## Friday, January 12, 2018

Adam Smith imagined two firms with a choice of production of two products*. The two firms' entrepreneurs are Alice and Bob. Alice and Bob have the choice of producing two independent goods - maybe apples and blackboards. The firms have what has become known as "constant technology", the ratio of their outputs and their inputs is a constant independent of said quantities (different for each firm and product to avoid indeterminacy). Unemployment is ignored in both inputs and outputs - all inputs are bought and all output sold (Say's Law). The law of one price holds.

Each entrepreneur is then left with only one decision: how much of each good shall I make?

Leonid Kantorovich

The fastest way to this answer is through the theory of linear programming. Start by denoting $$L_{firm,product}$$ the amount of labor Alice or Bob uses to make apples or blackboards. If $$L$$ is the amount of labor available in society, then we have as a constraint

$L_{Alice,apples} + L_{Bob,apples} + L_{Alice,blackboards} + L_{Bob,blackboards} = L$

The outputs are denoted $$Y_{firm,product}$$ and the prices are $$p_{product}$$ so that the total output is

$$Y = p_{apples} (Y_{Alice,apples}+Y_{Bob,apples})+p_{blackboards} (Y_{Alice,blackboards}+Y_{Bob,blackboards})$$.

Finally, the technical coefficients are $$a_{firm,product} = Y_{firm,product} / L_{firm,product}$$ . Our goal then is to maximize the above equation We know from LP theory* that only the vertices matter. Why? Well, the short answer is interior point optimization. Look at this picture:

The dark lines are the constraints and the darkened point is a guess at the optimum. The line through that point has a slope equal to the price ratio. It's obvious that more output by moving to a guess in the direction of the arrows that is still feasible. The triangle  made by the price ratio and the constraints (and its interior) are the "interior points". If an interior point set is one point, it must be the optimum. It's obvious that unless the price ratio is exactly the same slope as one of the boundaries, the only way to squeeze the interior point set down to one point is to chose one of the verticies.

Okay, so let's cycle through those verticies. The solution where no labor is used is the global minimum (the entire feasible set is the interior point set!). This also doesn't match the full employment constraint, so toss it.

Next, there's the where only one firm is buying labor to produce output (so that three $$L_{i,j}=0$$ and one $$L_{i,j}=L$$. We'll call this the $$Y_{one}$$ solution as in "Why one?". Mathematically, it is written

$Y_{one} = max_{i,j}(p_j a_{i,j}L)$

The next class of solutions is the instructive one. In these vertices, the non-negativity constraint is active on two possibilities - that is to say: labor is being hired for two reasons. But there's a problem. If both entrepreneurs  are making apples, then $$Y = p_{apples} (a_{Alice,apples} L_{Alice,apples} + a_{Bob,apples}L_{Bob,apples})$$. But if Alice is more efficient, then this can't be a maximum, because moving some labor from Bob's firm to Alice would increase output. This knocks out the two competitive verticies and leaves the four non-competitive verticies. Either one firm allocates labor two both products (and the other is dead) or two firms specialize in one product.

Read the above paragraph again, slowly. It's important to understand the next part. If you look at the verticies with three or all four labor hiring reasons, then the accounting for $$Y$$ always has at least one term like the above. For instance, one of the three term verticies is $$Y = p_{apples} (a_{Alice,apples} L_{Alice,apples} + a_{Bob,apples}L_{Bob,apples}) + p_{blackboards}L_{Bob,blackboards}$$ . But this can't be a maximum, because of the above paragraph - if Alice is more efficient in apple growing we can get more output by moving some labor from Bob's firm to hers.

There are therefore only possible three classes of extreme verticies: Alice or Bob make one thing, Alice or Bob make everything and finally Alice and Bob specialize.

David Ricardo

Phew! The great economist David Ricardo was uncomfortable with Smith's story. The vertices where one firm did everything and the other was non-existent troubled him.

Why would a non-existent firm trouble someone? Well, Ricardo was exploring an analogy between firms and nations. Nations have different technical coefficients - nobody needs to explain why California produces more wine than Utah or why Kazakhstan produces more uranium than Italy. But the idea that a country could "dominate" and produce everything was very troubling - not to mention that it seemed contrary to the facts.There was a political economy problem as well. Absolute advantage seems to suggest that global and local output could be at cross terms - global output might be maximized at the minimum of local output.

Ricardo "solved" this problem by ... restricting free trade! Yes, the classic argument for free trade assumes trade restrictions. Ricardo supposed that labor and capital couldn't move over national borders, but that consumption goods can.

John von Neumann

In the language of Linear Programming, Ricardo breaks the one labor constraint in the above problem into two:

$L_{Alice,apples} + L_{Bob,apples} =L_{Alice}$

$L_{Bob,apples} +L_{Bob,blackboards} = L_{Bob}$

Now the one firm produces everything equilbria are knocked out - they don't satisfy the above constraints. How they will specialize depends on the price ratio and technical coefficients, but they must always produce.

Comparative advantage is often thought of as a "long run" view - this is the supposed justification for the assumption of full employment and full consumption. But if capital and labor can flow over borders, it is a short run view that ignores unemployment and underconsumption.

Or is there a better interpretation of how comparative advantage works?

* The Adam Smith of my imagination.

## Thursday, December 28, 2017

### Who Cares About Ergodic Systems?

This is a quick teaching post. This stuff is high school level, but to make it formal can push it beyond the research level into high level philosophy.

The creator of Dr Pepper, Dr Alderton

First, some physical intuition. I pour some Dr Pepper into a cup. What shape does the fluid take on? There are really three fluids - Dr Pepper (largely water, basically incompressible), carbon dioxide (which takes the shape of tiny, interacting bubbles) and ambient air. There are many forces - solid resistance, buoyancy force, skin and interaction forces on the bubbles, gravity and (since the Dr Pepper and carbon dioxide are colder than the ambient temperature) thermal forces. The short run question of what happens to the fluids is complicated and depends on many tiny factors.

Despite this, the long run solution is easy - basic fluid statics tells us that the Dr Pepper will take the form of the cup and basic thermostatics tells us it will be the same temperature as the ambient atmosphere.

John von Neumann

How do we capture this intuition that - roughly - in the short run history matters but in the long run only structure matters? For many years, physicists and mathematicians have turned to Ergodic Theory to answer this question. Ergodic theory doesn't exactly have a great reputation.

Many people - including high powered top level experts - think that not only does ergodic theory require the formal manipulation skills of a von Neumann, the geometric insight of a Clerk Maxwell and the engineering experience of a Shannon - it doesn't even solve the problem.

But really ergodic theory is very simple - except for all the parts that are hard. Shannon's paper can be polished off in a couple days, and (with all due respect to Joe Doob) it's not clear that there is more to the theory than that.

You don't want to take a few days. Well, here's the few minutes version.

A connected and a disconnected network

We start with the intuitive idea of a network. We call the nodes the state. There are finitely many states and the each have a name. From a given state, there is a rule to transfer to one of the other states to which that node is connected. The rule and the network together are called the system. In theory the rule can be anything, for instance it might be "always go as far down as possible" where down is defined geometrically or topologically. The rules can be probabilistic.

A system is called "ergodic" if the long run amount of time spent at each node is independent of which node you start at. The idea of state gives us the short run detail dependence and the ergodicity gives us long run structure dependence.

For my deliberately dumb "go as far down as you can" rule on the above connected network, I have six possible runs

Pink, Purple, Purple, Purple...
Brown, Purple, Purple, Purple...
Blue, Black,  Purple, Purple, Purple...
Orange, Purple, Purple, Purple...
Black, Purple, Purple, Purple...
Purple, Purple, Purple...

No matter where I start, the long run relative frequency $$f_{purple} = 1$$ and all others are $$0$$. Therefore, this dumb system is ergodic. If we try the same thing on the disconnected network:

Black, Brown, Pink, Pink, Pink...
Brown, Pink, Pink, Pink...
Purple, Orange, Orange, Orange...
Blue, Orange, Orange, Orange...
Orange, Orange, Orange...

For the first two starting places the relative frequency of pink goes to one, for the second three, the relative frequency of orange goes to one. This dumb system is non-ergodic. But notice it is two ergodic pieces. In general, a non-ergodic system can be severed into ergodic components (in this case, the two connected subnetworks).

The underlying being connected isn't in general sufficient for being ergodic. On the above left graph, sever the Orange-Purple connection and follow the "go down" rule (question to check if you understand: what are the two ergodic subsystems?). It turns out* kinds of rules that are of physical interest are usually of the form "Given that I am on state N, I go down each connection NM with a certain probability $$p_{NM}\neq 0$$". For such a rule, being connected is sufficient for ergodicity**. So in this informal blog post I'll choose rules and networks such that connectedness and ergodicity are equivalent.

The ergodic distribution tells us the long run behavior of the system, but it also teaches us about the medium run behavior. We know that if the frequency at a state is "too low" (compared to the ergodic frequency), then we will see a flow into that state. This is more or less a definition of what a flow is!

This is all well and good, but what does it have to do with physics? A continuous system is ergodic if the one can cut up the possible states of the system into a discrete ergodic system. Let's make a pair of networks out of a physical model - a billiards model. I mentally divide a square billiard table into four regions A, B, C and D

Being in a region isn't sufficient to fix the dynamics - I need to know the velocities. I think that it's obvious that velocity digitizes into four chunks based on the number of regions away from the starting region you end up in after a time step. So the states are really:

A0, A1, A2, A3
B0, B1, B2, B3
C0, C1, C2, C3
D0, D1, D2, D3

Each 0 connects only with itself (remember, the billiard isn't necessarily staying still, it could be through all four blocks in one tick). There's a cycle A1 connects to B1 connects to C1 connects to D3 connects to C3 connects to B3 connects to A1. There are three cycles of length two, A2 connects to C2 connects to A2, B2 connects to D2 connects to B2 and A3 connects to D1 connects to A3. This is illustrated below

This particular digitization of the underlying continuous system isn't ergodic. If you start off with A0, then $$f_{A0} =1$$, if you start off with a non-zero velocity state, then $$f_{A0} =0$$. That's enough to show that this isn't an ergodic system.

It turns out that there is no nontrivial digitization of this system that is ergodic. That's because this system is exactly solvable... and I won't tell you why that's connected to ergodicity***.

Let's put a circular block in the middle of the square. Now the graph isn't disconnected. A particle that started four blocks per tick can have it's angle of attack by hitting the circular block to now be 3 blocks per tick (that is, it may be turned around because 3=-1). I don't know if this particular graph is really ergodic and I'm not going to check. In a tour de force, Yakov Sinai proved that this system has an ergodic digitization. This shows that the system is itself ergodic.

That means if the particle isn't in, say, C enough (compared to the ergodic distribution) we will see a flow towards C, just as in the discrete case. This is how ergodicity connects to physical quantities.

Finally: wasn't that Black Thought freestyle great?

*By the magic of symbolic dynamics
** By the magic of Markov Chains
*** I'll let wikipedia do it

## Thursday, December 21, 2017

The vast majority of Discourse about health is lies. Go to a supermarket and look at the "health shakes". Beyond the outright lies that are the overwhelmingly greater part there are the tentative part-truths - usually represented as absolute certainties. Supposedly there could logically could be definite truths. I have never seen them.

How do we talk about food? In episode 2 of Frasier, Frasier Crane discusses his preferred breakfast with his father. A "low fat, high fiber" breakfast with (terribly expensive) plain black coffee. There's a lot to be said about the implicit social views of eating in this scene. But instead imagine this: out of the woodwork I wander onto the show and say "Actually Dr Crane, you would be better off substituting that bran muffin for bacon.".

What does that mean? A calorie neutral substitution? A mass neutral substitution? A subjective substitution? The second is not a joke - it's clear (is it?) that a person who is fed via IV would "feel hungry" and attempt to eat whether the IV was sugar or oil. In the language of economics such a person would hit their first order conditions for diet optimality (I.e. calories would be right) but not their second order conditions. They're at a minimum utility. This cannot last. Believing in diet advice unstable to perturbations is unscientific - completely methodologically unsound. The third option is not necessarily unscientific - subjective feelings of fullness are related to the health relavent properties of food. Attempting to lose weight via a simplistic, objective calorie/mass accounting system may again put you at an unstable equilbrium. These kind of yo-yo unstable diets aren't obviously healthy.

The market for food is broken, possibly irrevocably broken. There has never been, in all the history of mankind, a society where farm labor is economically valued. As a result, all industrial societies prop up production - often in highly distortionary ways. It is obvious that, for instance, the US overproduces corn carbohydrates. This is a Bad Thing.

On the consumer side, it must be admitted by any person who desires to be taken seriously that branding and monopolistic competition more generally are real. Government intervention has been ineffective at policing this, even when it has been pointed in the right direction. This is unsurprising - Coase on the right and Stiglitz on the left are always fond of pointing out that the conditions in which governments/markets can fail are the exact conditions markets/governments can fail.

This brings us back to the first point. How do we talk about food? There is an enormous signaling problem here. Just as every prospective worker has an incentive to appear to be valuable to a prospective employer, so every prospective meal has an incentive to appear to be healthy to a prospective eater. (Healthiness & productivity of course defined variationally) Eaters therefore statistically discriminate, choosing foods with a few easily observed outward signs of "healthiness". Sugary cereals put photos of nutritious meals on the box. Having blueberries and a green container turns a malted into a diet food. More informative statistics - carbohydrate and protein and fat measures, total calories, ingredients, etc. - are buried in confusing, neutrally colored, small type statistical abstracts.

Adding exercise to a given diet is generally good, since exercise to a large extent determines the distribution of variable masses for a person of a given mass. We may not be indifferent between being Akebono and Bob Sapp. As Kimball notes above & every bodybuilder knows - total weight balances the mass of food that comes in and goes out. (Also, cardio health is good, even though the mass of the cardio system isn't particularly variable in mass) But there are huge difficulties here. First, it isn't methodologically sound to assume a person can vary exercise and not vary their diet (it assumes that their old equilibrium was unstable, which is exactly what it is not for an obese person who can't lose weight). The market for exercises is not obviously healthy. Like with food, every prospective exercise plan has an incentive to seem healthy & sustainable even if it is not. Survivor bias is endemic here - everyone who keeps up My Super Special Program long enough achieves their weight & weight distribution targets and everyone who doesn't leaves.

You can't sit around thinking about your diet all day. Simplistic accounting techniques can beat advanced techniques just because they're easier to understand. It may not be easy to account for carbohydrate, protein or fat intake simply because their are so many kinds of carbohydrates, proteins and fats that eating the "wrong" kind may not give the eater noticeable feedback. Fasting can then empirically outperform theoretically superior modes because it's easy to notice when you cheat.

These are only the simplest economic metaphors. Beyond this there are cultural and even political factors. But that'll have to wait for another post.

## Saturday, December 16, 2017

### Nozick On ... Inequality?

Robert Nozick was a Harvard philosopher, a political philospher among other things. He was an odd duck with an interesting sense of humor - speculating that autofellatio plaid a role in classical Hindu yoga was typical of his crass jokes. But he was very serious about one thing - Nozick sincerely believed in a philosophical theory of social desert - that one should be allowed to own all and only the goods entitled to you. Nozick traced this theory to the Lockean theory of production & distribution. Locke believed that one owned those goods which one mixed with one's labour (if you were European anyway).

Nozick brought his beliefs to so-called "libertarian" ends. If you are entitled to those goods to the extent which you mixed your labor into them, then it is not clear that you are entitled to any public goods at all. Nozick had an argument - the Utility Monster argument - that utilitarianism (the philosophical position that public policy should aim at some measure of aggregate happiness) could not be a priori true. Consider a society consisting of two consumer classes, one with decreasing returns to consumption (normal people) and one with increasing returns to comsumption (utility monsters). Holding a nations's output constant, the utilitarian political advisor says it is always worth it to tax the normal people and subsidize the utility monsters, which seems unjust a priori. Nozick says, shortly, that utilitarianism is false because it can excuse income inequality.

But here Nozick reaches an impasse. You see, it's not clear that a entitlement theory of desert avoids income inequality. In fact, Nozick argues that entitlement is true despite the fact that it can justify income inequality. His argument is not complex. First of all, he assumes that the set of just objects is closed - any outcome that is reached by individually just actions is just. Next he constructs a possible world where income inequality seems justified. In this imaginary Pittsburgh, everyone starts with $1. But one person is special - he's Wilt Chamberlin. If even one person pays 1¢ to see an exhibition from Wilt The Stilt then he becomes - through no fault of his own - the richest man in Pittsburgh. Why is this unjust? There are several responses to this. One is tu quoque - why is this income inequality obviously good but utilitarian income inequality obviously bad? But there are sharper critiques. Despite the appearance of dollars and cents, Nozick's example is not economic. The people of imaginary Pittsburgh are not given alternate uses for their money. Why would they hold cash? Why does Chamberlin hold cash? This points us to the deeper problem arising from Nozick's economic ignorance - income is a flow but he treats it as a stock. It is of interest to Nozick that Chamberlin's capital recieves dividends, but it's clear he hasn't placed those dividends in a society - his "possible world" is unworthy of the name. The choice of Wilt Chamberlin is careful rhetoric. Most capital that pays dividends as the Wilt Chamberlin case can be transferred - in the case of extreme regimes, land can be nationalized, factories can be seized, etc.. But Nozick clearly believes Chamberlin's God given talents are just that - in born natural talent (that Nozick thinks this about Chamberlin brings up the issue of race in ways I won't address) that can't be transferred. There might be a similar point about transferable utility in the utility monster case. The only way to redistribute the returns on Chamberlin-capital is to tax the income - right? Well, maybe maybe not. Most modern theorists on inequality concentrate on wealth inequality rather than income inequality. Unlike income inequality, wealth inequality doesn't correlate with Wilt Chamberlin like capital - not with things that a priori seem justified. Not IQ, for instance. This is a positive, not a priori case but it is still a hole in Nozick's point. Even if one believes that Chamberlin deserves the income he recieves from his non-transferable capital, that doesn't mean that one believes he deserves his stock of wealth. So Nozick's argument is all a little bit old fashioned. In sum, I think that Nozick's argument is unpersuasive for two reasons. It isn't obvious that he solves what he thinks are problems with other theories and even the internal coherence of his story is questionable. Still, Invariances is a good book. ## Wednesday, August 30, 2017 ### Price Gouging Is Bad J L Austin J L Austin once wrote a book How To Do Things With Words. One of the most important - or at least most analyzed - branch of words is "prices" - those signals that firms use to advertise their willingness to part with wares. Prices reflect many things: cost of production*, noise, willingness to purchase and the spatial, temporal & political relations between the seller(s) and the purchaser(s). Milton Friedman, Theodore Schultz & George Stigler There's an easy case. In the Heaven of "perfect competition", the price of a good balances two aspects: 1) the aggregate choices of all consumers and potential consumers is indifferent between purchasing and not purchasing an extra bit of that good and 2) the aggregate of all producers must be indifferent between manufacturing and not manufacturing an extra bit of good. In this topsy-turvy Never-Never Land, "price gouging" - sudden price rise immediately after a natural disaster - is Actually Good. Technocratically and morally good. A rising price reflects a greater need on the part of the consumer, which will be met by profit hungry producers**. Despite what Richard Posner tells you, we do not live in this place. Harold Hotelling It is simply not the case that the rise in price necessarily reflects a change in demand. The change in price can reflect a rise in the monopoly power of firms - a flood creates huge transaction costs. Recall the famous Hotelling Spatial Model of competition - one of the earliest completely specified monopolistic competition models. What happens when the transaction costs increase? We already know this. Transparency goes down, consumer surplus is consumed ... and profits go up. Exactly what is observed. Ed Chamberlin Neither is it the case that profit hungry firms can necessarily enter the market to meet demand. In order for a firm to enter a market, the long term expected profit to an entrepreneur must be non-negative. They must be able to overcome, for instance, fixed costs and compete with established firms with increasing returns. A flood creates higher fixed costs for entry - depressing the number of firms that can enter. A bit of price gouging is probably not enough to overcome this effect. Okay, but let's say you really want to believe in this "perfect competition" story. Maybe it isn't right in detail but you think it gives you the right ... laws of motion. Maybe not always but on average, in a broad sense of the term "average". This was Ol' Frank Knight's opinion on the nature of perfect competition predictions, so you're in good company. Yes, you admit that most people who hold this position are just contrarians who haven't thought beyond the textbook case. But you're not. You genuinely believe that local multipliers are generally strong enough that price increases - perhaps alongside government spending - generally returned devastated regions to the "status quo ante clades". This is a defensible position, econometrically. At least with small disasters, it seems to be true: every rainy day increases transaction costs - but they don't all destroy the city. Then how do you take into account that this isn't true in general? Do you think that the unregulated markets of the late 19th century just didn't gouge enough? I'll make my long story short: automatic disaster relief > price gouging. It's true that there should be changes in economic fundamentals: rain taxes, infrastructure investment, enforcing flood insurance laws, fixing zoning so that flood absorbing lands aren't eaten by sprawl (this is probably irreversible at this point - urban sprawl is one of worst ecological disasters in history but nobody does anything about it...) - but locally, around the disaster the important thing is to get spending back and let the multiplier work itself out. There's no a priori reason to think price gouging will help and not hurt. Prices are signals, words. Don't think those words can't be "Screw you!". *"cost" should be understood in a very wide sense. **It has been well established since Walras that "perfect competition" means constant returns, so small firms can always come up to meet demand in that mystic realm. ## Friday, August 18, 2017 ### About Armen Alchian Armen Alchian Armen Alchian - nicknamed by his friends "the Armenian Adam Smith" - was an economist at Stanford, UCLA and the RAND corporation. He participated in all the economic revolutions of the time - general equilibrium, economics of information, theory of the firm, utility theory and evolutionary economics. Though ideologically attracted to libertarianism, Alchian was a devout pluralist in his methods. He was a great writer - unlike most clear writers, Alchian comes by his clarity honestly rather than by covering up difficulties. Alchian did not participate in the usual tedious academic point scoring games. The economist who Alchian most resembles is Frank Knight, whose insightful but somewhat mystical Risk Uncertainty And Profit hovers behind much of Alchian's more innovative thinking. Alchian's thinking also engages with the even more mystical writing of Friedrich Hayek, especially his famous essay on knowledge. Alchian also wrote many important papers on inflation (that is - unpredictability in price level) on resource unemployment and therefore macroeconomics more generally. However, I haven't really read and engaged with these papers, so I will not be able to describe them here. I would recommend that you peruse Glasner's blog for that stuff. "... automobile makers may buy such things as finished seats from outside suppliers because their inspection is relatively easy. But automobile makers are hesitant to use outside sources to supply sheet metal parts, which are ordinarily only discovered to be out of tolerance only when a car body does not fit together correctly." - Armen Alchian on what is internal and external to the firm There used to be two major schools in thinking about the economics of organizations. There was the Lange-Coase school of high transactions costs. The main take away of this school is that the market cannot see into the firm because it is just too damn expensive to negotiate every little thing out. This school gave economists two avenues to improve life: 1. "Mechanism Design" of institutions with low transactions cost - from carbon markets to fight global warming to Lange's surreal technocratic communism 2. Use the courts to allocate goods optimally outside the market (this is regarded as an important insight - why has never been adequately explained to me) On the other side was the Mises-Pigou theory that the point of the firm was to organize the production process in a way that "internalized externalities" - made the production process as efficient as possible. This school too leaves us a bifurcated road 1. Reduce legislation so that firms can organize themselves as efficiently as possible 2. Use taxes to discourage firms from doing bad things. As the above quote shows, Alchian was on the side of Mises & Pigou in this conflict. The choice of what is internal to the firm is a market choice - the firm keeps close that which is expensive to monitor and lets loose what is cheap. Alchian's insight here is invaluable, but I find myself in mixed agreement with him. He often emphasizes the firm's ability to find a stable point in all this and the possible suboptimality of regulation - I demure. When Alchian says "Vertical integration identifies for consumers a single point of accountability for service quality", I think - when I am in a polite mood - "Maybe ...". Fortunately or not, Alchian never engaged in polemic over the conflict between his and Coase's points of view, probably because of their personal friendship and idealogical agreement. It's somewhat amusing to see Coase and Alchian talk as if their theories are in precise agreement when the generation of economists before them had raged as if the distinction was the most important thing in the world. "A football coach knows that the condition of winning is making more points than his opponent. Does knowing this imply that the coach can know what his team must do in order to win? Does the coach know how this can be done?" - Armen Alchian on the nature of profit maximization "There are no implications to 'profit maximization'..." - Armen Alchian on the positive implications of profit maximization Alfred Marshall called biology the "Mecca of Economics". His vision was to integrate economics into the vast System of Herbert Spencer. As one of the few people who have read the entirety of Marshall's Principles, I have to admit that his biology was kooky Victorian BS. But the reasons he had this goal were sound, even if Marshall's biology was nonsense. Alchian made a great stride into achieving a more realistic version of the evolutionary aim. He joined the fray in a rather kooky way - in response to a controversy over the meaning of "profit maximization". Economists have long reduced the firm to a single equation "Profit equals price times quantity minus costs". This is no idle slogan that can be tossed aside by a sophisticate. This equation contains (in a rather mysterious way) the whole of the neo-classical theory of production. A firm - the neoclassical says - may not change the price because then another firm could spring up and undercut them. Cost - they tell us - are an increasing function of quantity. If the profit in an industry is greater than zero, then another firm will leap in. Therefore - the neo-classical economist says sagely - the quantity the firm may produce is exactly the quantity that sets the price to the rate of cost increase. Brilliant! Or, perhaps, rather foolish. Nobody who studied the firm - from the unusual Thorstein Veblen to the very careful Richard Lester - could find any evidence that workers are paid "their" marginal product. This incomprehensible formulation is no straw man - it can be found in, for instance, Stigler. Alchian throws away such nonsense - anyone would have to. Instead, Alchian rethought what it means for a firm to "maximize profit". Profit maximization is meaningless in the presence of uncertainty. In this new interpretation, the competitive process is broken into two steps. In step one, a firm commits to producing a quantity and therefore paying a cost. In step two, consumers consume a quantity of produced goods. Firms are punished on the difference between the predicted demand for current production and the actual demand. Obviously, perfect prediction is an equilibrium of this system. Any dynamics that take you to this equilibrium will make economic analysis valid - no matter how stupid the dynamics are. Alchian gives the example of imitation dynamics (which are just about as 'dumb' as dynamics can get), Richard Day gave a more careful example which includes feedback. Alchian's theory makes sense of Frank Knight's speculations on the nature of entrepreneurial rent - if a man has the local knowledge to better forecast than his sister, then that man may charge her for his time. But more importantly, they give a comprehensible connection between the Neo-Classical theory and the screaming torrent we call reality - it is the equilibrium theory. However, Alchian's system isn't quite complete. He doesn't have a clear idea of what it is that punishes the badly predicting firms. Only later would Gary Becker explain correctly what Alchian meant to say - resource constraints on the firm were the whip punishing firms that missed their production quota. In other words - marginalism of the firm was just marginalism, scarce resources = scarce resources. Nothing else is involved. Though very intellectually satisfying, this is harmful to the idea of a positive theory of the firm. Alchian's ideas apply equally to a monopolistically competitive firm, a monopolist and a firm in a highly competitive market. Luckily, we now understand the mathematics of evolution much better than Alchian and have done much work in the area. For a tour I recommend Bowles & Gintis's fascinating speculations, John Sutton's bounds approach to monopolistic competition and any paper with Tit For Tat in the title. Incidentally, Alchian was one of the first people to play the iterated prisoner's dilemma. Alchian, unsurprisingly, played ungenerously. "The year before the H-bomb was successfully created ... we in the economics division at RAND were curious as to what the essential metal was—lithium, beryllium, thorium, or some other. The engineers and physicists wouldn’t tell us economists, quite properly, given the security restrictions. So I told them I would find out. I read the U.S. Department of Commerce Year Book to see which firms made which of the possible ingredients. For the last six months of the year prior to the successful test of the bomb, I traced the stock prices of those firms. I used no inside information. Lo and behold! One firm’s stock prices rose, as best I can recall, from about$2 or $3 per share in August to about$13 per share in December. It was the Lithium Corp. of America. In January, I wrote and circulated within RAND a memorandum titled 'The Stock Market Speaks'. Two days later I was told to withdraw it. The bomb was tested successfully in February, and thereafter the stock price stabilized."

- Armen Alchian inventing the Event Study

"We assert: 'All prices are Martingales.' And we conjecture a second proposition: 'No quantity variables are Martingales.'"

- Armen Alchian on flexible prices

"Nor do I find it warranted to call the stock market 'efficient' any in pertinent sense just because the present price is an unbiased estimate of the forthcoming price. I would rather call it unbiased."

-Armen Alchian on the efficient markets hypothesis

Because of his excellent writing, Alchian is often considered a 'literary' economist. But Alchian was capable of doing math. At RAND, Alchian wrote a paper (already linked) making sense of the notion of cost in a production process - cost is denoted in units of equity - this is the secret key between Alchian's analysis and the Becker analysis. Alchian kept a long interest in how the stock market moved information - one of his students, William Sharpe, got a Nobel Prize for stock market stuff.

But if I had to choose an Alchian paper with math, I would point to his more philosophical 1974 paper on the notion of a martingale as a sign of the depth of his thinking on these issues. This paper is not just about the idea of an unbiased random variable, but on the notion of what it means for relative prices to be flexible. He gives a picture of some imaginary time series that have clear patterns, but whose relative prices give no information.  An enormous amount of economics is dedicated to thinking through these kinds of models - all Tom Sargent's work for one.

The strict separation between price martingales and quantity martingales explains why neo-classicals placed such an emphasis on "price flexibility" - why the theorists like Stigler have such a horror of a price floor. If prices are a martingale, then their shifts don't disturb the underlying balance of goods and services - the scarce resources stay where they are. There is no unemployment, not really. The only "unemployment" comes from information frictions that cause the prices to not quite be martingales- they'd be "sticky".

Now, obviously, there is more unemployment that can be found from information friction. If it was just information friction, then unemployed workers could just reduce their so-called "reservation wage" (the lowest amount they'd work for, more or less) and always find a job. But recessions are real, and in a recession reducing the reservation wages raises unemployment. The price signals are confused.

Even when not in recession, unemployment is not just due to information frictions. Because unemployment is unpleasant, firms  can use it as a punishment for shirking workers. If workers are willing to work for less, this tool becomes less effective and unemployment must rise. Firms and workers can work out labour/"leisure" trade-off through the quantity channel too.

So Alchian misses important pieces of the market in his assumption that prices are martingales and quantities are not. With all due respect to Hayek, information is sent through the quantity channel, even if that channel is noisier than the price channel. But Alchian (and Hayek) deserve attention for putting the problem so clearly. Alchian deserves more respect, in my opinion, because he was less dogmatic on this issue - he cited and spread papers such as this one that featured "involuntary unemployment".

"Before condemning violence (physical force) as a means of social control, note that its threatened or actual use is widely practiced and respected—at least when applied successfully on a national scale. Julius Caesar conquered Gaul and was honored by the Romans; had he simply roughed up the local residents, he would have been damned as a gangster. Alexander the Great, who conquered the Near East, was not regarded by the Greeks as a ruffian, nor was Charlemagne after he conquered Europe. Europeans acquired and divided—and redivided—America by force. Lenin is not regarded in Russia as a subversive. Nor is Spain’s Franco, Cuba’s Castro, Nigeria’s Gowon, Uganda’s Amin, China’s Mao, our George Washington."

- Armen Alchian on alternative means of governance

"Incentives are the prizes in the game of life-the goals individuals seek - the carrots. Through the ages of Tutankhamen, Alexander, Caesar, Louis XIV, and the Atom, they have remained the same. Men want, and have always wanted, exorbitant wealth, tyrannical power, idolatrous prestige, lavish consumption, and undisciplined leisure. ... What does explain the disparities? Differences in the relations between costs and goals."

- Armen Alchian on the utility function

There is a mysticism to the work of many economists. An economist speaks of optimality more often than Dr Pangloss and stability in a world in constant torrent. One reason that I like the works of Alchain is that there is a sort of reality to them. They are philosophical, even when they are technical. "Cost is a choice" - dull. "Cost is a choice in different forms of equity" - interesting! "Firms, of course, evolve" - hogshew. "Selection on firms is on their ability to forecast" - neat!

Another example: Alchian's work Why Money? is a economic philosophy gem, deriving the existence of money from the network of trade. A good will become money if the costs of identifying the quality of a good are low for everyone - that's why kings put their stamp on it! This is obviously an important work and one that opened the door to fundamental new research.

I said I wouldn't quote Alchian on macro, but I can't resist one:

"Why would a cut in money wages provoke a different response than if the price level rose relative to wages – when both would amount to the same change in relative prices, but differ only in the money price level? Almost  everyone thought Keynes presumed a money wage illusion. However, an answer more respectful of Keynes is available. The price level rise conveys different information."

This is one of those things that is so obvious after it is pointed out to you. In many economists there is the magical thought - it all depends on the relative price level. But Alchian is more careful - prices of complex items adjust more slowly than prices of simple items and people are complex. In a crude Keyensian system,  a cut in money wages ahead of the business cycle would signal bad times and depress spending - a raise of the price level (with wages lagging behind) would signal good times and encourage spending. In the Classical & Neo-Classical system, this signalling does not occur, only the relative price levels matter. Yes, there really was a Keynesian Revolution.

This observation makes the difference between neoclassical and Keynesian economics at least partly testable! I know that Alchian has attempted these tests and generally hasn't found the desired lags. And I know many economists - such as Andolfatto - have pursued this avenue more deeply. But I don't know much beyond that and will therefore be silent.

## Monday, August 14, 2017

### A Productive Read: A Review of Pasinetti's Lectures on the Theory of Production

Luigi Pasinetti

This is a very fine book that should be read by anyone who has a sufficiently strong background knowledge of linear algebra. Pasinetti is a fine writer who brilliantly exposits all aspects of the so-called "Sraffian" or "neo-Ricardian" theory using algebra and numerical examples. Pasinetti paces the book musically, showing how the concepts of Sraffian theory illuminate the problem of disaggregated production. Pasinetti makes it easy to understand what I call the Central Sraffian Theorem and why it may be wise to place it at the center of economic analysis.

Piero Sraffa

Pasinetti begins with a chapter on the precursors to the linear Sraffian system. Unlike most of these kinds of chapters, Pasinetti keeps things worth reading by using simple mathematical models instead of tedious linguistic analysis. Pasinetti expounds the basics of the old Ricardian system in aggregate and disaggregated along with its Marxian gloss. This chapter also distinguishes the production coefficients of Walras - which assume constant returns to scale - and the distribution coefficients of Sraffa which Pasinetti will work with. In the next chapter, Pasinetti moves to the simplest analyses of the so-called "Input-Output" method, going over the primary practical difficulties and introducing concepts he will use throughout the book.

The meat of the book begins in chapter three, where Pasinetti develops the linear theory of production in the mode of Sraffa, again pausing to explain how the coefficients are not necessarily the static production coefficients of Leontief-Walras. In chapter four, Pasinetti completes the analysis using both basic algebra, linear algebra and numerical examples of the Leontief-Walras case where the distribution coefficients are also the static production coefficients. This chapter introduces the conditions on which a distribution matrix may correspond to a stable productive economy: that the Perron-Forbenius eigenvalue should be less than unity. Economically, this means that the "quantity of input" should be less than the "quantity of output" with linear algebra providing precise meaning to the words in quotes even in the case of complete disaggregation.

With those four chapters as introduction, Pasinetti begins his wonderfully clear exposition of the Sraffian system. Where Sraffa's exposition was brilliant but mysterious, Pasinetti lets the theory free with it's assumptions and their reasons completely out in the open.

Essentially, Sraffa's system is a very large production network, which you can think of as a directed graph with positive weights. Sraffa tries out a few conceptual/topological assumptions about the nature of the network of production - it should be connected, the weights should positive, etc.. Assuming that the network is constant in time, Pasinetti & Sraffa can use the Perron-Frobenious theorem to find the amount of surplus production. The division of the surplus (between workers and capitalists) might seem - at first - a difficult problem. If we want to find the wages* in terms of some numeraire - gold, dollars, corn - then changing the wage rate must decrease the quantity that goes to capitalists, but not necessarily in a simple manner.

This is unpleasant, because it is not necessarily convex. This means that a Bergson–Samuelson social welfare function would not guide a social planner - not even the distributed one that we call "the market". (History of economic thought fans will recall the tie between convexity and general equilibrium was first noticed by Joan Robinson)

I had to draw this one, Pasinetti would never draw something with Samuelson in the name.

Pasinetti was able to spot an assumption even I managed to miss when reading Sraffa - that Sraffa's construction of the so-called "standard commodity" requires the physical own-rate of reproduction of the "basic commodities" that enter into the production of every good (labor, etc) must be less than the the physical own-rate of reproduction of all other commodities. I will put the Central Sraffian Theorem carefully: If the assumptions on the production network hold and prices (or wages, at least) are stated in terms of the standard commodity, then the relationship between how much excess production goes to the workers (wages) and how much goes to capitalists (profit) is linear.

Since a line is convex, Bergson–Samuelson social welfare is deterministic again. Woo. Further, Pasinetti shows that this unit for wages makes wages exactly equal to the labor commanded by the system - just as Adam Smith tried to tell you 241 years ago. If the distribution around the network really is stable for all time, as Sraffa assumes and Pasinetti assumes for now, then one can expand the distribution backwards as the sum of the history of splits of surplus product - as Marx might have told you. The Central Sraffian Theorem is sufficient to show historical materialism is coherent (though not necessarily correct).

Pasinetti then goes on to consider Marx's infamous "transformation problem" in a very helpful and unpretentious way. Pasinetti suggests that Marx's garbled account is due to to Marx's habit of moving parameters around that don't affect the production network, solving this special case and then lastly declaring the general problem solved. For instance, if we want to know the maximum level amount of excess output, it doesn't matter if we set wages equal to zero. Analyzing the system with great care, Pasinetti is able to reformulate the Central Sraffian Theorem in this way "The rate of surplus value is inversely related to the wage." (well, Pasinetti is more precise, but this gives the flavor). If all the excess production of the economy is given to the workers, the surplus value is exactly zero. This makes sense of Marxist political economy (if Sraffa's assumptions hold).

Joan Robinson

Unfortunately, no book from this school is complete without the inevitable chapter on "reswitching". Reswitching is all about taking seriously the concept of a function - is A a function of B or is B a function of A. A breezy theory where everything is linear makes everything a function of everything else - but life is not so breezy.

Pasinetti is characteristically scintillating, spreading light over this darkened field. He starts by considering three sets of worlds, which adopt three different production networks for the creation of a product. The rate of profit for a capitalist is uniform across industries (remember how Pasinetti defines profit), so the capitalist would like to be in the world with the production network that minimizes cost. However, which world that is depends on the profit rate. Therefore, profit rate determines choice of technique but choice of technique does not determine profit rate. This is the only theoretical fine point in reswitching. Pasinetti goes on to consider special cases and the general case in turn, but the result is the same, choice of technique is a non-invertible function of profit.

The upshot of all this is that the solution to the problem implicit in the Central Sraffian Theorem is the fundamental problem of the economy. If you want to know how an economy is structured, you have to know how it divides its product between its people.

In the final chapter, Pasinetti considers exogenous growth in a disaggregated Sraffian growth network. Expressing the system in terms of a standard commodity, Pasinetti finds a linear trade-off between current consumption and growth rate - just as Irving Fisher et. al. would have told you. The level of discussion is not quite as high as the well known DOSSO textbook (esp. Chapter 12), which made the knife edge transversality problem of a growing economy rather clear (of course, this was for Leontief-Walras fixed coefficients case, not the more mysterious Sraffa case). One could look at the above picture and think - with Irving Fisher - that all you need is an indifference curve. Pasinetti closes his system instead with a hypothesis on savings rates - which, obviously, are the reverse of present consumption. The hypothesis is this: workers cannot save but capitalists can. So Marxist political economy is not quite saved in the Cambridge Neo-Ricardian system. Workers have more to lose than their chains - capitalists are their bank accounts. In general, any relation between current consumption and distribution of excess product will turn the analysis of the exogenous growth case back into the static case.

This book is short, clear and eye-opening. Anybody who reads this will come out with a better understanding of economics than when they went in - no matter how much they know now. The only two flaws of the book are: 1) sometimes vital assumptions are put in footnotes and 2) there is a bit too much point-scoring against Paul Samuelson for my taste. Also, I still find the meaning of the constants in the Sraffian distribution matrix mysterious in a growing economy, but this may be just me.

A+, ten stars, book's alright.

* Pasinetti calls "wages" and "profit" the distribution of the excess product to workers and capitalists respectively.