Tuesday, February 9, 2016

Production possibilities and Brownian motion

At the end of the previous post, I discussed how a bowed-out [1] production possibilities frontier (PPF) could arise from essentially Brownian motion -- and I mentioned that I'd do a simulation to demonstrate that idea. I took random paths (of length 1000) that correspond to king moves on an infinite chessboard, starting in the bottom left corner. These paths look like this (we'll get to the blue line, but the blue dot represents the average end point ... and for one path, that is just the end point):




What is the average end point of 1000 paths?


The average end point seems to coincide with that blue line -- that's because it's the diffusion length D for 1000 time steps of a random walk with a step size of (roughly) unit length. You don't need to worry about the fact that one of the steps sizes is ~ 1.4. Anyway, the level curves of the (smoothed) density histogram for the end points are bowed out (the diffusion length is roughly the level curve corresponding to the mean value of the end point):


So we'd generally expect a two-good economy to be approximately the diffusion length D away from the origin [2]. Since this is a radius, this forms a set of points (PPF) that is bowed-out relative to the line between (D, 0) and (0, D).

Note that if there was some kind of constraint -- e.g. budget constraint B -- that was close to D, it could have some impact on the shape of the level curves. If B >> D, then there is little impact and D determines the PPF; if B << D, then B determines the PPF. Additionally, there could be reasons that the diffusion might favor movement parallel to the axes, resulting in a "bowed-in" PPF (e.g. economies of scale).

But without other considerations or constraints, the default should be bowed-out PPFs and upward sloping demand supply curves.

...

Footnotes

[1] By "bowed-out", we mean what Nick Rowe means. Bowed-out PPFs mean upward sloping supply curves.

[2] If diffusion was asymmetric, you'd end up with an asymmetric diffusion length and quarter-ellipse PPFs.

13 comments:

  1. Jason,

    I really can't see the point of the exercise.

    It proves nothing at all. What other result would you expect from such a simulation?

    You must have a lot of time on your hands. :-)

    ReplyDelete
    Replies
    1. Jason,

      It seems Nick has a lot of time on his hands also.

      Henry

      Delete
  2. Jason, how would you relate this to Nick's long thing, North-South aligned island example, with apples growing best in the North and getting worse as you go South, and vice versa for bananas (good in the South, and not good in the North). This produces a bowed-out PPF as well, with any point on the PPF corresponding to a particular boundary between the apple producing North and the banana producing South. E.g. the PPF could be a 1/4 of circle, as in your figures above.

    I was imagining randomly assigning land to apple producers and banana producers, and perhaps land that was left unused, in such a way that any point in the area bounded by the three curves: the PPF, the x-axis, and the y-axis had equal probability of being the result. (The distribution is uniform over the 1/4 disk).

    Then I imagined adding a 3rd fruit which grew best at middle latitudes, making our PFF a surface (e.g. the surface of an eighth of a sphere). And then continuing on to higher dimensional hyperspheres. The PPF would be a constraint, not allowing any points outside it. But still I can see that the center of mass of these hyperspherical sections will migrate towards the PPF as the number of dimensions increases.

    However, I'm not sure how a random walk fits in (if it does in my example). Is a random walk similar to the randomly assigned land owners trading with each other at random, or perhaps starting to cultivate new land, or abandon old land at random?

    Maybe the random walk doesn't apply in my example.

    ReplyDelete
    Replies
    1. Lol, my first sentence: should read "Nick's long thin" not "thing."

      Delete
    2. I'll read what you suggested for anonymous above.

      Delete
    3. One thing you have to think about is that each point in the random walks represents an allocation of both products -- it is an explored state of the economy by the agents in that economy (think of millions of agents collectively rolling a ball around -- those points above -- by their choices to produce apples or bananas).

      Delete
    4. "each point in the random walks represents an allocation of both products"

      So if random land trades between agents, or random abandonment or settling of new land is significant enough (when all such events are combined over a time period) this *could* create a change in the allocation of one or both products (i.e. a step in a random walk).

      Delete
    5. You'd have to imagine every production or land decision by every agent being one of the water molecules that bumps the pollen (the thing undergoing brownian motion) -- they collectively lead to a random movement of the economic state (pollen grain). Since it is so complex, you have no idea which way it actually goes and randomness would be a good baseline assumption.

      Delete
  3. Jason: Typo (I think) "But without other considerations or constraints, the default should be bowed-out PPFs and upward sloping demand curves."

    Should be *supply* curves.

    BTW, here's an ecologist picking up on the same theme:

    https://dynamicecology.wordpress.com/2011/04/27/why-expect-trade-offs-in-ecology-and-evolution/

    ReplyDelete
    Replies
    1. Yes, it was a typo. Probably due to thinking upward sloping demand curves are more a more interesting case.

      That is a great article (as was your original).

      One thing that I think is interesting is that if the dimension of the space where that random walk takes place is high (say 1000 dimensions), then you are most likely to find the end states of those random walks almost as far as they can be from the start point. This is a basic property of high dimensional spaces -- nearly all of their points are near their surface.

      This leads to an observation: that allocations are near the PPF (and organisms have maximal relative fitness) simply because they are the result of randomly exploring a high dimensional space. Those 1000 different genes or 1000 different products mean 1000-dimensional spaces with the bulk of points near the surface of the explored area.

      I'll have to think about it some more to make it into a real argument.

      Delete
  4. O/T: Jason, apparently David Glasner was as entertained by your exchange with Avon as I was: he's quoted a large part of it in his latest post.

    ReplyDelete

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