# Intuition for effect of interest rate changes in New Keynesian model

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.

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.
• I understand that the lack of borrowing is an equilibrium outcome, and that if $i$ was set in the market it would adjust. I'm more interested in the case where $i$ is a policy instrument set by the central bank. – John Nov 9 '20 at 12:46
• @John but that was exactly my point. In equilibrium where $i$ would freely adjust there would be no borrowing but when monetary policy changes $i$ then you can have intertemporal substitutions of consumption – 1muflon1 Nov 9 '20 at 12:47
• If all the agents are identical, as in a representative agent model, how does that happen? At the new $i$, every agent wants to borrow, which means no one can. Who do they borrow from? – John Nov 9 '20 at 12:49
• or saving/borrowing because otherwise $i$ should not be in the Euler equation. If interest lower interest rate means that people save less and hence consume more in current period that still expands output. However, in the above you do not explicitly show what is happening but we can infer something like that must be happening depending on your Euler equation. For example, have a look at representative household consumption models in Romer Advanced Macroeconomics – 1muflon1 Nov 9 '20 at 12:55