## A Simple Model

Here is a simple way to think about monetary policy and inflation. This model is astoundingly simple (at least by economist standards) and is certainly not a complete macroeconomic model as it only considers one sector of the economy (investment/production) and ignores the demand for consumption. Putting the two together greatly complicates things. Because of this, this model is not useful for doing welfare analysis but it does provide some intuition for how expansionary monetary policy causes deflation in the long run.

Consider first the market for investment where firms buy units of investment I at a given (fixed) price P1 in one period. In the next period, output is determined by the function F(I) which is increasing and concave. In other words, the more they invest in period 1, the more they produce in period 2 but additional units of investment lead to smaller and smaller increases in future output. In order to invest, they have to borrow money at the nominal interest rate i which I assume is set by a central authority. In the second period the price P2 is determined by the inverse demand function P2=G(Q). Let Pe be the expected price in period 2.

The firm’s problem is to choose I to maximize total profits which equal total revenue in period 2 minus the cost of investment.

max PeF(I)-(1+i)I

This gives the first order condition: PeF'(I)=1+i

Notice that the amount of investment depends both on P2 and i. If i is lowered it will increase investment. But the more investment there is, the more output there will be in the second period. This increase in output will lower the price of the good in the future. This will not be a problem if firms accurately forsee it. However, if they expect future prices to be higher than what they end up being (for instance, if the Fed convinces them that they can cause perpetual single digit inflation at the same time that they are lowering interest rates) then they will invest too much and the lower price in the future will cause them to be less profitable than they had expected (and potentially go bankrupt, default on their loans etc.).

The above is a model of overinvestment. The reason for this is that we assumed the price of investment in the first period is fixed. This is most likely not the case (although this is the sort of thing that Keynesians tend to imagine). It is actually not clear that a lower nominal interest rate should cause additional investment in the aggregate. However, we can develop an equally simple model where we assume the quantity of investment stays the same and derive the same result, namely that lower interest rates will cause lower inflation.

To see this imagine a model similar to above except that the total quantity of investment goods in the economy in period 1 is fixed but firms compete over them. In this case, output in period 2 will be fixed by this amount and so will P2. The demand for investment will still be determined by the first order condition derived above. However, now since I is fixed it will be P1 that has to adjust to make the demand equal to the supply in period 1. The lower the interest rate, the higher the demand will be and *the higher will be P1*. This means that lowering the interest rate will cause the price level to rise in the short run but will cause the increase (decrease) in prices in the long run to be lower (higher). Again, if firms expect P2 to be higher than it actually will be, then they will bid P1 up too high and the unexpected deflation will be a shock (think of the housing bubble).

I will try to weave these concepts into a richer model but this gives the basic idea of how interest rates affect inflation.