What effect should a lower real interest rate have on the Hotelling path? The answer is that it should get flatter: investors need less price appreciation to have an incentive to hold gold.
Yesterday, in my high school economics class, I posed this question:
Suppose you have a fruit tree that you know you can sell for $100 in five years. The value of the fruit will be 5 percent of the value of the tree each year. The interest rate is 10 percent. What price should the tree sell for today?
I wanted them to use the formula:
profitability of buying = rental rate plus appreciation rate minus interest cost = 0
We set it equal to zero because there should be no excess profits from buying the tree. In that case, the annual appreciation rate should be 5 percent. So the price of the tree today should be about $73, allowing for annual compounding.
Next, suppose that the interest rate falls to 8 percent. Then the appreciation rate should be 3 percent, and the price of the tree should be about $84, or something like that.
I think that is the same story that Krugman is telling. There is an assumption that the long-term nominal price is unchanged, in spite of the deflationary shock that Krugman is arguing is at the heart of this. And my guess is that you need a really spectacular drop in real interest rates to get the sort of increase in the price of gold that we have seen over the past few years.
In the end, I think my preferred model (and perhaps Krugman's as well) is that there are two different sets of expectations at work. Buyers of Treasuries expect deflation. Buyers of gold expect inflation. It is well nigh impossible to arbitrage across the two. You could try shorting both in some combination, but, as the past year has shown, the market can stay irrational longer than you can stay solvent.