1.5. Consider the following two finite versions of the Cournot duopoly model. First, suppose each firm must choose either half the monopoly quantity, qm/2 = (a - c)/4, or the Cournot equilib- rium quantity, qc (ac)/3. No other quantities are feasible. Show that this two-action game is equivalent to the Prisoners' Dilemma: each firm has a strictly dominated strategy, and both are worse off in equilibrium than they would be if they cooper- ated. Second, suppose each firm can choose either qm/2, or qc, or a third quantity, q'. Find a value for q' such that the game is equivalent to the Cournot model in Section 1.2.A, in the sense that (qc, qc) is a unique Nash equilibrium and both firms are worse off in equilibrium than they could be if they cooperated, but neither firm has a strictly dominated strategy.

1.5. Consider the following two finite versions of the Cournot duopoly model. First, suppose each firm must choose either half the monopoly quantity, qm/2 = (a - c)/4, or the Cournot equilib- rium quantity, qc (ac)/3. No other quantities are feasible. Show that this two-action game is equivalent to the Prisoners' Dilemma: each firm has a strictly dominated strategy, and both are worse off in equilibrium than they would be if they cooper- ated. Second, suppose each firm can choose either qm/2, or qc, or a third quantity, q'. Find a value for q' such that the game is equivalent to the Cournot model in Section 1.2.A, in the sense that (qc, qc) is a unique Nash equilibrium and both firms are worse off in equilibrium than they could be if they cooperated, but neither firm has a strictly dominated strategy.

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

In the Cournot duopoly model, the analogous statement is that
the quantity pair (q₁,q) is a Nash equilibrium if, for each firm i,
q solves
max Ti(9₁,9₁)
0<9;<∞0
and
=
*
Assuming q <a − c (as will be shown to be true), the first-order
condition for firm i's optimization problem is both necessary and
sufficient; it yields
qi =
(1.2.1)
Thus, if the quantity pair (q†,92) is to be a Nash equilibrium, the
firms' quantity choices must satisfy
max qila - (qi+q₁) - c].
0<qi<∞0
92
1
-
2
1
q† = z (a − q ₂ − c )
2
(a - q - c).
1
-
₂ (a − 9₁ - c).
Solving this pair of equations yields
a C
91 = 92
gì =q2=ng?
3

==
1.5. Consider the following two finite versions of the Cournot
duopoly model. First, suppose each firm must choose either half
the monopoly quantity, qm/2 = (a − c)/4, or the Cournot equilib-
rium quantity, qc = (ac)/3. No other quantities are feasible.
Show that this two-action game is equivalent to the Prisoners'
Dilemma: each firm has a strictly dominated strategy, and both
are worse off in equilibrium than they would be if they cooper-
ated. Second, suppose each firm can choose either qm/2, or qc,
or a third quantity, q'. Find a value for q' such that the game is
equivalent to the Cournot model in Section 1.2.A, in the sense that
(qc, qc) is a unique Nash equilibrium and both firms are worse off
in equilibrium than they could be if they cooperated, but neither
firm has a strictly dominated strategy.

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