2023 EconS 425 Final Exam 1 Consider the following game about trust that has been played in | Assignment Collections

EconS 425

Final Exam

1. Consider the following game about trust that has been played in many experiments. The experimenter starts by giving player 1 ten dollars and player 2 zero dollars. The experimenter then asks player 1 how many dollars is he willing to give back to help player 2. If he chooses to give x dollars back

to the experimenter, then the experimenter gives player 3 3x dollars. Subsequently, player 2 has the

opportunity to give any or all (or none) of the money he has received to player 1.

(a) Assuming that the two players are risk neutral and care only about their own payo§ FInd the

subgame perfect equilibrium of this game. (The subgame perfect equilibrium, by the way, is not

what seems to happen in experiments. Usually player 1 gives some, but not all of the money back

to the experimenter.)

(b) Does the game have a Nash equilibrium in which the players receive higher payo§s?

(c) Suppose we modiFIed the game so that after the two stages above player 1 had the chance to

punch player 2. Suppose that this would reduce player 1ís utility by 1 dollar and reduce player

2ís utility by 5 dollars. Would this change your answers to parts (a) and (b)? How about if we

instead had the players play the game shown below after the second stage? Do these predictions

seem reasonable to you.

Player 2

A B

Player 1 A 5,5 -5,-5

B -5,-5 5,5

(d) An alternative explanation for the experimental results is that the players may be altruistic. Show

that the simplest representation of altruism ñeach player maximizing a weighted sum of his own

dollar payo§ and the other playerís dollar payo§ ñcannot account for the experimental regularity except for one very special (and not compelling) choice of the weights. Can you think of utility

functions that might be used to account for the experimental result?

2. Prove the second part of Proposition 12.1 (page 309) using the same procedure as the one used in the

proof of the FIrst part.

3. A monopoly o§ers a product for sale. The product costs c = $60 to produce. The product may fail

with probability 0:5, hence it is fully operative with probability  = 0:5. This probability is public

information in the sense that it is known to the seller and all buyers. The product can either be fully

functioning or totally defective. Consumers are willing to pay up to V = $240 for a fully-functioning

product. If the product is found to be defective, consumers do not gain any utility. Solve the following

problems.

(a) The monopoly provides a ìtwice-replacementîwarranty. That is, if the original purchase is found

to be defective, the consumer can have the product replaced free of charge. If the replacement

product is also found to be defective, it also gets replaced free of charged. However, the monopoly

will not replace the replacement of the replacement product if it also found to be defective.

Compute monopolyís proFIt-maximizing price and the resulting expected proFIt.

(b) Now suppose the monopoly provides a money-back guarantee (instead of the twice-replacement

warranty). Compute monopolyís proFIt-maximizing price and the resulting expected proFIt.

Microsoft and Apple produce di§erentiated goods, Microsoft Surface (MS) and iPad (iP), respectively.

There are two types of buyers in the economy, those who are FIrst time buyers (inexperienced consumers) and those who have purchased the product before (experienced consumers). The group of inexperienced consumers represents the 45% of the total group of consumers. Assume that the group of experienced  consumers is divided into two subgroups: those who prefer to purchase MS over brand iP, and those who prefer iP over MS. Let

4. The inverse market demand function for MP3 players is given by p = 240 -2Q. Initially, firm A and

firm B produce at equal unit cost, c0. After investing heavily in R&D, firm A has managed to reduce

its unit production cost to c1 = $40 < c0. For which values of c0, FIrm Aís innovation can be classiFIed

as drastic (major), and for which values of c0 the innovation is classiFIed as nondrastic (minor). Prove

your result using the definition.

5. The inverse market demand for Toaster-Phones in NYC is p =36 – Q. The manufacturer licenses

a single dealer to sell this brand in NYC. Therefore, the dealer acts as a monopoly in the NYC

market. The manufacturer sells each Toaster-Phone to the dealer for $d c, where c = $20 is the

cost of producing one Toaster-Phone. In addition, the manufacturer may levy a fixed fee of $‑ on the

dealership. Consider a two-stage game, in which in Stage I the manufacturer sets the per-unit price

charged to the dealer d, and the fixed fee. In Stage II the dealer determines the quantity sold as to

maximize the dealershipís profit.

(a) Compute the dealerís price p, quantity sold Q, and proFIt

d as a function of d and ‑.

(b) Suppose the manufacturer does not charge the dealer any FIxed fee, ‑ = 0. Compute the dealerís

price d which maximizes the manufacturerís proFIt. Then, compute the equilibrium consumer

price p, and the proFIts made by the manufacturer

m and the dealer

d

.

(c) Can the manufacturer set a di§erent contract with the dealer so that both, the manufacturer and

the dealer, make higher proFIts. Formally, FInd d and ‑ which would generate higher proFIt levels,

m and

d

, compared to the levels you computed in (6b).

6. The inverse demand function for QuasiSmart-Phones is P = 10-Q. Assume that a single FIrm initially

operates in this market (FIrm 1) and its marginal cost is c = 2. Firm 1 is facing the threat of entry,

since FIrm 2 is evaluating whether to join the market. There is a FIxed entry cost F > 0 and FIrm 2

has the same marginal cost than FIrm 1. First, the potential entrant observes FIrm 1ís output level and

then it decides whether or not to enter. If FIrm 2 enters, both FIrms compete a la Cournot. Find the

range of values of F for which FIrm 1 (incumbent) decides to accommodate entry.

7. Microsoft and Apple produce di§erentiated goods, Microsoft Surface (MS) and iPad (iP), respectively.

There are two types of buyers in the economy, those who are FIrst time buyers (inexperienced consumers) and those who have purchased the product before (experienced consumers). The group of inexperienced consumers represents the 45% of the total group of consumers. Assume that the group of experienced consumers is divided into two subgroups: those who prefer to purchase MS over brand iP, and those who prefer iP over MS. Let B, 0 < B < 1, be the fraction of MS-oriented consumers (among experienced consumers). Hence, (1-B) is the fraction of iP-oriented consumers (among experienced consumers). In addition, FIrms must decide between two advertising methods: Persuasive (P) or Informative advertising (I).

a) Following Assumption 11.1 (page 292), identify the proFIt level for each FIrm and the aggregate

proFIt under all four possible outcome.

(b) Discuss under which conditions both FIrms will use informative advertising. Discuss your results.

8. Following Brander and Spencer 1983 and 1985, consider two countries denoted by i = X; Y , each of

which has one FIrm producing a homogeneous product only for export, to be sold in the international

market. Both FIrms compete a la Cournot in the international market. The inverse demand function

in the international market is P = 15 – 0.5Q. In addition, assume that the preinnovation unit cost

of each FIrm is c = 4. Let ri denote the amount of R&D sponsored by the government in country i.

Assume that when the government i undertakes R&D at level ri, the unit production cost for the firm producing in country I is reduced to c-ri, I = X, Y. Finally, the total cost to government I of engaging in R&D at level ri, is TC

(a) Identify countries best response function.

6(b) What is the Nash equilibrium R&D level for each country? Discuss your results.

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