## A Bit More Finance

In this post I will try to explain the single most important “big picture” concept that is essential to understanding the macroeconomic model which I have been describing so far. It is the way that the price of an asset should change over time in a free market. Throughout this discussion it will be assumed that asset owners have perfect information about the future. In other words, they know what demand will be at all times in the future. Later, we will talk about what happens if this assumption does not hold and that will be a key to understanding what we are actually observing in the real world.

Begin by imagining a firm which holds some fixed amount of a valuable resource and imagine for now that they have a monopoly on this resource. It wants to sell the resource over time in such a way so as to maximize the present value of all the revenue it receives. It takes the interest rate (the rate at which future revenues are discounted) as given. If they face a downward sloping demand for their product in each period, then they face the following practical problem: do I want to sell more now which will lower the price now or save it until some later date, in which case, the revenue will be discounted more?

To get the intuition of this consider two periods (1 and 2) and two given prices (P1 and P2) which represent the prices in their respective periods. In this case, if you are choosing to sell in period 1 or 2 you simply compare P1 to (1/(1+r))P2 (the discounted value of P2). If the P1 is larger, you sell it all in period 1 and if the discounted value of P2 is larger, you sell it all in period 2. If this is not intuitive, notice that instead of selling a unit in the second period and receiving P2, you could sell it in the first period and invest the revenue (P1) which would earn the rate of return r.

Now if we shift to considering a monopoly firm who faces a downward sloping demand curve, they affect the prices in both periods. If they sell more in period 1 and less in period 2, P1 will decrease and P2 will increase (and vice versa). What this means is that if P1>(1/(1+r))P2, you would want to sell more in period 1 and less in period 2. As you do this P1 decreases and P2 increases and you will have an incentive to keep shifting sales from period 2 to period 1 until P1=(1/(1+r))P2. Similarly, if P1<(1/(1+r))P2 the firm will want to shift sales from period 1 to period 2. If there are more periods, the reasoning is the same. The result of all this is that you want to time your sales so that the price grows at rate r. This famous result is known as the Hotelling rule.

We can derive a similar rule when the resource is owned by many different firms/individuals in a competitive market. The logic is similar. If you owned some of the resource and the price today was greater than the discounted value of what you expect the price to be in the next period (or any other period in the future for that matter), then you would want to sell all that you have (and maybe sell some short). So if the market expected the price to be growing at a rate slower than r, there would be what is sometimes referred to as “predominant selling” and this would bring the price of the asset down. Conversely, if the market expected the price of the asset to increase faster than the rate of interest, people would want to buy the asset and this predominant buying would drive the price up. The equilibrium in this market today would therefore be precisely that price which leads the market to expect the price to increase at rate r.

Alright, so here is the takeaway. When the interest rate changes, it should have two distinct effects on the price of the asset.

1st effect: The price of the asset will move in the opposite direction of the interest rate. With perfect information, this would happen instantaneously.

2nd effect: The *growth rate* of the asset price will change along with the interest rate.

The following graph illustrates the time paths of an asset price for three given interest rates where r1>r2>r3.

Ok that concludes the lesson for today. This is pretty basic economics but its implications for monetary policy are not trivial. I will begin to get into that next.

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