11/29/13

One Horse Trick

A (Most Likely Incorrect) Thought Experiment (Ridiculously Wonkish)




My favorite way to approach economics (as I've written about here, here, and here) is through the equation of exchange, MV = PQ: money, multiplied by its velocity, is equal to price times the quantity of real output an economy produces. It's my favorite approach because it's both simple and complicated, obviously true but also open to interpretation. 

One of the common ways to interpret this deceptively simple equation is Milton Friedman's way, which is, in a nutshell, to assume that M has a one-to-one impact on P, and no impact on Q. Hence, monetarism. Milton based his interpretation partially on the interpretation of philosopher David Hume, who, writing before the relationship between money, velocity, price, and output had been formalized into an equation, argued that a change in M should only affect only P with a thought experiment: if everyone in England woke up one day with twice as many coins in their pocket, prices would change, but nothing else.


Thought experiments aren't particular popular nowadays, mainly because they are not particularly precise.  The enduring ones are rigorous in their assumptions and ruthless in their logic, and convincingly point out some fundamental truth or paradox, but they remain unclosed, unproven. They are like a one-sided equation, something we must satisfy ourselves with until we gain enough understanding to firmly establish meaning in the methodical language of math. However, they can come in handy when math is inadequate, like with the equation of exchange, or when, like me, you don't know enough math. 

Like all thought experiments, Hume's is limited, postulating a neutral relationship between M and Q under a very specific circumstance. Since Hume's heyday, the relationship between M, P, and Q has already been nicely clarified by expectations theory, but, as far as I understand, the relationship between V, P, and Q, is still largely up for grabs.  

Milton's monetarism is founded on the idea that M has a direct impact on P, but also usually assumes that V is a constant, or that V has a direct relationship with Q and can therefore be treated essentially like a constant when dealing with questions of money.  This approach has a certain logic; an increase in the velocity of money brought about by some real technological improvement, like, for example, a better highway system, should bring about a real increase in output. Conversely, John Maynard Keynes argued that V could impact P via changes in liquidity preference, but, like Friedman, tended to place more emphasis on the impact of V on Q, or the impact of liquidity preference on unemployment. Unfortunately, because of the frustrating ambiguity of the equation of exchange, and subsequent models, not much can really be proved either way. 

Of course all this is just an extended, unnecessary prelude to my thought experiment, the incredible, the fantastic, the confounding, One Horse Trick!!! (Warning, this is about to get even more boring). 

Imagine a three person economy:  at one end of the valley, there is a baker, at the other end, a brewer (As a vegetarian and a closet alcoholic, I always replace the butcher with a brewer), and constantly traveling in-between, a horseman.  The baker, baking at maximum capacity, is able to bake 15 loaves of bread a day, 5 of which he eats and 10 of which he trades. The brewer, brewing at maximum capacity, is able to brew 15 pints a day, 5 of which he drinks, and 10 of which he trades. The horseman can at maximum transport 10 items each day, delivering 5 loaves to the brewer, 5 pints to the baker, and taking 5 of each for himself. For money, the economy has 10 gold doubloons.


Never fear, the picture is here: 



Every morning in this economy, the horseman takes 5 doubloons and 5 loaves from the baker to deliver on his behalf to the brewer, and 5 doubloons for his fee; when he arrives at the brewer's at midday, he pays 10 doubloons for 10 pints, 5 for himself, which he immediately imbibes, and 5 for the brewer, and receives 10 doubloons back from the brewer for the loaves he has brought and for a fee to ferry 5 pints back to the baker. When he arrives back at the baker's at dusk, the horseman delivers the 5 already paid for pints to the baker, and pays the baker the brewer's 5 doubloon fee for the last 5 loaves of bread. (There is no credit in this economy). 


At the end of the day, the accounts are thus: 



In this economy, no one saves, so Investment = 0 = Saving, and Income = 30 = Spending, (ISn't it nice how that just IS all of the time?) Income and spending are also equal to price times quantity of real output, PQ, and each doubloon is spent 3 times, so:  

M = 10 
V = 3
P = 1 
Q = 30. 


Given this setup, you can immediately see that doubling the amount of doubloons will double P, so Hume's thought experiment holds.  The inverse condition, that doubling Q either by doubling the economy's population, or by giving the baker a better oven, the brewer a bigger brew kettle, and the horseman a second horse, would halve P, also holds:



(You have to assume that the horseman can direct his second horse remotely because the second horse is a magical telepathic unicorn. Ok ok, this isn't a perfect thought experiment.)

If only the baker or the brewer of the economy became more efficient, then the price level would fall by some fraction, and income would also reallocate to reward the more efficient producer at a some rate determined by the indifference curve between bread and beer. If only the horseman became more efficient through the acquisition of a second steed, nothing would happen, unless you assume that there is some benefit to receiving bread and beer at more regular intervals, so you can more consistently avoid both hunger and sobriety. In this case, one horse would start at one end of the valley, the other horse would start at the other end of the valley, and each would carry half of the items, and start with half of the money, the price level would fall, and presumably the horseman would receive some slight relative compensation for this added utility. 


Alternatively, you could replace the magical telepathic unicorn with a horse that can travel twice as fast:

Actual name: Rainbow Dash
What would happen here? Well, if the baker and the brewer don't get any more efficient, they still produce 15 loaves and 15 pints a day, respectively. There is still a marginal preference for more frequent trading, however, so instead of making one round trip across the valley, Rainbow Dash makes 2, carrying 2.5 items each trip, but also, because there is only one of her, all 10 doubloons. The accounts end thus:


Note that: 

M = 10 
V = 6
P = 3/2
Q = 40 

So, an increase in V increased both P and Q, disproving the neutrality of V in the Quantity Theory and also the independence of V and P in certain branches of Keynesian theory. 

WHOAAAA!!! REALLYY???? 


I don't know, it's a freaking thought experiment. I probably forgot or messed up a crucial assumption. And if, as both Keynes and Milton argued, their approach makes more sense empirically, then all the theory is for naught. But it does make you think... 





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