ECON 2350 Lecture Notes - Lecture 12: Best Response

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amaranthhamster412

27 Feb 2017

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One difficulty with the leader-follower model is that it is necessarily asymmetric: one firm is able to make its decision before the other firm. For example, suppose that two firms are simultaneously trying to decide what quantity to produce. Here each firm has to forecast what the other firm"s output will be in order to make a sensible decision itself. In this section we will examine a one-period model in which each firm has to forecast the other firm"s output choice. Given its forecast, each firm then chooses a profit-maximizing output for itself. We then seek an equilibrium in forecasts a situation where each firm finds its beliefs about the other firm to be confirmed. This model is known as the cournot model, after the nineteenth-century french mathematician who first examined its implications. We begin by assuming that firm 1 expects that firm 2 will produce ye 2 units of output.

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Consider a variant of the 𝑛-firm version of Cournot’s model in which each firm’s average cost function is U-shaped, rather than being constant. Suppose that there are infinitely many firms, all of which have the same cost function 𝐶. Assume that 𝐶(0) = 0, and that for 𝑞 > 0 the function 𝐶(𝑞)/𝑞 (the average cost function) has a unique minimizer q; denote the minimum of 𝐶(𝑞)/𝑞 by p. (See Figure 1.) Assume that the inverse demand function 𝑃 is decreasing.

Show that in any Nash equilibrium of the game the firms’ total output 𝑄^∗ satisfies

𝑃(𝑄^∗ + 𝑞) ≤ 𝑝 ≤ 𝑃(𝑄^∗). (1)

(That is, the price is at least the minimal value p of the average cost, but is close enough to this minimum that increasing the total output of the firms by q would reduce the price to at most p.)

To establish these inequalities, show that if 𝑃(𝑄^∗) < 𝑝 or 𝑃(𝑄^∗ + 𝑞) > 𝑝, then 𝑄^∗ is not the total output of the firms in a Nash equilibrium, because in each case at least one firm can deviate and increase its profit. (You need to identify the firm that can profitably deviate and the deviation it can profitably make.)

Figure 1: The Average Cost Function of Each Firm

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QUESTION 1

Stackelberg duopoly game is also known as the ________ model. If we change the Stackelberg ______ competition game to a simultaneous-move game, we get the ______ game results.

	A.	leader-follower, quantity; Cournot
	B.	Competitive fringe; price; backward induction
	C.	leader-follower, quantity; Bertrand
	D.	entry, price; Cournot

QUESTION 2

Comparing Stackelberg and Cournot competition results, we can say that the _____ is better off while the ______ is worse off under Stackelberg than under Cournot results. This result show that there is _______________ advantage.

	A.	entrant, incumbent, investment
	B.	leader, follower, first-mover
	C.	follower, leader, a size
	D.	incumbent, entrant, first-mover

QUESTION 3

Mark all the FALSE statements

	A.	An equilibrium is a collection of strategies (and a strategy is a complete plan of action), whereas an outcome describes what will happen only in the contingencies that are expected to arise, not in every contingency that might arise.
	B.	In games of complete but imperfect information, backward induction is still the strongest process to solve the model to get unique equilibrium.
	C.	All subgame perfect Nash equilibria (SPNE) are Nash equilibria (NE), but not all Nash equilibria is SPNE
	D.	We cannot apply the notion of Nash equilibrium to dynamic games of complete information if we allowed a player s strategy to leave unspecified actions in some contingencies.
	E.	A game can be of perfect information whenever Nature or Luck does not play, each information set does not necessarily need to have a single node.

QUESTION 4

True or false. Mark the correct sequence:
I. Simultaneity of moves means that these games have imperfect information.
II. Dynamic games of complete and perfect information do not necessarily need that a player observes all the previous moves, just part of them are fine.
III. Backward induction and subgame perfect equilibrium concept lead to the same result in games of incomplete information.
IV. Any game in extensive form is a subgame itself.

	A.	TFFT
	B.	TTTF
	C.	FTFF
	D.	TTFT

QUESTION 5

The subgame perfect Nash equilibrium is the equilibrium associated with ____________ outcome. Subgame perfect equilibrium ________ non-credible threats.

	A.	Maxmin; involves
	B.	the backward induction; does not involve
	C.	Iterative deletion of weakly dominated strategies; rules out
	D.	the backward induction; includes

I have the following questions for my Managerial Economics course.

A principal-agent problems occur when managerial decisions are not consistent with the firm's shareholders' interests.

ECON 2350 Lecture Notes - Lecture 12: Best Response

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