Monday, February 8, 2016

Production possibilities and the slope of the supply curve

There was a discussion on the blogs about teaching the Production Possibilities Frontier [PPF] (or curve) for two goods (say, Apples and Bananas) in introductory economics classes. Brad DeLong started it; Paul Krugman joined in. Then there was more from DeLong.

Nick Rowe went over it awhile ago in a post, and commented on DeLong's first post above, saying:
How can you (easily) explain (e.g.) why supply curves slope up without using a (curved) PPF?

Which is similar to what he said in his post:
Which means the PPF is now curved, and bowed out. ... Which means that the supply curve of apples will slope up.

Now I reconstructed the PPF using an information equilibrium model in this post based on Rowe's post. It turns out the PPF is a level curve of the production possibilities surface constructed from the quantity-weighted sum of the supply curves (surfaces) for the two goods. Here are the supply and demand diagrams (assuming the markets are independent and e.g. the supply curve line becomes a plane):

Here are the supply surfaces together:

And here is their quantity weighted sum -- the production possibilities surface [PPS] -- (with level curves, aka various PPF's):

If you take flat supply surfaces:

and take the quantity weighted sum, you get straight lines for your PPF's:

which is exactly how it works at Nick Rowe's post. But I did want to add a bit here. Nick Rowe's comment at DeLong's blog seems to suggest that a curved PPF "explains" the upward sloping supply curve. However, since those PPF's are level curves of the quantity-weighted sum of the two supply surfaces, the idea that "the PPF bows out" (the level curves of the PPS are bowed out) and the "supply curve for a single good slopes up" (i.e. the PPS has curvature) are not logically independent of each other. That is to say, they mean the same thing -- there is no knowledge added to a bowed out PPF to makes it lead to a upward sloping supply curve.

The curvature of the PPS is determined by weighting a locally linear supply curve 

P = a S + b

with a > 0 (i.e. upward sloping) by the quantity supplied, so we get

P × S = a S² + b S

Or in both directions:

P₁ × S₁ + P₂ × S₂ = a₁ S₁² + b₁ S₁ + a₂ S₂² + b₂ S₂

with a₁  and a₂  >  0 which is locally a paraboloid. Therefore the statements

P = a S + b

P₁ × S₁ + P₂ × S₂ = a₁ S₁² + b₁ S₁ + a₂ S₂² + b₂ S₂

are not logically independent. A "bowed out PPF" defines supply curves as upward sloping, and upward sloping supply curves defines the PPF as bowed out. A bowed PPF curve doesn't "explain" the supply curve any more than non-Euclidean geometry "explains" why the parallel postulate fails.

Now I think Rowe could mean that the bowed PPF curve is more intuitive than an upward-sloping supply curve -- and that I would completely agree with. Trying to explain why a supply curve slopes up is hard. Suggesting that the PPF might bow out? Easy. Think of filling exploring a state space with a random walk maybe with occasional jumps (instead of occupied, think visited) [1]:

Would exploring this space look more like a triangle, a bowed-out curve or bowed-in curve? You'd probably say "it depends" and you'd be right. But the choice that requires the least amount of additional assumptions is bowed-out (think of it as a quarter of a random walk starting at zero in an 2-dimensional space). The triangle essentially requires a budget constraint, and the bowed-in curve requires a reason to prefer the axes. If we think of this as two in n >> 1 dimensions, then the most likely place to find the "explorer" is near the PPF at the surface (since most points of a higher dimensional volume are near its surface) -- with a radius essentially given by the diffusion constant D.

I'll see if that works out with some numerical simulations in a future post. But that would explain why the supply curve slopes up: diffusion from zero leads to a circular regions bounded by the PPF with radius ~ √(D t), which is equivalent to an upward sloping supply curve.



[1] My extra explanations (visited vs occupied, jumps) are here just so I could re-purpose this figure I drew for what was going to be a purely MaxEnt description of the PPF -- which is a bit harder than I originally thought. You shouldn't think of the states inside the PPF as "occupied" (as in the diagram), but rather "explored". It is sketched out above, though.


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    1. Correction: the random walk gives me a reason to favor bowed-out vs straight.

      What would be interesting to see is the "bath" idea tied to the random walk.

  2. Jason: "That is to say, they mean the same thing -- there is no knowledge added to a bowed out PPF to makes it lead to a upward sloping supply curve."

    Almost, but not quite. To get a supply curve, we need to add the assumptions of profit-maximising sellers who take price as given. It's a behavioural relationship. A PPF is an engineering relationship.

    For example, a communist economy would have a PPF, but wouldn't have supply curves. The central planner just picks the point on the PPF he likes best. He don't need no steenking prices.

    But I do not follow your derivations of supply curves from PPF or vice versa.

    "But the choice that requires the least amount of additional assumptions is bowed-out (think of it as a quarter of a random walk starting at zero in an 2-dimensional space)."

    Interesting. Though I barely understand you. Is it related to this?:

    In that post I only talk about the negative slope, not about the curvature. But my shotgun would seem to give the same curvature as your random walk. Are we sort of on the same page?

    1. Hi Nick,

      Regarding the shotgun, yes, that is on the same page. Coincidentally, the pictures I just made even look like taking a shotgun to a piece of paper:

      Let me think about the first part a bit. The central planner violates the assumption of (partial) equilibrium prices -- so doesn't the PPF become the budget constraint in this situation? I'll think it over ..

    2. Nick, I enjoyed that shotgun post!

  3. In regard to supply curves and the assumption that each producer thinks that his production will not affect the price, and so holds price constant, that does seem to be grounded in experience. However, the idea that the supply curve tells us how producers will react to changes in price is more iffy. For instance, during the Great Depression, in the prelude to the Dust Bowl, as the price for wheat dropped, farmers in the mid-West did not decrease production, as an upward sloping supply curve might suggest, they increased production in an attempt to make enough money to survive and keep their farms. The resulting overproduction led to the collapse of wheat prices and piles of wheat rotting on the roadside. It also prepared the topsoil to blow away in the following years.

    1. The first part is very true -- it's part of the assumption that agents are "price takers". However the latter piece would be a shift *of* the supply curve which does lead to lower prices, not along it.

      If it was done slowly (in general equilibrium with increasing demand, rather than partial equilibrium with demand held constant), increased production would lead to economic growth and inflation.

    2. A change in price means a shift of the supply curve?

      OC, that would be so if there is a change in the equilibrium price, because different equilibria have different supply and demand curves. Is that what you mean?

    3. A change in production (as you mention above) is a shift of the supply curve.

      The farmers are price takers, as you say above, so they can't set the price. There are price changes but the cause is a supply shift.

      A price change could move the supply curve, the demand curve or both ... Depending on the situation. Monopoly can change prices ; so can government .

    4. As I understand it, the price drop preceded the increasing supply.

      Now, IIUC, the supply and demand curves are static in time. But the meaning of an upward sloping supply curve is that if the price were higher, producers would supply more, and if the price were lower, producers would supply less. Given that, is there not a suggestion that if the price does increase, producers will respond by supplying more and if the price does decrease, producers will respond by supplying less? OC, logically anything can happen.

    5. Hi Bill, have you read this post from Jason? The final paragraph explains why (from the ITM perspective) supply curves slope up, and even what a supply curve actually is. It's one of my favorite posts.

      Nick Rowe's explanation is also good and it relates to Jason's post here. He starts with a PPF (he uses a long thin island example, with apples growing best in the North and bananas best in the South). So I don't know what the proper terminology is, but if you were looking at the price of apples say, and more and more of the island were being committed to apple production (i.e. the boundary between the two zones moved further and further South), then more and more land is required per new apple produced. (The proper name for that concept no doubt has the word "marginal" in it). He goes on to draw more complex examples involving labor and capital, etc.

      What I haven't tried to tie together yet is Jason's explanation here (for upward sloping supply curves) with his explanation in that older post I link to above.

    6. Thanks, Tom. :)

      I think that Jason and I were talking past each other a bit. I think that he focused on the collapse of prices after the overproduction. I was focused on the overproduction resulting from the original price drops. Which I guess were the results of the depression, as consumers cut back on wheat products.

    7. In the post, "Is the supply curve flat?", Jason says, "a supply curve is what you get when you look at the partial equilibrium by holding supply constant"

      I don't think that this is the textbook definition. ;) Since supply is held constant, perhaps a good name might be an Iso-supply curve.

    8. Bill, yes, it took me a while to get me head around the way he defines it. I'm always helped by the ideal gas law analogy:

      Imagine a gas in an insulated cylinder (an isoentropic system) with a piston at one end:
      p = pressure is analogous to price
      V* = volume is analogous to supply
      W = work expended to compress gas (proportional to temperature with a fixed number of gas molecules), or work recovered from expanding gas against the piston is analogous to demand.

      I put a * on volume, because it uses a negative sign on its information index k (so it slopes down where a supply curve slopes up).

      The set of solutions (p, V, W) to this isoentropic case are analogous to general equilibrium solutions.

      Now if the cylinder's insulation is replaced with a bath, then we can hold T constant (equivalent to holding W constant *inside the gas*), and we get isothermal solutions. This is more directly analogous to the demand curve case (partial equilibrium with demand held constant). But what is W if T is held constant? (really delta W). It's the energy exchanged with the bath: so the energy of the gas really does stay constant, but the work done (say compressing the piston to decrease the volume) is absorbed by the bath. The opposite is true as well: if the volume is allowed to expand, work is recovered from the bath without changing the energy of the gas in the cylinder.

      You can think of the locus of possible isoentropic (G.E.) solutions prior to putting the cylinder in a bath. And the W being the work required to get back to that locus of points if the insulation is removed and the cylinder is placed in a bath before the piston is moved.

      I think of the supply and demand partial equilibrium (P.E.) curves the same way: the Delta S or the Delta D represents the supply or demand that would have to be added back to the system to get you back to your original locus of G.E. solutions, given that you're holding S or D constant, respectively.

    9. To me the key was the fact that the axes were labeled P vs. ΔD, ΔS, instead of P vs. D, S. Also, the numbers were normalized, which indicates that the graph is theoretical, not empirical. I expect that the supply and demand curves in economics textbooks are meant to indicate empirical data.

  4. The problem with ideas like clearing markets, price-taking suppliers, supply & demand curves and production frontiers is that they are mostly simplistic fictions in modern economies except perhaps for basic commodity products. Even for commodities, they don’t take into account the human factor. Three examples.

    First, commodities. If you are a commodity producer then you will, as per any supplier in any sector, want to earn a certain amount to support your desired standard of living. If the price moves against you i.e. moves down, it is a very human response to try to preserve your income. If the price is lower, you need to sell more to preserve your income, so you increase production. If everyone does this, the price reduces further and you have to sell even more to preserve your income, so you increase production again. If I’ve understood correctly, that is the point that Bill is making regarding the wheat market during the Great Depression. In such circumstances, the producers go on tilt and normal supply & demand curve rules go out of the window. There may well be something similar happening now in the crude oil markets. Interestingly, it would need OPEC to act as a cartel, and agree jointly to cut production, to fix this problem.

    Second, money. I recently read a conversation between Scott Sumner and Nick Rowe in the comments on Nick’s blog where they acknowledged that business demand for money (i.e. borrowing for investment) is higher when interest rates are high than when they are low. At first glance, this appears to contradict the laws of supply & demand as demand increases with price. You might expect that economists would at least debate this anomaly, to try to understand the implications of such behaviour, as it is central to conventional monetary policy. However, they just ignore it and carry on regardless even though there is a simple real-world explanation.

    Third, complex / digital products e.g. drugs, technology products, recorded music, movies. These products have huge upfront R&D / design / creative costs. Production costs are relatively trivial, particularly for drugs where they are trivial and for digital products where they are zero. The laws of supply and demand don’t work well for these products. These products are increasingly central to advanced 21st century economies.

    I could quote further examples relating to manufactured goods, and services, where it is almost impossible for demand and supply to match due to the need for suppliers to decide on production volumes / staffing levels in advance of any demand appearing in the marketplace. In some cases, mismatches are built into business processes e.g. supermarkets consistently overstock on food and throw away what they don’t sell. All of these mismatches are mostly invisible in macro measurements so economists tend to ignore them and retain their fictions.

    These perfect market fictions are right at the heart of the problems with economics. This is why economics students complain that their education does not prepare them for the real world. It’s also why, increasingly, central banks and businesses who employ economics graduates agree with the students.

    Surely the advantage of information transfer economics should be that it can work without these fictional micro assumptions. Otherwise, what is the point of it?

    1. Hi Jim,

      I don't hold the view that theories are either correct or fictions. There are shades of explanations. For example, you can describe an atomic nucleus as a hard sphere, a collection of protons and neutrons, nucleons and pions, constituent quarks and pions, and finally in terms of quarks and gluons (or maybe strings?). Everything above the quarks and gluons (or strings) is a "fiction" if you take it to be a binary. But the hard sphere was a useful fiction (effective theory) for Rutherford (and building radiation shielding today).

      The key is knowing when each apply.

      To that end, the information equilibrium view can represent the useful fiction of optimizing agents -- but it also tells us that fiction isn't always useful. In particular recessions seem to be cases of non-ideal information transfer where information equilibrium (and therefore emergent/effective rational agents) don't apply.

      I think that is useful application of information transfer economics -- even if it does have an effective description in terms of rational agents in some cases.


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