eco 444

Test 4

ECO 444

Konrad Grabiszewski

• This is an individual take-home test. Prepare the answers on your own.

• Deadline: deliver your exam in class on Monday, April 23, 2018.

• Show all your work for full credit. Providing only correct answers is not enough.

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Exercise 1 (30 points)

Consider the game in Exercise 3 (Test 2). However, instead of Ann and Bob making choices at

the same time, suppose that they are choosing in turns (i.e., dynamic game). In particular, in

stage 1, Ann chooses sA. In stage 2, Bob learns about sA and chooses sB. What is the sub-game

perfect Nash equilibrium?

Exercise 2 (30 points)

Consider the Stackelberg duopoly (Handout 12) but with different firms. In particular, the

marginal cost of Firm A (leader) is cA and the marginal cost of Firm B (follower) is cB with

0 < cA < cB. We also need to make the following additional assumptions (for the equilibrium to

make sense): (a) a > cA, (b) a > cB, and (c) a + 2cA − 3cB > 0. What is the sub-game perfect

Nash equilibrium?

Exercise 3 (40 points)

There are two firms – Firm A and Firm B – competing against each other in the innovation

race. Each Firm i ∈ {A, B} exerts effort si ∈ [0, 1]. Effort si indicates the probability of Firm

i being successful in creating an innovation. Note that 1 − si denotes the probability of Firm i

being unsuccessful. The cost of effort is s2i . If Firm i is successful and Firm j is unsuccessful,

then Firm i gets the prize of value 1 (one unit of utility). Otherwise, Firm i gets zero. That is,

even if two firms are successful, then each gets zero. In short, the utility function of Firm i is

ui(sA, sB) = si(1 − sj) − s2i . Suppose that you work as a consultant for Firm A. From the three

options listed below, which option would you recommend to Firm A? [Assume that Firm A is

rational and wants to maximize utility.]

1. Firms make choices at the same time.

2. Firms make choices in turns. First, Firm A chooses sA. Then, Firm B observes the choice

of Firm A and chooses sB.

3. Firms make choices in turns. First, Firm B chooses sB. Then, Firm A observes the choice

of Firm B and chooses sA.

Hint: for each option, you need to find the equilibrium (NE for static game, SPNE for dynamic

game) value of Firm A’s utility; then, compare these values.

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