Intuition for effect of interest rate changes in New Keynesian model

Question

I am trying to understand the intuition behind the effect of an interest rate change in a simple New Keynesian model (or really in any sticky price model).

As a simple example, I take (roughly) the model in https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxnYXV0aWVnZ2VydHNzb258Z3g6MTA1YTg1NTljNjE4ZjJhNg. In this model, a fraction $\gamma$ of firms sets their price optimally for that period, and the remaining firms index their price to last period. I have the following (log-linearised) equations:

1. Euler equation: $\hat{c}_t = E_t\hat{c}_{t+1} - \frac{1}{\sigma}\left(\hat{i}_t - E_t\hat{\pi}_{t+1}\right)$
2. Labour supply: $\hat{n}_t =\frac{1}{\phi}\left(\hat{w}_t - \hat{p}_t - \sigma \hat{c}_t\right)$
3. Optimal pricing: $\hat{p}_t^*-\hat{p}_t = \left(\hat{w}_t-\hat{p}_t\right) - \hat{a}_t$
4. Inflation: $\hat{\pi}_t = \frac{\gamma}{1-\gamma}\left(\hat{p}_t^* - \hat{p}_t\right)$
5. Output: $\hat{y}_t = \hat{a}_t + \hat{n}_t$
6. Goods market clearing: $\hat{y}_t = \hat{c}_t$

There is some redundancy because I am looking for intuition. The solution to this system is $$ x_t = E_t x_{t+1} - \frac{1}{\sigma}\left(\hat{i}_t - E_t\hat{\pi}_{t+1} - r_t^n\right) \\ \hat{\pi}_t = \kappa x_t $$ where $x_t = \hat{y}_t - \hat{y}_t^n$ is the output gap, $\hat{y}_t^n = \frac{1+\phi}{\sigma + \phi}\hat{a}_t$ is the natural level of output, $r_t^n = -\sigma\left(\hat{y}_t - E_t\hat{y}_t^n\right)$ is the natural rate of interest, and $\kappa = \frac{\gamma}{1-\gamma}\left(\sigma + \phi\right)$.

Now suppose the central bank reduces the nominal interest rate. Assume also that there is no change to the net supply of government bonds (which are the vehicle for borrowing and saving). I know that the outcome will be a positive output gap and increased inflation. I am trying to understand how that occurs.

The standard story is that there is consumption demand by the Euler equation, so firms have to increase output, which raises their costs. The firms that can do so will raise their prices. As a consequence, there is inflation and a positive output gap.

If this is the correct intuition, where does the increased demand faced by firms actually come from? Individuals do want to consume more (Euler eq, 1), but they cannot all borrow and they are already at their optimal intratemporal margin (labour supply, 2). Individuals can't actually consume more until their incomes/output is higher. They won't work more unless real wages rise. Firms won't increase output, wages, or prices unless there is more demand. What is the mechanism?

Thanks!

Edit - I clarified that individuals borrow and save via trading in government bonds, and that the net supply of government bonds is held constant.

Answer 1

Yes the 'standard story' you reference is the correct intuition here. Lower nominal interest rate encourage consumption which puts pressure on both output and prices.

Household can actually save and borrow in the model. However, because it is a representative household model in equilibrium there would not be any borrowing - the interest rate should just adjust in order to make agent's indifferent between consuming today and in the future - but the intertemporal consumption choice still depends on $i$ with higher $i$ incentivizing future consumption at the expense of the present one.

Also, even if in the description provided in question this is not shown the Euler equation has its micro foundation in model where households are allowed to save and borrow - otherwise the interest rate would not be part of the Euler equation in the first place, and I am not sure how it would even be modeled if there would be no link between present and future. For example, see the treatments of IS-LM models in lets say Woodford (2003). Interest and Prices. You will see that in dynamic models the Euler equation is typically derived from model of household behavior where households have access to saving/borrowing.