Philip Trammell | New Products and Long Term Welfare

PHILIP TRAMMELL: (00:00) I'm going to be sharing some preliminary thoughts about the relationship between new product introduction and long-term welfare, and a few other things too while we're at it. The main point I want to communicate is very simple and it’s summed up by this graph. Consider a society in which there's only one consumption good. Let's call this society the Golden Horde. This was a kingdom that followed Genghis Khan's Empire when it split up. It doesn't matter really. Let's call their one consumption good horse. If the economists of this society had tried to estimate the relationship between consumption and utility, they would probably have found, as people find today, that as consumption goes to infinity, utility rises but only toward an upper bound. Now, suppose you posit that there's this single quantity, this one-dimensional construct consumption, which is a coherent idea in which utility is defined over, when you say that this consumption thing can be compared across individuals at a time and across time or across societies with very different consumption baskets available, you'll be led into a paradox. Why? Because it sure seems that utility and consumption for a middle class American or member of the developed world, at least today, is higher than that achievable with arbitrarily many horses, but nothing else. Of course, we don't want to say that consumption for us today is more than infinite, but it seems we don't want to say that consumption for people in this past society or hypothetical society was zero either. So these are two things we have to reconcile somehow.

(02:12) Before proceeding, I think it's worth saying a few words about a red herring, I think, that sometimes comes up at this point in the conversation. This is the thought that maybe we aren't actually better off today than people in the past who just had lots and lots of basic goods like horses. The thought is, roughly, that to have a zillion horses is to be Genghis Khan or something and get to feel like you're on top of the world and that might offset all the benefits we get from modern medicine and technology and so forth. I think that another way of putting this line of thought is that if you did have a lot more horses, a lot more of the baskets available in the past than your compatriots, you would enjoy a lot of positional goods, the feeling of being on top or the benefits that come with being on top. But that's not what we're talking about today. We're talking about the relationship between consumption itself and utility, not between positional goods and utility. To separate these two, there are two things we might try. The first is to compare utility in consumption per se between the Golden Horde and the USA. We can consider the utility level for a middle-class Golden Horde member reached by scaling horse consumption for everyone in that society equally so that at the end of the day, we're comparing someone who is still middle class but has a million consumption units, 𝟤 million, 𝟦 million with someone who is middle class in the modern world. We're comparing people of equal social rank. Neither gets to feel like they're on top of the world and I think it's at least possible, it's certainly my intuition, that the middle-class member of the modern world is doing better.

(04:29) Something else we can do along a similar vein is to explicitly let that 𝑢(𝒸) I was putting on the vertical axis to the previous slide, represent not the utility function of a representative agent or anything like that, but a measure of social welfare per se. So as time goes on and society as a whole gets richer and more products become available, social welfare might increase or decrease in principle, but it wouldn't be affected by positionality in the way that growing rich within a single society would be.

(05:16) Now, when doing consumer theory it's worth remembering, just to close off this tangent, that we almost never assume that when a consumer is restricted to a subset of the available goods she can reach any indifference curve achievable with all of the goods. That's just not an axiom you make in Moscow or anywhere. So it seems equally ill-motivated to assume that a consumer can achieve any conceptually available indifference curve when restricted to the goods available in the past. Like today, if we could only consume things that were available in the past, why should we think that if we just had a lot of them we would be able to achieve any indifference curve we can with the full variety of goods we have available today?

(06:09) All right. On first encountering or wrestling with this line of thinking, some economists are drawn to the conclusion that we just need to rethink how we measure inflation and I think that's the wrong lesson to draw here. So here's what I'm going to call the common mistake in more detail. First, you'll notice that measured real consumption is only some multiple, say 𝟪𝟢 times higher today than it was in say, Italy in 𝟣𝟧𝟢𝟢. But, the observation continues, the basket of consumption goods we get today is far preferable to 𝟪𝟢 baskets of the goods available to an Italian in 𝟣𝟧𝟢𝟢. So we must have underestimated the real consumption increase. So goes the thinking. And so we must have overestimated inflation because we know the nominal consumption increase, we know how many dollars were spent on consumption goods, we know what GDP per capita was in 𝟤𝟢𝟤𝟣 America, and we know how many florins were spent somewhere in 𝟣𝟧𝟢𝟢 Italy, or we could in principle. So if the real consumption increase has been larger than our numbers had been telling us then it must be, the thinking goes, that we overestimated inflation. A dollar in 𝟤𝟢𝟤𝟣 must be more valuable than we had thought relative to a florin in 𝟣𝟧𝟢𝟢. And finally, the mistake concludes, once we've measured inflation properly, we will be able to make those welfare relevant comparisons of nominal consumption per capita across periods. We'll be able to say, this is the number of 𝟣𝟧𝟢𝟢 florins that would have been an equivalent amount of consumption to 𝟦𝟢,𝟢𝟢𝟢 USD or whatever.

(08:27) So here's the crispest example I could find of an economist going through what I'll be calling the common mistake and saying that this is in fact, common, at least. So "The common presumption... Nordhaus, 𝟤𝟧 years ago, said that…

"The common presumption among most economists is that…

(08:50) Indeed.

… price indexes fail to deal adequately with quality change and new products"…

(08:27) And the presumption is that…

…this failure consists of…"an upward bias of prices (or inflation) over time... [Some] products…

(09:09) He concedes.

…do expand the range of service characteristics spanned by available commodities.”

(09:17) So even I would concede, for instance, that if the new product that's come out this year is a loaf of bread that's twice as big as the old loaves of bread but doesn't have anything different about it, it's not going to be pushing up the utility ceiling, right? Whatever you could have gotten with this new product you could have gotten last year by just buying twice as much of the old product. The interesting cases are products that expand the range of service characteristics spanned by available commodities. And as is clear and as Nordhaus notices, some products do this.

“[When constructing] price indexes for products that represent new service characteristics, the appropriate technique… he proposes… is to estimate the value of the new-characteristic commodity.”

(10:09)  Okay. The smartphone introduced to the Golden Horde.

“by determining the reservation income at which consumers would be indifferent to the choice between the budget set without the new-characteristic commodity.”

(10:26) Okay. Just give the Golden Horde member money and just say he can only buy horses, but that he's not allowed to buy this new thing.

“and the actual income…  

(10:38) No extra money.

… with the new-characteristic commodity.”

(10:41) Now he can buy a smartphone if he's willing to give up some horses.

(10:48) The reason why this is, at least in some cases, insufficient as a method to coming up with a definition of consumption that can be compared across periods is hopefully pretty straightforward by now. It's that when new products expand the range of available utility levels, there will be at least cases in which there is no such reservation income. Infinite income with which to buy nothing but horses is still dispreferred to some finite income with a wider product range. In short, the problem is with the whole idea of a 𝑢(𝒸) not with any particular approach to measuring 𝒸 in so-called real terms over time.

(11:36) So what should we do instead? Well, there's many conceivable alternatives. But here's one that I think is particularly simple and well-suited to capturing the consideration that I'm proposing. So suppose there's a continuum of goods from 𝟢 to 𝐿. There's also a continuum of corresponding good-specific consumption levels for some consumer. Utility as defined over these good-specific consumption levels, is simply the integral across goods of good-specific utility, 𝑉(𝐶_𝓁) here, where good-specific utility is isoelastic in good-specific consumption. You need the max 𝟢 thing because we don't want to say that if 𝐿 rises and new goods get introduced, you have to consume a little bit of them or else utility falls. In fact, it falls  to negative infinity, as you would conclude, if you maintain this 𝜂 > 𝟣 condition and you didn't have the max 𝟢. So that's just a technical detail. Basically, we're integrating across goods and we're saying that utility is isoelastic in each good. Now, let's say the prices of the goods are equal and without loss of generality, let's say they're equal to 𝟣. What this implies is that given some budget, call it 𝒸, the consumer will choose to consume 𝒸/𝐿 of each good. That will be the consumption density of each good and this in turn, will imply that we can write utility not as a function of a whole continuum of consumption levels but of two parameters, aggregate consumption and product range, where the function is isoelastic here in consumption per product because that's how much you're consuming each one times 𝐿, which is just the size of the range of products. So that's just what we're doing. We're integrating from 𝟢 to 𝐿. So here's a width of length 𝐿 and for each point on that continuum from 0 to 𝐿, we get utility that's isoelastic in 𝒸/𝐿, consumption per product. This condition here is necessary because if 𝒸 is low enough, there are some products you won't want to consume at all because of this max 𝟢 condition discussed before. So this is just the technical detail carried forward, but we won't worry about it. We'll just assume that consumption is high enough that we're in an interior solution.

(14:58) Here's how this utility function can be represented graphically. The utility curve on the bottom is the one we saw on the first graph. As 𝐿 rises, the utility ceiling rises and the utility curve gets sort of stretched out here. The ceiling will always equal this term 𝐿/𝜂 – 𝟣 and this line truncates the bottoms of the utility function so that we're never led into the paradoxical conclusion that if consumption is low enough, raising the product range lowers the utility level.

(15:40) You might notice that the utility function I introduced two slides ago has something in common with what are called expanding varieties models. So at least since Romer's 𝟣𝟫𝟫𝟢 growth paper, it's been standard to embed new product introduction into growth models using a constant elasticity of substitution consumption aggregator, which looks like this. Utility is undefined over 𝒸, which is the aggregation of the consumption levels for all the available goods. Note that the goods here are gross substitutes. The elasticity of substitution is greater than 𝟣 because of the assumption that 𝜌 > 𝟢. What this means is that if you're just restricted to a subset of the goods, you can still reach an aggregate consumption level ant that’s the utility level that you could achieve with a wider range of goods. So if 𝜌 = ½ say, you're just getting the square root of your consumption of goods, 𝟢 to 𝐿/𝟤, if only half the range is available and then you're integrating and you're squaring it. Clearly, there's no reason why you can't achieve any given level of 𝒸 and therefore, any conceivable level of 𝑢(𝒸) just because you don't have access to go to 𝐿/ 𝟤 up to 𝐿. By contrast, the utility function defined in line (𝟥) posits that the goods are gross complements. The elasticity of substitution between them turns out to be 𝟣/𝜂 < 𝟣 because 𝜂 > 𝟣 and so, no amount of a subset of the goods can make up for not having any of the remaining goods. This was the whole point.

(17:50) Now in practice, of course, some goods are gross complements and some are gross substitutes. But the interesting point is that at least some of them are gross complements, pardon, and as those get introduced over time, they really do seem to push up the utility ceiling, or at least, it's conceivable that they would. And so 𝐿 in my model might be interpreted as the number of goods that are genuinely different enough from each other that they're not very substitutable and 𝐿 rises when we introduce new goods that really do raise the utility ceiling. But it doesn't when we just introduce products like bigger loaves of bread.

(18:42) In any event, let's now explore an implication of this model that might at first seem tangential but I think helps us understand how it works and ultimately perhaps, even to calibrate it. So, the conventionally assumed preferences of a consumption smoothing household assumed by typical economists look like this. So here's the standard framework. People have time-separable utility that's isoelastic in consumption alone, or at least it stays in the context of the growth model. This would be the most common supposition. And people are assumed to maximize their expected utility. So they maximize the expectation at the time 𝟢 of the integral of discounted flow utility, or flow utility is isoelastic in flow consumption. In a model like this 𝜂  captures two things at once. It's the coefficient of relative risk aversion. It captures how averse a consumer is to risk at a given time and the inverse elasticity of intertemporal substitution. So it captures how averse the consumer is to unequal consumption across periods. So it's just the curvature of the utility function, full stop. A problem with taking this model too seriously and literally is that empirically, people seem to exhibit more risk aversion than aversion to unequal consumption across periods. And this is a big effect.

(20:49) This discrepancy underlies several financial puzzles of which the most notable one has got to be the equity premium puzzle. So concretely, when deciding how much expected return to give up in exchange for lower risk, people act as if 𝜂 were high. They're willing to invest in bonds instead of stocks, even though the returns are much higher for the stocks because it seems that this would be motivated by extremely concave utility functions, very high 𝜂. But when deciding how much to invest for the future, taking advantage of positive interest rates, even though people expect to be richer in the future anyway, people act as if 𝜂 were low. Sure, I'll be richer in the future, but it's still worth saving even more for my future because interest rates are positive and what does it matter if I'll be a bit richer in the future. That won't push down my marginal utility and consumption that much.

(21:51) So okay. There's sort of two classes of solutions to this issue. The first is to posit preferences that are not time separable. So yeah, you can't get your total utility just by integrating across time, your flow utility at each time. Most common class of these sorts of preferences are Epstein-Zin  preferences. But the second class of solutions, one that seems a bit less ad hoc and better motivated to my mind, is to keep time separability, but to say that there's something changing over time, some trend going on in the background, such that even though the consumers consumption level is higher in the future, her marginal utility and consumption has not fallen by as much as if her consumption suddenly increased in the present.

(22:53) So one example of a model of this style is the “Keeping Up with the Joneses” model of Gali '94. Here, the thing that's changing over time is other people's consumption. If to some extent, you care not just about your own consumption, but about how yours compares to others, sort of touching on the positional goods point we mentioned earlier, then you can see how this would work. Maybe suddenly, getting three times richer today, would leave you without much reason for an extra dollar. But you anticipate that even though you might expect to be three times richer in 𝟦𝟢 years, other people will also have gotten three times richer in 𝟦𝟢  years. And so there's still value in having that extra dollar, there's still some competition to be done.

(23:52) Second class of models of this... A second example of models in the class are habit formation models. Here, the thing that's changing over time is one's own previous period consumption. Once again, if your consumption suddenly multiplied by three today, you'd hardly know what to do with the money, but you expect to develop expensive tastes and habits over time that mean that you think you'll value a dollar in the future even if you're three times richer. Value it a lot. And now maybe it's clear where the new product thing fits in. The product range could also be a thing that's growing over time and that raises the marginal utility in consumption in the future at any given level of consumption even if it's quite high. So I'm not saying that this model is probably the reason for the equity premium puzzle and all the other adjacent financial puzzles but it is nice to observe that it predicts a difference between these two parameters, CRRA and IEIS in effect and in the observed direction.

(25:13) But if we do take it kind of literally or if we embedded a new products framework in a model that tried to account for whatever other factors of the previous slide, we believe that then it gives us a way… This sort of insight gives us a way to calibrate something that would otherwise be financials, and that's 𝘨_𝐿, the growth rate of complementary, at least, new products.

(25:45) So, suppose consumers maximize their discounted flow utility, as in the conventional framework but where utility is defined as in equation (3) a few slides ago. It's a function over both consumption and product range. To satisfy a consumer’s intertemporal Euler equation, the interest rate has to satisfy this sort of tweaked version of the Ramsey formula. It has to just compensate the consumer for their pure time preference 𝛿 and for the fact that as time goes on, the marginal utility of consumption is changing and in the generic case, you might expect, falling. Given the utility function of equation (3), the second term will equal 𝜂 times the growth rate of consumption per product because the marginal utility in consumption, how much utility you can get by consuming a bit more of any given product, depends on consumption per product. If the product range doubled and consumption also doubled, so you're at the same point in the utility curve for each product. Marginal utility in consumption will have stayed the same. This in turn implies that the interest rate will equal the following because the growth rate of consumption per product is just the difference between the growth rate of consumption and the growth rate of the product range. And if we believe the Kaldor Facts, that in the long run the interest rate and the consumption growth rate tend to be constant. This implies that 𝘨_𝐿 here will also be constant and it'll equal this term. Right, it's just algebra.                                               

(27:51) In this framework, the curvature of the utility function within a period, the actual CRRA, is going to equal 𝑟 – 𝛿 divided by this difference here in the growth rates. This also just follows by rearranging one in (8). We're trying to get 𝜂, so we do 𝑟 – 𝛿 divided by this difference. By contrast, let ῆ denote the inverse elasticity of intertemporal substitution that would be mistakenly estimated by some economists that was assuming the standard Ramsey formula. So assuming that utility was just defined over 𝑐, consumption, which could be straightforwardly compared across periods and therefore that the interest rate equals 𝛿 + ῆ𝘨_𝑐. So 𝜂 here, this ῆ over here, this estimate of the curvature equals 𝑟 – 𝛿/𝘨_𝑐  and this is lower than this because we're dividing by something smaller here. So okay, so this is a concrete way in which the model predicts 𝜂 (curvature at a time) > ῆ sort of a quasi curvature across time, willingness to substitute across time, as observed.

(29:32) If a model of this sort is roughly right, it has a long list of implications. It's changing something that's at the foundation of so many castles of economic reasoning. But I'll just list a few of what seemed to me like the most important and/or at least obvious. So first, just to emphasize, this model implies that economic growth is much more important than it might otherwise have seemed. In particular, if what we mean by economic growth is growth in both consumption and product range, then it allows for, at least in theory, in principle, unbounded growth in welfare per person. By contrast, a one-dimensional 𝑢(𝒸) type framework with 𝜂 > 𝟣 implies that all economic growth can do is bring us closer to the welfare ceiling that we approach as consumption goes off to infinity. And if you think that we're not far from the ceiling today, which I think is actually a very plausible conclusion to draw, then there isn't much value to be produced, at least in terms of welfare per capita, via the consumption benefits of economic growth. So I'm proposing a much more optimistic view on that front.

(31:05) Another implication I won't dwell on much, but hopefully, it's sort of straightforward, is that the optimal savings rate is much higher than it might seem. In the future, sure we'll be richer, but there'll be more goodies to spend the wealth on. And so it's worth saving up to use resources when we have more utility productive ways to do so.

(31:32) Thirdly, perhaps more subtly, this model allows us to reconcile two different concerns that we might have that otherwise seem at odds. So first is, we might think that it’s worth paying a lot to avoid future climate damages even on the assumption, even in the scenario, that society is much richer in the future. So it's common in climate IAM, Integrated Assessment Models, to assume that economic growth will continue. We'll be much richer in the future and the damages from climate change will basically just take the form of like, some percent loss in GDP or in the capital stock at the time. And the only way to really care about that, to be willing to pay a lot to avoid that, is if you think that marginal utility doesn't diminish much in consumption as you get richer and even if it diminishes a lot, anyway we'll be way richer in the future, so why does it matter if we'll lose some of our GDP or some of our capital? So if you want to retain concern for future climate damages on the assumption we'll be richer in the future, you have to say, well okay, I guess marginal utility doesn't diminish much in consumption even as you get richer, it's like close to linear. Utility is close to linear in consumption. But if you make that choice, you would have to conclude there isn't much benefit in redistributing today from the rich to the poor because, again, utility is close to linear in consumption. But in the framework I've produced, these intuitions are totally compatible. Redistribution is great today because marginal utility diminishes sharply within periods. But it's also worth paying a lot to avoid future climate damages because people will be richer, but there will be more opportunities to do good with wealth at the time. I'm not saying this is right, I'm just pointing out that it reconciles these intuitions.

(33:48) And finally, because new product introduction raises the marginal utility of consumption, for any given level of consumption, it weakens a link proposed by Aschenbrenner's 𝟤𝟢𝟤𝟢 paper Existential Risk and Growth between economic growth and existential risk. The link he proposes is that as people get richer, they want to spend more on x-risk reduction because in effect, what else is there to spend the money on? In the model those are the only two things, consumption and x-risk mitigation. Consumption raises x-risk, mitigation lowers it . And as people get richer, there is almost no value in further consumption, so they start pouring all their wealth into x-risk mitigation. So yeah, we might not want new product introduction because it would give our wealthy descendants ways to have fun with their wealth and that might continue to raise x-risk damage. It will not force them as quickly to sort of reconsider that. So longtermists may actually want to slow new product introduction, sort of ironically enough, at least for some period, even though it's a main driver of long-run welfare growth in this model. So many other implications, but I will leave it there.