Why Consumer Surplus is Negative the Integral of the Demand Function
Having just spent too many hours on this, I’m posting it so I won’t forget it. Maybe it’ll save some other folks some time too.
I’ve learned that, up to a constant, consumer surplus is the integral of the demand function. That’s wrong. It’s negative the integral of the demand function. The reason has to do with the fact that what we call the demand function–that which translates price into quantity–is not what we draw on a graph. Thanks to Alfred Marshall, the custom is to draw the inverse demand function, with price on the vertical axis and quantity on the horizontal axis.
To show formally that consumer surplus is the negative of the integral of demand (or that demand is negative the derivative of consumer surplus), one can use the rule for integrating inverses. To do so, let
- Q denote quantity and P denote price,
- Q = D(P) be the demand function,
- and the equilibrium quantity and price be Q0 and P0, respectively.
Then, consumer surplus (CS) is
where the first line is the definition of consumer surplus, the second follows from the rule of integrating inverses and the facts that Q=D(P) and P=D-1(Q), and the third holds if PD(P) is zero when P is infinity.
This is important for the very reason I spent hours trying to find or work out the above explanation. If one is working with discrete choice models (logit, nested logit, conditional logit, etc.) and wishes to derive the expression for consumer surplus, one might do so by recalling two things:
- The functional form of consumer surplus is the log of the denominator of the discrete choice probability model, which is also the demand function. That is, consumer surplus has a log-sum of exponentials form.
- The derivative of consumer surplus is negative the demand function, i.e. the discrete choice model.
Recalling point 1 and not 2, which is what I did, one is tempted to put the wrong sign on consumer surplus. That leads to very incorrect (exactly backwards) results. One is tempted to scrutinize one’s code for bugs. But the problem isn’t the code, it’s Alfred Marshall’s graphing convention.
Google Reader and Economic Welfare
I’m on a blog break this week so all posts are reruns until the new year. This post originally appeared on 19 May 2009 on The Finance Buff. If you wish to leave comments, please do so on the original post.
What follows is an illustrative sketch of a classical economic welfare analysis using Google Reader as an example. It is intended for an audience with no economics background. At the risk of disappointing readers, I confess that, due to lack of data, I make up all the numbers. However, anyone with better estimates of the components could improve the calculation by following the steps outlined. Those familiar with Google Reader may wish to skip the next paragraph.
One way to read blogs is to visit each one’s website. Wouldn’t it be better if each could send new posts to one consolidated place for you? Well, nearly all can; the way to take advantage of this efficiency is using an aggregator like Google Reader. Blog sites, like The Finance Buff, advertise this service with icons that say “subscribe,” “RSS,” or “Atom,” or with a symbol like this. By clicking on these icons, you can add blog “feeds” to Google Reader (or another aggregator). The result is akin to a receive-only e-mail experience in which blog posts are listed in your reader and updated automatically.
Google Reader has enhanced my life, changing how I work and spend leisure time. I would not easily keep up with blogs of interest without Google Reader or something like it. (The blogs to which I subscribe are listed in this spreadsheet.) Google Reader is free, but it provides more than zero dollars worth of value to me. If it were not free I would pay something for the service.
For the sake of argument, let’s say I’d be willing to pay $100 per year for Google Reader. This represents the gross value of Google Reader to me. Since I actually pay $0, then the net value I receive from Google Reader is $100 – $0 = $100 per year. This is just like $100 in my pocket because I’d have paid that amount more for Google Reader than I had to. That $100 represents my individual annual consumer surplus.
I don’t know how many Google Reader users there are, but let’s say there are 10 million. Suppose the average individual user is like me, with a consumer surplus of $100 per year. Then the annual total consumer surplus of Google Reader is $100 x 10 million = $1 billion per year. That’s a lot of value. Of course, I made the numbers up.
There is also value that Google receives beyond its costs of producing Google Reader. Google does not receive revenue directly from its Reader right now. Let’s assume it attributes revenue to its Reader because it draws users to other (ad-based) revenue-producing Google products. Ad revenue earned by Google subsidizes the price consumers would otherwise pay for its services in general and the Reader in particular. A market in which producers use revenue from one group (e.g., advertisers) to subsidize another (e.g., viewers, readers) is called a two-sided market. Broadcast TV is another example of a two-sided market.
Let’s pretend that ad-based revenue Google imputes to its Reader is, on average, $11 per user per year. Let’s also assume that the average marginal cost of providing Google Reader to each of the 10 million users is $1 per user per year. So, Google nets $11 – $1 = $10 per year for an average user and, therefore, $100 million per year for all users. This $100 million is the annual total producer surplus associated with Google Reader.
Total surplus is the sum of consumer and producer surplus, and is an economic measure of welfare. In our example total surplus is $1.1 billion, $1 billion for consumers and $100 million for the producer.
Typically an economic welfare analysis of a market includes comparisons of surplus values for different market configurations. Doing so leads to conclusions about which group, consumers or producers, is made better or worse off in one setting versus another. It is possible for both producers and consumers to be made better off (or both worse off) via a structural change in the market. Total surplus is maximized in the circumstance of perfect competition, an ideal situation which is actually quite rare.
As an example, we could consider the RSS aggregator market with and without the participation of Google Reader. The analysis above is almost complete for the market with Google Reader. What it lacks is the consumer (producer) surplus associated with RSS aggregator users (producers) who do not use (supply) Google Reader that can be attributed to its presence in the market. It is plausible that its presence in the market causes other aggregators to be less expensive or of higher quality. Thus, users of other aggregators receive consumer surplus and their producers lose producer surplus due to lower prices and higher marginal cost of higher quality.
A separate analysis is required to compute the surplus of consumers and producers in the RSS aggregator market in the absence of Google Reader. To conduct this analysis, one would need a model that predicts what current Google Reader users would do in this case. If Google Reader left the market, how would aggregator market shares adjust? How many Reader users would forego aggregator use altogether? I’ll leave suggestions of ways of making such predictions to a future post.
One thing the above explicitly illustrates is that both producers and consumers are made economically better off through market transaction. It isn’t just that producers get revenue and consumers get products. Producers earn profit and consumers receive value beyond the sticker price. Since most of us are in the role of consumer frequently but seller hardly ever we tend to begrudge the firm its profit and pay little notice to the additional value beyond price we receive in return. Google Reader is not the only bargain. Given the enjoyment and convenience obtained by the multitude of products we use it’s a wonder how little of that full value we actually pay. The rest is consumer surplus.
Let’s Talk Price
In Why Are Some Companies Hated By Consumers?The Finance Buff (TFB) describes the reasons why he hates AT&T, Apple, and BlackBerry. He finds frustrating the constraints they impose on users about which devices can be used on which network. He laments the higher prices charged for service plans that offer more features.
Can I get a data-only plan on the BlackBerry? Yes, AT&T has one for $35 a month, but I can’t get the BlackBerry to synchronize with my work e-mails, calendar, and address book. AT&T’s $35-a-month data plan is called BlackBerry Personal, which only gives you access to personal e-mails and web browsing. To use it with corporate e-mails and calendar, you have to buy AT&T’s BlackBerry Enterprise service, which is $50 a month. Come on, data is data.
I feel his pain. I find myself getting angry sometimes when the thing I want costs more than what I want to pay for it, that is when my reservation price is below the market price. Yet there are lots of things I’d like that I think are too costly. Am I angry that I can’t buy a personal jet? That I can’t have a yacht? That I can’t afford live-in staff? No. Why not? What’s going on?
First of all, I never want to pay more than marginal cost. I want every market in which I’m a buyer to be perfectly competitive. If that were the case then TFB could get the BlackBerry Enterprise service for less than the $50/month he doesn’t want to pay. Since, as he says, “data is data,” then in a perfectly competitive market he would only need to pay the marginal cost of transmitting the data. It wouldn’t matter whether it is corporate or personal data.
Most markets aren’t perfectly competitive. So long as the lack of competition isn’t brought about by anti-competitive practices then there is nothing illegal about it. (Whether or not that is the case in the markets TFB described I do not know.) Imperfect competition arises for many legitimate reasons. Markups above marginal cost are to be expected in imperfectly competitive markets and are related to patterns of consumer demand (as I explained in What Is the Source of Price Setting Power).
When I find the market price of a good or service upsetting it helps me to think about the economics. I am calmed by the understanding that the above-marginal-cost price serves a useful role in allocating the limited resource to those who value it most. When my reservation price is below the market price it means I don’t want it as much as someone who’s reservation price is above the market price. Put another way, I cannot extract positive consumer surplus (a measure of satisfaction) from the good or service at the market price while others can (for more on consumer surplus see An Illustrative Welfare Analysis of Google Reader). Shouldn’t those who can extract higher consumer surplus get the good?
So why does it not upset me that I can’t buy a jet or yacht or live-in staff? It isn’t necessarily that my reservation price is below market price. It is the fact that I cannot afford them. That’s a different problem. I’m not upset by things I cannot afford. When I forget the economics I might get angry when something I can afford is literally over-priced for me. I think that may be what is bothering TFB too. It isn’t that the market price is too high (presuming no illegal anti-competitive behavior), it is that his reservation price is too low. He doesn’t value the BlackBerry Enterprise service enough. So he doesn’t buy it. That’s a perfectly reasonable outcome even in an imperfectly competitive market.
An Illustrative Welfare Analysis of Google Reader
This post originally appeared on The Finance Buff and was cited by The Carnival of Personal Finance #206.
What follows is an illustrative sketch of a classical economic welfare analysis using Google Reader as an example. It is intended for an audience with no economics background. At the risk of disappointing readers, I confess that, due to lack of data, I make up all the numbers. However, anyone with better estimates of the components could improve the calculation by following the steps outlined. Those familiar with Google Reader may wish to skip the next paragraph.
One way to read blogs is to visit each one’s website. Wouldn’t it be better if each could send new posts to one consolidated place for you? Well, nearly all can; the way to take advantage of this efficiency is using an aggregator like Google Reader. Blog sites, like The Finance Buff, advertise this service with icons that say “subscribe,” “RSS,” or “Atom,” or with a symbol like this. By clicking on these icons, you can add blog “feeds” to Google Reader (or another aggregator). The result is akin to a receive-only e-mail experience in which blog posts are listed in your reader and updated automatically.
Google Reader has enhanced my life, changing how I work and spend leisure time. I would not easily keep up with blogs of interest without Google Reader or something like it. (The blogs to which I subscribe are listed in this spreadsheet.) Google Reader is free, but it provides more than zero dollars worth of value to me. If it were not free I would pay something for the service.
For the sake of argument, let’s say I’d be willing to pay $100 per year for Google Reader. This represents the gross value of Google Reader to me. Since I actually pay $0, then the net value I receive from Google Reader is $100 – $0 = $100 per year. This is just like $100 in my pocket because I’d have paid that amount more for Google Reader than I had to. That $100 represents my individual annual consumer surplus.
I don’t know how many Google Reader users there are, but let’s say there are 10 million. Suppose the average individual user is like me, with a consumer surplus of $100 per year. Then the annual total consumer surplus of Google Reader is $100 x 10 million = $1 billion per year. That’s a lot of value. Of course, I made the numbers up.
There is also value that Google receives beyond its costs of producing Google Reader. Google does not receive revenue directly from its Reader right now. Let’s assume it attributes revenue to its Reader because it draws users to other (ad-based) revenue-producing Google products. Ad revenue earned by Google subsidizes the price consumers would otherwise pay for its services in general and the Reader in particular. A market in which producers use revenue from one group (e.g., advertisers) to subsidize another (e.g., viewers, readers) is called a two-sided market. Broadcast TV is another example of a two-sided market.
Let’s pretend that ad-based revenue Google imputes to its Reader is, on average, $11 per user per year. Let’s also assume that the average marginal cost of providing Google Reader to each of the 10 million users is $1 per user per year. So, Google nets $11 – $1 = $10 per year for an average user and, therefore, $100 million per year for all users. This $100 million is the annual total producer surplus associated with Google Reader.
Total surplus is the sum of consumer and producer surplus, and is an economic measure of welfare. In our example total surplus is $1.1 billion, $1 billion for consumers and $100 million for the producer.
Typically an economic welfare analysis of a market includes comparisons of surplus values for different market configurations. Doing so leads to conclusions about which group, consumers or producers, is made better or worse off in one setting versus another. It is possible for both producers and consumers to be made better off (or both worse off) via a structural change in the market. Total surplus is maximized in the circumstance of perfect competition, an ideal situation which is actually quite rare.
As an example, we could consider the RSS aggregator market with and without the participation of Google Reader. The analysis above is almost complete for the market with Google Reader. What it lacks is the consumer (producer) surplus associated with RSS aggregator users (producers) who do not use (supply) Google Reader that can be attributed to its presence in the market. It is plausible that its presence in the market causes other aggregators to be less expensive or of higher quality. Thus, users of other aggregators receive consumer surplus and their producers lose producer surplus due to lower prices and higher marginal cost of higher quality.
A separate analysis is required to compute the surplus of consumers and producers in the RSS aggregator market in the absence of Google Reader. To conduct this analysis, one would need a model that predicts what current Google Reader users would do in this case. If Google Reader left the market, how would aggregator market shares adjust? How many Reader users would forego aggregator use altogether? I’ll leave suggestions of ways of making such predictions to a future post.
One thing the above explicitly illustrates is that both producers and consumers are made economically better off through market transaction. It isn’t just that producers get revenue and consumers get products. Producers earn profit and consumers receive value beyond the sticker price. Since most of us are in the role of consumer frequently but seller hardly ever we tend to begrudge the firm its profit and pay little notice to the additional value beyond price we receive in return. Google Reader is not the only bargain. Given the enjoyment and convenience obtained by the multitude of products we use it’s a wonder how little of that full value we actually pay. The rest is consumer surplus.
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