# Government spending dynamics

I am currently reading Woodford (2010) and I have understood it for the most part. However, as I am new to solving these kind of models I have a question regarding a specific type of monetary policy rule. Namely one where the domestic central bank pegs domestic rates to foreign interest rates (due to e.g a currency peg).

The examples I have working with involved a Taylor rule of the form:

$i_{t} = \bar{r} + \phi \pi_{t}$ I then use the method of undetermined coefficients and assume a solution on the form:

$i_{t} = \bar{r} + \gamma_{i} \hat{g}_{t}$, $\pi_{t} = \gamma_{\pi}$ $\hat{g}_{t}$ , $y_t = \gamma_y \hat{g}_t$

and do the appropriate substitution in to the IS function:

$\hat{y_{t}} = E_t \hat{y}_{t+1} + \sigma^{-1}(i_{t} - E_t \pi_{t+1} - \bar{r})$

So the term inside the brackets become:

($\phi$ $\gamma_{\pi}$ $\hat{g}_{t}$ - $\gamma_{\pi}$ $\hat{g}_{t+1}$) and after assuming a functional form of $g_{t}$ I am able to solve the model.

However, now when the monetary policy rule is to peg the interest rate I am unsure how to guess the form of the solution.

My monetary policy rule is:

$i_{t} = i_{t}^{*}$ where $i_t^*$ is the interest rate of the foreign country. My initial thought was to do something like:

$i_{t}^* = \gamma_{i^*} \hat{g}_{t}$ but then when I try to solve it I end up with two equations (PC and IS ) and three unknowns ($\gamma_\pi,\gamma_i^*,\gamma_y$). TO clarify what I mean, consider again the term in brackets from the IS function:

$(\gamma_i^*\hat{g}_t - \gamma_\pi \hat{g}_{t+1})$, it's the $\gamma_i$ that is causing me problems. With the Taylor rule I was able to deal with it because the policy rule implies:

$\gamma_i = \phi \gamma_\pi$

Any ideas where I am going wrong?

• I am not very familiar with undetermined coefficients but to me, $i_{t}^* = \gamma_{i^*}\hat{g}_{t}$ doesn't make much sense. Surely shocks to domestic government spending does not affect foreign interest rates? – BenBernke Nov 24 '17 at 18:18
• Thank you Ben for the reply. What you say seems logical but I can't really elaborate further because I have no idea how else I could proceed. Do you know of any other methods for finding the multiplier? – Kronecker Nov 24 '17 at 19:35
• It is very likely I am wrong here (someone please correct me) but if you peg interest rates, shouldn't that fix the path of expected inflation due to the Fisher reaction? I.e $\bar{r} = i_{t} - E_t \pi_{t+1}$, if $i_t = i_t^*$, then $E_t \pi_{t+1}$ should be a constant. Again, don't take this as an answer, I am most likely wrong. – BenBernke Nov 24 '17 at 20:16
• I see. I'll try using the Fisher equation and see if I get an answer. Thanks once again for trying. – Kronecker Nov 25 '17 at 16:53