Econ 387L: Macro II
Spring 2008, University of Texas
Instructor: Dean Corbae
Answers Problem Set # 9
Problem 2
Consider a version of the environment studied by Stokey (1989) that was taught in class. Let
(
x, X, y
)
be
the choice variables available to a representative agent, the market as a whole, and a benevolent government,
respectively, where
x, X
∈
X
=
{
x
L
, x
H
}
and
y
∈
Y
=
{
y
L
, y
H
}
. Let per period payo
ff
s to the government
be denoted
u
(
x
i
, X
j
, y
k
)
. In the case where
x
j
=
X
j
let payo
ff
s be given as in the table
u
(
x
i
, X
j
, y
k
)
X
L
X
H
y
L
0*
20
y
H
1
10*
The values of
u
(
x
i
, X
j
, y
k
)
not reported in the table are such that the competitive equilibria (i.e. ones where
x
=
X
=
h
(
y
)
) are the outcome pairs denoted by the asterisk.
1
Objective
:
The objective of this problem is to help you understand how we may take a Ramsey problem
(a sequential game) and imbed it in a dynamic simultaneous move game.
Using a trigger strategy (i.e., a
threat), the government can be induced to play the best outcome even though in a single period, its incentive
is to deviate.
We will also see that many strategies may be played that satisfy the trigger strategy and that
there exists a tension between the agent’s discount factor and the punishment strategy  as the pain of the
punishment increases, the discount required to support the optimal strategy decreases.
Conversely, if I care
more about the future then the punishment in
fl
icted upon me in the future matters more.
(1) De
fi
ne a Ramsey plan and a Ramsey outcome for a oneperiod economy. Find the Ramsey outcome.
Answer:
In a Ramsey equilibrium, government goes
fi
rst and the public responds to that decision, playing a best
response which is necessarily a competitive equilibrium. Assuming that government’s objective is to maximize
public utility, a Ramsey equilibrium solves
max
y
u
(
h
(
y
)
, h
(
y
)
, y
)
(1)
which in our example is
(
y, x
) = (
y
h
, x
h
)
.
(2) De
fi
ne a Nash equilibrium (in pure strategies) for the oneperiod economy.
Answer:
A NE is a triplet
¡
x
NE
, X
NE
, y
NE
¢
such that this set solves
max
x,y
u
(
x, X, y
)
(2)
where
¡
x
NE
, X
NE
, y
NE
¢
is a competitive equilibrium.
(3) Show that there exists no Nash equilibrium (in pure strategies) for the one period economy.
Answer:
1
The problem imposes consistency (
x
=
X
)
in the set up, so the solutions will also impose this consistency.
A more complete
setup would include payo
ff
s that allow agents to play strategies di
ff
erent than the average agent. For completeness assume
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 Spring '07
 CORBAE
 Game Theory, Subgame perfect equilibrium, Ramsey, yl, Ramsey Plan

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