Cheap Talk

A blog about economics, politics and the random interests of forty-something professors

# Maximizing Consumer Surplus Part II: Compromising

April 18, 2011 in Uncategorized | Tags: economics, food and wine, incentives, vapor mill

Restaurants, touring musicians, and sports franchises are not out to gouge every last penny out of their patrons.  They want patrons to enjoy their craft but also to come away feeling like they didn’t pay an arm and a leg.  Yesterday I tried to formalize this motivation as maximizing consumer surplus but that didn’t give a useful answer. Maximizing consumer surplus means either complete rationing (and zero profit) or going all the way to an auction (a more general argument why appears below.)  So today I will try something different.

Presumably the restaurant cares about profits too.  So it makes sense to study the mechanism that maximizes a weighted sum of profits and consumer’s surplus. We can do that.  Standard optimal mechanism design proceeds by a sequence of mathematical tricks to derive a measure of a consumer’s value called virtual surplus.  Virtual surplus allows you to treat any selling mechanism you can imagine as if it worked like this

1. Consumers submit “bids”
2. Based on the bids received the seller computes the virtual surplus of each consumer.
3. The consumer with the highest virtual surplus is served.

If you write down the optimal mechanism design problem where the seller puts weight $\alpha$ on profits and weight $1 - \alpha$ on consumer surplus, and you do all the integration by parts, you get this formula for virtual surplus.

$\alpha v + (1 - 2\alpha) \frac{1 - F(v)}{f(v)}$

where $v$ is the consumer’s willingness to pay, $F(v)$ is the proportion of consumers with willingness to pay less than $v$ and $f(v)$ is the corresponding probability density function.   That last ratio is called the (inverse) hazard rate.

As usual, just staring down this formula tells you just about everything you want to know about how to design the pricing system.  One very important thing to know is what to do when virtual surplus is a decreasing function of $v$. If we have a decreasing virtual surplus then we learn that it’s at least as important to serve the low valuation buyers as those with high valuations (see point 3 above.)

But here’s a key observation: its impossible to sell to low valuation buyers and not also to high valuation buyers because whatever price the former will agree to pay the latter will pay too.  So a decreasing virtual surplus means that you do the next best thing: you treat high and low value types the same. This is how rationing becomes part of an optimal mechanism.

For example, suppose the weight on profit $\alpha$ is equal to $0$. That brings us back to yesterday’s problem of just maximizing consumer surplus. And our formula now tells us why complete rationing is optimal because it tells us that virtual surplus is just equal to the hazard rate which is typically monotonically decreasing. Intuitively here’s what the virtual surplus is telling us when we are trying to maximize consumer surplus. If we are faced with two bidders and one has a higher valuation than the other, then to try to discriminate would require that we set a price in between the two. That’s too costly for us because it would cut into the consumer surplus of the eventual winner.

So that’s how we get the answer I discussed yesterday.  Before going on I would like to elaborate on yesterday’s post based on correspondence I had with a few commenters, especially David Miller and Kane Sweeney. Their comments highlight two assumptions that are used to get the rationing conclusion:  monotone hazard rate, and no payments to non-buyers.  It gets a little more technical than usual so I am going to put it here in an addendum to yesterday (scroll down for the addendum.)

Now back to the general case we are looking at today, we can consider other values of $\alpha$

An important benchmark case is $\alpha = 1/2$ when virtual surplus reduces to just $v$, now monotonically increasing.  That says that a seller who puts equal weight on profits and consumer surplus will always allocate to the highest bidder because his virtual surplus is higher.  An auction does the job, in fact a second price auction is optimal.  The seller is implementing the efficient outcome.

More interesting is when $\alpha$ is between $0$ and $1/2$. In general then the shape of the virtual surplus will depend on the distribution $F$, but the general tendency will be toward either complete rationing or an efficient auction.  To illustrate, suppose that willingness to pay is distributed uniformly from $0$ to $1$. Then virtual suplus reduces to

$(3 \alpha - 1) v + (1 - 2 \alpha)$

which is either decreasing over the whole range of $v$ (when $\alpha \leq 1/3$), implying complete rationing or increasing over the whole range (when $\alpha > 1/3$), prescribing an auction.

Finally when $\alpha > 1/2$ virtual surplus is the difference between an increasing function and a decreasing function and so it is increasing over the whole range and this means that an auction is optimal (now typically with a reserve price above cost so that in return for higher profits the restaurant lives with empty tables and inefficiency.  This is not something any restaurant would choose if it can at all avoid it.)

What do we conclude from this?  Maximizing a weighted sum of consumer surplus and profit yields again yields one of two possible mechanisms: complete rationing or an auction.  Neither of these mechanisms seem to fit what Nick Kokonas was looking for in his comment to us and so we have to go back to the drawing board again.

Tomorrow I will take a closer look and extract a more refined version of Nick’s objective that will in fact produce a new kind of mechanism that may just fit the bill.

Addendum: Check out these related papers by Bulow and Klemperer (dcd: glen weyl) and by Daniele Condorelli.

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## 8 comments

Yup, Jeff, this covers it. I’ll send this to Jeremy and Paul. They might want to include something about it in their paper.

I admire Nick’s empathy with his customers, but I don’t think consumer surplus should be a component of Nick’s objective at all. Isn’t the difference between the price the consumer pays and the minimum price they are willing to pay (summed across all consumers) the consumer surplus? And as long as the expectation of the quality of food/service is met in the price they pay, the consumer will be satisfied, although the lower the price they would pay, the more satisfied they would be.

The auction system resolves this issue by asking consumers to set the prices. An individual won’t pay anything more than the benefit they would receive. You could design a system that auctions tickets for different times of the day.

With an auction, the price settled upon of a ticket is a perception issue… of the consumers perceiving the quality of the restaurant! They’re WILLING to pay. If they don’t think the price of the auctioned ticket was worth it, they wouldn’t go back. Those that are satisfied, would. If the perception is of a lower quality, prices would reflect that. If they think Nick’s restaurant is golden, prices would reflect that too.

Nick wants his customers to be satisfied, but not so satisfied that they perceive they got a huge bargain. The auction hits it on the nose. The only question with the auction is how to construct one that can maximize profit for different size parties, for location of tables, etc. I mean, for a larger party, you may want to set a higher reserve price, and the same for table locations.

But with regards to my original point, is there something I’m missing??

this is the point i will address tomorrow. i believe if you stare at Nick’s comment you will see what he is getting at and that it makes good sense.

[...] Next in Chicago and proprietors Grant Achatz and Nick Kokonas new ticket policy.   In the previous two installments I tried to use standard mechanism design theory to see what comes out when you feed in [...]

Incidentally, the rationing result features in McAfee and McMillan’s paper on bidding rings. When transfers are not feasible the ring allocates the object randomly at the reserve price.

i see, thanks.

[...] blogged many times about the Next restaurant’s innovative ticket scheme. Potential restaurant goers had to sign up to [...]

[...] solution to be a rationing system with a price below market clearing. I devoted a series of posts to this point last year. The basic idea is that the price that the efficiency gains you get from [...]
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