Economic Models - Convex Optimization

84 Andrew Hughes Hallet
Given this new interpretation, we can now solve for πt e,π t, and y t from
Eqs. (3) and (4). This yields the following reduced forms:
[
π t (g t ,m t ) = (1 + αβ) −1 αβm t + αγg t + m e t + γ ]
β ge t + αε t + u t (6)
y t (g t ,m t ) = (1 + αβ) −1 [βm t + γg t − βm e t − γge t + ε t − βu t ]. (7)
Solving for τ t using Eqs. (5) and (7), then yields:
τ t (g t ,m t ) = [s(1 + αβ)] −1 [(1 + αβ + sbβ)m t − (1 + αβ − sbγ)g t
− sbβm e t − sbγge t + sb(ε t − βu t )]. (8)
5.3. Government and Central Bank Objectives
In our formulation, we allow for the possibility that the government and
an independent central bank may differ in their objectives. In particular,
we assume that the government cares about inflation stabilization, output
growth, and the provision of public services (and hence the size of the public
sector deficit or debt); whereas the central bank, if left to itself, would be
concerned only with the first two objectives, 15 and possibly only the first
one. We also assume that the government has been elected by majority
vote, so that the government’s loss function reflects society’s preferences
to a significant extent.
Formally, the government’s loss function is given by:
L g t = 1 2 (π t −ˆπ) 2 − λ g 1 y t + λg 2
2 [(b − θ)y t − τ t ] 2 (9)
where ˆπ is the government’s inflation target, λ g 1
is the relative weight or
importance that the government assigns to output growth, 16 and λ g 2
is the
relative weight, which it assigns to the debt or deficit rule. The parameter θ
represents the target value for the debt or deficit to the GDP ratio, which the
government would like to reach: hence (b − θ)y t becomes the target for its
15 Since the central bank has no instruments to control the debt itself, it can only react to
poor fiscal discipline indirectly: e.g., to the extent that its inflation objective is compromised,
where it does have an instrument.
16 Barro and Gordon (1983) also adopt a linear output target. In the delegation literature, the
output component in the government’s loss function is usually represented as quadratic to
reflect an output stability objective. In our model, the quadratic term in debt/deficits allows
monetary and fiscal policy to play a stabilization role as well as pick a position on the
economy’s output-inflation trade-off.