Economics of small changes

by paulfchristiano

Consider the following three actions:

  1. I bury $20 in my back yard.
  2. I sell a pound of strawberries at a farmer’s market.
  3. I accept an engineering job I’m offered.

Each of these actions has some obvious local effects:

  1. I’m $20 poorer.
  2. I have some money; my customer has some strawberries and they won’t buy strawberries from anyone else.
  3. I have an engineering job; my employer has an engineer and will not hire anyone else.

But each of these actions also has a more subtle long-term effect:

  1. The value of a dollar is slightly increased; everyone else is slightly richer (or the government is slightly richer, depending on their monetary policy).
  2. The price of strawberries is slightly decreased; worldwide strawberry production will decrease to substantially offset my contribution (and those resources will be used for other things).
  3. Engineering salaries are slightly reduced.

These long-term effects are very small but affect a very large number of people. They are mediated by complicated chains of cause and effect with which I have no direct experience. There are many causal steps between burying money in the ground and my local supermarket decreasing the price of milk very slightly. This seems like a brittle picture of how the world works, which could well be completely inapplicable to the real world even if it applies to an idealized efficient market.

Nevertheless, I think that these indirect effects are quite robust in the face of uncertainty and non-ideal behavior. As a corollary, I think the efficient-market hypothesis is almost always a much better approximation of the world than the oblivious-market hypothesis which is often implicitly invoked,  and that this approximation becomes better, not worse, as the situation becomes more complicated and uncertain. I think these points are widely accepted or assumed amongst economists, but I seem to often run into confusion about them.

The unifying characteristic of these scenarios is that they involve very small changes, in a domain where a large (but still marginal) change would have a predictable effect via a well-understood mechanism. I claim that the expected impact of each of these small changes is the same as the average impact of the associated large (but marginal) changes. Sometimes we can point to a particular small change and say “the effect of this particular change will be atypical,” but this is increasingly unlikely as the world gets more complicated and uncertain.

Local linearity

If a thousand extra people sell strawberries for 20 years, it is pretty easy to see that the normal laws of supply and demand will apply. In fact, you can look at economic data and estimate the effect those extra thousand people would have based on very closely analogous empirical evidence.

Now suppose that instead one extra person shows up every day for 20 years. What will the effect be? The important observation is that collectively the 1000 strawberry vendors have a pretty small effect on market conditions. So if we imagine adding them to the counterfactual one by one, each of them is going to have basically the same effect in expectation. And so the effect of the very first strawberry vendor is just 1/1000 of the total effect.

Similarly, if I scale down “20 years” to “one day,” I again scale down the expected impact by a factor of 7300. Of course, the effects of selling strawberries for 20 years are likely to be highly non-linear on me (I learn how to sell strawberries, people start to expect me to be there, etc.) but the effects of the actual strawberries I sold on the global strawberry market will be linear. 

So: if we can make a robust statement about the effects of a big change (e.g. based on relevant empirical evidence, or simple considerations), and a change has an approximately linear effect, then we can make a robust statement about what would happen due to a small change, even if the actual mechanisms of that change are nebulous. This didn’t rely on strong assumptions about efficient markets; it only relied on the assumption that significant increases in strawberry production would have a significant effect on strawberry prices. And given the state of the empirical evidence, calling that an “assumption” is pretty uncharitable.

I think this argument goes a long way towards justifying the use of classical economic equilibrium analysis in general, even when classical models are an extremely crude description of economic activity. I would expect such models to produce good approximations as long as the large-scale long-run behavior of a market is approximately efficient. Often it will be possible to significantly improve on such crude estimates (for example by having better models, or empirical evidence that supersedes theoretical speculation), but I think the efficient-market approximation is almost always much better than the no-market approximation.

Postscript: Long-run or short-run elasticity?

If I exogenously increase the supply of X, the price of X will fall, more people will buy X and fewer other people will sell it, and the market will still clear. If the supply is much more sensitive than the demand to changes in price, then the price will drop until the supply has decreased by 1, offsetting the exogenous change. If the demand is much more sensitive to changes in price,  then the price will drop until the demand has increased by 1, meaning that total consumption of X increases by 1. In between we have intermediate outcomes. The elasticity of supply (resp. demand) measures this sensitivity: an elasticity of p means that a 1% increase in price results in a p% increase (resp. reduction) in supply (resp. demand).  If the supply elasticity is s and the demand elasticity is d, then the total change in consumption is [d / (s + d)].

But this description is very incomplete: when the price changes, the reaction of the market is spread out over time. In general, the supply of good X won’t react immediately to a change in price. In the short run it is often difficult to reallocate capital or people to a different purpose, while in the long run there is more flexibility. So if I look up numbers on the price and demand elasticity of goods, I will typically find different values for the short run and the long run. If I’m considering the long-term effect of a permanent change in the supply of good X, then (by definition) I care about the long run elasticities. But what if I’m considering the effect of a temporary boost in the supply of good X?

The above analysis suggests you should care about the long run elasticity, even if the change is temporary (at least if you mostly care about the long run). This is just because the effects of changes in supply are probably linear, and a temporary change is just a small fraction of long-term change. I implicitly appealed to this principle in my earlier post about replaceability.

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