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## Why your otherwise smart professor is a socialist

I saw a video the other day from the American Enterprise Institute about the morality of capitalism. Capitalism, to paraphrase, clears access to the satisfaction that comes from achieving something. Being given the same thing brings us far less happiness. Government, then, takes something from someone to whom it brings a lot of joy and gives it to someone to whom it brings very little. Further, it removes the motivation for those receiving welfare to seek the joy of production and achievement. Yuck! How can we be so heartless?

It occurred to me as I was watching that, while I’ve achieved many things in my life, the American Enterprise Institute might scoff that them.  You see, I work for big companies or worse, universities, where I do research that never makes the New York Times and won’t be featured in a product next year–or next decade. Even the work I do for private companies is often funded by public grants given either to my company or to our customers. Like the villains in the video, I’m often not pleasing “customers” so much as the government committees who review grant applications.

### Gittin’ ‘er done, collectively

When I achieve things, I share credit with thousands of people. Can this collective achievement be as satisfying or valuable as the individual achievement described in the video? Your professor and I think so. We are used to being a small cog in an absolutely enormous machine and we recognize that some things can only be done this way. My grandfather had a tiny role in the early flights of the Space Shuttle–a very big deal that improves your life whenever you turn on your GPS or check the weather report. But if you listed the contributors to that project, Grandpa would surely be buried somewhere in the back with the dolly grip and craft services. He didn’t mind that at all. Whether it’s better play a small part in our mission to space or a large part in a Jamba Juice franchise can certainly be debated.

These collective works we’re about are far-reaching and critically important. One day we’ll announce cold fusion and a cure for cancer, saving the planet and literally millions of lives. Someone who put together the last piece of the puzzle will be on the cover of Time. But behind her, there’ll be legions of scientists who will sit back in their easy chairs with a self-satisfied grin, knowing they’ve done good work and ever so glad they didn’t take their uncle’s advice to drop this ivory tower nonsense and become a day trader.

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## Spontaneous order is always awesome

As I take aim at Friedrich Hayek, on a site called thisweekinstupid, I do it with some trepidation. Hayek was a well-spoken, skilled and innovative economist. That doesn’t mean he didn’t occasionally get it wrong. And in the unfortunate case I’ll discuss today, Hayek is found contributing to a potent and damaging piece of stupid that characterized much of the late 20th century–the cult of the invisible hand.

### Pros and cons of spontaneous order

In the 1950s natural sciences like physics and especially biology began to notice that large systems made from simple parts could work together to create surprising and miraculous results. The brain is the most exciting example of this. Although some neurologists will likely disagree, the dynamics of a single neuron are simple. On receiving a pulse of energy from a nearby neuron through its dendrites, it sends a pulse to other neurons through its axon. This pulse is then received by the dendrites of other neurons. No one would look at that simple system and guess that a collection of those interactions would produce human thought. That miracle of complex macrodynamics from a multiplicity of simple microsystems is what Hayek called “spontaneous order.” Hayek and others believed fervently in the power of spontaneous order to improve people’s lives. Hayek called it a “fatal conceit” to imagine that a designed system could match a spontaneously ordered system for efficiency.

During the Goldwater/Reagan revolution, this became the justification for opposing government economic interference in almost any form. Any top-down tweaking by government moves the economy away from the spontaneous order, which is assumed to be the most efficient possible. It was also a convenient defense against the primary ideological foe of the United States–the Soviet Union. To those of the Austrian school, the economic failure of the Soviet Union was definitive proof of Hayek’s idea.

But on closer examination, the assumption that spontaneous order is always elegant or beneficial seems to come from nowhere, and certainly not from any of the natural sciences. As we look at other examples, we find spontaneous order is, indeed, powerful. But sometimes spontaneous order can be fatal. A herd of cattle can be thought of as a complex system made of simple parts. We could describe the behavior of cows quite simply: Move toward grass; avoid obstacles. But, spurred by the wrong external stimulus, those simple dynamics can cause a stampede as one cow starts to run enticing others to run to get out of its way. Here, the order that arises spontaneously is certainly unexpected in that it does not follow in a straightforward way from the micro behavior. In this, a herd of cattle is like a snowflake or a brain or an ecosystem. But in the case of cows, the macro behavior is not beneficial. Although the microdynamics were about avoiding injury, the resulting stampede can cause cattle to be trampled and killed.

### Spontaneously ordered transportation

So, which kind of spontaneous order is our modern economy? Here’s modern-day libertarian John Stossel extolling spontaneous order and its wisdom in leading America away from transportation by train in favor of cars.

At last month’s State of the Union, President Obama said America needs more passenger trains. How does he know? For years, politicians promised that more of us will want to commute by train, but it doesn’t happen. People like their cars. Some subsidized trains cost so much per commuter that it would be cheaper to buy them taxi rides.

The grand schemes of the politicians fail and fail again.

By contrast, the private sector, despite harassment from government, gives us better stuff for less money—without central planning. It’s called a spontaneous order.

Cars may be the right answer for many communities, but transportation innovations can be a very clear example of the failure of spontaneous order. That is to say, the order arises, it’s just not helpful. Examine the problem of electric cars. My conservative friends have posted pictures to Twitter and Facebook of four or five completely unused car charging stations, usually at government buildings. “Typical government waste,” they’ll say.

They think the market has spoken, and maybe it has. But the other side of the story is that the least convenient aspect of owning an electric car is finding a place to charge it. This certainly reduces the number of electric cars on the road. When a car buyer (one simple part in our complex system) is shopping, she, hypothetically, considers an electric, but since there are no charging stations where she works, she decides on internal combustion. Meanwhile, someone at her work proposes installing charging stations in the parking lot. They take a stroll through the parking lot and find that very few employees own electric cars. So they decide against the charging stations. And around and around we go. More electric cars and more charging stations might be the optimal solution, but the individual actors, pursuing their own interest, can’t get there. Certainly the company can take a chance and build the charging stations hoping more employees are enabled to buy the electric cars they want, but that risk undeniably reduces the chance of us getting there.

For some other examples of the inefficiencies of spontaneously ordered system, check out my post on public goods.

### Lessons from simulation science

In simulating complex systems, we call this a local minimum. Very often a complex system can find itself in a configuration that is not the global best configuration, but from which any small change looks worse. This is a local minimum. When considering electric cars, the status quo (no electric cars and no charging stations) is better than either a) some electric cars with no charging stations or b) no electric cars and some charging stations. So each individual player sees it in their interest to stay right where they are.

Consider this ball rolling on an odd-shaped surface.

The lowest potential energy configuration for the ball–the place it “wants” to be–is at the bottom of the valley marked 3, but in some places on the curve, point 2 for example, the ball sees a hill on either side. It’s in a “stable equilibrium.” If I want to move the ball to the true lowest energy state, it needs a push up the hill. It needs to be moved toward higher energy in order to find a better state.

Our electric car economy is the same (or might be). The economy of transportation is sitting at point 2. Everyone’s myopic view tells them unilateral action is wasteful. Charging stations installed at libraries and government buildings are an attempt to push us up the hill to see if we’ll fall into a better global minimum. It looks like “typical government waste” because we’re not looking at the whole curve. All we see is the hill in from of us. It might work or it might not. Only a global view could hope to predict. But only a fool concludes that the order found organically is always best. In game theory, this kind of stable, non-optimal state is called a Nash equilibrium after John Forbes Nash, Jr. profiled in A Beautiful Mind.

Simulating large systems is what I do for a living. Hayek didn’t have the benefit of huge supercomputers to predict what complex systems will do, but from my experience, assembling a system of millions of interacting parts, turning it on and expecting it to organize itself into an optimal configuration in a reasonable time without any help from me is insanity. When we want to optimize complex systems like static fluid flows or magnetic materials, we have to nudge them to pop them out of local minima or steer them speedily through what would otherwise be a slow spiral toward optimality. We try solving pieces of the problem independently, then stitching them together. Sometimes we reset things to an alternate starting configuration and see if that leads to a better place. (The economic implications of that should keep wealthy capitalists up nights). From where I sit, to expect something as complex as a national economy to optimize its resources without any help demonstrates profound ignorance of the dynamics of complex systems.

We should neither discount the power of spontaneous order, nor place unwarranted faith in it. That’d be stupid.

Categories

## Governor Stupid: “Warren Buffett should write a check and shut up”

We turn our attention to Chris Christie, who, in truth, is my kind of Republican. But nobody’s perfect. Back in 2012, he repeated a familiar refrain in the GOP, although he stated it more forcibly than some. In response to Warren Buffett’s revelation that he pays a lower tax rate than his secretary, Christie said that Buffett should “just write a check and shut up.” That is, if Buffett wants to pay more taxes, he can. What Christie resents is that Warren Buffett wants to force others to pay more taxes to support his favorite programs.

William F. Buckley once said something related

Liberals,  it has been said, are generous with other peoples’ money, except when it comes to questions of national survival when they prefer to be generous with other people’s freedom and security.

Liberals, including thisweekinstupid, disagree. Once again, an ignorance or underestimation of a basic market failure causes us to talk past one another. Government agencies, including the Department of the Interior, NOAA, the FAA, the CDC, and the SEC,  produce what are called “public goods”–goods that you don’t have to own to enjoy. A more thorough discussion of public goods (now featuring math!) is in our first ever thisweekinstupid appendix.

### Public goods for non-economists

In brief, Investopedia defines a public good as a product or service that one individual can consume without reducing its availability to another individual and from which no one is excluded.… National defense, sewer systems, public parks and basic television and radio broadcasts could all be considered public goods.

Left to the free market, public goods will be under-produced. That is, the dollars I voluntarily spend on public goods return a benefit spread over a large group, so, I tend to lean toward spending on private goods, the benefit of which I can secure to myself. This is rational and efficient on a personal level, but irrational and inefficient on a societal level. This is known as the “free riders” problem.

### An example

Imagine five homes in a cul-de-sac. Each of the homes is for rent by a different landlord. A local landlord’s association reports that a $30 per month lawn service increases the rental value of a property by, on average,$40. Further, $100 worth of landscaping of a median in the middle of the cul-de-sac increases the rent for each of the houses by the same$40. A house with a lawn service and a landscaped median can get $55 more than without any landscaping at all. What should I as a landlord, do? Considering only myself, a lawn service for my own lawn is a good deal, so I sign up. But, if I can get everyone to chip in, landscaping the median is even better. If everyone contributes, we can get the same benefit for only$20 each. Imagine a few of us try to pull the money together for a landscaped median. But, as good libertarians, we’re not going to compel anyone to contribute.  Unfortunately, one owner sees the opportunity for a free ride. If she pays only for her own lawn and the rest of us pay for the median, she can make $55 more in rent for only$30 in lawn care. The rest of us are left with a choice. Pay $25 each to landscape the median, or pay$30 to landscape our own yards. The median is still a better deal, but less so than if we had enforced a contribution, for example through home owner’s association dues.  Public goods work the same way in a larger community. In the absence of public funding for arts, parks, public highways, defense, etc. the temptation to free-ride causes us to spend less than is most efficient. Failing to properly value public goods and take steps to produce them, wastes resources. Where public goods are concerned, government spending via taxation can lead to more wealth, efficiency and happiness (more in the appendix).

### Public good skepticism

Intelligent conservatives know this, of course and still want to spend less than me on many government programs. The disagreement is sometimes rooted in the different value conservatives and progressives assign to public goods. Environmental issues are an easy place to see this. In discussions with Republicans and Libertarians on this topic it becomes quickly clear that, compared to myself, they underestimate the differential benefit between an intact ecosystem and one damaged by oil exploration, air pollution or deforestation. This can be either because, compared with me,

1. they underestimate the ecological damage; or
2. they undervalue the things a healthy ecosystem provides.

I find that it’s often a combination of these. Climate change skeptics often don’t think carbon emissions cause warming to a great extent, but even if it did, they don’t think warming is so bad. Connecting it to the lawn care analogy, some owners might doubt that tenants care about a landscaped median. Or they might not trust the community to choose the right landscaper and deliver the promised value. Any one of these factors may cause them to undervalue the public good and reject the cooperative solution.

And so it goes with taxation. Compared with me, Chris Christie underestimates our connectedness. I believe that undereducated or unhealthy children and impoverished families cause great harm to us all and so education, public health care and anti-poverty programs are a public good of great value. Further, I believe we’re economically connected enough that the presence of a public safety net is a great economic benefit to the whole country, and especially to the people with assets to lose. If we removed the public safety net, some of us would put money in the collection plate to cover rent and medical care for the poor, but to get really rich, a better strategy would be to keep your money, buy cheap apartments and rent them to the poor, collecting the charity they’re given by everyone else. Free riders win again.

### Role reversal

The other half of William Buckley’s statement concerns the military, which is also a public good. You can’t let someone opt out of funding the military because protecting every third house from the Red Dawn is just as expensive as protecting the whole country. Opters-out become free riders. But, in the case of defense spending, Republicans simply value that public good higher than I do. I think our military is big and strong enough to defend our country from any (terrestrial) enemy several times over. I regard the marginal benefit of an additional dollar in the defense budget to be zero (possibly less). So, I’m quite tired of Chris Christie and William Buckley and their set being quite so generous with my money toward military contractors.

Armed with an understanding of public goods and free riders, liberals are not naive enough to remove public services, cut taxes, and wait for libertarians to plow their tax savings into private programs to keep the water clean, track diseases, make sure the planes don’t crash, get the mentally ill off of the street, monitor trading of securities and pay the rent and medical bills of the elderly poor. On the contrary, sensible economic thinking says the market will be far too profligate with other people’s health, air quality and safety.

Instead, I propose the libertarians go first. As soon as there’s a private program tracking the evolution of new diseases, we can defund the CDC. When private charity clinics provide the medical care for all the poor, there’ll be no need for Medicaid. And the great thing about some of these programs is that no one need “go first.” Shriners Hospital provides all kinds of free medical care to children. Every surgery in a Shriners Hospital is one that doesn’t have to be done elsewhere. If the child being treated is covered by Medicaid, that’s savings and deficit reduction right there. In a very organic way, private charities could take over for public services. As I’ve detailed above, I have my own reasons for believing it wouldn’t happen that way, but I’d love to be proven wrong. And this is certainly no less reasonable than the opposite proposition–defunding these programs without a viable replacement and blithely hoping one will appear.

So, Governor Christie, if you want smaller government, call up the Shriners, “write a check, and shut up.”

Categories

## Appendix: Public Goods

Our discussion of public goods seemed incomplete without at least a little math to back it up. But no one wants to alienate the mathophobes, so we parked this tidbit in the “appendix.”

Let’s analyze a few public goods. Imagine, again, 5 farmers. Last year, one farmer contracted with a beekeeper to manage a hive of bees near his field. He paid $2500 for the year. The next year, his harvest increased by$4500. A farmer in a nearby neighborhood paid for even more hives and boosted his profits even more, although the first $2500 is the most effective in this regard. But he’s not the only one who benefits. Each of this neighbors also increased their crop yields. After doing some research, he discovers that, in a community such as theirs, for each of your neighbors hiring a$2500 bee hive, you can get $1500 more crop even without spending a cent on your own bees. Mathematically, let’s conjecture that the value of the increased yield is $\sqrt{\alpha S_0 + \sum \beta S_i}$ and the profit from that extra spending is $\mbox{Profit}= \sqrt{\alpha S_0 + \sum \beta S_i} – S_0$ where$latex S_0$is the amount I spend on bees and$latex S_i$are my neighbors’ spending. The constant (exogenous) parameters$latex \alpha$and$latex \beta$control just how much benefit I get from my own and from my neighbors’ spending, respectively. The reason for the square root is the idea of “diminishing marginal utility.” Spending$6000 on bees is still better than spending $3000, but something less than twice as good. Your first dollar is more important than any subsequent dollar. That said, the utility curve does not have to be a square root. Now, if all of the neighbors are equally interested in their property values, they might agree to all spend the same on bees, perhaps in a written agreement. If we constrain all of$latex S_i$to be the same value, we have a formula of just one variable and can plot the profit function: From the plot, we can see that, if the neighbors all cooperate, they can each get$6000 more crop from spending just $3000 on bees. This is the most efficient level of spending. But here we run into the free rider problem. Suppose the neighbors do not cooperate and one of the neighbors does the calculation without considering anyone else. If everyone else continues spending$3000, how much should I spend to maximize my profit? When we fix $latex S_i$ at $3000 and allow just$latex S_0$to vary, the new plot looks like this: The neighbors cooperating can make a$3000 profit, but from the blue curve we  see that the selfish neighbor can spend just $500 on bees and make a$3500 profit. In this situation, the self-interest of the individual hinders the prosperity of the community.

The situation gets even worse for goods which are more public. Perhaps the road leading between the town and the farms needs fixing and they’re freestaters, so the government won’t do it. They’re on their own. This fix will reduce wear on the trucks that take crops to the markets, etc.. Mathematically, $latex \alpha = \beta$. In that case the plot looks like this:

A free-rider, in this case, would have almost no incentive to contribute. That’s not exactly true since even the free riders understand they’re gaming the system and that others will be inclined to seek the same deal. When you incorporate this, you find that, in fact, neighbors are willing to spend just $200 on paving, leaving$3200 of profit on the table. (For more details on this aspect, check out the appendix to the appendix, Galt-ifying Public Goods.) The problem only gets worse as the community grows. If you’re paving a street that benefits 20 people with the same utility function, people are willing to contribute just $47 and will miss out on more than$20,000 profit.

Public goods are not always so easily quantified and it’s usually there that disagreements arise, but a little math goes a long way toward appreciating that opinions about public spending are a continuum and that pretending the market is always right (or always wrong) has real consequences.

Not enough math yet? Check out the appendix to the appendix, where we ask the mathematical question, how much more efficient would John Galt’s community (from the novel Atlas Shrugged) have to be to make up for refusing to subsidize public goods.

Categories

## The Market Will Set the Right Price for Health Care

I don’t suppose the interweb needs another discussion of the failings of market forces in health care markets. And yet the idea persists that setting the price of health care should be left to the unregulated market and that this will lead to maximum efficiency. The omnipotence of the market is a comforting concept.

Demand-side troubles

But the market for anti-retroviral drugs bears little resemblance to golf clubs or top sirloin. At risk of sounding condescending, a typical demand curve looks like this

As the price of a good or service rises, the quantity demanded shrinks. Lots of people will buy a pair of basketball shoes for $6. A few would pay$140.

But this doesn’t often work for medical services. I recently had my appendix removed. I’m told a typical appendectomy costs $25,000–a lot of money no matter who you are. But whether it had cost$100 or $100,000, I still would have bought exactly one appendectomy. The demand curve for appendectomies looks more like this: There is, perhaps, some fall in demand. A 96-year-old might decline a quarter million dollar appendectomy. In some cases, there may be more than one treatment for an ailment so that one good or service may be substituted for another. But, in general, demand for health care services, especially the very expensive, life-saving treatments, tends to be highly inelastic. The quantity demanded is affected very little by the price. So, can providers of health care charge whatever they like? Supply-side salvation? Not necessarily. There may be some help on the supply side of this market. But, one reason markets are so effective and robust is the interplay of supply and demand. Without both working properly, the market can misallocate. When the supply side of the equation breaks down, for example in the case of a monopoly, we know that even with a healthy demand side, we’ll run into inefficiencies. So, we already have cause to worry. A truly effective market needs both a healthy demand side and supply side. Switching to the supply side of things, if the price of a good is very high, more people will be willing to supply it. A supplier that overcharges will find herself undercut by a competitor willing to supply the good at a lower price. So, even with highly inelastic demand curves, there’s an equilibrium price at the point where the supply and demand curves meet. So, it all might work out just fine, as long as actors can’t manipulate the supply curve. Unfortunately, two of the easiest ways to do that are both, of necessity, highly active in health care markets. The first is patents. A patent grants a single company the exclusive right to produce a product for a certain period. A supplier with a patent cannot be undercut by a cheaper competitor. When you are the only supplier of a lifesaving procedure, the market will not place any limit on the price (although public opinion or your own morality might). For this reason, it’s more useful to think about the supply curve for health care research than for particular treatments. If the payout for developing a new drug is very high, more people will be willing to do research toward that end. So, even where patents are applied, there is a functioning supply side curve at work. But in such a market, price signals can take longer to move through the market. When I need medication today, it’s slim comfort to know that the exorbitant price I pay for patent-protected drugs is providing the impetus for a robust market for drug research. Another common way to move a supply curve is through licensing. Most people who treat you in the hospital are licensed, some of them very licensed, which is wonderful. It comforts me immensely that the person holding the scalpel has undergone years of training and scrutiny. But, the effect of this is to reduce the supply of doctors, nurses and other medical professionals. The FDA’s approval processes provide the same type of scrutiny for medications, equipment and treatments, with the same effect. I wouldn’t have it any other way, of course, but the effect of this licensing is to shift the supply curve downward, increasing the price of goods. The further the supply is reduced, the higher the price. The inelasticity of the demand curve (and also the supply curve) multiplies this effect. You can see how influence of these licensing processes could be very lucrative for suppliers. Doctors, for the most part, do important work for sincerely good reasons, but putting the AMA in charge of licensing doctors is a lot like asking the fox to guard the hen house. The tendency of almost everyone is to highly value their own work and the incentive for doctors is to limit the supply of doctors, raising their own salaries. Similarly, if pharmaceutical or medical supply companies can delay or scuttle approval for competing drugs and equipment, they also stand to make lots of money. I’m not for a minute suggesting we do away with licensing of doctors or patents for drugs. Health care markets can’t be effective without these things. But, perhaps it’s a good idea to think hard about the markets for health care rather than blithely assuming the miraculous market will allocate everything just right. Without some advocates for consumers of health care, rising, inelastic demand will push prices out of reach and make life-saving care an unaffordable luxury. Categories ## Selfishness is Good For our first real post, we’ll turn to a deeply entrenched conservative belief. You might expect this on a site called ThisCentury(and Last)InStupid. It’s the idea that selfishness isn’t so bad after all. In fact, we could use more of it. Occasionally, someone comes out and states this explicitly, but not usually. More often its the implied justification of cruel policy. Or it appears as an allegory as in Paul Ryan’s favorite ode to greed, Atlas Shrugged. As will be familiar to many, Atlas Shrugged is a novel by uber-capitalist and self-professed narcissist Ayn Rand about a fictional America in which all the captains of industry tire of being disrespected, regulated, demonized and (most egregiously) taxed and decide to flee society to a secret mountain enclave and live together in capitalist utopia. The rest of the country, bereft of its “engine” grinds to a halt. Poverty and violence ensue. This vision is the engine behind Republican policies that, sometimes quite overtly, enrich the wealthy at the expense of everyone else. One fine recent example is the refusal of Republican Governors across the country to expand Medicaid in their states. Although it would cost their governments very little, and benefit the poor quite a lot, the principle of giving as little to the poor as you can get away with is so ingrained that they’d rather leave the poor without health care and let them be treated in emergency rooms and the cost of their unpaid care absorbed into the premiums of the rest of the state. Or there’s the fact that we can’t close the “carried interest” loophole by which the income of hedge fund managers is taxed as if it were long-term capital gains or dividends (on assets they don’t own!). No matter how you feel about lower rates on capital gains, these are clearly income. So, you have people paying 15% on their multi-million dollar income. The people emptying their trash pay a higher rate. There are efforts at very high levels to fix this, but none very successful. Agitation against sensible policies to alleviate poverty are often accompanied by mumbled pseudo-economic arguments about how discouraging the money-making activities of the wealthy will harm us all or how giveaways to the poor create dependency. But, simmering beneath the surface of these attempts to make economics support fiscal austerity is an audacious Republican hope–one buoyed by this Randian vision. The wish at the heart of modern conservatism is that selfishness, in the end, will turn out to be altruism—that, when the score is all tallied, the best thing we can do for the poor is to stick it to them just as hard as we can. Thus conservative guilt is swept away by one swift stroke of Milton Friedman’s pen. This miraculous moral alchemy whereby, if two wrongs don’t make a right, nonetheless, two million do, is unchallenged on the Right since Adam Smith despite its manifest stupidity. This is partly because it’s unstated but also because it’s so damned useful. It was useful during the Cold War marriage of what today are called social and fiscal conservatism by severing the link between Christian charity and social welfare. It continues to be useful in justifying the neglect or animosity toward the social contract by the upper and middle classes today. It makes conservatism easier on the soul. Now, some conservatives will try to separate personal and public morality saying that while public welfare programs harm the poor, private programs are another matter altogether. Of course, the original source of the charity can’t likely save the poor from the dependency-producing effects of free stuff. This view is likely more closely related to another pervasive dose of stupid up for discussion in a later post : “The government can’t do anything right.” So, you heard it here first: Selfishness isn’t good. It isn’t noble or necessary or even inevitable. It wrecks communities and nations and, in a cosmically ironic twist, makes it’s practitioners the most miserable of all. Categories ## Appendix: Galt-ifying public goods ### See how deep the rabbit hole goes… In our first appendix, we wandered a little into the math behind public goods. That post has less algebra and more graphs. If what you read here is moving too fast, that might be a good place to start. Public goods, in brief, are goods which you don’t have to own to enjoy. These are things like public parks, an army, or your neighbor’s front lawn. In that post, we assumed a particular form for a utility function for public goods. $\mbox{Utility} = \sqrt{\alpha S_0 + \sum\beta S_i}$ where$latex S_0$is the amount I spend on the good,$latex S_i$are my neighbors’ spending. The parameters,$latex \alpha$and$latex \beta$determine how much benefit I get from my own spending and my neighbors’ spending, respectively. There’s more on why this specific functional form was chosen here. Other forms are certainly valid. In this model, the profit is the utility minus my expenditure,$latex S_0$$\mbox{Profit} = \sqrt{\alpha S_0 + \sum\beta S_i} – S_0$. If an overbearing government (or home owners association, or business) enforced the same spending on public goods by all beneficiaries, we can determine the optimal spending by equating$latex S_0$and$latex S_i$and taking the derivative of profit with respect to this single variable: $\frac{dP}{dS} = \frac{\alpha+(n-1)\beta}{2\sqrt{S(\alpha+(n-1)\beta)}} – 1$ Here we’ve introduced n, the total number of participants. Equating this derivative to zero, we find that the optimum level of spending is $S = \frac{\alpha + (n-1)\beta}{4}$ for a profit of $P = \frac{\alpha+(n-1)\beta}{4}$. per person. If, on the other hand, we leave it up to each individual to contribute what she thinks is best for herself, we differentiate the profit equation with respect to changes in$latex S_0$alone. $\frac{\partial P}{\partial S_0} = \frac{\alpha}{2\sqrt{\alpha S_0+\beta \sum S_i}} – 1$ Then, since all the participants are assumed to be 100% rational (and will therefore make the same decision), we equate$latex S_0$and$latex S_i$, set the derivative equal to zero and discover that spending becomes $\frac{\alpha^2}{4(\alpha + (n-1)\beta)}$ and profit is $\frac{\alpha}{2}-\frac{\alpha^2}{4(\alpha+(n-1)\beta)}$. In the other post on public goods,we reported that, for our simple model of paving the street leading to a cul-de-sac with 5 houses, our homeowners would only be willing, on their own, to spend$2 on paving rather than the ideal $50, which reduced their profit from$50 to $18 each. Now, let’s introduce a “Galt factor,” G, representing the increased productivity of workers in John Galt’s mountain enclave due to the absence of moochers. How high would the Galt factor have to be for the Galt society to have more efficient lawn care than us regular shmoes? Both$latex S_0$and$latex S_i$should be multiplied by G since the labor and other input in the Galt society is that much more potent. So, we multiply all$latex S_0$and$latex S_i$by G in the profit for the non-cooperative profit and equate that to the (unmodified) profit for the cooperative system. $\frac{G\alpha}{2} – \frac{G\alpha^2}{4(\alpha+(n-1)\beta)} = \frac{\alpha+(n-1)\beta}{4}$ Solving for G, we find that the Galt factor necessary for the Galt society to operate more efficiently than a “moocher” society which recognizes and subsidizes the public good in question is $\frac{(\alpha+(n-1)\beta)^2}{\alpha(\alpha+2(n-1)\beta)}$ Here are some sample results using numbers from the beekeeping and road paving examples in the last post. In the gardening example, most of the benefit of keeping up my lawn goes to me. Only a little goes to my neighbors which is why$latex \alpha$is greater than$latex \beta$in the first two rows. The final 3 rows are based on a good that’s completely public–everyone benefits equally no matter who paid. Last post we used the example of paving the road that leads to the cul-de-sac. Since everyone enjoys the benefits of the spending equally,$latex \alpha=\beta$.$latex \alphalatex \betalatex nlatex G$80 10 5 1.17 80 10 100 7.0 40 40 5 3.27 40 40 20 10.7 40 40 100 50.7 We could even simplify the above expression by constructing the “public-ness” of a good, P. $P = \frac{(n-1)\beta}{\alpha+(n-1)\beta}$ Public-ness ($latex P$) is the fraction of the total benefit that goes to the community as whole. The Galt factor in terms of$latex P$is $G= \frac{1}{1-P^2}$ Now, if we had any reason to believe our model this would be a very useful formula. Should a particular service be left to the market or paid for with public funds? Just figure P, make a guess as to how much more efficient your private market would be and compare. For goods that are completely shared, P is just$latex 1- 1/n$with$latex n$being the size of the community. For a cul-de-sac of 5, the Galt factor must be 25/9 = 2.78. For a small town of 10,000, P = .9999 and G = 100,000,000. So, for this adorably naive model the Galt society could likely manage to outdo us in gardening, as long as their cul-de-sacs didn’t get much bigger than 5 houses, but when it comes to truly public goods consumed by hundreds of people (like a public park or a security guard at your apartment complex) they’ll be hard pressed to accomplish in one hour what takes the rest of us a full work week. For goods shared by hundreds of thousands of people, the Galters might as well not try. In many cases, at least some of the benefit can be secured exclusively for the benefit of the producer and her customers. For example, a road can be built with toll booths or the radio station can interrupt your music with commercials. These actions are effectively changing$latex \alpha$and$latex \beta$. These efforts almost always lead to a reduction in both (i.e. it’s less useful to everyone), but also reduces$latex P\$ making the private option more attractive and enticing people to invest in production. If we assume that private systems are more efficient than public ones, we can even reach the conclusion that adding obstacles (like commercials, patents and toll booths) can boost efficiency.

It’s fair to say that a model this crude isn’t likely to be accurate to any degree, but it gives an idea of some of the qualitative factors that effect evaluation of public goods. Too often our political discussion never gets past pithy platitudes. When we say, “private markets are more efficient,” it’s important to specify, even with a crude estimate, just how much more efficient. Liberals should not pretend the Galt factor is 1, but conservatives should not pretend it’s infinite. And, we should recognize that neither private nor public production is the right solution for all good in all markets. Math matters.

Do you have corrections or comments? How would you improve the model? Can you think of some goods for which the utility functions could be measured? Send me an email.