Tax incidence revisited, part 2: How taxes affect prices

JDN 2457341

One of the most important aspects of taxation is also one of the most counter-intuitive and (relatedly) least-understood: Taxes are not externally applied to pre-existing exchanges of money. Taxes endogenously interact with the system of prices, changing what the prices will be and then taking a portion of the money exchanged.

The price of something “before taxes” is not actually the price you would pay for it if there had been no taxes on it. Your “pre-tax income” is not actually the income you would have had if there were no income or payroll taxes.

The most obvious case to consider is that of government employees: If there were no taxes, public school teachers could not exist, so the “pre-tax income” of a public school teacher is a meaningless quantity. You don’t “take taxes out” of a government salary; you decide how much money the government employee will actually receive, and then at the same time allocate a certain amount into other budgets based on the tax code—a certain amount into the state general fund, a certain amount into the Social Security Trust Fund, and so on. These two actions could in principle be done completely separately; instead of saying that a teacher has a “pre-tax salary” of $50,000 and is taxed 20%, you could simply say that the teacher receives $40,000 and pay $10,000 into the appropriate other budgets.

In fact, when there is a conflict of international jurisdiction this is sometimes literally what we do. Employees of the World Bank are given immunity from all income and payroll taxes (effectively, diplomatic immunity, though this is not usually how we use the term) based on international law, except for US citizens, who have their taxes paid for them by the World Bank. As a result, all World Bank salaries are quoted “after-tax”, that is, the actual amount of money employees will receive in their paychecks. As a result, a $120,000 salary at the World Bank is considerably higher than a $120,000 salary at Goldman Sachs; the latter would only (“only”) pay about $96,000 in real terms.

For private-sector salaries, it’s not as obvious, but it’s still true. There is actually someone who pays that “before-tax” salary—namely, the employer. “Pre-tax” salaries are actually a measure of labor expenditure (sometimes erroneously called “labor costs”, even by economists—but a true labor cost is the amount of effort, discomfort, stress, and opportunity cost involved in doing labor; it’s an amount of utility, not an amount of money). The salary “before tax” is the amount of money that the employer has to come up with in order to pay their payroll. It is a real amount of money being exchanged, divided between the employee and the government.

The key thing to realize is that salaries are not set in a vacuum. There are various economic (and political) pressures which drive employers to set different salaries. In the real world, there are all sorts of pressures that affect salaries: labor unions, regulations, racist and sexist biases, nepotism, psychological heuristics, employees with different levels of bargaining skill, employers with different concepts of fairness or levels of generosity, corporate boards concerned about public relations, shareholder activism, and so on.

But even if we abstract away from all that for a moment and just look at the fundamental economics, assuming that salaries are set at the price the market will bear, that price depends upon the tax system.

This is because taxes effectively drive a wedge between supply and demand.

Indeed, on a graph, it actually looks like a wedge, as you’ll see in a moment.

Let’s pretend that we’re in a perfectly competitive market. Everyone is completely rational, we all have perfect information, and nobody has any power to manipulate the market. We’ll even assume that we are dealing with hourly wages and we can freely choose the number of hours worked. (This is silly, of course; but removing this complexity helps to clarify the concept and doesn’t change the basic result that prices depend upon taxes.)

We’ll have a supply curve, which is a graph of the minimum price the worker is willing to accept for each hour in order to work a given number of hours. We generally assume that the supply curve slopes upward, meaning that people are willing to work more hours if you offer them a higher wage for each hour. The idea is that it gets progressively harder to find the time—it eats into more and more important alternative activities. (This is in fact a gross oversimplification, but it’ll do for now. In the real world, labor is the one thing for which the supply curve frequently bends backward.)

supply_curve

We’ll also have a demand curve, which is a graph of the maximum price the employer is willing to pay for each hour, if the employee works that many hours. We generally assume that the demand curve slopes downward, meaning that the employer is willing to pay less for each hour if the employee works more hours. The reason is that most activities have diminishing marginal returns, so each extra hour of work generally produces less output than the previous hour, and is therefore not worth paying as much for. (This too is an oversimplification, as I discussed previously in my post on the Law of Demand.)

demand_curve

Put these two together, and in a competitive market the price will be set at the point at which supply is equal to demand, so that the very last hour of work was worth exactly what the employer paid for it. That last hour is just barely worth it to the employer, and just barely worth it to the worker; any additional time would either be too expensive for the employer or not lucrative enough for the worker. But for all the previous hours, the value to the employer is higher than the wage, and the cost to the worker is lower than the wage. As a result, both the employer and the worker benefit.

equilibrium_notax

But now, suppose we implement a tax. For concreteness, suppose the previous market-clearing wage was $20 per hour, the worker was working 40 hours, and the tax is 20%. If the employer still offers a wage of $20 for 40 hours of work, the worker is no longer going to accept it, because they will only receive $16 per hour after taxes, and $16 isn’t enough for them to be willing to work 40 hours. The worker could ask for a pre-tax wage of $25 so that the after-tax wage would be $20, but then the employer will balk, because $25 per hour is too expensive for 40 hours of work.

In order to restore the balance (and when we say “equilibrium”, that’s really all we mean—balance), the employer will need to offer a higher pre-tax wage, which means they will demand fewer hours of work. The worker will then be willing to accept a lower after-tax wage for those reduced hours.

In effect, there are now two prices at work: A supply price, the after-tax wage that the worker receives, which must be at or above the supply curve; and a demand price, the pre-tax wage that the employer pays, which must be at or below the demand curve. The difference between those two prices is the tax.

equilibrium_tax

In this case, I’ve set it up so that the pre-tax wage is $22.50, the after-tax wage is $18, and the amount of the tax is $4.50 or 20% of $22.50. In order for both the employer and the worker to accept those prices, the amount of hours worked has been reduced to 35.

As a result of the tax, the wage that we’ve been calling “pre-tax” is actually higher than the wage that the worker would have received if the tax had not existed. This is a general phenomenon; it’s almost always true that your “pre-tax” wage or salary overestimates what you would have actually gotten if the tax had not existed. In one extreme case, it might actually be the same; in another extreme case, your after-tax wage is what you would have received and the “pre-tax” wage rises high enough to account for the entirety of the tax revenue. It’s not really “pre-tax” at all; it’s the after-tax demand price.

Because of this, it’s fundamentally wrongheaded for people to complain that taxes are “taking your hard-earned money”. In all but the most exceptional cases, that “pre-tax” salary that’s being deducted from would never have existed. It’s more of an accounting construct than anything else, or like I said before a measure of labor expenditure. It is generally true that your after-tax salary is lower than the salary you would have gotten without the tax, but the difference is generally much smaller than the amount of the tax that you see deducted. In this case, the worker would see $4.50 per hour deducted from their wage, but in fact they are only down $2 per hour from where they would have been without the tax. And of course, none of this includes the benefits of the tax, which in many cases actually far exceed the costs; if we extended the example, it wouldn’t be hard to devise a scenario in which the worker who had their wage income reduced received an even larger benefit in the form of some public good such as national defense or infrastructure.

The scissors of supply and demand

JDN 2457299 EDT 17:03

In recent posts I talked about demand and then I talked about supply. Now it’s time to talk about both at once–which is where the real magic happens. Alfred Marshall famously compared supply and demand to the lower and upper blades of a pair of scissors:

We might as reasonably dispute whether it is the upper or the under blade of a pair of scissors that cuts a piece of paper, as whether value is governed by utility or cost of production. It is true that when one blade is held still, and the cutting is effected by moving the other, we may say with careless brevity that the cutting is done by the second; but the statement is not strictly accurate, and is to be excused only so long as it claims to be merely a popular and not a strictly scientific account of what happens.

~Alfred Marshall, Principles of Economics

Before Marshall, it was actually rather common to debate whether prices are determined by supply or by demand. Actually there seems to be a certain branch of Marxists today who insist upon the “labor theory of value” that seems to rest upon a similar sort of confusion, basically saying that the real value of something is entirely determined by its cost of supply. If the value of something were strictly determined by the labor put into making it, there would be literally no reason to ever make anything. If the value you get from a good is precisely equal to the labor put into it, there is no net benefit to ever making any goods. At most, embodying labor in a product might allow you to transfer labor from one person to another; but there would be no such thing as real economic growth. In order to have real economic growth, products must end up being worth more than what it cost to make them—that is, their value of demand must exceed their cost of supply.

Toward the other end of the political spectrum, we have “Say’s Law”, which says that “supply creates its own demand”; that is, that there is never any such thing as too much or too little overall demand in an economy, because supplying a good automatically makes that good available to trade for something else. I hate to even call it a “law” because isn’t even like the Pirate Code; it’s not even useful as a guideline, it’s just flat wrong. There is absolutely no reason that making something would make someone else want to buy it from you. You can make all sorts of things that nobody wants to buy; the possibilities are endless, really. Balls of lint dusted with powdered sugar, broken ballpoint pens dipped in motor oil, burnt-out lightbulbs covered in melted Swiss cheese. It’s possible that someone might want to buy such bizarre items (call them “postmodernist found art” or something), but there clearly isn’t a large market for such goods, even if you should decide to manufacture thousands of them. Even in an aggregate sense, there’s also no particular reason to think that we can’t have an economy where millions of products pile up on shelves because no one can afford to buy them; indeed, that’s basically what happens in a recession.

In fact, the converse, “demand creates its own supply”, is considerably closer to true. It’s still not strictly true—centuries of searching for the elixir of immortality have failed to produce it, though modern genetic engineering just might finally succeed where all else has failed. (After all, every new technology is impossible… until it isn’t.) But in the long run, this converse law (it doesn’t have a name so far as I know) does contain an important grain of truth: If people want something badly enough, they will spend enormous resources in order to find a way to get it. If you know that a lot of people want something that no one is supplying, it behooves you to find a way to provide it—it might just make you a billionaire. Over centuries of technological advancement, humanity has found ways to provide many goods and services that were previously thought impossible, and one of the central benefits of a capitalist economy is that it provides powerful economic incentives for entrepreneurs to innovate and find ways to provide goods that people have always wanted but never had. Yet, even so, it isn’t true that demand creates its own supply—certainly not in the short run.

Neither supply or demand on its own does much of anything. You can have insatiable demand for something nobody can supply (the aforementioned elixir of immortality), and it still won’t be sold. You can have endless supply of something nobody demands (vacuum?), and it will remain worthless. It’s only when you have both supply and demand that a market becomes possible.

One of the central insights of modern economics is that prices and quantities in a capitalist market are determined simultaneously by supply and demand. In general, both supply and demand are constantly changing in response to events in the world, and thus the prices and quantities of goods shift from one equilibrium to another. In order to predict exactly how they will shift, we would need to know how both supply and demand have changed.

As Marshall alludes to in the above quotation, in some cases we can take either supply or demand as fixed and then the other one is what matters; but these are only special cases. In general, both supply and demand are subject to the winds of changing markets, and we need to keep track of both at once. If that sounds really difficult, that’s because it is—most of what economists do in the real world ultimately amounts to finding ways to distinguish supply effects from demand effects in various situations. Even most statistical methods in econometrics were basically designed as means of separating out demand-related causes from supply-related causes.

A lot of policy questions ultimately depend upon whether supply or demand is the dominant factor: If the business cycle is primarily driven by changes in demand, it makes sense to use monetary and fiscal policy to stabilize the economy (short version: it is, and it does). If it were instead driven by supply (“supply-side economics”), it would instead be better to make structural changes that reduce costs of production. (Why is this obviously wrong? Because there weren’t sudden increases in production costs in 2008—but there was a sudden collapse of consumer buying power. Maybe the 1973 recession can be explained by a sudden increase in oil prices, but there was no such supply shock in 2008.) If the labor market is primarily driven by demand, we need to find ways to get business to hire more people; but if it’s primarily driven by supply, we need to find ways to get people to get off their butts and try to find work. (Again, I think it’s pretty obvious that the former is true, not the latter—since at least 2000 there have never been as many job openings in the US as there were unemployed people.)

In the above policy questions the liberal view is the demand-side and the conservative view is the supply-side, but that need not be the case. Regarding renewable energy, for example, the more liberal view is that lots of people would want to buy electric cars and solar panels, if they were made available, but they aren’t—we are supply-constrained. The more conservative view is that the reason they aren’t selling more is that nobody particularly wants them and trying to force them on us is a fool’s errand—we are demand-constrained. Likewise when it comes to banking, liberals generally think that the reason there isn’t more credit is that banks refuse to supply loans, while conservatives (particularly from the banks themselves) usually argue that it’s because people aren’t willing to take the risk of taking out more loans.

The point, however, is that a lot of policy debates ultimately hinge upon the question of whether demand or supply is more important in driving a particular market—and since sometimes they are both important, sometimes the policy solution requires a combination of different approaches. One of the advantages of quantitative economic analysis is that we can determine exactly how much the costs and benefits of each policy option will be, and thereby choose the one that is most cost-effective.

In this way, “supply or demand?” is a lot like “nature or nurture?”; the answer is always “both”, but there are times when one factor or the other is more important for the policy question at hand.

Elasticity and the Law of Supply

JDN 2457292 EDT 16:16.

Today’s post is kind of a mirror image of the previous post earlier this week; I was talking about demand before, and now I’m talking about supply. (In the next post, I’ll talk about how the two work together to determine the actual price of goods.)

Just as there is an elasticity of demand which describes how rapidly the quantity demanded changes with changes in price, likewise there is an elasticity of supply which describes how much the quantity supplied changes with changes in price.

The elasticity of supply is defined as the proportional change in quantity supplied divided by the proportional change in price; so for example if the number of cars produced increases 10% when the price of cars increases by 5%, the elasticity of supply of cars would be 10%/5% = 2.

Goods that have high elasticity of supply will rapidly flood the market if the price increases even a small amount; goods that have low elasticity of supply will sell at about the same rate as ever even if the price increases dramatically.

Generally, the more initial investment of capital a good requires, the lower its elasticity of supply is going to be.

If most of the cost of production is in the actual marginal cost of producing each new gizmo, then elasticity of supply will be high, because it’s easy to produce more or produce less as the market changes.

But if most of the cost is in building machines or inventing technologies or training employees which already has to be done in order to make any at all, while the cost of each individual gizmo is unimportant, the elasticity of supply will be low, because there’s no sense letting all that capital you invested go to waste.
We can see these differences in action by comparing different sources of electric power.

Photovoltaic solar power has a high elasticity of supply, because building new solar panels is cheap and fast. As the price of solar energy fluctuates, the amount of solar panel produced changes rapidly. Technically this is actually a “fixed capital” cost, but it’s so modular that you can install as little or as much solar power capacity as you like, which makes it behave a lot more like a variable cost than a fixed cost. As a result, a 1% increase in the price paid for solar power increases the amount supplied by a whopping 2.7%, a supply elasticity of 2.7.

Oil has a moderate elasticity of supply, because finding new oil reserves is expensive but feasible. A lot of oil in the US is produced by small wells; 18% of US oil is produced by wells that put out less than 10 barrels per day. Those small wells can be turned on and off as the price of oil changes, and new ones can be built if it becomes profitable. As a result, investment in oil production is very strongly correlated with oil prices. Still, overall production of oil changes only moderate amounts; in the US it had been steadily decreasing since 1970 until very recently when new technologies and weakened regulations resulted in a rapid increase to near-1970s levels. We sort of did hit peak oil; but it’s never quite that simple.

Nuclear fission has a very low elasticity of supply, because building a nuclear reactor is extremely expensive and requires highly advanced expertise. Building a nuclear power plant costs upward of $35 billion. Once a reactor is built, the cost of generating more power is relatively trivial; three-fourths of the cost a nuclear power plant will ever pay is paid simply to build it (or to pay back the debt incurred by doing so). Even if the price of uranium plummets or the price of oil skyrockets, it would take a long time before more nuclear power plants would be built in response.

Elasticity of supply is generally a lot larger in the long run than in the short run. Over a period of a few days or months, many types of production can’t be changed significantly. If you have a corn field, you grow as much corn as you can this season; even if the price rose substantially you couldn’t actually grow any more than your field will allow. But over a period of a year to a few years, most types of production can be changed; continuing with the corn example, you could buy new land to plant corn next season.

The Law of Supply is actually a lot closer to a true law than the Law of Demand. A negative elasticity of supply is almost unheard of; at worst elasticity of supply can sometimes drop close to zero. It really is true that elasticity of supply is almost always positive.

Land has an elasticity near zero; it’s extremely expensive (albeit not impossible; Singapore does it rather frequently) to actually create new land. As a result there’s really no good reason to ever raise the price of land; higher land prices don’t incentivize new production, they just transfer wealth to landowners. That’s why a land tax is such a good idea; it would transfer some of that wealth away from landowners and let us use it for public goods like infrastructure or research, or even just give it to the poor. A few countries actually have tried this; oddly enough, they include Singapore and Denmark, two of the few places in the world where the elasticity of land supply is appreciably above zero!

Real estate in general (which is what most property taxes are imposed on) is much trickier: In the short run it seems to have a very low elasticity, because building new houses or buildings takes a lot of time and money. But in the long run it actually has a high elasticity of supply, because there is a lot of profit to be made in building new structures if you can fund projects 10 or 15 years out. The short-run elasticity is something like 0.2, meaning a 1% increase in price only yields a 0.2% increase in supply; but the long-run elasticity may be as high as 8, meaning that a 1% increase in price yields an 8% increase in supply. This is why property taxes and rent controls seem like a really good idea at the time but actually probably have the effect of making housing more expensive. The economics of real estate has a number of fundamental differences from the economics of most other goods.

Many important policy questions ultimately hinge upon the elasticity of supply: If elasticity is high, then taxing or regulating something is likely to cause large distortions of the economy, while if elasticity is low, taxes and regulations can be used to support public goods or redistribute wealth without significant distortion to the economy. On the other hand, if elasticity is high, markets generally function well on their own, while if elasticity is low, prices can get far out of whack. As a general rule of thumb, government intervention in markets is most useful and most necessary when elasticity is low.

Is marginal productivity fair?

JDN 2456963 PDT 11:11.

The standard economic equilibrium that is the goal of any neoclassical analysis is based on margins, rather than totals; what matters is not how much you have in all, but how much you get from each new one. This may be easier to understand with specific examples: The price of a product isn’t set by the total utility that you get from using that product; it’s set by the marginal utility that you get from each new unit. The wage of a worker isn’t set by their total value to the company; it’s set by the marginal value they provide with each additional hour of work. Formally, it’s not the value of the function f(x), it’s the derivative of the function, f'(x). (If you don’t know calculus, don’t worry about that last part; it isn’t that important to understand the basic concept.)

This is the standard modern explanation for Adam Smith’s “diamond-water paradox“: Why are diamonds so much more expensive than water, even though water is much more useful? Well, we have plenty of water, so the marginal utility of water isn’t very high; what are you really going to do with that extra liter? But we don’t have a lot of diamonds, so even though diamonds in general aren’t that useful, getting an extra diamond has a lot of benefit. (The units are a bit weird, as George Stigler once used to argue that Smith’s paradox is “meaningless”; but that’s silly. Let’s fix the units at “per kilogram”; a kilogram of diamonds is far, far more expensive than a kilogram of water.)

This explanation is obviously totally wrong, by the way; that’s not why diamonds are expensive. The marginal-utility argument makes sense for cars (or at least ordinary Fords and Toyotas, for reasons you’ll see in a minute), but it doesn’t explain diamonds. Diamonds are expensive for two reasons: First, the absolutely insane monopoly power of the De Beers cartel; as you might imagine, water would be really expensive too if it were also controlled by a single cartel with the power to fix prices and crush competitors. (For awhile De Beers executives had a standing warrant for their arrest in the United States; recently they pled guilty and paid fines—because, as we all know, rich people never go to prison.) And you can clearly see how diamond prices plummeted when the cartel was weakened in the 1980s. But Smith was writing long before DeBeers, and even now that De Beers only controls 40% of the market so we have an oligopoly instead of a monopoly (it’s a step in the right direction I guess), diamonds are still far more expensive than water. The real reason why diamonds are expensive is that diamonds are a Veblen good; you don’t buy diamonds because you actually want to use diamonds (maybe once in awhile, if you want to make a diamond saw or something). You buy diamonds in order to show off how rich you are. And if your goal is to show how rich you are, higher prices are good; you want it to be really expensive, you’re more likely to buy it if it’s really expensive. That’s why the marginal utility argument doesn’t work for Porsches and Ferraris; they’re Veblen goods too. If the price of a Ferrari suddenly dropped to $10,000, people would realize pretty quickly that they are hard to maintain, have very poor suspensions, and get awful gas mileage. It’s not like you can actually drive at 150 mph without getting some serious speeding tickets. (I guess they look nice?) But if the price of a Prius dropped to $10,000, everyone would buy one. For some people diamonds are also a speculation good; they hope to buy them at one price and sell them at a higher price. This is also how most trading in the stock market works, which is why I’m dubious of how well the stock market actually supports real investment. When we’re talking about Veblen goods and speculation goods, the sky is the limit; any price that someone can pay is a price they might sell at.

But all of that is a bit tangential. It’s worth thinking about all the ways that neoclassical theory doesn’t comport with reality, all the cases where price and marginal value become unhinged. But for today I’m going to give the neoclassicists the benefit of the doubt: Suppose it were true. Suppose that markets really were perfectly efficient and everything were priced at its marginal value. Would that even be a good thing?

I tend to focus most of my arguments on why a given part of our economic system deviates from optimal efficiency, because once you can convince economists of that they are immediately willing to try to fix it. But what if we had optimal efficiency? Most economists would say that we’re done, we’ve succeeded, everything is good now. (I am suddenly reminded of the Lego song, “Everything is Awesome.”) This notion is dangerously wrong.

A system could be perfectly efficient and still be horrifically unfair. This is particularly important when we’re talking about labor markets. A diamond or a bottle of water doesn’t have feelings; it doesn’t care what price you sell it at. More importantly it doesn’t have rights. People have feelings; people have rights. (And once again I’m back to Citizens United; a rat is more of a person than any corporation. We should stop calling them “rats” and “fat cats”, for this is an insult to the rodent and feline communities. No, only a human psychopath could ever be quite so corrupt.)

Of course when you sell a product, the person selling it cares how much you pay, but that will either trace back to someone’s labor—and labor markets are still the issue—or it won’t, in which case as far as I’m concerned it really doesn’t matter. If you make money simply by owning things, our society is giving you an enormous gift simply by allowing that capital income to exist; press the issue much more and we’d be well within our rights to confiscate every dime. Unless and until capital ownership is shared across the entire population and we can use it to create a post-scarcity society, capital income will be a necessary evil at best.

So let’s talk about labor markets. If you’ve taken any economics, you have probably seen a great many diagrams like this:

supply_demand2

The red line is labor supply, the blue line is labor demand. At the intersection is our glorious efficient market equilibrium, in this case at 7.5 hours of work per day (the x-axis) and $12.50 an hour (the y-axis). The green line is the wage, $12.50 per hour. But let’s stop and think for a moment about what this diagram really means.

What decides that red labor supply line? Do people just arbitrarily decide that they’re going to work 4 hours a day if they get paid $9 an hour, but 8 hours a day if they get paid $13 an hour? No, this line is meant to represent the marginal real cost of working. It’s the monetized value of your work effort and the opportunity cost of what else you could have been doing with your time. It rises because the more hours you work, the more stress it causes you and the more of your life it takes up. Working 4 hours a day, you probably had that time available anyway. Working 8 hours a day, you can fit it in. Working 12 hours a day, now you have no leisure at all. Working 16 hours a day, now you’re having trouble fitting in basic needs like food and sleep. Working 20 hours a day, you eat at work, you don’t get enough sleep, and you’re going to burn yourself out in no time. Why is it a straight line? Because we assume linear relationships to make the math easier. (No, really; that is literally the only reason. We call them “supply and demand curves” but almost always draw and calculate them as straight lines.)

Now let’s consider the blue labor demand line. Is this how much the “job creators” see fit to bestow upon you? No, it’s the marginal value of productivity. The first hour you work each day, you are focused and comfortable, and you can produce a lot of output. The second hour you’re just a little bit fatigued, so you can produce a bit less. By the time you get to hour 8, you’re exhausted, and producing noticeably less output. And if they pushed you past 16 hours, you’d barely produce anything at all. They multiply the amount of products you produce by the price at which they can sell those products, and that’s their demand for your labor. And once again we assume it’s a straight line just to make the math easier.

From this diagram you can calculate what is called employer surplus and worker surplus. Employer surplus is basically the same thing as profit. (It’s not exactly the same for some wonky technical reasons, but for our purposes they may as well be the same.) Worker surplus is a subtler concept; it’s the amount of money you receive minus the monetized value of your cost of working. So if that first hour of work was really easy and you were willing to do it for anything over $5, we take that $5 as your monetized cost of working (your “marginal willingness-to-accept“). Then if you are being paid $12.50 an hour, we infer that you must have gained $7.50 worth of utility from that exchange. (“$7.50 of utility” is a very weird concept, for reasons I’ll get into more in a later post; but it is actually the standard means of estimating utility in neoclassical economics. That’s one of the things I hope to change, actually.)

When you add these up for all the hours worked, the result becomes an integral, which is a formal mathematical way of saying “the area between those two lines”. In this case they are triangles of equal size, so we can just use the old standby A = 1/2*b*h. The area of each triangle is 1/2*7.5*7.5 = $28.13. From each day you work, you make $28.13 in consumer surplus and your employer makes $28.13 in profit.

And that seems fair, doesn’t it? You split it right down the middle. Both of you are better off than you were, and the economic benefits are shared equally. If this were really how labor markets work, that seems like how things ought to be.

But nothing in the laws of economics says that the two areas need to be equal. We tend to draw them that way out of an aesthetic desire for symmetry. But in general they are not, and in some cases they can be vastly unequal.

This happens if we have wildly different elasticities, which is a formal term for the relative rates of change of two things. An elasticity of labor supply of 1 would mean that for a 1% increase in wage you’re willing to work 1% more hours, while an elasticity of 10 would mean that for a 1% increase in wage you’re willing to work 10% more hours. Elasticities can also be negative; a labor demand elasticity of -1 would mean that for a 1% increase in wage your employer is willing to hire you for 1% fewer hours. In the graph above, the elasticity of labor supply is exactly 1. The elasticity of labor demand varies along the curve, but at the equilibrium it is about -1.6. The fact that the profits are shared equally is related to the fact that these two elasticities are close in magnitude but opposite in sign.

But now consider this equilibrium, in which I’ve raised the labor elasticity to 10. Notice that the wage and number of hours haven’t change; it’s still 7.5 hours at $12.50 per hour. But now the profits are shared quite unequally indeed; while the employer still gets $28.13, the value for the worker is only 1/2*7.5*0.75 = $2.81. In real terms this means we’ve switched from a job that starts off easy but quickly gets harder to a job that is hard to start with but never gets much harder than that.

elastic_supply

On the other hand what if the supply elasticity is only 0.1? Now the worker surplus isn’t even a triangle; it’s a trapezoid. The area of this trapezoid is 6*12.5+1/2*1.5*12.5 = $84.38. This job starts off easy and fun—so much so that you’d do it for free—but then after 6 hours a day it quickly becomes exhausting and you need to stop.

inelastic_supply

If we had to guess what these jobs are, my suggestion is that maybe the first one is a research assistant, the second one is a garbage collector, and the third one is a video game tester. And thus, even though they are paid about the same (I think that’s true in real life? They all make about $15 an hour or $30k a year), we all agree that the video game tester job is better than the research assistant job which is better than the garbage collector job—which is exactly what the worker surplus figures are saying.

What about the demand side? Here’s where it gets really unfair. Going back to our research assistant with a supply elasticity of 1, suppose they’re not really that good a researcher. Their output isn’t wrong, but it’s also not very interesting. They can do the basic statistics, but they aren’t very creative and they don’t have a deep intuition for the subject. This might produce a demand elasticity 10 times larger. The worker surplus remains the same, but the employer surplus is much lower. The triangle has an area 1/2*7.5*0.75 = $2.81.

elastic_demand

Now suppose that they are the best research assistant ever; let’s say we have a young Einstein. Everything he touches turns to gold, but even Einstein needs his beauty sleep (he actually did sleep about 10 hours a day, which is something I’ve always been delighted to have in common with him), so the total number of work hours still caps out at 7.5. It is entirely possible for the wage equilibrium to be exactly the same as it was for the lousy researcher, making the graph look like this:

inelastic_demand

You can’t even see the top of the triangle on this scale; it’s literally off the chart. The worker had a lower bound at zero, but there’s no comparable upper bound. (I suppose you could argue the lower bound shouldn’t be there either, since there are kinds of work you’d be willing to do even if you had to pay to do them—like, well, testing video games.) The top of the triangle is actually at about $90, as it turns out, so the area of employer surplus is 1/2*(90-12.5)*7.5 = $290.63. For every day he works, the company gets almost $300, but Einstein himself only gets $28.13 after you include what it costs him to work. (His gross pay is just wage*hours of course, so that’s $93.75.) The total surplus produced is $318.76. Einstein himself only gets a measly 8.9% of that.

So here we have three research assistants, who have very different levels of productivity, getting the same pay. But isn’t pay supposed to reflect productivity? Sort of; it’s supposed to reflect marginal productivity. Because Einstein gets worn out and produces at the same level as the mediocre researcher after 7.5 hours of work, since that’s where the equilibrium is that’s what they both get paid.

Now maybe Einstein should hold back; he could exercise some monopolistic power over his amazing brain. By only offering to work 4 hours a day, he can force the company to pay him at his marginal productivity for 4 hours a day, which turns out to be $49 an hour. Now he makes a gross pay of $196, with a worker surplus of $171.

monopoly_power

This diagram is a bit harder to read, so let me walk you through it. The light red and blue lines are the same as before. The darker blue line is the marginal revenue per hour for Einstein, once he factors in the fact that working more hours will mean accepting a lower wage. The optimum for him is when that marginal revenue curve crosses his marginal cost curve, which is the red supply curve. That decides how many hours he will work, namely 4. But that’s not the wage he gets; to find that, we move up vertically along the dark red line until we get the company’s demand curve. That tells us what wage the company is willing to pay for the level of marginal productivity Einstein has at 4 hours per day of work—which is the $49 wage he ends up making shown by the dark green line. The lighter lines show what happens if we have a competitive labor market, while the darker lines show what happens if Einstein exercises monopoly power.

The company still does pretty well on this deal; they now make an employer surplus of $82. Now, of the total $253 of economic surplus being made, Einstein takes 69%. It’s his brain, so him taking most of the benefit seems fair.

But you should notice something: This result is inefficient! There’s a whole triangle between 4 and 7.5 hours that nobody is getting; it’s called the deadweight loss. In this case it is $65.76, the difference between the total surplus in the efficient equilibrium and the inefficient equilibrium. In real terms, this means that research doesn’t get done because Einstein held back in order to demand a higher wage. That’s research that should be done—its benefit exceeds its cost—but nobody is doing it. Well now, maybe that doesn’t seem so fair after all. It seems selfish of him to not do research that needs done just so he can get paid more for what he does.

If Einstein has monopoly power, he gets a fair share but the market is inefficient. Removing Einstein’s monopoly power by some sort of regulation would bring us back to efficiency, but it would give most of his share to the company instead. Neither way seems right.

How do we solve this problem? I’m honestly not sure. First of all, we rarely know the actual supply and demand elasticities, and when we do it’s generally after painstaking statistical work to determine the aggregate elasticities, which aren’t even what we’re talking about here. These are individual workers.

Notice that the problem isn’t due to imperfect information; the company knows full well that Einstein is a golden goose, but they aren’t going to pay him any more than they have to.

We could just accept it, I suppose. As long as the productive work gets done, we could shrug our shoulders and not worry about the fact that corporations are capturing most of the value from the hard work of our engineers and scientists. That seems to be the default response, perhaps because it’s the easiest. But it sure doesn’t seem fair to me.

One solution might be for the company to voluntarily pay Einstein more, or offer him some sort of performance bonus. I wouldn’t rule out this possibility entirely, but this would require the company to be unusually magnanimous. This won’t happen at most corporations. It might happen for researchers at a university, where the administrators are fellow academics. Or it might happen to a corporate executive because other corporate executives feel solidarity for their fellow corporate executives.

That sort of solidarity is most likely why competition hasn’t driven down executive salaries. Theoretically shareholders would have an incentive to choose boards of directors who are willing to work for $20 an hour and elect CEOs who are willing to work for $30 an hour; but in practice old rich White guys feel solidarity with other old rich White guys, and even if there isn’t any direct quid pro quo there is still a general sense that because we are “the same kind of people” we should all look out for each other—and that’s how you get $50 million salaries. And then of course there’s the fact that even publicly-traded companies often have a handful of shareholders who control enough of the shares to win any vote.

In some industries, we don’t need to worry about this too much because productivity probably doesn’t really vary that much; just how good can a fry cook truly be? But this is definitely an issue for a lot of scientists and engineers, particularly at entry-level positions. Some scientists are an awful lot better than other scientists, but they still get paid the same.

Much more common however is the case where the costs of working vary. Some people may have few alternatives, so their opportunity cost is low, driving their wage down; but that doesn’t mean they actually deserve a lower wage. Or they may be disabled, making it harder to work long hours; but even though they work so much harder their pay is the same, so their net benefit is much smaller. Even though they aren’t any more productive, it still seems like they should be paid more to compensate them for that extra cost of working. At the other end are people who start in a position of wealth and power; they have a high opportunity cost because they have so many other options, so it may take very high pay to attract them; but why do they deserve to be paid more just because they have more to start with?

Another option would be some sort of redistribution plan, where we tax the people who are getting a larger share and give it to those who are getting a smaller share. The problem here arises in how exactly you arrange the tax. A theoretical “lump sum tax” where we just figure out the right amount of money and say “Person A: Give $217 to person B! No, we won’t tell you why!” would be optimally efficient because there’s no way it can distort markets if nobody sees it coming; but this is not something we can actually do in the real world. (It also seems a bit draconian; the government doesn’t even tax activities, they just demand arbitrary sums of money?) We’d have to tax profits, or sales, or income; and all of these could potentially introduce distortions and make the market less efficient.

We could offer some sort of publicly-funded performance bonus, and for scientists actually we do; it’s called the Nobel Prize. If you are truly the best of the best of the best as Einstein was, you may have a chance at winning the Nobel and getting $1.5 million. But of course that has to be funded somehow, and it only works for the very very top; it doesn’t make much difference to Jane Engineer who is 20% more productive than her colleagues.

I don’t find any of these solutions satisfying. This time I really can’t offer a good solution. But I think it’s important to keep the problem in mind. It’s important to always remember that “efficient” does not mean “fair”, and being paid at marginal productivity isn’t the same as being paid for overall productivity.