Featured Comments: Week of 2013 August 18

There was one post this past week that got a few comments, so I'll repost one of those.

Cap and Trade and Soda

After an odd irritable initial pair of ranting comments, an anonymous reader followed up with this: "i apologize for my snide remarks before. i was just irritated. okay, so maybe there are a few other ingredients (like oils). it might be worth your while to check out OpenCola, which (or so i've heard online) tastes pretty close to Coke. http://en.wikipedia.org/wiki/OpenCola_(drink)".

Thanks to that commenter for that (and especially also for apologizing about the previous comments). This coming week I will have a few more posts about quantum field theory out, but no reviews as I had promised, as that will likely have to wait until after I get back to campus next week. In the meantime, if you like what I write, please continue subscribing and commenting!

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Cap and Trade and Soda

A few days ago, my family and I went on vacation. On the way back, my family and I were discussing various things including some matters of politics. One thing that came up was some of New York City mayor Michael Bloomberg's recent actions. I expressed the view that the ban on soft drinks larger than 16 ounces seemed rather heavy-handed. (After reading a little more about the exceptions for fruit drinks along with sales at grocery stores, I'm a little more happy to see that, but I still feel the ban was heavy-handed.) I then heard the argument that even if it is heavy-handed, it does help combat the obesity crisis by reducing access to drinking 16 ounces of soda at a time, because even if it is still technically possible for someone to fill up an 8-ounce cup twice, human psychology is such that said person would only fill up once, because for many people the convenience of filling up once trumps the desire to have as much as possible. I then wondered what other alternatives could be considered. The simplest alternative would seem to be a tax akin to taxes on cigarettes; if the large sodas are taxed heavily at such venues, people would naturally be discouraged from drinking as much. I have taken the class 14.03 — Microeconomic Theory and Public Policy, though, so I have seen that in many cases a Pigovian tax scheme like that may not achieve the most efficient outcome because it is difficult to adjust tax rates to control quantities precisely. Then I also remembered learning about cap and trade schemes to control quantities. Would that work? Let's take a look after the jump.

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Expected Utility in Quantum States

Last semester, 14.04 — Intermediate Microeconomic Theory covered choice theory under uncertainty; at the same time, I was taking 8.05 — Quantum Physics II, where we had talked about 2-state systems and the formalism of quantum states being complex vectors in a Hilbert space, and as choice under uncertainty discusses how consumers make choices based on states of the world, I thought it would be cool to extend it to quantum states, but I wasn't sure how to do that at that time. Now that 14.03 — Microeconomic Theory and Public Policy is talking about choice under uncertainty as well this semester, and now that I have had some time for the stuff about 2-state systems to simmer in my head, I think I have a slightly better idea of how to think about extended the states of the world to include the superpositions as allowed by quantum mechanics.

One of the simplest examples would be the 2-state system. In the class I am taking now, a typical 2-state system would be a fair coin which can either take on values of heads or tails (each with probability $\frac{1}{2}$). What might this look like in quantum mechanics? We could replace the coin with the spin $S_z$ of an electron. A state of the electron corresponding to being measured as $|\uparrow_z \rangle$ or $|\downarrow_z \rangle$ each with equal probability could be labeled as $|\psi \rangle = \frac{1}{\sqrt{2}} \left(|\uparrow_z \rangle + |\downarrow_z \rangle\right)$. This indeed gives equal probabilities of being measured as spin-up or spin-down in the $z$-direction. The problem is that this state is exactly $|\psi \rangle = |\uparrow_x \rangle$, so the probability of the state being measured in the $x$-direction as spin-up is unity. This leads to issues in trying to reconcile the interpretation of payoffs for different states of the world; this particular state would pay $w(|\uparrow_z \rangle)$ or $w(|\downarrow_z \rangle)$ with equal probabilities until $S_z$ is measured, but would pay $w(|\uparrow_x \rangle)$ with certainty as $S_x$ has essentially already been measured, collapsing a previously unknown state into this one. So there seems to be an issue with trying to stuff the interpretation of probabilistic measurement of a state of the world into the idea of superposing quantum states, as certain measurable states of the world do not commute ($[S_i, S_j] = i\hbar \epsilon_{ijk} S_k$ for instance). So what can be done now?

Recall that each state of the world $j$ has an associated probability $\mathbb{P}_j$. Yet, once a state is measured, those probabilities are meaningless, because a state becomes observed or not observed; it cannot be "observed in fraction $\mathbb{P}_j$" or anything like that. Taking the frequentist view, these probabilities are only meaningful in the large ensemble limit; after a large number of observations, the fraction of those observations corresponding to the state $j$ converges to $\mathbb{P}_j$. This is exactly the same rationale behind the density matrix, which describes the state of an ensemble of systems which are usually not all prepared in the same quantum state, but in the large number limit, the fraction of those prepared in the state $|\psi_j \rangle$ is $\mathbb{P}_j$ such that $\sum_j \mathbb{P}_j = 1$. It is then defined as \[ \rho = \sum_j \mathbb{P}_j |\psi_j \rangle \langle \psi_j | . \] Note that in general $\langle \psi_j | \psi_{j'} \rangle \neq \delta_{jj'}$ because the states do not in general need to be orthogonal. Furthermore, different state preparations can have the same density matrix, in which case the "different" state preparations are actually physically indistinguishable if a quantum mechanical measurement is made.

So what does this have to do with the states of the world? If the example of a fair coin flip is again translated into measuring each eigenvalue of $S_z$ with equal probability, then the density matrix becomes $\rho = \frac{1}{2} \left(|\uparrow_z \rangle \langle \uparrow_z | + |\downarrow_z \rangle \langle \downarrow_z |\right)$. Now it is much more plausible to define a wealth $w(|\psi_j \rangle)$ dependent on the state of the world.

For example, let us now consider an unfair "coin" represented by the density matrix $\rho = \frac{1}{3} \left(|\uparrow_z \rangle \langle \uparrow_z | + 2|\downarrow_z \rangle \langle \downarrow_z |\right)$. What is the probability of measuring this system in the state $|\uparrow_x \rangle$? That would be $\langle \uparrow_x | \rho | \uparrow_x \rangle = \frac{1}{2}$. Similarly, $\langle \downarrow_x | \rho | \downarrow_x \rangle = \frac{1}{2}$. It is interesting to note that equal probabilities are achieved for spin-up and spin-down in the $x$-direction (and $y$-direction as well) for this system; the difference is that now there are off-diagonal terms in the density matrix describing the degree of interference between the states of $S_x$ in the two populations of spins.

What can we do with this? Payoffs now need to be defined over every plausible noncommuting observable in the Hilbert space (because commuting observables yield the same state, and the payoff is really defined for the state). Thankfully this is fairly simple for spins, as the noncommuting observables in question are the components of $\vec{S}$.

Let us return to the previous unfair "coin". Classically (so it would be a coin), the result of the coin flip would be the end of the story. Let us suppose that the consumer had a risk-neutral utility $V(w) = w$. Let us also suppose that there was a game which payed off 20 if heads and 0 if tails and cost 10 to enter. The expected payoff, accounting for the cost of entering, would be $\mathbb{E}(w) = -\frac{10}{3}$, so the consumer would prefer to not play.
Classically that would be the whole story. Quantum mechanically, though, there is more to the story. If the owner of the game decided to measure $S_z$, then the result would be the same as the classical result. If the owner decided instead to measure $S_x$ or $S_y$, then $\mathbb{E}(w) = 0$ so the consumer would be indifferent between playing and not playing, which is certainly a different outcome from preferring not to play. In fact, if the consumer does not know for sure what the owner decides to measure, then probabilities could be assigned to the measurement choice itself, and these could then be used to find the expectation value of payoff or utility over all possible choices, where each of those choices will have an expected payoff or utility as well.

One thing to look further at is how to extend this to include continuous ensembles. The example in class was how a state of the world might be the outdoor temperature measured in some range. The quantum mechanical equivalent might be having a continuum of systems, each infinitesimal one having a position inside a given range of allowed positions; the only issue with this is that matter is not continuous, so I don't think it is possible to have a density matrix in the form of $\rho = \int |\psi(s) \rangle \langle \psi(s)| \, ds$ for some continuous index $s$. However, that can certainly be further investigated, and maybe I'll do that next time. The point is that states of the world in analyzing expected utility can easily be generalized to include quantum states through density matrices, and the question of which observable to measure in a noncommuting set brings out some interesting behavior not seen in classical states.

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Frictions, Subsidies, and Taxes

One of the things I learned in my high school AP Microeconomics class was that a tax causes the supply curve to shift to the left, making the equilibrium quantity decrease and price increase. Consumer and producer surplus both decrease, but while government revenue can account for some of the loss in total welfare, some part of total welfare gets fully lost, and this is what is known as deadweight loss. I didn't have a very good intuition for how this worked at the time (though I was able to get through it on homework, quizzes, tests, and the AP exam). At the same time, though, I thought that a tax should be fully reversible by having the government subsidize producers, and that as this would be the opposite of a tax, supply would shift to the right, the equilibrium quantity would rise and price would fall, and there would be a welfare gain.

Then, when I took 14.01 — Introduction to Microeconomics, we again discussed the situation with a tax. Then we talked about subsidies, but I was confused because the mechanism seemed to be in providing a subsidy to consumers rather than to producers. My intuition at that point was that taxes were creating deadweight loss because producers who wanted to produce and consumers who wanted to consume near the original equilibrium could not do so after the tax, so some transactions were essentially being prohibited. However, I still didn't quite understand why a subsidy would create deadweight loss, because it seemed to me like consumers who wanted to consume more and producers who wanted to produce more than the original equilibrium quantity could now do so, meaning it seemed to me like more transactions were being made possible. That said, I did understand why the government would never subsidize producers: unless the market is perfectly competitive, producers would rather collude and pocket their subsidies while keeping prices high when they can. On the other hand, consumers prefer consuming, so subsidizing consumers is a more surefire way of increasing the equilibrium quantity, even though the price would go up rather than down.

(In 14.04 — Intermediate Microeconomic Theory, we barely touched on deadweight loss in the way that it is covered in more traditional microeconomics classes.) Now, in 14.03 — Microeconomic Theory and Public Policy, I think I better understand the intuition behind deadweight losses stemming from taxes and subsidies, and why a subsidy is not the opposite of a tax. In a tax, the government might try to target some new equilibrium quantity below the original one, so the tax revenue collected, which increases total welfare, is the difference between the willingness of consumers to pay and the willingness of producers to accept at that quantity multiplied by that quantity. Consumer and producer surplus both decrease, and the tax revenue contribution to the increase in total welfare is not enough to offset these two, so there is an overall deadweight loss. A completely isomorphic way of picturing this is by considering the tax falling on consumers so that the demand shifts to the left; in both cases, the equilibrium quantity drops, the government collects its revenue, surpluses drop, so deadweight losses appear.
Meanwhile, for a subsidy, the government might target a higher quantity than the original equilibrium. The spending on that subsidy is the difference between the willingness of producers to accept and the willingness of consumers to pay at that quantity multiplied by that quantity. Consumer and producer surpluses both increase, but together they do not increase enough to offset government spending which is an overall drain on total welfare, so there exists a deadweight loss.

It's interesting that taxes and subsidies are not opposites. The intuition is that for a tax, the revenue is not enough to compensate for the welfare losses of consumers and producers because the new equilibrium quantity is lower. By contrast, for a subsidy, the spending is too high compared to the welfare gains of consumers and producers because the new equilibrium quantity is higher. It looks like it is not possible to spend money given by tax revenue to undo the effects of a tax; instead, the government can only overshoot and overspend. It reminds me very much of how friction works: moving in one direction on a surface with friction causes energy loss, while turning around to move in the other direction on that same surface most certainly does not cause energy gain. Essentially, in this model, the market is frictionless, and the government introduces friction.

Of course, this essentially contradicts Keynesian models of government taxation and spending and their respective effects. That's why care must be taken when putting microeconomic models in a macroeconomic perspective. This also doesn't consider externalities, less than perfectly competitive market structures, et cetera. Anyway, I hope my musings on this may help give other people some intuition on simple issues of deadweight loss in microeconomic theory.

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More on 2012 Fall

Last semester, I was taking 8.05, 8.13, 8.231, and 14.04, along with continuing my UROP. I was busy and stressed basically all the time. Now I think I know why: it turns out that the classes I was taking were much closer to graduate classes in material, yet they came with all the trappings of an undergraduate class, like exams (that were not intentionally easy). Let me explain a little more.

8.05 — Quantum Physics II is where the linear algebra formalism and bra-ket notation of quantum mechanics are introduced and thoroughly investigated. Topics of the class include analysis of wavefunctions in 1-dimensional potentials, vectors in Hilbert spaces, matrix representations of operators, 2-state systems, applications to spin, NMR, continuous Hilbert spaces (e.g. position), the harmonic oscillator, coherent & squeezed states as well as the representation of photon states and the electromagnetic field operators forming a harmonic oscillator, angular momentum, addition of angular momenta, and Clebsch-Gordan coefficients. OK, so considering that most of these things are expected knowledge for the GRE in physics, this is probably more like a standard undergraduate quantum mechanics curriculum rather than a graduate-level curriculum. That said, apparently this perfectly substitutes for the graduate-level quantum theory class, because I know of a lot of people who go right from 8.05 to the graduate relativistic quantum field theory class.

8.13 — Experimental Physics I is generally a standard undergraduate physics laboratory class (although it is considered standard in the sense that its innovations have spread far and wide). The care and detail in performing experiments, analyzing data, making presentations, and writing papers seem like fairly obvious previews of graduate life as an experimental physicist.

8.231 — Physics of Solids I might be the first class on this list that actually could be considered a graduate-level class for undergraduates, also because the TAs for that class have said that it is basically a perfect substitute for the graduate class 8.511 — Theory of Solids I, allowing people who did well in 8.231 to take the graduate class 8.512 & mdash; Theory of Solids II immediately after that. 8.231 emphasized that it is not a survey course but intends to go deep into the physics of solids. I would say that it in fact did both: it was both fairly broad and incredibly deep. Even though the only prerequisite is 8.044 — Statistical Physics I with the corequisite being 8.05, 8.231 really requires intimate familiarity with the material of 8.06 — Quantum Physics III, which is what I am taking this semester. 8.06 introduces in fairly simple terms things like the free electron gas (which is also a review from 8.044), the tight-binding model, electrons in an electromagnetic field, the de Haas-van Alphen effect, and the integer quantum Hall effect, and it will probably talk about perturbation theory and the nearly-free electron gas. 8.231 requires a good level of comfort with these topics, as it goes into much more depth with all of these, as well as the basic descriptions of crystals and lattices, reciprocal space and diffraction, intermolecular forces, phonons, band theory, semiconductor theory and doping, a little bit of the fractional quantum Hall effect (which is much more complicated than its integer counterpart), a little bit of topological insulator theory, and a little demonstration on superfluidity and superconductivity.

14.04 &; Intermediate Microeconomic Theory is the other class I can confidently say is much closer to a graduate class than an undergraduate class, because I talked to the professor yesterday and he said exactly this. He said that typical undergraduate intermediate microeconomic theory classes are more like 14.03 &; Microeconomic Theory and Public Policy (which I am taking now), where the constrained optimization problems are fairly mechanical, and there may be discussion on the side of applications to real-world problems. By contrast, 14.04 last semester focused on the fundamentals of abstract choice theory with a lot more elegant mathematical formalism, the application of those first principles to derive all of consumer and producer choice theories, partial and general equilibrium, risky choice theory, subjective risky choice theory and its connections to Arrow-Debreu securities and general equilibrium, oligopoly and game theory, asymmetric information, and other welfare problems. The professor was saying that by contrast to a typical such class elsewhere, 14.04 here is much closer to a graduate microeconomic theory/decision theory class, and the professor wanted to achieve that level of abstract conceptualization while not going too far for an undergraduate audience.

At this point, I'm hoping that the experiences from last semester pay off this semester. It looks like that has been working so far!

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Economic Thoughts About Chubby Pageant Toddlers

I haven't gotten a good time to write this until now, despite how short this will be. Anyway, before the semester got busy, I remember browsing around some YouTube videos, and before one of them started, I saw a longer version of the television advertisement for the show "Here Comes Honey Boo Boo".

Like most people, I thought the show represented yet another new low in yet another new dimension of reality television. But then I also realized that this family is perfectly happy doing their own thing, while the TLC show executives are essentially manipulating them to gain viewers, and the family is doing this for TV only because they need the money that badly. That's worse!

So then I wondered about two economic possibilities. Given the multiplier effects present from consumption and investment in the economy, how would the growth of the US economy change if instead of spending as much money as they are on filming the antics of this family, the TLC show executives were to spend that much to help give this family and many others like it a decent education and steady jobs? Or, what would happen to the economy if instead of making a show that essentially mocks this family, TLC were to make a show specifically following this family in getting them a decent education and steady jobs? This way, when the show ends, even if the family ends up squandering all their money, they will still have steady paychecks coming in.

What do you think? Let me know in the comments below. Anyway, I'm going to get back to...uh...my own education (i.e. the never-ending stream of problem sets and other classwork).

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One Outrageously Uninformed Yahoo! Article

I was browsing through my Yahoo! homepage and stumbled upon this article. I looked through the 6 outrageously overpriced items, and I do agree with all of the choices. I suppose that for an article targeted at total laypeople, I can understand why things like "paying for the label" might not make sense at first; that said, I am still astounded as to how a financial article doesn't (appear to) understand why these prices are actually so high. Let's start from the top.

Movie Theater Popcorn

The article here asks why the markup for a bag of popcorn at the movie theater is a factor of six compared to the price for a similar product at a supermarket. It claims to not know the answer to this.
The answer is that in a movie theater, there is nowhere else to get popcorn, and moviegoers like the convenience of buying already-cooked popcorn (as opposed to having to cook and pack their own popcorn). The theater owners have a near-monopoly over the popcorn market inside the theater (as enough people would rather pay more than spend time making their own popcorn), so theater popcorn will of course be more expensive.

Greeting Cards

This follows a similar vein as the popcorn case, though here the card sellers don't have quite as close to a monopoly over the market. For ordinary people, time is precious, and they believe the higher price is a worthy tradeoff for the time saved compared to making a card by hand (after all, one has to buy similar quality paper and pens and think of a witty message (and maybe even a drawing)).

College Textbooks

This is one place where I'm also not sure why the products are so expensive, given the presence and ubiquity of used-book shops (both online and brick-and-mortar) and the rising influence of book rental companies. That said, do consider that prices of goods like cars have experienced similar increases over the same period of time (according to this, the average price of a Ford Mustang in 1985 was $7286, while the current MSRP of a Ford Mustang (2011) is just above $22000). While the increases may be for different reasons, I think it can be partly attributable to plain-old inflation.

Bottled Water

Despite the numerous reports that bottled water is usually packaged unfiltered tap water, perceptions do linger, and the leading perception is still that bottled water is of higher quality than tap water (and, by extension, that tap water is of poor quality). Hence, people would rather pay for the convenience and higher quality of bottled water than spend time filtering tap water and refilling a reusable bottle.

Printer Ink

There are two reasons for this one. One is that there's no alternative to printer ink sold (except to buy another printer). The other is that printer manufacturers lose money on every printer sold, so they recoup it by putting huge markups on printer cartridges. This is also why video games on current consoles still cost $60 even though the disks themselves cost next to nothing to actually produce.

Brand-Name Fashions

Isn't this one obvious? Most of these labels have established reputations, so customers really are willing to pay for the label. It's Microeconomics 101.

I also find it shocking that the material for this article was provided by Investopedia. Why are they writing articles?

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Book Review: "The Undercover Economist" by Tim Harford

(CC-BY-NC-SA Das U-Blog by Prashanth)
This is also a borrowed book (from the same relative that lent me the book Mokshagundam Visvesvaraya). That said, I will say right off the bat that this is a far superior book in terms of writing style and quality.
I took an AP Economics course this past year, so I was somewhat familiar with most of the concepts presented in this book beforehand. However, when I read the book, I realized what I had been missing.
My class's questions didn't have any apparent relation with what was going on in the world. Why should I care if XYZ Corporation's average variable cost is higher than the market price of the good it sells? One is truly an economic theorist if one weeps at the thought of a hypothetical company shutting down to minimize costs. On the other hand, this book relates economic concepts to things that matter, like why coffee is sold at different prices and why different land areas fetch different prices.
Harford first talks about the ideas of rents and profits, continuing with the various functions of prices as signals and the various means companies try to extract different prices from different customers (for the same product). He then talks about externality taxes and subsidies, continuing with market failures occurring due to the presence of inside information (i.e. information that one side but not the other doesn't). He then continues with a discussion on the stock market and on bidding wars. He concludes with discussions on why poor countries remain poor and on the effects of globalization on various countries, both rich and poor.
In short, I am convinced. By using relevant, practical examples in explanations of every concept introduced, he has convinced me that while the free market can solve most problems, government intervention is generally required when externalities show up (and these can't be solved by market negotiations); this is a very reasonable-sounding idea, without resorting to the ideologies of either "government is always bad" or "markets are always harmful".

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How the Economics of Rewards Relates to Open Source

As I just noticed, this is post #10000000_2, #80_16, or #200_8! (The underscore separates the number (on the left) from the base system (on the right).)
I got the idea for this post from a video embedded in an article on the website The Linuxologist. It deals with how incentives affect people's performance - specifically the incentive of a higher reward for better performance. Follow the jump to read the rest.
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