The (In)efficiency of Perfect Price Discrimination

I have always thought, and taught, that Perfect Price Discrimination leads to an efficient allocation of resources. I now think that is wrong. It only seems to work if we use partial equilibrium reasoning, for a single monopolist that practices PPD, holding constant consumers' income and the monopolist's Marginal Cost curve. It doesn't work in general equilibrium. And the easiest way to see this is to construct a counterexample where all consumption goods are produced by monopolists who practice PPD, while the labour market is perfectly competitive.

A PPD monopolist captures 99.99% of Consumers' Surplus by setting a different price for each apple for each consumer, that is 99.99% of that consumer's Willingness To Pay for that apple, down to where the marginal price equals the Marginal Cost of producing an extra apple. So the consumer buys all the apples up to the point where his marginal Willingness to Pay (Marginal Benefit) equals Marginal Cost, but is almost at the point of walking away from the deal and not buying any apples at all. It looks like it works to get an efficient allocation of resources (where Marginal Benefit equals Marginal Cost) if we think partial equilibrium.

But it doesn't work in general equilibrium. If all producers of consumption goods practice PPD, this is like the government imposing a 99.99% marginal tax rate on labour income. It reduces the marginal incentive to work extra hours to earn extra income to buy extra consumer goods to enjoy extra Consumer Surplus by 99.99%, because the monopolists will raise your individual price schedules in response to your increased income to capture 99.99% of your increased Consumer Surplus.

TLDR: if you make the rich pay more, because they are willing to pay more, nobody will do any work to produce anything.

It is important to note that the source of the inefficiency is a substitution effect on the individual's consumption-leisure trade-off. It is not an income effect, because the monopolists' profits are transferred to the households that own the firms, just like where the government's revenue from taxes is transferred as lump-sum payments to households. With a large number of households, each individual household will take its income from profits as exogenous when deciding whether to work an extra hour.

The model I have in mind is similar to the standard New Keynesian macro model, with n firms, each of which has a monopoly on producing 1 consumption good. And the consumption goods are produced by labour supplied by households, and labour is homogenous so the labour market is perfectly competitive. And firms' profits are transferred to households, who own shares in all the firms. To keep it simple, we can consider a separable utility function:

V = U(24-L) + U(C1) + U(C2) + ....U(Cn)  where U(24-L) is the utility from leisure, and L is labour.

For each of the n consumption goods, one unit of output requires one unit of labour.

A social planner would allocate resources so that the Marginal Utility of consumption for each of the n goods equals the marginal utility of leisure. That is the requirement for efficiency.

If the monopolists were forced to set a single price equal to Marginal Cost (which equals the wage) the decentralised allocation would be the same as the planner's solution. Because the individual household that considers working an extra hour would capture 100% of the extra consumer surplus when its extra wage income shifts its demand curve for the consumer goods to the right. But with Perfect Price Discrimination, the firm knows that the household has extra income, and knows that its demand curve has shifted right, and raise the price schedule for that household accordingly, to capture 99.99% of the extra Consumer Surplus.

It's like if you won the lottery, but everyone knew you had won the lottery, and would be prepared to pay more for the goods you buy, and so raised prices to capture nearly all of your extra consumer surplus. Your lottery ticket wouldn't be worth very much. Only you have to work extra hours to get a winning lottery ticket. So it's not worth it, because you don't like work. So nobody works. So nothing gets produced. So it's not efficient.

Here's a picture:

That red shaded area show the extra consumer surplus you would get under Marginal Cost pricing if you worked an extra hour to earn extra income so your demand curve for one of the consumption goods shifted right. That's your private Marginal Benefit to working an extra hour, which you compare to your private Marginal Cost from losing an hour of leisure. But under Perfect Price Discrimination, the monopolist charges you higher prices to grab 99.99% of that red area. So you lose 99.99% of your incentive to work that extra hour.

And by the way, in a barter economy, where households directly swap their labour for consumption goods, this inefficiency would not arise. Firms would offer households a deal where households would be just willing to swap the efficient amount of labour for the efficient amount of consumption goods. You can't have monopoly in the market for consumption goods and perfect competition in the market for labour, because those are the same market. But in a monetary exchange economy you can have monopoly in the market where consumption goods are swapped for money and perfect competition in the market where labour is swapped for money. Clower and all that.

[Thanks to Chris Surro on Twitter for letting me know about this paper. But I'm afraid the math is too hard for me, so I don't understand what it's saying.]

Comments

[dlr]

why isn't this just dependent on using a willingness to pay definition that is perhaps unnecessarily and rigidly ex-post? think about a normal partial equilibrium ppd case in an industry where the consumption comes first and the labor second, like a for profit university. do we say that first degree price discrimination fails here because firms will charge a price that causes all students to work less tomorrow and buy less education? or can wtp mean the firms explicitly consider the leisure/labor trade off in their ppd price setting? what does the wtp foresight embedded in ppd even mean in general equilibrium if everyone could do it and everyone knew everyone could do it?

[Nick Rowe]

dlr: suppose there were some exogenous individual-specific characteristic E that perfectly predicted the cross-section distribution of wage income in equilibrium. If PPD firms made prices contingent on E, rather than on observed income Y, there wouldn't be any distortion. Just like if the government put a 99.99% tax on E instead of Y there wouldn't be any distortion. Yes.

But notice that the government, being a large player, has an incentive to make taxes contingent on E rather than Y. An individual monopolist, being a small part of the whole economy, has no such incentive, because its choosing Y rather than E will have only a very small effect on the wage it pays in equilibrium. And even then, actual governments sets taxes contingent on Y not E, because E is hard to observe.

[dlr]

makes sense, thank you.

[Nick Rowe]

dlr: if it makes sense to you, then I'm more confident than I was that it really does make sense!

[Chris Surro]

I've thought about this some more and at first I thought it depended on your definition of willingness to pay (similar to dlr's comment above). In partial equilibrium, willingness to pay should include all the information about what could have been purchased in every other market. For example, if each apple gives me 2 units of utility and each banana gives me 3 and they are the same price, isn't my marginal benefit of buying an apple actually negative (because I lose the 3 units of utility from the banana)? In the same sense, we could think of leisure as just another good and so consumer surplus should actually be (utility from the good - disutility from working long enough to buy it) and that is really what should be set to zero.

But I think your comment above explains why this intuition doesn't work. I can't make decisions about how much to buy and how much to work at the same time. Once I have earned the income, I can't then buy back my leisure time back so my only option is to buy one of the goods which have perfect price discrimination and so I can't get any of the utility I lost from working back. Therefore total expected consumer surplus ex ante will always be negative. If consumers could somehow write a contract to lock in a price before they made a decision on how much to work we could reach the efficient outcome, but without that nobody would have any incentive to work at all.

Is that right?

[Nick Rowe]

Chris: I *think* that's right. But notice it all works differently if we barter labour for apples. Then we would easily escape my PPD equilibrium where nobody works: the firm would offer one household just enough apples to persuade them to accept a take-it-or-leave-it offer of apples for labour to produce those apples plus maximum apple profits leftover for the firm.

[Chris Surro]

I can't fully wrap my head around it, but even in barter it seems like there could be inefficiency if people want to consume more than one good. Unless they are working for each type of firm and getting paid in goods at each one, it seems like we would run into the same problem. I would have no incentive to get paid apples to trade for bananas because the price of bananas in terms of apples would be pushed so that my surplus was still zero. So I would work, but I would never work longer than my desired consumption of whatever good I am being paid in.

But then it gets tricky because I'm not sure what happens to the apple profits. I have to think more.

[Nick Rowe]

Chris: you could get efficiency if either:

1. Each household works for each of the n firms, and gets wages paid in kind, so there are n exchanges, each with 2 agents (firm and household). That's barter.

2. There's one big central Walrasian market where each household meets with all n firms at the same time, and the n+1 parties to the exchange.

The trouble with micro is that they are a bit sloppy about specifying how many markets there are, and what gets swapped by who in each market. (Macro aren't much better).

[Chris Surro]

But even with working at each of the n firms, is it always efficient? What if workers have different productive abilities. So one guy is really good at producing apples but would also like to consume bananas. Workers should produce according to comparative advantages, but if the only way to get a good is to produce it and get paid those wages then it seems like this specialization wouldn't happen.

[Nick Rowe]

Chris: I think you're right on that. With differing types of labour, they would need to use some sort of medium of exchange.

[Matthew Young]

This is the progressive tax model, you first buy one progressively priced unit of government good then go shopping for regular stuff.

So, the government agent stands outside of WalMart, and they make you buy your quota of government good before you enter the store. But, you need twice as many government agents as you have Walmart clerks, because the government agents need a complete sample of customers queues before they measure the income adjusted price. They have to process customers at least twice the rate as Walmart clerks.

Another option is to have everyone buy a government good in January, then they are free of government for 11 months. That solution will cycle, the government agents cannot keep up.

The third option is TBTF banking. Any agent can maintain a triple entry accounting system, and bank directly with the central bank. Because they agree to cover the government share, they get low rates on overnight money. This third option gets you a stable split between high powered money and retail money.

[Epiphyte]

Have you ever tried a mangosteen before? I vaguely remember trying one a while back and really liking it. Problem is that they are relatively rare here in California. Personally, I'd be willing to pay a lot more for a mangosteen than for an apple. What would happen if everybody paid their max WTP for mangosteens? Mangosteen farmers would sure be happy. Except, this would mean more mangosteen farmers, and more mangosteens. As they became more common, people's max WTP for them would decrease accordingly.

Buying mangosteens has an inherent contradiction. On the one hand, I want to pay the smallest amount possible (maximize consumer surplus)... but on the other hand, I also want the supply to be optimal. These two desires are mutually exclusive. The better the deal that I get, the less accurate the signal that my payment sends.

A good way to bump the intuition is to eliminate the buying aspect entirely. Imagine if Netflix subscribers could decide for themselves how they divide their subscription dollars among all the content. They'd still have unlimited access to the content. In this case, the subscribers aren't buying the content. They are simply and solely using their subscription dollars to direct society's resources. If you perceive that there's a shortage of nature shows, then you'd spend more of your subscription dollars on nature shows. The amount that you spent would accurately signal your perception of the size of the shortage. There'd be absolutely no point in trying to get a deal. It's not like you'd have the option to spend your subscription dollars outside of Netflix.