Economic Models - Convex Optimization

86 Andrew Hughes Hallet
5.4. Institutional Design and Policy Choices
We characterize the strategic interaction between the government and the
central bank as a two-stage non-co-operative game in which the structure of
the model and the objective functions are common knowledge. In the first
stage, the government chooses the institutional parameters δ and λ cb . The
second stage is a Stackelberg game in which fiscal policy takes on a leadership
role. In this stage, the government and the monetary authority set their
policy instruments, given the δ and λ cb values determined at the previous
stage. Private agents understand the game and form rational expectations
for future prices in the second stage. Formally, the policy game runs as
follows:
5.4.1. Stage 1
The government solves the problem:
{ 1
min ELg (g t, m t, δ, λ cb ) = E
δ,λ cb 2 [π t(g t ,m t ) −ˆπ] 2 − λ 2 1 [y t(g t ,m t )]}
+ λg 2
2 E[(b − θ)y t(g t ,m t ) − τ t (g t ,m t )] 2 (11)
where L g (g t ,m t ,δ,λ cb ) is Eq. (9) evaluated at (g t ,m t ,δ,λ cb ), and E
denotes expectations.
5.4.2. Stage 2
(a) Private agents form rational expectations about future prices πt e, before
the shocks u t and ε t are realized.
(b) The shocks u t and ε t are realized and observed by both the government
and the central bank.
(c) The government chooses g t , before m t is chosen by the central bank,
to minimize L g (g t ,m t , ¯δ, ¯λ cb ) where ¯δ and ¯λ cb are at the values determined
at stage 1.
(d) The central bank then chooses m t , taking g t as given, to minimize:
L cb (g t ,m t , ¯δ, ¯λ cb ) = (1 − ¯δ)
[π t (g t ,m t ) −ˆπ] 2
2
− (1 − ¯δ)¯λ cb [y t (g t ,m t )] + ¯δL g (g t ,m t , ¯δ, ¯λ cb )
(12)