Wednesday, February 17, 2016

The value of diversity and upward sloping supply curves

Let me expand on something I said in this post:
There's another possible explanation of the bowed-out [Production Possibilities Frontier] curve. In the information equilibrium model, information entropy is equivalent to aggregate demand. Therefore the states with higher information entropy (i.e. states with more equal probability of finding apples and bananas) have higher AD relative to states with lower entropy (i.e. states with higher probability of finding either apples or bananas). Therefore AD near the middle of the PPF is slightly higher. This leads to a bowed-out PPF and upward sloping supply curves.

In these two posts ([1] and [2]), I show how to account for a contribution to output due to entropy (in terms of economic potentials, analogous to thermodynamic potentials). That is to say nominal output = sum of goods and services + entropy of goods and services. We don't know what the coefficient of the second term is exactly so I let it vary in the simulation below. We take the quantity of goods and services to be limited by a budget constraint (i.e. more X → X + dX means less Y → Y – dX), but allow that budget constraint to have a contribution due to entropy. There is a "real" budget constraint -- one more Xylophone means one less Yak -- but the nominal value of 5 Xylophones and 5 Yaks is greater than the nominal value of 10 Yaks or 10 Xylophones. By how much? I let that vary from zero to "a lot" in the simulation. One other thing to note is that I discussed this idea here in the context of Diane Coyle's review of Cesar Hidalgo's book Why Information Grows.

So here is the simulation. I generated 10,000 allocations of up to 50  yaks and xylophones (X + Y = 50) and added a constant (ranging from zero to "a lot") times the entropy of the resulting allocation to the total value of the yaks and xylophones ... and then normalized everything because the specific numbers don't matter. Here's the result (blue dots are the 10,000 allocations, the dashed straight line is the "real" budget constraint X + Y = 1 ... i.e. the prices of X and Y are equal, and the dashed curved line is the "real" budget constraint plus the entropy = PPF):

You can see that the entropy term creates a bowed-out PPF -- and thus upward sloping supply curves. The entropy term measures the value of diversity ... as Diane Coyle put it: a knife, a fork and a spoon is worth more than three spoons.


  1. Very interesting. Can we relate this to the reason for upward sloping supply curves you discuss here?:

    1. I imagine you'd proceed down the path where one good is the supply or demand bath for the other good.

      But as that post doesn't depend on entropy being output, there is no reason for the two to be related.

    2. Also the supply curve doesn't have to slope up in general.

  2. Jason, in your animation, when the blue dotted line representing the PPF is maximally bowed-out, it intersects the points (0,1) and (1,0) with positive slopes. Is that actually possible for a PPF?

    I'm thinking of Nick's long skinny island again, with apples doing better in the North and bananas doing better in the South.

    Positive slopes at both (1,0) and (0,1) means that starting from an island 100% dedicated to growing one fruit, and then dedicating an incremental amount to the other fruit, means production of the 1st fruit increased! Thus, how could dedicating the island 100% to one fruit have ever have been on the PPF to begin with?

    This brings up the notion of the benefits of "crop rotation" to me (if we somehow make it an intertemporal PPF).

    1. I don't see how Nick's land use model is completely general and describes all possible production functions.

      Don't get confused thinking there is a reason behind the entropy description ... That it is more efficient to grow bananas in the warmer part and apples in the colder part of the island.

      There is literally no reason for the entropy gain besides there simply being more information entropy in an allocation that has equal amounts of two things.

      That is to say: there doesn't have to be any just so stories for supply curves to slope up. They naturally do if entropy is proportional to output.

    2. I agree that Nick's curve isn't general. I was just curious why PPFs with a positive slope seem so rare. Yours is the only one I've seen.

      Nick answers me here.

    3. I finally see what he's saying: free disposal means no positive slope on the PPF. People that draw PPFs must always assume free disposal.

    4. Crop rotation would also do it. You need to plant some bananas in order for apples to grow (but not necessarily one for one).

      But again, this is pure entropy here. Everything is just worth more if there are other goods around.

      Imagine going up and to the right from (0,1) is always accompanied by economic growth where everyone wins.

    5. It is a pure entropic benefit with no microeconomic explanation.

  3. It seems to me that this is a good corrective to the maximum utility fetish. :)


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