The difference between price, cost, and value

JDN 2457559

This topic has been on the voting list for my Patreons for several months, but it never quite seems to win the vote. Well, this time it did. I’m glad, because I was tempted to do it anyway.

“Price”, “cost”, and “value”; the words are often used more or less interchangeably, not only by regular people but even by economists. I’ve read papers that talked about “rising labor costs” when what they clearly meant was rising wages—rising labor prices. I’ve read papers that tried to assess the projected “cost” of climate change by using the prices of different commodity futures. And hardly a day goes buy that I don’t see a TV commercial listing one (purely theoretical) price, cutting it in half (to the actual price), and saying they’re now giving you “more value”.

As I’ll get to, there are reasons to think they would be approximately the same for some purposes. Indeed, they would be equal, at the margin, in a perfectly efficient market—that may be why so many economists use them this way, because they implicitly or explicitly assume efficient markets. But they are fundamentally different concepts, and it’s dangerous to equate them casually.

Price

Price is exactly what you think it is: The number of dollars you must pay to purchase something. Most of the time when we talk about “cost” or “value” and then give a dollar figure, we’re actually talking about some notion of price.

Generally we speak in terms of nominal prices, which are the usual concept of prices in actual dollars paid, but sometimes we do also speak in terms of real prices, which are relative prices of different things once you’ve adjusted for overall inflation. “Inflation-adjusted price” can be a somewhat counter-intuitive concept; if a good’s (nominal) price rises, but by less than most other prices have risen, its real price has actually fallen.

You also need to be careful about just what price you’re looking at. When we look at labor prices, for example, we need to consider not only cash wages, but also fringe benefits and other compensation such as stock options. But other than that, prices are fairly straightforward.

Cost

Cost is probably not at all what you think it is. The real cost of something has nothing to do with money; saying that a candy bar “costs $2” or a computer “costs $2,000” is at best a somewhat sloppy shorthand and at worst a fundamental distortion of what cost is and why it matters. No, those are prices. The cost of a candy bar is the toil of children in cocoa farms in Cote d’Ivoire. The cost of a computer is the ecological damage and displaced indigenous people caused by coltan mining in Congo.

The cost of something is the harm that it does to human well-being (or for that matter to the well-being of any sentient being). It is not measured in money but in “the sweat of our laborers, the genius of our scientists, the hopes of our children” (to quote Eisenhower, who understood real cost better than most economists). There is also opportunity cost, the real cost we pay not by what we did, but by what we didn’t do—what we could have done instead.

This is important precisely because while costs should always be reduced when possible, prices can in fact be too low—and indeed, artificially low prices of goods due to externalities are probably the leading reason why humanity bears so many excess real costs. If the price of that chocolate bar accurately reflected the suffering of those African children (perhaps by—Gasp! Paying them a fair wage?), and the price of that computer accurately reflected the ecological damage of those coltan mines (a carbon tax, at least?), you might not want to buy them anymore; in which case, you should not have bought them. In fact, as I’ll get to once I discuss value, there is reason to think that even if you would buy them at a price that accurately reflected the dollar value of the real cost to their producers, we would still buy more than we should.

There is a point at which we should still buy things even though people get hurt making them; if you deny this, stop buying literally anything ever again. We don’t like to think about it, but any product we buy did cause some person, in some place, some degree of discomfort or unpleasantness in production. And many quite useful products will in fact cause death to a nonzero number of human beings.

For some products this is only barely true—it’s hard to feel bad for bestselling authors and artists who sell their work for millions, for whatever toil they may put into their work, whatever their elevated suicide rate (which is clearly endogenous; people aren’t randomly assigned to be writers), they also surely enjoy it a good deal of the time, and even if they didn’t, their work sells for millions. But for many products it is quite obviously true: A certain proportion of roofers, steelworkers, and truck drivers will die doing their jobs. We can either accept that, recognizing that it’s worth it to have roofs, steel, and trucking—and by extension, industrial capitalism, and its whole babies not dying thing—or we can give up on the entire project of human civilization, and go back to hunting and gathering; even if we somehow managed to avoid the direct homicide most hunter-gatherers engage in, far more people would simply die of disease or get eaten by predators.

Of course, we should have safety standards; but the benefits of higher safety must be carefully weighed against the potential costs of inefficiency, unemployment, and poverty. Safety regulations can reduce some real costs and increase others, even if they almost always increase prices. A good balance is struck when real cost is minimized, where any additional regulation would increase inefficiency more than it improves safety.

Actually OSHA are unsung heroes for their excellent performance at striking this balance, just as EPA are unsung heroes for their balance in environmental regulations (and that whole cutting crime in half business). If activists are mad at you for not banning everything bad and business owners are mad at you for not letting them do whatever they want, you’re probably doing it right. Would you rather people saved from fires, or fires prevented by good safety procedures? Would you rather murderers imprisoned, or boys who grow up healthy and never become murderers? If an ounce of prevention is worth a pound of cure, why does everyone love firefighters and hate safety regulators?So let me take this opportunity to say thank you, OSHA and EPA, for doing the jobs of firefighters and police way better than they do, and unlike them, never expecting to be lauded for it.

And now back to our regularly scheduled programming. Markets are supposed to reflect costs in prices, which is why it’s not totally nonsensical to say “cost” when you mean “price”; but in fact they aren’t very good at that, for reasons I’ll get to in a moment.

Value

Value is how much something is worth—not to sell it (that’s the price again), but to use it. One of the core principles of economics is that trade is nonzero-sum, because people can exchange goods that they value differently and thereby make everyone better off. They can’t price them differently—the buyer and the seller must agree upon a price to make the trade. But they can value them differently.

To see how this works, let’s look at a very simple toy model, the simplest essence of trade: Alice likes chocolate ice cream, but all she has is a gallon of vanilla ice cream. Bob likes vanilla ice cream, but all he has is a gallon of chocolate ice cream. So Alice and Bob agree to trade their ice cream, and both of them are happier.

We can measure value in “willingness-to-pay” (WTP), the highest price you’d willingly pay for something. That makes value look more like a price; but there are several reasons we must be careful when we do that. The obvious reason is that WTP is obviously going to vary based on overall inflation; since $5 isn’t worth as much in 2016 as it was in 1956, something with a WTP of $5 in 1956 would have a much higher WTP in 2016. The not-so-obvious reason is that money is worth less to you the more you have, so we also need to take into account the effect of wealth, and the marginal utility of wealth. The more money you have, the more money you’ll be willing to pay in order to get the same amount of real benefit. (This actually creates some very serious market distortions in the presence of high income inequality, which I may make the subject of a post or even a paper at some point.) Similarly there is “willingness-to-accept” (WTA), the lowest price you’d willingly accept for it. In theory these should be equal; in practice, WTA is usually slightly higher than WTP in what’s called endowment effect.

So to make our model a bit more quantitative, we could suppose that Alice values vanilla at $5 per gallon and chocolate at $10 per gallon, while Bob also values vanilla at $5 per gallon but only values chocolate at $4 per gallon. (I’m using these numbers to point out that not all the valuations have to be different for trade to be beneficial, as long as some are.) Therefore, if Alice sells her vanilla ice cream to Bob for $5, both will (just barely) accept that deal; and then Alice can buy chocolate ice cream from Bob for anywhere between $4 and $10 and still make both people better off. Let’s say they agree to also sell for $5, so that no net money is exchanged and it is effectively the same as just trading ice cream for ice cream. In that case, Alice has gained $5 in consumer surplus (her WTP of $10 minus the $5 she paid) while Bob has gained $1 in producer surplus (the $5 he received minus his $4 WTP). The total surplus will be $6 no matter what price they choose, which we can compute directly from Alice’s WTP of $10 minus Bob’s WTA of $4. The price ultimately decides how that total surplus is distributed between the two parties, and in the real world it would very likely be the result of which one is the better negotiator.

The enormous cost of our distorted understanding

(See what I did there?) If markets were perfectly efficient, prices would automatically adjust so that, at the margin, value is equal to price is equal to cost. What I mean by “at the margin” might be clearer with an example: Suppose we’re selling apples. How many apples do you decide to buy? Well, the value of each successive apple to you is lower, the more apples you have (the law of diminishing marginal utility, which unlike most “laws” in economics is actually almost always true). At some point, the value of the next apple will be just barely above what you have to pay for it, so you’ll stop there. By a similar argument, the cost of producing apples increases the more apples you produce (the law of diminishing returns, which is a lot less reliable, more like the Pirate Code), and the producers of apples will keep selling them until the price they can get is only just barely larger than the cost of production. Thus, in the theoretical limit of infinitely-divisible apples and perfect rationality, marginal value = price = marginal cost. In such a world, markets are perfectly efficient and they maximize surplus, which is the difference between value and cost.

But in the real world of course, none of those assumptions are true. No product is infinitely divisible (though the gasoline in a car is obviously a lot more divisible than the car itself). No one is perfectly rational. And worst of all, we’re not measuring value in the same units. As a result, there is basically no reason to think that markets are optimizing anything; their optimization mechanism is setting two things equal that aren’t measured the same way, like trying to achieve thermal equilibrium by matching the temperature of one thing in Celsius to the temperature of other things in Fahrenheit.

An implicit assumption of the above argument that didn’t even seem worth mentioning was that when I set value equal to price and set price equal to cost, I’m setting value equal to cost; transitive property of equality, right? Wrong. The value is equal to the price, as measured by the buyer. The cost is equal to the price, as measured by the seller.

If the buyer and seller have the same marginal utility of wealth, no problem; they are measuring in the same units. But if not, we convert from utility to money and then back to utility, using a different function to convert each time. In the real world, wealth inequality is massive, so it’s wildly implausible that we all have anything close to the same marginal utility of wealth. Maybe that’s close enough if you restrict yourself to middle-class people in the First World; so when a tutoring client pays me, we might really be getting close to setting marginal value equal to marginal cost. But once you include corporations that are owned by billionaires and people who live on $2 per day, there’s simply no way that those price-to-utility conversions are the same at each end. For Bill Gates, a million dollars is a rounding error. For me, it would buy a house, give me more flexible work options, and keep me out of debt, but not radically change the course of my life. For a child on a cocoa farm in Cote d’Ivoire, it could change her life in ways she can probably not even comprehend.

The market distortions created by this are huge; indeed, most of the fundamental flaws in capitalism as we know it are ultimately traceable to this. Why do Americans throw away enough food to feed all the starving children in Africa? Marginal utility of wealth. Why are Silicon Valley programmers driving the prices for homes in San Francisco higher than most Americans will make in their lifetimes? Marginal utility of wealth. Why are the Koch brothers spending more on this year’s elections than the nominal GDP of the Gambia? Marginal utility of wealth. It’s the sort of pattern that once you see it suddenly seems obvious and undeniable, a paradigm shift a bit like the heliocentric model of the solar system. Forget trade barriers, immigration laws, and taxes; the most important market distortions around the world are all created by wealth inequality. Indeed, the wonder is that markets work as well as they do.

The real challenge is what to do about it, how to reduce this huge inequality of wealth and therefore marginal utility of wealth, without giving up entirely on the undeniable successes of free market capitalism. My hope is that once more people fully appreciate the difference between price, cost, and value, this paradigm shift will be much easier to make; and then perhaps we can all work together to find a solution.

Tax incidence revisited, part 4: Surplus and deadweight loss

JDN 2457355

I’ve already mentioned the fact that taxation creates deadweight loss, but in order to understand tax incidence it’s important to appreciate exactly how this works.

Deadweight loss is usually measured in terms of total economic surplus, which is a strange and deeply-flawed measure of value but relatively easy to calculate.

Surplus is based upon the concept of willingness-to-pay; the value of something is determined by the maximum amount of money you would be willing to pay for it.

This is bizarre for a number of reasons, and I think the most important one is that people differ in how much wealth they have, and therefore in their marginal utility of wealth. $1 is worth more to a starving child in Ghana than it is to me, and worth more to me than it is to a hedge fund manager, and worth more to a hedge fund manager than it is to Bill Gates. So when you try to set what something is worth based on how much someone will pay for it, which someone are you using?

People also vary, of course, in how much real value a good has to them: Some people like dark chocolate, some don’t. Some people love spicy foods and others despise them. Some people enjoy watching sports, others would rather read a book. A meal is worth a lot more to you if you haven’t eaten in days than if you just ate half an hour ago. That’s not actually a problem; part of the point of a market economy is to distribute goods to those who value them most. But willingness-to-pay is really the product of two different effects: The real effect, how much utility the good provides you; and the wealth effect, how your level of wealth affects how much you’d pay to get the same amount of utility. By itself, willingness-to-pay has no means of distinguishing these two effects, and actually I think one of the deepest problems with capitalism is that ultimately capitalism has no means of distinguishing these two effects. Products will be sold to the highest bidder, not the person who needs it the most—and that’s why Americans throw away enough food to end world hunger.

But for today, let’s set that aside. Let’s pretend that willingness-to-pay is really a good measure of value. One thing that is really nice about it is that you can read it right off the supply and demand curves.

When you buy something, your consumer surplus is the difference between your willingness-to-pay and how much you actually did pay. If a sandwich is worth $10 to you and you pay $5 to get it, you have received $5 of consumer surplus.

When you sell something, your producer surplus is the difference between how much you were paid and your willingness-to-accept, which is the minimum amount of money you would accept to part with it. If making that sandwich cost you $2 to buy ingredients and $1 worth of your time, your willingness-to-accept would be $3; if you then sell it for $5, you have received $2 of producer surplus.

Total economic surplus is simply the sum of consumer surplus and producer surplus. One of the goals of an efficient market is to maximize total economic surplus.

Let’s return to our previous example, where a 20% tax raised the original wage from $22.50 and thus resulted in an after-tax wage of $18.

Before the tax, the supply and demand curves looked like this:

equilibrium_notax

Consumer surplus is the area below the demand curve, above the price, up to the total number of goods sold. The basic reasoning behind this is that the demand curve gives the willingness-to-pay for each good, which decreases as more goods are sold because of diminishing marginal utility. So what this curve is saying is that the first hour of work was worth $40 to the employer, but each following hour was worth a bit less, until the 10th hour of work was only worth $35. Thus the first hour gave $40-$20 = $20 of surplus, while the 10th hour only gave $35-$20 = $15 of surplus.

Producer surplus is the area above the supply curve, below the price, again up to the total number of goods sold. The reasoning is the same: If the first hour of work cost $5 worth of time but the 10th hour cost $10 worth of time, the first hour provided $20-$5 = $15 in producer surplus, but the 10th hour only provided $20-$10 = $10 in producer surplus.

Imagine drawing a little 1-pixel-wide line straight down from the demand curve to the price for each hour and then adding up all those little lines into the total area under the curve, and similarly drawing little 1-pixel-wide lines straight up from the supply curve.

surplus

The employer was paying $20 * 40 = $800 for an amount of work that they actually valued at $1200 (the total area under the demand curve up to 40 hours), so they benefit by $400. The worker was being paid $800 for an amount of work that they would have been willing to accept $480 to do (the total area under the supply curve up to 40 hours), so they benefit $320. The sum of these is the total surplus $720.

equilibrium_notax_surplus

After the tax, the employer is paying $22.50 * 35 = $787.50, but for an amount of work that they only value at $1093.75, so their new surplus is only $306.25. The worker is receiving $18 * 35 = $630, for an amount of work they’d have been willing to accept $385 to do, so their new surplus is $245. Even when you add back in the government revenue of $4.50 * 35 = $157.50, the total surplus is still only $708.75. What happened to that extra $11.25 of value? It simply disappeared. It’s gone. That’s what we mean by “deadweight loss”. That’s why there is a downside to taxation.

equilibrium_tax_surplus

How large the deadweight loss is depends on the precise shape of the supply and demand curves, specifically on how elastic they are. Remember that elasticity is the proportional change in the quantity sold relative to the change in price. If increasing the price 1% makes you want to buy 2% less, you have a demand elasticity of -2. (Some would just say “2”, but then how do we say it if raising the price makes you want to buy more? The Law of Demand is more like what you’d call a guideline.) If increasing the price 1% makes you want to sell 0.5% more, you have a supply elasticity of 0.5.

If supply and demand are highly elastic, deadweight loss will be large, because even a small tax causes people to stop buying and selling a large amount of goods. If either supply or demand is inelastic, deadweight loss will be small, because people will more or less buy and sell as they always did regardless of the tax.

I’ve filled in the deadweight loss with brown in each of these graphs. They are designed to have the same tax rate, and the same price and quantity sold before the tax.

When supply and demand are elastic, the deadweight loss is large:

equilibrium_elastic_tax_surplus

But when supply and demand are inelastic, the deadweight loss is small:

equilibrium_inelastic_tax_surplus

Notice that despite the original price and the tax rate being the same, the tax revenue is also larger in the case of inelastic supply and demand. (The total surplus is also larger, but it’s generally thought that we don’t have much control over the real value and cost of goods, so we can’t generally make something more inelastic in order to increase total surplus.)

Thus, all other things equal, it is better to tax goods that are inelastic, because this will raise more tax revenue while producing less deadweight loss.

But that’s not all that elasticity does!

At last, the end of our journey approaches: In the next post in this series, I will explain how elasticity affects who actually ends up bearing the burden of the tax.

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.