# A Liquidity Trap in an AS/AD Framework (Part I)

First post in a while. This part just sets the basic AS/AD model up. Part II will introduce the zero lower bound (ZLB), which is the fact that interest rates can’t fall below zero, and show that allowing for the ZLB leads to some interesting & surprising changes to the basic model. Part III will show how an economy can become trapped in a deflationary spiral (with inflation and output both falling forever) when the Fed hits the ZLB. Part IV will recommend policies to help avoiding the ZLB.

Before we talk about the ZLB, let’s discuss the basic math of the AS/AD model as presented in the intermediate textbook by Jones (2014). We will eventually end up with three (and then two) equations, which I discuss below in more detail. These three equations are:

IS: $\tilde{Y}_t = \bar{a} - \bar{b}(R_t-\bar{r})$
MP: $R_t = \bar{r} + \bar{m}(\pi_t - \bar{\pi})$
AS: $\pi_t = \pi_{t-1} + \bar{v} \tilde{Y}_t + \bar{o}$

The IS Curve describes how output, $\tilde{Y}_t$, changes in response to a change in the real interest rate, $R_t$. It is given by: $\tilde{Y}_t = \bar{a} - \bar{b}(R_t-\bar{r})$. Here, $\bar{a}$ is a demand shock, $\bar{r}$ is the marginal product of capital (you can think of this as the return that a businesses in the economy will receive if they buy one more machine). A good way to think about the real interest rate, $R_t$, is the return that businesses would receive if they put their money in a savings account. Therefore if the real interest rate increases, then businesses will be more likely to put their money in a savings account rather buying a new machine, and investment will decline.

The second equation is the monetary policy (MP) curve, which describes how the central bank changes the real interest rate in response to changing inflation: $R_t = \bar{r} + \bar{m}(\pi_t - \bar{\pi})$. Here, $\pi_t$ is the actual inflation rate this year, and $\bar{\pi}$ is the central bank’s inflation target. This rule says that as inflation increases, the central bank will raise interest rates (with the intention of decreasing investment, and therefore cooling the economy off).

The third equation is the aggregate supply (AS) curve. This equation describes how firms change prices, and therefore it describes how inflation changes over time. It is given by: $\pi_t = \pi_{t-1} + \bar{v} \tilde{Y}_t + \bar{o}$. Here, $\bar{o}$ is an inflation shock, and $\pi_{t-1}$ is the inflation rate last year. This equation says that if short-run output, $\tilde{Y}_t$, increases, then businesses are faced with a lot of extra demand, so they will raise prices. The story that we tell is that if businesses have a lot of demand, then they can raise prices by more than usual without fear of losing customers. If businesses are raising prices, then there is more inflation.

Taken together, we have:
IS: $\tilde{Y}_t = \bar{a} - \bar{b}(R_t-\bar{r})$
MP: $R_t = \bar{r} + \bar{m}(\pi_t - \bar{\pi})$
AS: $\pi_t = \pi_{t-1} + \bar{v} \tilde{Y}_t + \bar{o}$

We can combine the IS & MP curves to get the aggregate demand curve. Now we have our two equations:
AD: $\tilde{Y}_t = \bar{a} - \bar{b} \bar{m}(\pi_t - \bar{\pi})$
AS: $\pi_t = \pi_{t-1} + \bar{v} \tilde{Y}_t + \bar{o}$

Aggregate demand (AD) combines two relationships. First, if inflation rises, the central bank will increase interest rates. Second, if interest rates rise, investment will fall, leading to a decrease in output. Therefore, as we can see in the AD curve, if inflation increases, output will fall. Note that this is not just a law of nature – the reason that output falls when inflation rises is due to the policy of the central bank. This will be very important in future posts.

Graphically: This summarizes the model as presented in Jones (2014). However, this analysis completely neglects the zero-lower bound, i.e. the fact that nominal interest rates on savings accounts can’t go below zero (if interest rates were negative, then people would just keep their money at home rather than put it in the bank). Therefore, if the MP curve calls for the central bank to set a very negative real interest rate, the central bank will not be able to do it – this problem is called a liquidity trap, and it’s where policymakers around the world have found themselves stuck ever since 2009.

Because the model, as currently derived, does not take this possibility into account, it features a misspecified monetary policy curve (and therefore a misspecified AD curve, since MP is used to derive AD). How to fix the MP curve to allow for the fact that nominal interest rates can’t go negative is the subject of my next post.