ECOS3012 Strategic Behaviour
Tutorial 2
• You should try your best to solve these questions before coming to the tutorial.
• For the easier questions, you will be asked to share your answers with your classmates.
• For the other questions, you will be invited to share your thoughts on them (even if you
hadn’t figured out the final solution). We will discuss these questions in detail.
Q1. The tragedy of the commons
• Players: 20 farmers
• Strategy: each farmer i chooses the number of goats gi to graze on the village
green
• Payoff: for each farmer i,
– the cost of owning a goat is 10
– Let the total number of goats grazed on the green be G = g1 +g2 + …+g20.
For each farmer’s each goat, the value of grazing that goat on the green is
v(G) = 10000−G2
(a) In the Nash equilibrium, what is the total number of goats grazed on the village
green in the Nash equilibrium? What is the profit earned by each farmer?
(b) Find the total number of goals on the village green, G, that maximises the total
sum of all farmer’s profit. This is called the “socially optimal” allocation. If G is
divided evenly among the farmers, how much profit does each farm earn? Discuss
the difference between your answers in (a) and (b).
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Q2. Cournot competition with infinitely many homogeneous firms
There are n firms that produce homogenous goods. Each firm i chooses to produce at
quantity qi ≥ 0. The total cost to produce qi is Ci(qi) = 10qi for each firm i. When the
total quantity produced by all firms is Q = q1 + q2 + …+ qn, the market price for the
good is P(Q) = 100−Q.
(a) Find the Nash equilibrium of this game. Express optimal quantities as functions of
n.
(b) What is the market price in the Nash equilibrium? What is each firm’s profit? How
does the price and the profit change with n?
(c) Calculate the limit equilibrium price and profit as n converges to infinity.
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Q3easier Bertrand competition with heterogeneous firms
There are two firms that produce heterogeneous goods. Each firm i chooses to sell its
products at price i. The quantity that consumers demand from firm i is
qi(pi, p j) = 100− pi + p j
(Firm i’s good is a substitute for Firm j’s good, so an increase in p j makes more
consumers choose Firm i.)
There are no fixed costs of production and marginal costs are constant at 10. The firms
announce their prices simultaneously.
Find the Nash equilibrium in this game. Calculate the equilibrium profits for each firm.
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