Elasticity of Substitution or Why Simple Tools Teach Us Tons

I enjoy simple methods in economics. For economic history, which is my field of specialization, its often by constraint that I have to use them. Because of that, one has to be creative. In the process, however, one spots how well-used simple methods can be more powerful (both in terms of pedagogy and explanatory uses) than more advanced methods. Let me show you an example from Canadian history: the fur trade industry.

Yes, Canada’s mighty beaver! Generally known for its industriousness, the beaver has been mostly appreciated for its pelt which was the main export staple from Canada during the 17th and 18th centuries. In fact, if one is pressed to state what they think of when they think about Canada, fur pelts come in the top 10 (if not the top 5). It is thus unsurprising that there are hundreds of books on the business history of the fur trade in Canada.

One big thesis in Canadian economic history is that the fur trade was actually a drag on economic development (here and here and, most importantly, here with a wikipedia summary here). The sector’s dominance meant that the colony was not developing a manufacturing sector or other industries such as the timber, cod fishing, agriculture or potash. Political actors were beholden to a class of fur merchants who dominated. In a way, it looks a lot like the resource curse argument. And, up to 1810-1815, the industry represents the vast majority of exports (north of 60% always and generally around 75%). During the French colonial era, they represented 20% of GDP at some ponts.

Its only after 1815 that furs collapse as a staple — and quite rapidly. It represented less than 10% of exports and less than 2% of GDP by 1830. To explain the rapid turnaround, most of the available work has focused on demand for the industry’s output (see here) or internal industry factors. In a weird way, the industry is taken in isolation.

And that is where a simple tool like the elasticity of substitution between inputs becomes useful. First, I want you to notice the dates I invoked for the turning point: 1810-1815. These are not trivial years. They mark the end of the contest at sea between Britain and France and the beginning of the former’s hegemony on the sea. This means few trade interruptions due to war and insecurity at sea. Before 1815, the colonies in North America would have experienced nearly one year out of two.

What does that have to do with the fur trade’s dominance and elasticity of substitution? Well, it could be that war affects industry differently. Lets look at isoquants for a second to see how that could be the case. Imagine a constant elasticity of substitution function of the following shape:

Where L and K are your usual terms for labor and capital and r is the elasticity. Now, for the sake of argument, let us imagine what happens to the isoquant of a production function as r tends to infinity. As it tends to infinity, the marginal rate of technical substitution between L and K approaches zero if L > K. This means that there is a form of pure complementarity between inputs and no substitution is possible to produce the same quantity of output. The isoquant looks like this.

As r tends to infinity

On the other hand, if r tends to -1, there is perfect substitutability between both L and K. The isoquant then looks like this.

As r tends to -1

What if the fur industry’s isoquant looked more like the latter case while other industries looked like the former? More precisely, what if wars affected the supply of one input more than another? With a simple element like our description of the production function above, we see that if wars did not evenly affected the supply of one input, then one industry would be forced to contract output more than another. In our case, this would be the timber, potash, cod and agricultural sectors versus the fur trade.

Does that fit with the historical evidence? We know that the fur industry frequently changed the inputs it used in trading with the First Nations of Canada to buy furs. Whatever was deemed most valued by the natives would be what would be used. It could be alcohol, clothing, firearms, furnishings, silverware, tobacco, spices, salt, etc. This we get clearly from the work of Ann Carlos and Frank Lewis (a book linked to above). There was great ability to substitute. In contrast, other industries could not shift as easily. Take the timber industry which needed to import axes, saws, hoops, iron and nails from France or the United Kingdom for most of the 18th century. If wars disrupted the supply of these capital goods from Europe, there was very little substitution available which meant that the timber industry would have to contract output considerably to reflect the higher cost of these items. The same thing applies to the cod fishing industry whose key input was salt. No salt, no drying of the cod for preservation and export, thus no cod exports. And salt needed to be imported. In wartime, salt prices tended to jump much faster than other goods because its supply was entirely imported. Thus, wartime meant that the cod industry had to contract its output quite importantly.

The cod fishing industry is an amazing example of this if you take the American revolutionary war. During the war, the colony of Quebec (which represented 85% + of Canada’s population at the time) was invaded by the Americans and the French’s alliance with the Americans jeopardized trade between Quebec and Britain (its mother country at that point). The result was that salt prices jumped rapidly compared to all other goods and the output of the cod industry contracted. In contrast, the fur trade sector was barely affected. Look at this graph of the exports of beaver skins and codfish. Codfish output collapses whereas beaver skins barely show any sign of a major military conflagration.

In a longer-run perspective, its easy now to understand why the industry was dominant. It was the only industry that was robust to wartime shocks. All other industries would have had quite large shifts in factor prices causing them to contract and expand output in a very volatile manner. Now you may think this is just a trivial re-arranging of the argument. It is not because it invalidates the idea that the colony was poor or developed slowly because of the dominance of the fur industry. Rather, it shifts the burden on wartime shocks. Wars, not the dominance of the fur trade itself, meant that the economy was heavily mono-industrial.

A simple tool, the elasticity of substitution (which we can derive from the marginal rate of technical substitution), changes the entire interpretation of Canadian economic history. Can you see what I mean by the claim that simple tools combined with simple empirical observations can lead to powerful explanations? I hope you do! 

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