Solving Leeper (1991) model

Question

I am attempting to solve a variation of Leeper's (1991) model, which deals with the FTPL. This is what I have done so far:

The utility function is $\log(c_t)+\delta\log⁡(M_t/p_t)$.

I obtain two first order conditions:

(1) $\frac{1}{R_t}=β E_t \left[\frac{1}{\pi_{t+1}} \right]$ and

(2) $m_t = \delta c \left[ \frac{R_t}{R_t - 1} \right]$.

Where $R_t$ is the gross nominal interest rate, $\pi_t$ is the gross inflation rate and $c$ the deterministic steady state value of consumption.

Suppose the monetary authority follows the following rule:

$R_t = α0+ απ_t+ θ_t R_t$, where $θ_t = ρ_1 θ_{t−1} + ϵ_{1t}, |ρ_1|<1$.

The fiscal authority follows:

$τ_t = γ_0 + γ b_{t−1} + ψ_t$, where $ψ_t = ρ_2 ψ_{t−1} + ϵ_{2t}$.

I am having trouble seeing how Leeper was able to take all the given information so far and produce the following two equations:

(3) $E_t \hat{π}_{t+1} = αβ\hat{\pi}_t + βθ_t$

(4) $φ_1 \hat{π}_t + \hat{b}_t + φ_2 \hat{π}_{t−1} - (β^{−1} − γ)\hat{b}_{t−1} + φ_3 θ_t + ψ_t + φ_4 θ_{t−1}=0$

where the varphi's are the steady state constants:

$\varphi_1 = \frac{\delta c}{\bar{R} - 1} \left[ \frac{1}{\beta \bar{\pi}} - \frac{\alpha}{\bar{R} - 1} \right] + \frac{\bar{b}}{\beta \bar{R}}$

$\varphi_2 = - \frac{\alpha}{\bar{\pi}} \left[ \frac{\delta c}{\left(\bar{R} - 1\right)^{2}} - \bar{b} \right] $

$\varphi_3 = \frac{\delta c}{\left(\bar{R} - 1\right)^{2}}$

$\varphi_4 = - \frac{1}{\bar{\pi}} \left[ \frac{\delta c}{\left(\bar{R} - 1\right)^{2}} - \bar{b} \right] $

Note: I was able to solve for (3); however, I am still unsure why $\theta_t$ does not have a hat on top of it. If someone can walk me through how to reach (4), I would very much appreciate it!

Answer 1

You have the government's flow budget constraint (re-written in real terms):

$b_{t} + m_{t} + \tau_{t} = g + \frac{m_{t-1}}{\pi_{t}} + R_{t-1}\frac{b_{t-1}}{\pi_{t}}$ (1)

Now all you need to do is substitute (2) and the policy rules (also, I don't think Leeper's utility function had a $\delta$ but that's not important) and linearise.

$b_{t} +δc[\frac{R_{t}}{R_{t}-1}] + (\gamma_{0} + \gamma_{1}b_{t-1} + \psi_{t}) = g +\frac{δc[\frac{R_{t-1}}{R_{t-1}-1}]}{\pi_{t}} + (\alpha_{0} + \alpha_{1} \pi_{t-1} + \beta \theta_{t-1}) \frac{b_{t-1}}{\pi_{t}}$ (2)

Note: I left some $R_{t}$ and $R_{t-1}$ in there but you will have to substitute the monetary policy rule there as well. Then all you do is a Taylor approximation around the deterministic steady state. Just a warning, the algebra will be very hairy, it took me several tries to get it right.

OP requested some more reading on this topic so I'll add a list here:

[1] Loyo (1999), Tight Money Paradox on the Loose: A Fiscalist Hyperinflation

[2] Woodford (1998), Public Debt and the Price Level

[3] Woodford (2000), Fiscal Requirements for Price Stability

[4] Leeper & Leith (2016), Understanding Inflation as a Joint Monetary-Fiscal Phenomenon

I have many more but start with these.