<p>You have the government's flow budget constraint (re-written in real terms):</p>
<p><span class="math-container">$b_{t} + m_{t} + \tau_{t} = g + \frac{m_{t-1}}{\pi_{t}} + R_{t-1}\frac{b_{t-1}}{\pi_{t}}$</span> (1)</p>
<p>Now all you need to do is substitute (2) and the policy rules (also, I don't think Leeper's utility function had a <span class="math-container">$\delta$</span> but that's not important) and linearise.</p>
<p><span class="math-container">$b_{t} +δc[\frac{R_{t}}{R_{t}-1}] + (\gamma_{0} + \gamma_{1}b_{t-1} + \psi_{t}) = g +\frac{δc[\frac{R_{t-1}}{R_{t-1}-1}]}{\pi_{t}} + (\alpha_{0} + \alpha_{1} \pi_{t-1} + \beta \theta_{t-1}) \frac{b_{t-1}}{\pi_{t}}$</span> (2)</p>
<p>Note: I left some <span class="math-container">$R_{t}$</span> and <span class="math-container">$R_{t-1}$</span> 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.</p>
<p>OP requested some more reading on this topic so I'll add a list here:</p>
<p>[<a href="http://sims.princeton.edu/yftp/Loyo/LoyoTightLoose.pdf" rel="nofollow noreferrer">1</a>] Loyo (1999), Tight Money Paradox on the Loose: A Fiscalist Hyperinflation</p>
<p>[<a href="http://www.columbia.edu/%7Emw2230/BOE.pdf" rel="nofollow noreferrer">2</a>] Woodford (1998), Public Debt and the Price Level</p>
<p>[<a href="http://www.columbia.edu/%7Emw2230/jmcb.pdf" rel="nofollow noreferrer">3</a>] Woodford (2000), Fiscal Requirements for Price Stability</p>
<p>[<a href="https://bfi.uchicago.edu/sites/default/files/research/Leeper-Leith-Handbook-Jan2016-Final.pdf" rel="nofollow noreferrer">4</a>] Leeper &amp; Leith (2016), Understanding Inflation as a Joint
Monetary-Fiscal Phenomenon</p>
<p>I have many more but start with these.</p>

CC BY-SA 3.0
2017-08-29T16:59:32.963Z
Thanks for replying! I guess my confusion comes from when do I substitute the $R_t$ and $R_{t-1}$ into the monetary policy rule. Do I take your (2) right now and try to linearise or should I just go ahead and replace the $R-_t$ and $R_{t-1}$ now?
sageofspades
CC BY-SA 3.0
2017-08-29T17:55:32.400Z
Correct. Substitute the $R_{t}$ and $R_{t-1}$ now and linearise. Because as you can see in (4), there are no R tilde's.
user11767
CC BY-SA 3.0
2017-08-29T18:34:47.547Z
Awesome. The only reason why I ask is because of the $\bar{R}$'s in the varphi's.
sageofspades
CC BY-SA 3.0
2017-08-29T20:53:12.070Z
I was able to figure it out! It was definitely not the prettiest algebra exercise, but I arrived at the same results as Leeper.
sageofspades
CC BY-SA 3.0
2017-08-29T22:07:26.667Z
Great! Nice to see people taking an interest in that paper :) And regarding the R's in the varphi's, they are steady state nominal interest rates as opposed to deviations from the steady state which is denoted by a tilde in the paper.
user11767
CC BY-SA 3.0
2017-08-29T22:57:39.693Z
Yeah, as soon as I was able to solve the issue I noticed it was just simpler to keep the R's. I am trying to do research for my dissertation in monetary/fiscal policy interactions. Do you have any other paper suggestions by chance?
sageofspades
CC BY-SA 3.0
2017-08-29T23:12:26.067Z
I have a lot of papers. I will try to provide links that aren't paywalled.
user11767
CC BY-SA 3.0
2017-08-30T13:37:37.377Z
Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/64826/discussion-between-benbernke-and-sageofspades).
user11767
CC BY-SA 4.0
2017-08-29T11:56:12.150Z
2022-10-24T19:09:05.237Z
2022-10-24T19:09:05.237Z
user11767
<p>I am attempting to solve a variation of <a href="http://mypage.iu.edu/~eleeper/Papers/LeeperJME1991.pdf" rel="nofollow noreferrer">Leeper's (1991)</a> model, which deals with the FTPL. This is what I have done so far:</p>

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

<p>I obtain two first order conditions:</p>

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

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

<p>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.</p>

<p>Suppose the monetary authority follows the following rule:</p>

<p>$R_t = α0+ απ_t+ θ_t R_t$, where $θ_t = ρ_1 θ_{t−1} + ϵ_{1t}, |ρ_1|&lt;1$.</p>

<p>The fiscal authority follows:</p>

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

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

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

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

<p>where the varphi's are the steady state constants:</p>

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

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

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

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

<p>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!</p>

CC BY-SA 3.0
2017-08-30T01:28:18.143Z
It might be nice (for search purposes and style) to expand the acronym FTPL to the fiscal theory of the price level.
jmbejara
<academic-graduate><difference-equations><rational-expectations>
Solving Leeper (1991) model
sageofspades
# Solving Leeper (1991) model

(Asked by: username_0 on 2017-08-29)

I am attempting to solve a variation of [Leeper's (1991)](http://mypage.iu.edu/~eleeper/Papers/LeeperJME1991.pdf) 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!

**Tags:** academic-graduate, difference-equations, rational-expectations

> **Comments:**
> **username_1**: It might be nice (for search purposes and style) to expand the acronym FTPL to the fiscal theory of the price level.  

---

## 1 answers

### Answer by username_2 on 2017-08-29. Score: 3

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](http://sims.princeton.edu/yftp/Loyo/LoyoTightLoose.pdf)] Loyo (1999), Tight Money Paradox on the Loose: A Fiscalist Hyperinflation

[[2](http://www.columbia.edu/%7Emw2230/BOE.pdf)] Woodford (1998), Public Debt and the Price Level

[[3](http://www.columbia.edu/%7Emw2230/jmcb.pdf)] Woodford (2000), Fiscal Requirements for Price Stability

[[4](https://bfi.uchicago.edu/sites/default/files/research/Leeper-Leith-Handbook-Jan2016-Final.pdf)] Leeper & Leith (2016), Understanding Inflation as a Joint
Monetary-Fiscal Phenomenon

I have many more but start with these.

> **Comments:**
> **username_0**: Thanks for replying! I guess my confusion comes from when do I substitute the $R_t$ and $R_{t-1}$ into the monetary policy rule. Do I take your (2) right now and try to linearise or should I just go ahead and replace the $R-_t$ and $R_{t-1}$ now?  
> **username_2**: Correct. Substitute the $R_{t}$ and $R_{t-1}$ now and linearise. Because as you can see in (4), there are no R tilde's.  
> **username_0**: Awesome. The only reason why I ask is because of the $\bar{R}$'s in the varphi's.  
> **username_0**: I was able to figure it out! It was definitely not the prettiest algebra exercise, but I arrived at the same results as Leeper.  
> **username_2**: Great! Nice to see people taking an interest in that paper :) And regarding the R's in the varphi's, they are steady state nominal interest rates as opposed to deviations from the steady state which is denoted by a tilde in the paper.  
> **username_0**: Yeah, as soon as I was able to solve the issue I noticed it was just simpler to keep the R's. I am trying to do research for my dissertation in monetary/fiscal policy interactions. Do you have any other paper suggestions by chance?  
> **username_2**: I have a lot of papers. I will try to provide links that aren't paywalled.  
> **username_2**: Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/64826/discussion-between-benbernke-and-username_0).  

---

