finding that Rm(SB)  > Rm(FB)  is called credit rationing, meaning that some projects can potentially generate positive net present value but are not funded even when there is credit available in the mar- ket. The rest of the question attempts to identify the source of credit rationing in this model. 1  There are two potential sources of inefficiencies here: the entrepreneur’s moral hazard and the assumption that the bank is a monopoly in the credit market in our current model.  We will try to isolate the two factors. In this part, we examine what happens when there is no moral hazard: repeat the first three parts assuming that c(e) = 0 for every e ∈ [0, ]. Is there still credit rationing?

(e)  [10 marks for ECON3152,  5 marks for ECON4453/8053] Considering competing banks is more

complicated than considering Betrand competition between two sellers. A standard approach is as follows:  we only model one bank explicitly (as we have been doing so far) but assume that the entrepreneur’s payoff is some u ≥ L when he is not offered a loan contract or he rejects the offer. This u is interpreted as his payoff from going to a competing bank. The credit market is perfectly competitive when the u is so high that the bank in our model cannot receive any positive payoff. Find the minimum value of R for which there exists a u ≥ L and a subgame perfect equilibrium such that the project is implemented.

2.  (Monopoly pricing of durable goods) There is a monopoly seller of durable goods and two consumers. The goods are sold in two period of time, t = 1 and t = 2; the idea is that the good becomes obsolete after the second period and is no longer sold.  The main result of this problem generalizes to the case

with more than two periods.  Each consumer chooses either to buy one unit of the goods or to buy nothing. Consumer 1 enters the market at t = 1 and may buy in either period, while Consumer 2 enters the market at t = 2 and may only buy at t = 2. The cost of producing a unit is c ≥ 0 and each consumer i has an independent type θi  ∈ [θ , θ¯] where θ < c < θ¯. The probability distribution of each θi is described by a continuous cumulative distribution function F, which means that (i) the probability that θi  ∈ (a,b] is F(b) − F(a) for all a ≤ b and (ii) the probability that θi  = a is zero for every a ∈ [θ , θ¯].  Item (ii) means that we do not need to worry about what a consumer would do when indifferent between two

1 There are other models suggesting different sources of credit rationing; in the real world, credit rationing may be due to a combination of several factors and the different models help us understand the factors separately before we try to combine them in a more complicated mathematical framework.

options in the entire analysis, as this indifference happens with probability zero. In what follows, that the equilibrium is essentially unique means that the equilibrium is unique up to the consumers’behavior when indifferent.

Every player discounts payoff in Period 2 by a factor 6 ∈ (0, 1).  Denote by pt  the unit price at Period t set by the seller. Then the seller’s payoff is n1 (p1  − c) + 6n2 (p2  − c) where n1  is the number of units sold in Period 1 and n2  is the number of units sold in Period 2.  If Consumer i buys in Period 1, her payoff is θi − p1 ; if she buys in Period 2, her payoff is 6(θi − p2 ); if she buys nothing, her payoff is zero. Here θi  should be interpreted as the discounted total value from using the durable good.  We consider two different timings.

Case 1  Committed seller.

(1) Seller announces p1  and p2 .

(2) Nature chooses θ 1  ∈ [θ , θ¯].

(3) Consumer 1 observes θ 1 , p1  and p2  and decides whether and when to buy.

(4) Nature chooses θ2  ∈ [θ , θ¯] (independent of θ1 ).

(5) Consumer 2 observes θ 1 , Consumer 1’s action, θ2 , p1  and p2  and decides whether to buy in

Period 2.

Case 2  Adaptable seller.

(1) Seller announces p1 .

(2) Nature chooses θ 1  ∈ [θ , θ¯].

(3) Consumer 1 observes p1  and θ 1  and decides whether to buy in Period 1.

(4) Without observing θ 1  or whether Consumer 1 bought, the seller announces p2 .2

(5) Consumer 1 decides whether to buy in Period 2 observing p2 .

(6) Nature chooses θ2  ∈ [θ , θ¯] independent of θ 1 .

(7) Consumer 2 observes θ 1 , Consumer 1’s actions, θ2 , p1  and p2  and decides whether to buy in

Period 2.

The timing in the second case is framed so that the resulting extensive form game has as many subgames as possible. Throughout the entire question, the solution concept is subgame perfect equilibrium. The following assumptions will be made on F :

• F is strictly increasing on [ θ , θ¯] with F(θ  ) = 0 and F(θ¯) = 1.

• F is continuously differentiable.

2 The seemingly weird assumption that the buyer cannot observe whether Consumer 1 bought in Period 1 is due to the fact that our Consumer 1 represents many consumers and in reality the fraction of Consumers 1 buy in Period 1 is not (very) random due to the Law of Large Numbers. Ignore this if you find the footnote difficult to understand and simply accept the assumption as given.

• The function that maps p ∈ [θ , θ¯] to F(p)(p − c) is strictly convex.

(a)  [10 marks for ECON3152; 5 marks for ECON4453/8053] Write down each player’s set of strategies

or describe their typical strategies in the case of adaptable seller.  (We restrict p1  and p2  to be in [θ , θ¯] as unit prices outside this interval do not make sense.)

(b)  [5 marks] Determine Consumer 1’s equilibrium strategy in the subgame following each (p1 ,p2 ) in

Case 1.

(c)  [5 marks] Assuming that the seller sets p2  = g(p1 ) in Case 2 for some function g known to Consumer 1, determine Consumer 1’s best response in the subgame following each p1  in Case 2.

(d)  [10 marks for ECON3152; 5 marks for ECON4453/8053] (Challenging) Without solving either case, determine in which case the seller receives higher expected payoff (profit) using the following argu- ment.  Start with the subgame perfect equilibrium of the adaptable case that favors the seller the most, assume that the seller’s expected payoff in this case is πA , try and find a committed seller’s strategy that gives the committed seller the same payoff πA , and explain why this fact answers the question.  (This line of argument is widely used in many game theoretic and contract theoretic problems; next time it shows up, there will be no hint.)

(e)  [10 marks] We focus on Case 1 in this and the next two parts.  By now you should realize that

Consumer 1 behaves differently depending on which between p1  and p2  is higher. We solve the game in Case 1 by treating the cases p2   ≥ p1  and p2   ≤ p1  separately and comparing the answers.  In this part, consider the case where p2  ≥ p1 : try and solve the game in Case 1 giving the seller the additional constraint that p2   ≥ p1  and show that there is an essentially unique subgame perfect equilibrium, and p1  = p2  in that equilibrium. Denote this unit price by pc .

(f)  [10 marks for ECON3152; 5 marks for ECON4453/8053] Now consider the case where p2  ≤ p1 . Solve

the game in Case 1 giving the seller the additional constraint that p2  ≤ p1 . (Hint: it might be useful to pose the seller’s problem at the beginning of the game as choosing r =  and p2  and writing p1  as a function of r and p2 .)

(g)  [5 marks] Sum up the previous two parts to obtain all the subgame perfect equilibria without giving

the seller any additional constraint.

(h)  [5 marks] In this and the next part, focus on Case 2. Our goal is to show that the seller will not be

able to maintain prices at pc  or higher in both periods when he is adaptable. Consider the subgame following the choice of a p1  ∈ [pc , θ¯). Show that the subgame has no equilibrium in which p2  ≥ p1 .

(i)  [10 marks for ECON3152; 5 marks for ECON4453/8053] Show that in every subgame perfect equi- librium of the subgame following a p1  ∈ [pc , θ¯), the seller chooses a p2  strictly lower than pc .

3.  (ECON4453/8053 only, monopolistic competition) There are n sellers of different goods and a consumer. Let pi be the unit price of the good sold by Firm i; then the consumer chooses her consumption (x1 , ...,xn ) of the n goods to solve the following problem:

maxx1 , . . . ,xn

s.t.

\ 之(n) xj s/(1+s)  (1+s)/s ;

xj  ≥ 0, for j = 1, ...,n;

n

之 pj xj  ≤ y .

j=1

Here y > 0 and s > 0 are known numbers. The n firms choose their unit prices simultaneously; Firm i’s payoff is its profit

πi (pi ,p−i) = xi(∗)(p)(pi − ci ),

where xi(∗)(p) is the consumer’s optimal consumption of Good i and ci  ≥ 0 is Firm i’s cost of production, for i = 1, ...,n. We treat the pricing problem as a strategic form game among the n firms. Each firm is a monopoly of the good it sells as it can sets its unit price at will, but the firms are competing for higher quantities.  Let us recall that in solving the consumer’s problem, we are allowed to perform a strictly monotonic transformation of her objective (utility) function prior to any further analysis.

(a)  [3 marks] It makes no sense for Firm i to choose a pi   ≤ ci .  Therefore, we assume that Firm i’s

set of strategies is (ci , ∞).  This means that each firm’s profit is positive.  In general, taking the logarithm of the payoff function of each player is a NO-NO; we are making this dangerous move here because all the results of supermodular games remain valid when the payoffs are always positive and increasing difference condition is replaced with

 ≥  ,  for all p ≥ pi  and pi  ≥ p −i .

Now show that log πi (pi ,p−i) has increasing differences in pi  and p −i  for i = 1, ...,n.

(b)  [5 marks] Show that for every i, Firm i’s marginal profit can be written as

∂πi

where gi  is always positive and fi (pi ,p−i) = cipi(−)s  − (之j i ps ) (spi  − (s + 1)ci ), which is strictly

decreasing in pi .

(c)  [5 marks] Let  = max{c1 , ...,cn } and p¯ = s −1  (s + 1 + ) . For every i, show that BRi (p−i) ≤ p¯ if pj  ≤ p¯ for all j  i.

(d)  [5 marks] The previous part implies the existence of a Nash equilibrium; you do not need to prove