Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
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# The End of Economic Growth? Unintended Consequences of a Declining Population

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Video: https://www.youtube.com/watch?v=wTXf2oMOIyg&t=0s


## Talk (00:00:00 – 00:22:37)

[00:00:02] hi so this is a 20-minute presentation of a recent paper mine called the end of economic growth unintended consequences of a declining population so a large class of growth models works the following way the people produce ideas  
[00:00:16] and those ideas are the engine of economic growth so the original classic papers by romer again how at grossman heltman all have this mechanism at their heart also the semi-endogenous growth papers that i worked on sam cordon and paul segerson worked on  
[00:00:30] also work this way and basically many many idea-driven growth models have this mechanism at their heart now in all these models the population is assumed to be their constant or growing exponentially and from a historical standpoint that seems like a  
[00:00:45] reasonable assumption initially question arises when you think about what's the future of global population and my view before i started working on this project was kind of that the conventional wisdom was that the global population was likely to stabilize at around 8 or 10  
[00:00:59] billion people just last year though bricker and ibbitson published this book called empty planet where they challenged the conventional view and kind of made a point that i think ex-post is is completely obvious and relatively compelling  
[00:01:13] and that is that maybe the future is one of a declining population i.e negative population growth rather than a stable global population and the evidence for this is that high-income countries all around the world  
[00:01:26] already have fertility rates that are consistent with the declining population fertility's below replacement so to see this evidence take a look at the table in the in the top right of your screen now this is the total fertility rate and you'll recall that to keep  
[00:01:41] the population stable women on average need to have two children right so uh what we see in the united states for example is that the fertility rate is about 1.8 below two in high income countries as a whole it's around 1.7  
[00:01:55] china 1.7 germany 1.6 japan italy and spain actually 1.3 or 1.4 so in the rich countries of the world fertility rates are already below replacement that fertility is consistent with a declining population rather than a stable population  
[00:02:10] in the graph what you see is is the fact that i think we all know which is fertility rates have been declining throughout much of the world so take india as a classic example in the 50s and 60s the average woman in india had about six kids and since then that's been  
[00:02:25] falling from five to four to three down to about two and a half and it's not implausible to think that if we wait another decade or two fertility in india will fall below to that is india's population will start to decline  
[00:02:38] instead of continue to to rise and so this is kind of the bricker and evidence and evidence that we may actually need to take very seriously the possibility that population in the future may decline rather than stabilize or continue to  
[00:02:53] rise and kind of the question at the heart of this paper is what happens to economic growth in those models that i began with if the population's declining i.e if population growth is negative instead of populations that are stable or growing  
[00:03:07] and what the paper shows is kind of two two classes of results in the first half of the paper i look at exogenous population growth and suppose populations exogenously uh declining and there i derive what i call the empty planet result  
[00:03:21] and this is that if population growth is negative and you feed it into an idea-based growth model then you get the end of economic growth living standards stagnate and because population growth is negative the population vanishes and so that's what i'm going to  
[00:03:36] call the empty planet result the situation where living standards stagnate they stabilize and then the population vanishes potentially to to zero arbitrarily low values and this contrasts with the standard result in these growth  
[00:03:51] models which i'm going to call uh by comparison the expanding cosmos result where we get exponential growth for an exponentially growing population so these are two very starkly different results okay then the second half of the paper  
[00:04:05] endogenizes the population growth process itself and derives what i think is kind of a surprising and maybe even paradoxical result but i'll try to explain it by the end of this this presentation and that is that a social planner  
[00:04:18] prefers the expanding cosmos who wants exponential growth for an exponentially growing number of people that social planner can actually get trapped in the empty planet and how that works is an interesting thing that i'll get to later  
[00:04:32] okay in the interest of keeping this short let me with apologies skip the literature review this is uh it can be found in the paper um so let me start with the first you know half of the paper this this empty planet result and and to get there let me go back  
[00:04:46] to kind of a simplified version of the roamer again grossman helpman model so production of goods features increasing returns to scale this was kind of roamer's insight so it's constant returns to rivalrous inputs here just people  
[00:04:59] and therefore increasing returns to people and ideas together where the parameter sigma measures the degree of increasing returns to scale uh production of ideas the growth rate of knowledge is proportional to the population  
[00:05:12] and then again to start let's let's revisit the classic result assume the population is constant okay then you combine these equations and what you see is well first of all income per person is proportional to the stock of ideas  
[00:05:27] and then since the stock of ideas is growing because of the population income per person grows exponentially at a rate that's proportional to the size of the population so that's sort of the classic roamer grossman helping again and how to resolve now the question is what happens if  
[00:05:40] population growth is negative so we're just changing that third equation to say let's suppose population declines exogenously at a rate eta okay so eta could be half a percent per year one percent per year if you combine these second and third  
[00:05:55] equations you get this law of motion for the stock of knowledge and it says the growth rate of knowledge is itself falling exponentially and it's easy to integrate this equation and get the result  
[00:06:07] that the stock of knowledge asymptotes to a constant stock of knowledge is bounded from above and therefore income per person is bounded from above and you can calculate the values of these bounds right so this is what i mean by you know  
[00:06:22] the stock of knowledge stagnates and therefore living standards stagnate when population growth is negative all right that's the empty planet result what i do next is i say well let's go to  
[00:06:35] the semi-endogenous growth models and you may recall that when you've got a constant population or growing population the fully endogenous growth models and the semi-endogenous growth models yield very different results interestingly with negative population  
[00:06:49] growth the the two classes of models yield very similar results so that's the point i want to go through now so again let's start with the classic semi endogenous result we've got the same production function for goods the production function for ideas introduces this a to the minus beta term  
[00:07:03] this just says ideas are getting harder to find it's getting harder and harder to make proportional improvements in productivity and beta kind of captures the degree to which ideas are getting harder to find let's feed in constant positive population growth  
[00:07:17] and then again you get the classic result that the growth rate of income per person is proportional to the rate of population growth where the factor of proportionality is the overall degree of increasing returns to scale in the economy  
[00:07:31] right this is the classic semi-endogenous growth result and this is the expanding cosmos result right we've got exponential growth in living standards for a population that itself is growing exponentially  
[00:07:44] okay what happens if we change the population growth equation so that now instead of population growing the population declines exponentially again at rate eta well combine the second and third equations again you get this equation  
[00:07:58] and you can already see here the growth rate of knowledge is declining exponentially as it was before but even faster than exponentially and so you kind of know if it was already bounded if the stock of knowledge was asymptotically bounded in the roamer model  
[00:08:12] it's going to be bounded here as well again you can you can integrate this equation quite easily and and find the value for the upper bound basically both knowledge and living standards asymptote to an upper bound that's  
[00:08:24] finite y star and a star so again we have the empty planet result that living standards stabilize for a population that vanishes and that's kind of that empty planet result is the first key result and  
[00:08:38] you know just to review it i would say historically fertility's been trending downward from five kids to four kids to three to two and even less in rich countries and there's an interesting point here which is for an individual family from the microeconomic standpoint  
[00:08:52] there's nothing special about fertility being slightly above two versus slightly below two but the macroeconomics of the problem makes this distinction absolutely critical if fertility is slightly below two then we get negative population growth  
[00:09:07] and as i've just shown that condemns us to stagnation on an empty planet in contrast if fertility is slightly above two then we get exponential growth in population exponential growth and living standards  
[00:09:20] and we get the expanding cosmos result okay so that's kind of the first result in the paper all right what i want to do next is shift to a model of endogenous fertility let's endogenize the population growth rate itself and there's a big literature on  
[00:09:34] different ways to do this what i'm doing here is i'm trying to take the simplest possible model i can come up with that makes the point that i want to make which is namely that even the social planner can get caught in this empty planet so so this is a subtle result and so it's really helpful to make the model as  
[00:09:49] simple as possible so what am i doing here i introduce one economic decision which is how much time do people spend working versus having kids let's script l denote the time people spend having kids and so one minus l is going to be the  
[00:10:04] time people spend producing so that enters the production function uh population growth is just births minus deaths and births are proportional to the amount of time you spend investing in having kids right so b bar is the maximum  
[00:10:19] fertility rate and so b bar minus delta is the maximum population growth rate and if you have fewer kids population growth can be even slower the idea production function is the semi-endogenous growth model generation zero's utility depends on  
[00:10:32] their consumption and their number of offspring right so people uh care about how much they eat and how many kids and grandkids and great-grandkids they have and to keep things simple we specify flow utilities depending on log  
[00:10:46] consumption and log offspring okay so the log consumption is needed so that substitution and income effects cancel this will mean that with a growing wage with economic growth we can have a constant fertility right this is sort of a classic you know result from the  
[00:11:00] business cycle literature for example um the log uh n tilde the log number of offspring that log form is very convenient for getting understanding the transition dynamics you can relax  
[00:11:13] that assumption and and write down more general utility functions that's done in the paper as well and then finally there's no capital consumption is just output per person okay so um uh notice here all people generate ideas  
[00:11:28] uh in equilibrium here's how again the model is going to be set up rather than model monopolistic competition the way sort of you know models often do and instead to keep things simple assume ideas are just external okay so we have kids because we love  
[00:11:43] them we ignore the fact that our kids might actually create ideas that benefit everyone in the world okay so we're going to completely ignore that the planner therefore is going to desire higher fertility in this setup because the planner values the ideas  
[00:11:57] that our kids might have this is entirely a modeling choice and other results are possible but again to illustrate the point i want to make this is going to be convenient okay important point this i've already made uh  
[00:12:10] already but it's important to see how it fits in with endogenous fertility so this is a graph of the steady-state growth rate of knowledge if population growth is positive then in the semi-endogenous growth model uh the growth rate of knowledge is proportional to the rate of population growth that's the  
[00:12:25] expanding cosmos if population growth is negative then the stock of knowledge ultimately stagnates so the steady-state growth rate is zero notice that there's a kink here at zero and when i show you the results related to you know how the planner can get  
[00:12:39] trapped at the empty planet you're going to see multiple steady states and if you ask where does that multiple steady state result come from ultimately it's from this kink this kink is what gives rise to multiple  
[00:12:51] steady states and and two regimes okay uh rather than walk you through the details of the equilibrium and the optimal allocation let me just uh in the interest of keeping this down to 20 minutes let me just show you how it works so the fertility equation  
[00:13:05] in both the equilibrium and the optimal location is b bar l minus delta where l is how much time you spend having kids in both the optimal and equilibrium allocation the amount of time you spend having kids  
[00:13:16] is one minus one over v where v is the utility value of people okay and v differs between equilibrium allocation and the optimal allocations the same functional form but the value of v is going to differ  
[00:13:31] and so population growth is b bar minus delta minus 1 over v so in equilibrium we we value kids because we love them so definitely v equilibrium is just epsilon divided by rho and if i go back to my utility function you see  
[00:13:44] epsilon is how much we love our kids and you adjust for row because we're we're taking the present value so in equilibrium v bar v equals epsilon divided by rho and it turns out to be constant there's no transition dynamics in v so  
[00:13:59] population growth is going to be constant notice that if you plug in the epsilon over rho here this becomes rho over epsilon if epsilon is sufficiently small then we can have an equilibrium that features negative population growth so that's kind of the first result of the  
[00:14:13] equilibrium model you can actually support an equilibrium with negative population growth okay next what about the planner's v well the planner values kids because we love them epsilon over row but the planner also values  
[00:14:26] the ideas that our kids produce mu times a dot a dot is the flow of ideas mu is is kind of the price of the ideas and notice a dot depends on n it depends on population growth and so this this equation gives rise to  
[00:14:39] a v of n equation so we've got n of v and v of n and those two equations and two unknowns determine uh what happens so in this graph for example here's here's the v event equation faster growth makes  
[00:14:53] people more valuable because of the ideas that they're producing and here's the end of the equation so first go back here to the equilibrium remember the equilibrium occurs where v equals epsilon over rho and therefore n is given by reading off the blue line  
[00:15:07] so i've drawn it here for a case where there's positive population growth at equilibrium and then the optimal allocation is where the v of n and n of v schedules intersect and here this occurs at a higher population growth so this is exactly  
[00:15:21] what you'd expect from a model like this the planner wants more kids because the kids produce ideas and the equilibrium ignores that so the planner wants a higher population growth rate and the planner wants a higher steady state in some sense than equilibrium  
[00:15:35] okay the interesting thing that happens though is if we make the equilibrium fertility rate negative if we make the equilibrium population growth rate negative if we shift the blue curve down to intersect at epsilon over rho where n is negative  
[00:15:50] well then you can see because of this kink that occurs at zero and this is ultimately related to the kink that i just showed you the blue lines and the green lines intersect at three points there's still this expanding cosmos steady state  
[00:16:03] but now the equilibrium the empty planet result is also a steady state of the planner's problem and there's a third steady state in between okay let me i'll explain why why this occurs and how to understand it  
[00:16:18] in a second you sort of see mathematically a bit from this graph why it's happening okay to understand looking back at this picture you know given we start somewhere where does the economy go does it go to the expanding cosmos steady state does it go to the low steady state  
[00:16:31] what is what happens for that we need to look at the transition dynamics and to solve the transition dynamics the natural state variables here are the number of people and the stock of knowledge but it turns out to solve for the transition dynamics it's convenient to  
[00:16:45] change the state to redefine a new state like variable which is x which is a to the beta divided by n to the lambda which i'm going to ignoring the exponents referred to as knowledge per person and now the state variables are  
[00:16:58] population and knowledge per person why is this convenient well notice that you know with this definition the growth rate of knowledge is just 1 over x so you can see if x is constant the growth rate of knowledge is going to be constant so in any steady state  
[00:17:12] x is going to need to be constant the key reason this is useful is that you can show that optimal fertility depends only on x and not on n separate from x and so this means we can  
[00:17:26] solve things in a one-dimensional kind of problem rather than a two-dimensional problem and that makes things just just much easier to solve okay all right so to start with let me show you the transition dynamics for the equilibrium and these are all numerical results so  
[00:17:40] i'm picking uh parameters such that the equilibrium population growth rate is negative a half a percent per year and notice on this axis we've got knowledge per person and on the vertical axis we've got population growth so knowledge per person again is a to  
[00:17:55] the beta over the lama ignore the exponents and it looks like now it's reversing okay so what happens if we start with knowledge per person equal to 50 well the equilibrium wants negative population growth so look at x so a is rising and going to  
[00:18:09] some asymptote n is declining so x is rising so x continues to rise and that actually happens forever right and remember a dot over a is 1 over x so 8 out of a is going to zero  
[00:18:22] as x goes to infinity and so the empty planet result is sort of an asymptotic result in this graph as x goes to infinity and fertility stays at negative half a percent what does the optimal allocation look like here well first look at the high  
[00:18:37] steady state the expanding cosmos steady state this is saddle path stable meaning if you start off with you know 25 ideas per person or 400 ideas per person no matter where you start close to the expanding cosmos steady state the  
[00:18:50] planner wants to move there okay interestingly though there's this middle steady state which is unstable and never reached and then the empty planet steady state is also a steady state of the planner problem  
[00:19:05] so let me talk about each of these in turn the middle steady state again i'm going to skip over it turns out it's an unstable spiral and so you actually never reach it you can see it's unstable here it's called a schema point technically  
[00:19:18] the paper discusses a lot of these technicalities you get this jump but the jump doesn't violate any first order condition because um uh consumption never jumps you know on the path you go you go one way or the other um  
[00:19:33] next though let me discuss the the low steady state and why that's a steady state and to see that let's suppose we start with uh on the equilibrium with x equals to 25 we start with 25 ideas per person and we start at equilibrium  
[00:19:47] so population growth is negative the planner of course would love you to move up here and go to the expanding cosmos steady state but suppose we don't adopt the good policies so we stay on the equilibrium path  
[00:20:00] eventually we get to 400 ideas per person the planner would still like you to jump up here and have higher population growth rates and go to the expanding cosmos steady state the interesting thing that happens is suppose we keep waiting well  
[00:20:13] x keeps rising and if we wait long enough if we wait till say x is 2000 then an interesting thing occurs okay so x is very high so we've got very high ideas per person  
[00:20:26] and what that means is that the value of the ideas that the extra person produces is very low remember one over x is a dot over a so a dot over a is very low and so the  
[00:20:38] the value that the planner attaches the idea value that the planner attaches to people is very low so certainly at 2000 the planner wants more kids than the individual but since the individual wants negative population growth  
[00:20:51] if if the planner wants insufficiently high numbers of kids even the planner can have negative population growth right once you've got negative population growth remember the dynamics of x take you off to the right because a is you know asymptoting to a constant  
[00:21:06] and n is falling and so ideas per person continues to increase and that's what takes us to the empty planet steady state so the interesting thing here is that if society delays  
[00:21:18] adopting the good policies so if we stay on the equilibrium for too long if we wait too long to adopt the good policies even the planner can get trapped by the empty planet result and that's  
[00:21:31] really um where i want to end up so so to conclude you know fertility matters a lot more uh than i thought in these growth models right so fertility rates have been trending downward from an individual standpoint there's nothing special about above two or below two  
[00:21:45] but critically from the from the growth standpoint this distinction matters a lot if we've got negative population growth we may get stuck in this empty planet result where we have the end of economic growth we have living standards that stagnate  
[00:21:59] for a population that disappears a population that vanishes and if instead fertility were just a little bit higher we could have this wildly different result of exponential growth in living standards for an exponentially growing  
[00:22:12] number of people that you know fills those the cosmos potentially right and again the surprise is that even the planner can get trapped by this empty planet result if we wait too long um okay so uh that's it i hope you enjoyed it i  
[00:22:26] hope you'll uh uh if you're interested you can go to my webpage and and see more in in the paper thanks very much