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Hume’s Sailing Ships correcting Samuelson’s Icebergs

Summary:
Paul Samuelson said (PITA gated) that David Hume's price-specie flow mechanism (ungated) was wrong. And I am saying, nervously biting my fingernails, that Samuelson was wrong. Assume that durable sailing ships are costly to build, but have low (or zero for simplicity) operating costs. Assume apples are the only tradeable good, and one ship can transport one apple per year across the English Channel between Britain and France (the only countries in the world). Let P be the price of apples in Britain, P* be the price of apples in France, and R be the annual rental of a ship, (all prices measured in gold), then R=ABS(P*-P). In this model, the Law of One Price (P=P*) will only hold if the volume of exports of apples (in either direction) is unconstrained by the existing stock of ships, so

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Paul Samuelson said (PITA gated) that David Hume's price-specie flow mechanism (ungated) was wrong. And I am saying, nervously biting my fingernails, that Samuelson was wrong.

Assume that durable sailing ships are costly to build, but have low (or zero for simplicity) operating costs. Assume apples are the only tradeable good, and one ship can transport one apple per year across the English Channel between Britain and France (the only countries in the world). Let P be the price of apples in Britain, P* be the price of apples in France, and R be the annual rental of a ship, (all prices measured in gold), then R=ABS(P*-P).

In this model, the Law of One Price (P=P*) will only hold if the volume of exports of apples (in either direction) is unconstrained by the existing stock of ships, so rentals on ships are driven to zero. But then no ships would be built to export apples if ship rentals were expected to be always zero, which is a contradiction of the Law of One Price because arbitrage is impossible without ships. But an existing stock of ships represents a sunk cost (sorry) and they keep on sailing even as rentals approach zero. They sail around Samuelson's Iceberg model (sorry) of transport costs.

If you are already familiar with Hume's essay, and have a good economic intuition, the rest of this post should be obvious.

Start with zero exports, zero ships, and P=P*. Then suppose, like Hume, that some of the gold in Britain magically disappears. (And unlike Hume, just to keep it simple, suppose that gold magically reappears in France.) The price of apples in Britain drops, the price of apples in France rises, and so the rent on a ship is now positive because you can use it to export apples from Britain to France. If that rent is big enough, and expected to stay big long enough, some ships will be built, and Britain will export apples to France in exchange for gold. Gold will flow from France to Britain, so the stock of gold will slowly rise in Britain and slowly fall in France, and the price of apples will likewise slowly rise in Britain and fall in France, so ship rentals will slowly fall, and the price of ships (the Present Value of those rents) will eventually fall below the cost of production, so no new ships will be built. But the ships already built will keep on sailing until rentals fall to zero or they rot (whichever comes first). Some clever grad student can do the math and figure out the parameter values for which the model converges all the way (rather than part way) back to the original equilibrium before all the ships rot. (My guess is that it depends on whether ships depreciate slower or faster than the rate of interest.) Should be a fun topic, combining irreversible investment with international trade and finance, plus there's data (see below).

The flow of exports and hence the flow of specie is limited by the stock of ships. And only a finite number of ships will be built. So we observe David Hume's price-specie flow mechanism playing out in real time.

This bugs me. Because it's all sorta obvious really. And my Sailing Ship model of transportation costs is a helluva lot more realistic than Samuelson's Iceberg model, where ships cost nothing to build but are fueled by burning apples (OK, the ship's crew could be fueled by eating apples). And it would have been a lot more likely to have been sorta similar to whatever was at the back of David Hume's mind when he wrote:

"What nation could then dispute with us in any foreign market, or pretend to navigate or to sell manufactures at the same price, which to us would afford sufficient profit?"

See that word "navigate"? Hume's gotta be thinking about sailing ships, right? And the Baltic Dry Index (equivalent to R in my model-sketch) fluctuates a lot, especially in times of international monetary disequilibrium, for the obvious reason that the stock supply of ships is very inelastic in the short run. It takes time to build new ships, and ships last a long time.

Prices don't just arbitrage themselves. Even if we take the limit of my model, as the cost of building ships approaches zero, we need to explain what process ensures the Law of One Price holds in equilibrium. Suppose it didn't...then people would buy low and sell high.....you know the rest.

Would introducing international finance (borrowing and lending gold) change the story? Yes, but not qualitatively. France would lend Britain gold, so prices would deviate less initially, so ship rentals would be lower, fewer ships would be built initially, and initial exports would be smaller, so we would observe Hume's price-specie flow mechanism for longer.

It doesn't have to be sailing ships. Any one-time investment cost of setting up in a new foreign market or expanding the volume of exports to an existing market could play the same role. Goods don't just export themselves. Hume's two century old essay is just as relevant today.

The volume of exports will not be a simple function of the current real exchange rate. It will be much more sensitive to changes in the real exchange rate that are expected to be longer-lasting than temporary. And there will be hysterisis, because the current stock of sailing ships will depend on a distributed lag of past expectations of future real exchange rates.

Thanks to C Trombley and David Glasner for letting me know about Samuelson's classic article. And to my daughter for helping me actually read it. This criticism of that article, and defence of Hume, can't be original, can it? Someone else should have thought of it earlier, shouldn't they?

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