- Discretionary Policy

Appendix 3 - Discretionary Policy

(1) Deriving the Bellman equation The first problem we face is in formulating a recursive problem when our

model contains expectations of the future value of variables, in particular con- sumption, E g

t c t +1 and inflation, E t π t +1 . However, since we have a linear- quadratic form for our problem we can hypothesize a solution for these en- dogenous variables of the form,

c g t = CS t −1 π t = FS t −1

where C = c

1 c2 c3 c4 c5 and F = f 1f2f3f4f5 ⎡ are two ⎤

bb t

c g t ⎤ 1x5 vectors of undefined constants and S t = ⎢ N ⎥

the vectors of state and control variables respectively. 13

Using the former of these we can write the equations describing the evolution of the state variables 14 as,

(1 + ϕ) where B0 1,1 =β − σβc1, B0 1,2 = (σ(1 −ρ a β) − (σ − 1)(1 − β)) σ+ϕ − σβc2

B 0 1,3 = −(σ(1 − ρ N β) − (σ − 1)(1 − β)) σ+ϕ − σβc3 wN τ

B 0 1,4 =

b − σβc4, and, B0 1,5 = σβ(1 −ρ ξ ) − σβc5

13 We treat the consumption gap as a control variable since the monetary authorities have perfect control of this variable by varying interest rates.

14 In this section we make the empirically plausible assumption that debt is denominated in nominal terms.

B B (ϕθ + σ)( B −(1 + γ) where B2 1,1 =

Y − (1 − θ)) − θ(1 − θ) − (1 − β)θ Y + βϕθ Y

B 2 1,2 = Y )((1 − β) Y + (1 − θ))

B ( B −(2 − θ) − (1 − β + γ)

B and,

which allows us to rewrite the equation of motion for the state variables as,

Similarly we can write the evolution of inflation as follows,

− β Leading equation (16) forward one period and utilising the equation describing

the evolution of the state variables, we can write, FD1S t −1 + FD2u t = A1π t + A2u t

Solving for inflation,

π t = C1S t −1 + C2u t

where

C1 ≡ [A1] −1 [FD1]

and,

C2 ≡ −[A1] −1 [A2 − FD2]

These allow us to derive equation (14) in the main text

(2) Solving the Bellman equation The first-order conditions with respect to the control variables from solving

(14) are then given by,

∂V (S 2C2 0 R t π t + (Q + Q 0 )u ) t + βD2 0 E t =0

∂S t

Note that since Q has a middle row and column of zeros, the focs will not contain any terms in the tax instrument, such that we effectively have three

g focs in four unknowns, π g t ,y t, g t and E t ∂V (bb t+1 ) ∂bb t+1 . These can be arranged as the following linear target criteria,

where H = ⎣ Q 2,1 +Q 1,2 Q 2,3 +Q 3,2 βD2 2,1 ⎦ and Q i,j denotes the ele- Q 1,3 +Q 3,1

2Q 33 βD2 3,1

ment contained in row i, column j of matrix Q. This can be solved to yield the following target criteria under discretion,

(ϕ+σ)σθ((1 −θ)+(1−β) B Y B +1+ γ)

(θ(ϕ+σ)+σ(β(1+ϕ) X −1) Y )ε

σ(ϕ+σ)((1 −θ)+(1−β) B +1+ B γ)

2( −γ B Y

Y (1 −σc1−f1)+f1(2−θ)+f1 B Y (1 −β))ε

γ((1 −θ)+(1−β) B +1+ Y B Y γ)

Note that the first two elements do not depend upon our ‘guess’ parameters,f1..f5 and c1..c5, and imply that there is a linear relationship between the consump- tion gap and inflation and between the government spending gap and inflation under discretion. The implications of equation (19) for how policy instruments move debt over time are discussed in the main text.

The first order conditions with respect to the state variables are given by,

∂V (S t −1 ) ∂V (S t )

= 2C1 0 R π t + βD1 0 E t

∂S t −1 ∂S t Since the state variables relating to the shock process are exogenous, we can

focus on the first row of these first-order conditions to obtain,

Using the focs (i.e applying the envelope theorem),

∂V (bb t ) = Wπ t

∂bb t

where W ≡[2C1 2,1 R + βD1 2,1 X 3,1 ].

Leading this one period and applying expectations, ∂V (bb t +1 )

= WE t π t +1

∂bb t +1 = WFS t

= WF[D1S t −1 + D2u t ]

substituting back into the focs, 2C2 0 R π t + (Q + Q 0 )u t + βD3WF[D1S t −1 + D2u t ]=0

⎡ ⎤ D2 1,1

where D3 = ⎣ D2 1,2 ⎦. Eliminating inflation, D2 1,3

2C2 0 R [C1S t −1 + C2u t ] + (Q + Q 0 )u t + βD3WF[D1S t −1 + D2u t ]=0 and solving for control variables,

− [U1] −1 U2S t

where U1 =[2C2 0 RC2 +[Q+Q 0 ]+βBD3WFD2] and U2 = [2C2 0 RC1 +βD3WFD1]. The solution for inflation is now given as,

π = [C1

− C2[U1] U2 ]S t −1

However, this solution is a function of the undetermined coefficients, f 1.. f 5 and c1..c5, which can be derived by equating coefficients,

F = C1

− C2[U1] −1 U2

C =X 1,1 F

Figure 1: Coefficient G[1,1] as a function of the debt/gdp ratio and degree of price stickiness.

Response to a 1% Technology Shock Under Commitment and Discretion (Notes to Figure: Time period in graph in bottom left is 200 periods rather

than 10.)

The contribution of alternative policy instruments to debt adjustment under

discretion.