How much am I willing to pay for a new good? If I consume the good immediately when I buy it, then we can just use standard techniques to estimate the demand curve and calculate the consumer surplus after the introduction of the good (I particularly like Hausman (1997) on the value of Honey Nut Cheerios). But some goods require time to be consumed. For example, the internet’s costs represent only .2% of average expenditure in the US, but time spent on the internet is something like 10% of non-waking hours. In this short article (in this year’s AER P&P), Goolsbee and Klenow discuss how we might estimate consumer surplus when time matters.

The problem is as follows. Rather than giving a consumer a budget in dollars, and letting her choose how to spend that money, instead give a consumer an endowment of time (as in Becker’s famous 1965 paper), and have them allocate that time to leisure, to work whose earnings are devoted to the composition good and the internet, and time spent using the internet. We can then use variance in wages to estimate the elasticity of substitution between internet consumption and other goods, noting that time must spent to consume the internet. Note that such a model has pretty firm implications: people with higher wages will use the internet less, for instance, since the number of hours in the day is the same for rich and poor people, but rich people can work and earn a lot of money instead of using the internet. Such an implication turns out to be true in the data.

Using data on internet usage and incomes to estimate the elasticity of substitution between internet and other goods, and the estimating the demand curve with respect to time, rather than with respect to money, gives an estimate of the consumer surplus equal to over a thousand dollars per household. The exact number isn’t really important, as the choice setup is really really limited – there is no substitution allowed to other time-intensive goods like TV, wage income is assumed to be constant as hours increase, etc. – but the method of modeling choice for goods with a time component is interesting. Actually, there are a bunch of applications to copyright policy, for instance, that could certainly use such methods.

I also wanted to write about this paper as a backdoor way to discuss a mistake I’ve seen twice in other papers just this week. I have a rule on this site that I only discuss papers I like; you won’t find any offhand insults of other people’s work here, so I don’t want to write about those papers in particular. The mistake doesn’t appear in Goolsbee and Klenow, so we’re safe. The problem I’m referring to is the meaning of willingness-to-pay/consumer surplus. No term is used more informally by economists than WTP, and such lack of formality causes errors in analysis.

Consider the following question. Cutting the price of a good in half is shown to increase consumer surplus by .4. Cutting the price of another good in half also increases consumer surplus by .4. It would cost society .45 for each good to invent the process which cuts cost of production in half. Are these investments socially beneficial? The correct answer is “it depends”. For instance, let the goods give utility in Cobb-Douglas form, with both parameters equal to .5, the price of both goods equal 1, and income equal to 1. I’ll compute equivalent variation instead of consumer surplus for simplicity, but the basic point is true with either measure. Before the price change, you can solve to see .5 of each unit are bought, giving utility (.5^.5)*(.5^.5)=.5. Cutting the price of good 1 in half changes consumption to 1 of good 1 and .5 of good 2, giving utility .7. This is symmetrically true for good .7. To get utility .7 under old prices requires consuming .7 of each good, which costs 1.4, so equivalent variation is .4. If both prices are cut in half, utility is 1. To get utility 1 under old prices requires consuming 1 of each good, which costs 2, so equivalent variation is 1. That is, it would be worthwhile to invent the new cost-saving technologies as long as we invent both of them.

How does this example apply to the papers referenced above? Willingness-to-pay measures the price at which I no longer consume an object, *fixing all other prices and available products*. But it is not transitive! You cannot compute willingness-to-pay for two different goods, or two different attributes of a good, individually, then sum up the willingness-to-pay figures to get a “willingness-to-pay for both attributes”. I suppose there’s nothing surprising about this: everything has general equilibrium effects. But the problem here, when attributes or goods are substitutes or complements for each other, is particularly important, and making the above error will lead to wildly wrong estimates of welfare impact.

http://faculty.chicagobooth.edu/austan.goolsbee/research/timeuse.pdf

If I understand this correctly:

There exist situations in which a single good has multiple dimensions (which individually have utility) summing up WTP measures is inadequate.

There’s some coming together of attributes then, in that the whole is greater than the sum of the parts.

Is this right?

Not necessarily greater. Consider right shoes and left shoes, which are basically perfect complements. Given the status quo of having a pair of shoes worth $60 together, my willingness to pay for a right shoe is $60 (since otherwise the left shoe is worthless), and my willingness to pay for the left shoe is $60 (since otherwise the right is worthless). But I can’t then sum these up and say I’m willing to pay $120 for a pair of shoes.

To carry the argument a bit further:

This would appear to be a conditional statement; conditional on having either a left or right shoe.

If you ask “if you don’t have a single shoe, how much is your WTP for a right shoe?” well that’s going to be worthless, so you should say “zero”. You can carry out a similar exercise with a left shoe and if you sum then you get…zero $ for both shoes!

this is obviously absurd. I don’t think I ever thought about this before. Which papers do you think make this mistake?

The problem is that the word “willingness to pay” is used loosely. The correct interpretation of WTP is from Hicks, but from demand data you can compute WTP only for one good, as far as I’m concerned, properly WTP means “how much would you have to pay me for me to sell you a marginal unit of this good, conditional on keeping the rest of my current bundle, and conditional on the prices and availability of all other goods in the market.” Now, IF you linearize utility, such that u(x)=ax(1)+bx(2)+…, then of course WTP for a change in bundle can be computed by summing WTP for each good.

As for what papers make the mistake…I’m a lowly PhD student, so I don’t like to call out papers on this blog. But the answer is: a lot of them, particularly in empirical innovation and marketing, but also elsewhere…

(One more thing: with continuous utility, and computing the value of a single good, there is no problem with WTP. The shoe example is a bit contrived because of the discontinuity at 1 of each shoe. But the point about computing WTP for bundles that have differing complementarity or substitutability still stands. The problem, empirically, shows up because people estimate McFadden/Manski random utility models with OLS, hence they must linearize the attributes of the goods (or the goods in the bundle).)

I’m a lowly phd student too…

as far as linear utility goes, the random utility model makes this assumption and I always find it somewhat troubling because it assumes all the given characteristics of the consumption of any good or service are essentially substituting for each other.

If, on the other hand, there is a complementarity relation then this is somewhat odd. It’s been a while since I read about this; but I wonder how weak complementarity speaks to this.

anyway, I really like your website.