Understanding the New Normal: The Role of Demographics

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Understanding the New Normal: The Role of Demographics Etienne Gagnon, Benjamin K. Johannsen, and David Lopez-Salido 2016-080 Please cite this paper as: Gagnon, Etienne, Benjamin K. Johannsen, and David Lopez-Salido (2016). “Understanding the New Normal: The Role of Demographics,” Finance and Economics Discussion Series 2016-080. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2016.080. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Understanding the New Normal: The Role of Demographics∗ Etienne Gagnon Benjamin K. Johannsen David Lo´pez-Salido October 3, 2016 Abstract Since the Great Recession, the U.S. economy has experienced low real GDP growth and low real interest rates, including for long maturities. We show that these developments were largely predictable by calibrating an overlapping-generation model with a rich demographic structure to observed and projected changes in U.S. population, family composition, life expectancy, and labor market activity. The model accounts for a 1¼–percentage-point decline in both real GDP growthandtheequilibriumrealinterestratesince1980—essentiallyallofthepermanentdeclines in those variables according to some estimates. The model also implies that these declines were especially pronounced over the past decade or so because of demographic factors most-directly associated with the post-war baby boom and the passing of the information technology boom. Our results further suggest that real GDP growth and real interest rates will remain low in coming decades, consistent with the U.S. economy having reached a “new normal.” JEL Codes: E17, E21, J11. Keywords: new normal, equilibrium real interest rate, GDP growth, demographics. ∗The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or any other person associated with the Federal Reserve System. We are grateful to Carter Bryson and Kathryn Holston for excellent research assistance, and to seminar participants at the Bank of Canada, the Einaudi Institute for Economics and Finance, and the Board of Governors for helpful feedback. Comments and suggestions can be directed to etienne.gagnon@frb.gov, benjamin.k.johannsen@frb.gov, or david.lopez-salido@frb.gov. 1

1 Introduction To support the economic recovery from the Great Recession, the Federal Reserve held the federal funds rate near zero for over seven years and acquired large holdings of longer-term securities. Despite these extraordinary measures, real GDP has grown at only a modest pace during the recovery. Meanwhile, long-term interest rates have declined whereas measures of longer-term inflation expectations have been relatively stable, resulting in lower real long-term interest rates, as shown in Figure 1. Some observers, such as Rogoff (2015), trace these developments to persistent, but ultimately transitory, debt deleveraging and borrowing headwinds in the wake of the global financial crisis. Others, like Summers (2014) and Eggertsson and Mehrotra (2014), see these developments as symptomatic of “secular stagnation,” by which a confluence of changes in the structure or functioning of the economy is leading to weak GDP growth and low interest rates of an enduring rather than cyclical nature.1 Commentators and policymakers have described this combination of low growth and low interest rates as a “new normal” for the U.S. economy.2 This paper seeks to understand how much of the new normal can be explained by demographic factors in the United States. The United States, like other advanced economies, is undergoing a dramatic demographic transition related to the unfolding of the post-war baby boom.3 As a consequence, the growth rate of the labor force has declined and should remain low for the foreseeable future.4 In this paper, we investigate the extent to which demographic changes, especially those related to the baby boom, canexplainthecurrentlylowlevelsofrealinterestratesandGDPgrowth. Webuildanoverlapping generation (OG) model that is consistent with observed and projected changes in fertility, labor supply, life expectancy, family composition, and international migration. The model allows us to explore the extent to which demographic changes, in and of themselves, can explain the timing and 1Theexpression“secularstagnation”waspopularizedduringtheGreatDepressionbyHansen(1939),whowrote: “[this] is the essence of secular stagnation—sick recoveries which die in their infancy and depressions which feed on themselves and leave a hard and seemingly immovable core of unemployment.” One potential contributor to secular stagnationinthe21stcenturyisastepdownintechnologygrowth,withauthorssuchasFernald(2015)andGordon (2016)arguingthatthehighproductivitygainsenjoyedbeforetheearly1970sandduringtheinformationtechnology boom of the mid-1990s to early 2000s are unlikely to be repeated. 2For an early use of the expression “new normal,” see El-Erian (2010). 3The baby-boom generation consists of the cohorts born after World War II through the mid-1960s; see Hogan et al. (2008) for a statistical definition. 4ColbyandOrtma(2015)describethesalientfeaturesoftheU.S.demographictransition. Aaronsonetal.(2014) discuss the effects of the demographic transition on recent and future movements in the aggregate labor supply. 2

magnitude of movements in real interest rates and real GDP growth during the post-war period and beyond. Because we are interested in longer-term trends, we do not attempt to model business cycle variation, abstract from nominal rigidities, and posit that factor markets are perfectly competitive. We also assume no intentional bequest motives for our households; without such motives, the finite lifespan of households allows demographic transitions to influence real interest rates. By contrast, in the standard real business cycle (RBC) model in which households have infinite planning horizons, demographic transitions may have no such effect on real interest rates. We find that demographic factors alone can account for a 11–percentage-point decline in the 4 equilibrium real interest rate in the model since 1980—much, if not all, of the permanent decline in real interest rates over that period according to some recent time-series estimates, such as Johannsen and Mertens (2016b) and Holston et al. (2016). The model is also consistent with demographics having lowered real GDP growth 11 percentage points since 1980, primarily through 4 lower growth in the labor supply; this decline is in line with changes in estimates of the trend of GDP growth over that period. Interestingly, the model also implies that these declines have been most pronounced since the early 2000s, so that downward pressures on interest rates and GDP growth due to demographics could be easily misinterpreted as persistent but ultimately temporary influences of the global financial crisis. When we enrich our model with total factor productivity (TFP) growth consistent with the observed historical trend, the predicted declines in GDP growth and the equilibrium real rate between 1980 and the present are essentially unchanged but the pace at which the economy has been transitioning to the new normal over the past decade is more dramatic. Inthemodel, thedynamicsofinterestratesandGDPgrowtharemostdirectlyconnectedtothe consequencesofthepost-warbabyboom. Asthebaby-boomgenerationreachedworkingageinthe 1960s, the aggregate labor supply, GDP growth, and interest rates all increased. The abundance of labor was accentuated by the fact that the baby boomers had far fewer children than their parents did, leading the United States to reap a “demographic dividend” by which the number of workers relative to the number of dependents climbed to historically high levels by the turn of the 21st century. The demographic situation stimulated aggregate capital formation as members of an unusually large cohort with few dependents simultaneously supplied labor and aimed to save ahead of retirement. The low fertility rates of the baby-boom generation also supported the accumulation 3

of capital by facilitating greater labor-force participation of women and by freeing resources that families would have otherwise allocated to consumption by children. Furthermore, steady gains in health and life expectancy have increased the amount of time households expect to spend in retirement, in turn boosting their desire to save. The baby-boom generation has since begun to retire and the growth rates of the aggregate labor supply and aggregate output have accordingly slowed. From our model’s perspective, these factors have led to a current abundance of capital relative to labor, depressing the return on capital and also causing aggregate investment to decline, a phenomenon that we see as consistent with puzzlingly low rates of capital investment in the recovery from the global financial crisis. Going forward, the model predicts that the capital-labor ratio will remain elevated despite low rates of aggregate investment in capital because the growth rate of the labor supply will also below, so that both realinterest rates andGDP growth will linger near their current low levels. One remarkable feature of our model is that the macroeconomic effects of the demographic transition have been largely predictable. In making this point, we use the model to show that the most consequential demographic changes for interest rates and GDP growth today took place before the 1980s. Had demographic variables held constant at their 1960 levels, both real interest rates and GDP growth today would be similar to their levels in 1980. In sharp contrast, had demographic variables held constant at their 1980 levels, the declines in the equilibrium real rate and real GDP growth would have been largely as predicted under the baseline calibration of the demographic variables. We also use these simulations to parse out the relative contributions of each of the demographic factors to those declines. In the case of real GDP growth, the sharp drop in fertility rates in the 1960s and 1970s, which ultimately led to a slowing in the growth rate of the labor supply, is the predominant contributor to the decline since 1980. In the case of real interest rates, we find that falling fertility rates and increased employment rates in the 1960s and 1970s have each contributed ½ percentage point to the decline in the equilibrium real rate between 1980 and the present, with longer life expectancies accounting for the residual. Taken together, these and our earlier results show that demographic factors tend to reinforce each other in the transition to a new normal for the U.S. economy. In an earlier contribution, Krueger and Ludwig (2007) use an OG model to explore the consequences of demographic transition in advanced economies for capital accumulation and rates of 4

return from 2005 to 2080. Our study focuses on the United States and traces the extent to which historical demographic changes can account for both past and future changes in interest rates and GDP growth as well as a seemingly rapid transition to the new normal. Our results also complement those of Ikeda and Saito (2014) for Japan and Carvalho et al. (2016) for the OECD. These studies focus on the distinction between working-age adults and retired adults, and show that a rise in the proportion of retired adults lowers real interest rates. Our model is richer, taking into account differences in demographic characteristics over the life time and across birth cohorts. We argue that the consideration of a rich model matters for obtaining reliable empirical estimates of the size and timing of the macroeconomic effects of demographics. In particular, the model of Carvalho et al. (2016) predicts a sharp decline in the equilibrium real interest rate in the 1990s driven by increased life expectancy, which we show, within the context of our larger model, has had relatively little effect on interest rates over the past several decades. The paper is organized as follows. Section 2 presents our model economy. Section 3 details the demographic data and other parameters of our calibration, as well as our approach to computing the model’s dynamic general equilibrium. Section 4 shows the main results under our baseline calibration and gauges the relative importance of demographic factors through the consideration of a number of alternative scenarios. Section 5 offers a series of robustness checks and observations pertainingtoourchoiceofatechnologygrowthprocess, theelasticityofsubstitutioninproduction, the role of dependents in household preferences, and open-economy considerations. Finally, section 6 concludes. 2 Model economy OurOGmodelborrowsfromthevastliteratureexploringthelong-runfiscalandmacroeconomic implications of demographic transitions and social security programs. See, among many related contributions, Yaari (1965), Fehr et al. (2003), or, more recently, Reichling and Smetters (2015). Theeconomyiscomprisedofhouseholdsandrepresentativefirms. Householdsareheadedbyadults who are representative of their birth cohort in terms of life-cycle characteristics. Each period, these adults inelastically supply their labor endowment, which varies over the life cycle. Additionally, these adults decide how much they and their children who have yet to reach adulthood consume, and how much capital to save for retirement. Representative firms operate a constant returns to 5

scale production technology that uses capital and labor inputs. Given our focus on low-frequency dynamics, we abstract from business cycle shocks and aggregate uncertainty. Instead, we assume that adults and firms have perfect foresight with respect to aggregate and demographic variables, and that they face no idiosyncratic risk besides the uninsurable risk of death. The timing of events is as follows: At the beginning of each period, children are born with no capital holdings according to fertility rates that depend on the age and birth cohort of the adults. After children are born, some household members die, with probabilities conditional on age and birth cohort, and orphans are redistributed among surviving adults belonging to the same cohort as the deceased parent. The capital held by dead adults is then uniformly redistributed to surviving adults. Finally, surviving adults make their consumption/saving decisions and firms produce. 2.1 Households We assume that adults derive utility from their children’s consumption as in the recent work of Curtis et al. (2015a) and Curtis et al. (2015b). Consider a cohort-representative adult aged a in period t. This adult maximizes A (cid:88) −a βsΓ (cid:32) (C a+s,t+s )1−ν +(cid:15)(n )η (cid:0) C a c +s,t+s (cid:1)1−ν(cid:33) , (1) a,t,t+s a+s,t+s 1−ν 1−ν s=0 where C is the adult’s consumption, Cc is the average consumption of each of her dependent a,t a,t children, and n is the number of her dependent children. We allow the number of dependent a,t children to be fractional. That is, n need not be an integer value. Allowing for a fractional a+s,t+s number of dependent children allows us to match population statistics without sacrificing the assumptionofrepresentativeadultcohorts. Theparameter(cid:15) ∈ [0,1]controlstheweightthatadults place on their children’s consumption, the parameter η ∈ [0,1] captures the effects of family size on household utility, and the parameter ν captures the curvature of the preferences (it corresponds to the inverse of the elasticity of intertemporal substitution). Our specification implies that adults derive utility only in periods in which they are alive and that they have no intentional bequest motive. The probability that an adult aged a in period t survives through the end of the next s periods is given by the cohort-specific function Γ . We a,t,t+s assume that no adult survives past some finite length of life, A. Households face the following 6

sequence of budget constraints C +n Cc +K = (K +φΞ ) (cid:0) RK +1−δ (cid:1) +e W . a+s,t+s a+s,t+s a+s,t+s a+s+1,t+s+1 a+s,t+s t+s t+s a+s,t+s t+s (2) Here, K is the age a adult’s capital holdings at the beginning of the period, e is the age a a,t a,t adult’s labor endowment, RK is the real rental rate of capital, and W is the real wage rate. We t t assume that only a fraction φ ∈ [0,1] of the capital held by adults who die, Ξ , at the beginning t of the period is redistributed among surviving adults, with the remainder being lost. In short, the budget constraint states that family consumption and net investment (in capital units) cannot exceed the adult’s income from renting her capital units (both previously saved and inherited) and supplying labor. We assume that adults receive a labor endowment in each period so the total amount of labor supplied is exogenous in the model. Adults optimally choose consumption per dependent child be proportional to their own consumption so that Cc (cid:16) (cid:17)1 a+s,t+s = (cid:15)nη−1 ν . (3) C a+s,t+s a+s,t+s The assumed utility function is flexible in that the factor of proportionality can be scaled by the parameter (cid:15) and can exhibit curvature in the number of children through the choice of the parameter η. This proportionality allows us to express the household objective function in terms of adult consumption only, A (cid:88) −a βsΓ κ (C a+s,t+s )1−ν , (4) a,t,t+s a+s,t+s 1−ν s=0 where κ a+s,t+s ≡ (cid:16) 1+(cid:15)ν 1 (n a+s,t+s )1+η− ν 1(cid:17) . (5) Adults take fertility and mortality rates as exogenous. So, Γ and κ are taken as given a,t,t+s a+s,t+s by the household. Thus, the optimization problem of representative adults can be written so that they maximize Equation (4) subject to a sequence of budget constraints given by κ C + K = (K +φΞ ) (cid:0) RK +1−δ (cid:1) + e W . (6) a+s,t+s a+s,t+s a+s+1,t+s+1 a+s,t+s t+s t+s a+s,t+s t+s 7

Total household consumption for a household headed by an adult aged a is given by Ch = C + a,t a,t n Cc , and the inter-temporal Euler equation of the adult can be written as a,t a,t (cid:32) (cid:33)ν Ch κ a+1,t+1 = βΓ a+1,t+1 (cid:0) 1+RK −δ (cid:1) . (7) Ch a,t,t+1 κ t+1 a,t a,t Equation (7) illustrates how life-cycle elements affect the consumption and saving decisions of the household. The effective discount factor of adults depends on the usual discount factor, β, adjusted by the probability of surviving to enjoy consumption in the next period, Γ . In a,t,t+1 the absence of a bequest motive, adults therefore become increasingly impatient as they approach the end of their lives and the probability of dying rises. To the extent that the probability of surviving is close to 1 in the early decades of adulthood, consumption growth is fairly steady because the young are relatively patient.5 The term κ /κ , is increasing with the projected a+1,t+1 a,t increase in family size, making the household act as if it were relatively patient. As an illustration, consider a situation in which the number of children is projected to increase between the current and next periods. In this case, the household would like to transfer resources to the next period, so as to smooth adult consumption and to provide for a larger number of dependent children. This mechanism helps produce hump-shaped consumption profiles over the life cycle of a typical household with dependent children. Thus, as was the case with mortality, changes in family size over time enter the household decision directly through their effects on the effective rate at which households discount future household consumption. 2.2 Firms The representative firm rents capital and labor inputs in perfectly competitive factor markets. It uses those factors in a constant-elasticity-of-substitution (CES) production function to produce final output, Y . The functional form of the production technology is given by t Y t = (α(K t )ρ+(1−α)(Z t L t )ρ)ρ 1 , (8) in which the elasticity of substitution between capital and labor in production is given by 1/(1−ρ), 5Consistentwiththisprediction,BrowningandEjrnæs(2009)provideevidencethattwo-adulthouseholdswithout children—forwhichhouseholdconsumptionroughlyequalsadultconsumption—havefairlyflatconsumptionprofiles over their midlife. 8

and Z is the level of labor-augmenting technology. Firm are atomistic and rent capital and hire t labor in perfectly competitive factor markets so as to maximize per-period profits, which are given by (α(K t )ρ+(1−α)(Z t L t )ρ)ρ 1 −L t W t −R t KK t . (9) Perfect competition and constant returns to scale imply that the representative firm pays a rental rate on capital equal to the marginal product of capital, (cid:18) Y (cid:19)1−ρ RK = α t , (10) t K t and a real wage rate equal to the marginal product of labor, (cid:18) Y (cid:19)1−ρ t W = Z (1−α) . (11) t t Z L t t 2.3 International migration, aggregation, and market clearing Inadditiontobirthsanddeaths,weallowthepopulationtochangethroughinternationalmigration, afeaturethathelpsustoensurethat,inoursimulations,thesizeandcompositionofthepopulation matches official estimates and projections of the U.S. population. At the beginning of each period, a measure m of individuals aged a enters the economy a,t on net. For simplicity, we assume that these immigrants (or emigrants, if m is negative) are a,t identical in terms of asset holdings, labor endowments, and demographic characteristics to the resident population of age a.6 Let µ be the number of individuals aged a at the beginning of period t after accounting for a,t net migration and births, but before deaths have occurred. Given our assumptions, this variable evolves according to µ = µ Γ + m for all a > 0, and µ = births + m for a,t a−1,t−1 a−1,t−1 a,t 0,t t 0,t newborns. Equilibrium in the labor market requires that A (cid:88) L = µ Γ e , (12) t a,t a,t,t+1 a,t a=Ac+1 that is, the economy-wide labor supply is equal to the total labor endowment of surviving adults. 6This assumption echoes a similar approach by Fehr et al. (2003). 9

Equilibrium in capital markets requires A A (cid:88) (cid:88) K = µ Γ K +φ µ (1−Γ )K . (13) t a,t a,t,t+1 a,t a,t a,t,t+1 a,t a=Ac+1 a=Ac+1 Theaggregatecapitalstockiscomposedoftwosummations. Thefirstsummationisthecapitalthat surviving adults brought into the period. The second summation is the redistributed capital from adultswhodieintheperiod. Finally,goods-marketclearingrequiresthatproductionbeequaltothe sum of household consumption and gross investment aggregated across the cohort-representative adults, A A (cid:88) (cid:88) Y = µ γ Ch + µ (K −(1−δ)(K +φΞ )). (14) t a,t a,t a,t a,t t+1,a t,a t a=Ac+1 a=Ac+1 Importantly,thepresenceofimmigrationdoesnotchangetheoptimalityconditionsofthehousehold as the effects of immigration will show up in prices, which the household takes as given. 2.4 Solution method We consider a perfect-foresight equilibrium such that, in each period, (a) goods and factor markets clear, (b) factor prices correspond to their marginal products, (c) the bequests received by surviving adults sum up to the capital held by adults who passed away at the beginning of the period (net of any destroyed capital), and (d) household optimality conditions are satisfied. While uncertainty may influence household decision making, we are interested in the low frequency effects of demographic changes, which are predictable far in advance. Thus, we view the perfect foresight assumption as an innocuous abstraction for the questions addressed by this paper. We describe our solution method to find the perfect-foresight general equilibrium in the appendix. In short, we conjecture the paths of aggregate variables over the simulation period and solve the problem of the cohort-representative adults. The solution to this problem is conditional on those paths and on the household’s demographic variables (namely the mortality rates, fertility rates, and labor endowments conditional on age and birth cohort), net migration, and the initial conditions at the start of the simulation period in 1900. We next compute new paths for the aggregate objects by aggregating across the individual investment, saving, and labor supply decisions 10

of the cohort-representative adults and by imposing the first-order conditions of the representative firms. If these paths differ from our initial guesses, then we update the paths and iterate an additional time. Once we have reached numerical convergence, our search for the equilibrium stops. Because the equilibrium is conditional on an initial distribution of capital holdings and population at the start of 1900, all aggregate objects, including the equilibrium real rate, depend on these distributions. However, 1900 is sufficiently far in the past that reasonable choices of the initial distributions have little influence on equilibrium outcomes in the post-war period. The solution method is sufficiently rapid that we solve the model at a quarterly frequency. 3 Model calibration and demographic data We take as inputs to our model demographic data, as well as labor-market participation rates. In particular, we use data on mortality to calibrate Γ for each age-specific cohort. We use t,t,t+a population data to calibrate the number of dependent children per adult, n , for each represena,t tative adult aged a at each time t. These data pin down the effective discount rate in Equation (7) for each household aged a at every date. Furthermore, the data on population ensure that the correct proportions of old and young households obtain at every point in time, so that the rental rate of capital and real interest rates reflect the relative supply of savers and borrowers. Finally, we use labor market data to ensure that the supply of labor by age and over time is consistent with the data. In this section, we describe the historical demographic data that we utilize to calibrate our model as well as the projections we utilize to calculate household expectations about the future. We also discuss the calibration of the model’s parameters. A key feature of our calibration is that the economy eventually converges to a balanced growth path, which facilitates the computation of a solution. The simulation period runs from 1900 to 2400; such a long period recognizes that lifecycledecisionsatanypointintimetakeintoaccounttheevolutionoffactorpricesanddemographic variables over both the past and the next several decades. 3.1 Mortality Our individual life expectancy data are derived from mortality rate estimates and projections reported by Bell and Miller (2005) from the U.S. Social Security Administration. These estimates 11

and projections cover every decade from 1900 to 2100 and are reported separately for men and women. We aggregate mortality rates by age and birth cohort across genders in proportion to the number of surviving males and females up to the period. We interpolate mortality rates conditional on age and cohort within inter-census periods and age groups using a cubic spline. In our applications, we assume that the mortality rates in 1900 prevailed in all earlier periods and, similarly, that the mortality rates in 2100 prevail in all subsequent periods.7 Figure 2 shows a dramatic drop in infant mortality in the early 20th century, along with a realized and projected steadyincreaseinadultlifeexpectancy. Themostdramaticgainsinlifeexpectancywereapparently achieved early in the 20th century. 3.2 Fertility and household composition To determine family composition over time, we combine our data on mortality rates with data on the number of live births, female fertility, and yearly infant population estimates. Data on live births are published by the National Center for Health Statistics (NCHS) starting in 1972.8 Prior to that year, we construct live birth estimates by adjusting infant population estimates from the Census Bureau for mortality. We note that, in recent decades, yearly observations from the NCHS series essentially coincide with corresponding five-year data from the United Nations’ World Population Prospects (WPP): 2015 Revision. We use the WPP data, which extend through 2100, toconstructbirth-rateestimatesbeyond2014. Ofnote, bytheendofthecurrentcentury, theWPP data project that live birth rates have essentially stabilized. Using this observation, we impose that the number of live births is constant from 2100 onward. Further details regarding the construction of fertility rates are available in our appendix. We obtain a measure of the number of dependent children per representative adult in two steps. We first estimate the number of births per mother of a given age. The main data source is a set of fertility tables published by Hamilton and Cosgrove (2010) from the Center for Disease Control (CDC). The tables provide estimates of the number of live births in each calendar year conditional 7Thecohortbornin1900hadalowerprobabilityofdeathatallagesthanpersonsbornearlierinthe19thcentury. Forexample,theprobabilityofdyingatage65conditionalonhavingsurvivedtoage64was3.6percentforthecohort of men born in 1900, whereas 4.2 percent of men who turned 65 years in 1900 died that year. The corresponding gap for women is similar, suggesting that the steady fall in mortality rates around that time was not primarily due to lingering consequences of the American Civil War. 8WeuseannualestimatesoflivebirthsfromNCHS.TheNCHSalsomakesavailablemonthlydataonlivebirths, however these data contain apparent seasonal patterns. 12

on the age (in years) of the mother. We then adjust the age of the mother by an estimate of the age difference between married men and women to obtain the age of the representative parent. This adjustment is based on our interpolation of historical information on the median ages of men and women at time of their first marriage published by the Census Bureau. The data are available from 1890 through 2015. Figure 4 shows that the median age gap between men and women has been somewhat stable over the past several decades, at between 1.5 and 2.5 years. In our application, we assume that the average age difference observed in recent years will prevail beyond 2015. We then fit a polynomial to the historical and projected values to obtain a smooth series. A question arises with teen pregnancies because young parents may still be dependent children themselves. In the absence of detailed information, we follow a somewhat simple procedure. Whenever a child is born to a representative parent whose average age is less than 18 years old, we temporarily allocate the child to her representative grandparent until the representative parent turns18yearsold, atwhichpointtherepresentativeparentbecomesfullyresponsibleforherchild’s consumption. With all of the above fertility data we can allocate children at the end of 1899 to their parents andcalculatetheaveragenumberofdependentsenteringtheparents’consumptionfunction. Figure 5 shows the share of births by age of the mother. We then iteratively calculate the population and family structure over the full sample period by letting the population evolve according to births, deaths, and net migration. Finally, we note that our procedure works only with averages rather than actual family cells. For example, deaths of adults and children do not result in discrete jumps inthenumberofsurvivingparentsordependents. Figure6showstheaveragenumberofdependent children by the age of their parents for a number of birth cohorts. Throughoutthepaper, weassumethatadultsaged18yearsormoremakealllife-cycledecisions for themselves and their dependents. Once a child turns 18 years old, he or she begins life as a decision-maker. 3.3 Employment rates Wecalibratethelaborendowmentsofadultsinthemodelusingdataontheemployment-population ratios (EPRs) by age groups published by the BLS since 1948. We complement these data with 13

a forecast of the EPRs for 2024 based on sectoral employment projections published by the BLS. Given that we do not model labor force participation, we interchangeably refer to the EPRs as the “employment rates,” although our EPR measures do not correspond to official employment rates, which are conditional on labor force participation and thus less suited for measuring the aggregate labor supply. We assume that employment rates by age before 1948 and after 2024 correspond to the employment rates by age in those years. Cyclical fluctuations in employment rates are evident in the raw data; given our focus on explaining longer-term trends in interest rates and GDP growth, we use filtered versions of the published series that strip out these movements.9 We set the employment rates of people aged less than 18 years to zero so as to conform with the model’s treatment of dependent children. Figure 7 shows our fitted employment rates by age. A few patterns are apparent. Most notably, employment rates have risen overall since the 1950s, a consequence of women participating in the laborforceingreaternumberssoastomorethanoffsetcorrespondingdeclinesinmaleemployment. Theemploymentratesrisesteeplyoveranindividual’slateteensandearly20s, butthenplateaufor the next few decades. Some dynamics are also apparent around the time of retirement. In the early 1960s,workerstypicallyremainedactiveinthelabormarketuntillaterinlifethaninthesubsequent decades. Theemploymentratesofolderworkersinitiallydeclinedforacoupleofdecades,atendency that subsequently reversed. Our projections are consistent with a continuation of the trend toward increased participation of older workers in the labor force in the coming decade.10 3.4 Other parameters Most of the remaining parameters of the model pertain to the representative adult’s utility function and to the representative firm’s production function. We set the elasticity of substitution in the utility function to unity (ν = 1). We follow Curtis et al. (2015b,a) in setting η = 0.76 and (cid:15) = 0.65 to relate the consumption of adults with that of their dependent children. When we consider a version of the model without dependent children, we set (cid:15) = 0. In our baseline model, 9To extract the trends in quarterly EPR series, we use the Hodrick–Prescott filter with a Lagrange multiplier of 25,000. This value is much higher than the typical value of 1,600 used in the business cycle literature. However, we found that the typical multiplier is much too low to filter out the cyclical effects of the Great Recession. See our technical appendix for further discussion. 10Thesepatternareconsistentwithseveralandattimesoffsettingfactors,includingincreasedaccesstopublicand private savings and pension funds, improved health and access to health insurance, and longer life expectancy. For a recent review of the relevant literature on the labor supply of older workers, see Coile et al. (2016). 14

we set the elasticity of substitution in the production function to 1, so that the function is of the Cobb-Douglas form, and set the share of capital, α, to 0.35. We also set φ = 1 so that no capital is destroyed in the inheritance process. To isolate the effects of demographic factors on the aggregate objects of interest, our baseline calibration features no technology growth (that is, Z /Z = 1). In some simulations, we will t t−1 augment the model with a TFP series matching low-frequency movements in productivity growth torefineouranalysisofthetransitiontothenewnormal. OurmainTFPdatasourceisanupdated version of the data from Fernald (2014), whose data sample runs from 1947:Q2 to 2016:Q1. Before 1947, we assume that TFP grew at an annual rate of 2 percent, which is in the middle of the range of estimates surveyed by Shackleton (2013). After 2016, we assume that TFP is growing at a rate of 1.1 percent, which is consistent with the projections of Gordon (2015). As with employment rates, we filter our TFP series so as to strip out cyclical fluctuations.11 Thelastparameterthatweneedtopindownisthehousehold’sdiscountfactor, β. Wecalibrate this parameter such that the average level in the 1980s of the real short-term interest rate in the model—which we refer to as the equilibrium real rate—matches the average point estimate of the real interest rate expected in the long run calculated by Johannsen and Mertens (2016b) for that decade. We compare the model’s path for the short-term real interest rate to the estimate from Johannsen and Mertens (2016a) of the real interest rate expected in the long run. We see these authors’ estimate as a good benchmark because, like our model, it seeks to capture low-frequency movements in real interest rates that are unrelated to business cycle developments. Moreover, the contouroftheestimateaccordswithlow-frequencymovementsinobservedrealinterestrates, being low in the 1960s, somewhat higher in the 1970s and 1980s, and low again in the recent period. We will return in section 5 to the discussion of our benchmark for the equilibrium real rate. For now, we note that changing β shifts the level of the equilibrium real rate path in our simulations but otherwise has little effect on the contribution of demographic factors to observed changes in real interest rates and real GDP growth over time. 11We again use the Hodrick–Prescott filter with a Lagrange multiplier of 25,000 on quarterly data. 15

4 Main results Our dynamic simulations track the evolution of the U.S. economy from 1900 to 2400. However, our discussion focuses on the implications of the U.S. demographic transition for the recent and next few decades, especially as they relate to a possible new normal for the U.S. economy. One reason to discount the model results for the early decades of the 20th century is that the initialization of the simulation is subject to some uncertainty. Our sample begins only 35 years after the end of the U.S.CivilWar, whichsawthedestructionofasubstantialamountofcapitalandthetragicdeathof 2.5percentofthepopulation—mostlymenwhowouldhavebeenpartofthelaborforce.12 Thefirst halfofthe20thcenturyalsoencompassestwoworldwarsandtheGreatDepression,whoseeconomic consequences are outside of our framework. The uncertainty surrounding demographic projections and future technology growth similarly grows with the time horizon. That said, knowledge of the current distribution of population, mortality rates, birth rates, and immigration rates imply that the evolution of the population over the next few decades is reasonably certain. 4.1 Dynamics of real interest rates Figure 8 shows the model’s dynamic equilibrium real rate over the 1960–2030 period under our baseline specification with dependent children. At a qualitative level, the model generates a path of the equilibrium real interest rate that resembles that of Johannsen and Mertens (2016b) over the period: The equilibrium real rate rises through most of the 1960s and 1970s, reaches a peak around 1980, and falls thereafter. Time-series estimates of the equilibrium real rate are inherently uncertain, as the band in the panel highlights. Nonetheless, we find the similarity in the timing of the rise, peak, and decline in the equilibrium real rate between these authors’ estimates and our baseline estimates rather remarkable. Since 1980, Johannsen and Mertens (2016b) estimate that the long-run equilibrium real rate has fallen 3 percentage point, whereas our model predicts 4 a modestly larger fall, at 1¼ percentage points. Put differently, demographic factors appear to be an economically important contributor to low-frequency movements in the equilibrium real rate, 12Moreover,the1900Censuscountsbyageareratherimprecise,displayingjumpsinthenumberofpersonsreporting that their age in years ends with either a 0 or a 5. The quality of Census data appears to have improved quickly thereafter, making us confident that our key findings are insensitive to reasonable deviations from the initialization of demographic data. 16

possibly explaining all of the permanent declines in real interest rates in recent decades. Asthefigureshows,theresultsarebroadlysimilarwhenweassumethatchildren’sconsumption doesnotenterenterthehouseholdutilityfunction,acommonassumptioninmacroeconomicmodels. Whendependentsareignored,thefallintheequilibriumrealratesince1980occurssomewhatmore gradually. By contrast, our model with dependent children has the steepest declines in real interest rates shortly after the year 2000, and thus predicts that the equilibrium real rate was falling especially rapidly during the Great Recession and its aftermath. Figure 9 offers some guidance to understand the demographic forces that are driving our simulation. The top panel shows that the economy-wide employment rate increased consistently as the baby-boom generation entered the labor market, peaking about the year 2000. The bottom panel shows that the number of dependent children per adult dropped precipitously as the baby-boom generation reached the age of maturity. Additionally, the number of inactive adults relative to employed adults—a measure of the importance of older dependents—similarly reached a turning point around the year 2000 and is projected to increase in the next few decades as the baby-boom generation retires.13 In this sense, the U.S. economy has had a demographic dividend as of late, in that the number of workers relative to the total population has been historically high. 4.2 Dynamics of GDP growth and other variables Figure 10 shows the predictions from our baseline model for real GDP growth. Our baseline model abstracts from productivity growth, yet GDP grows at different rates over time for a variety of reasons. Most importantly, movements in the total number of working-age adults and in their employment rates affect the total labor supply, and thus have direct effects on real GDP growth. Demographic factors also affect the accumulation of capital, although the corresponding contribution to movements in GDP growth is small. Our model predicts that demographic factors caused real GDP growth to rise about a percentage point from 1960 to 1980. From 1980 to 2015, the transition toward lower growth in the labor supply erased that increase: Initially, the decline was modest but picked up around the year 2000. Figure 11 provides information about the effects of the demographic changes in our data on some economic variables. By construction, the real rental rate of capital in our model (shown 13We calculate this ratio as (1−e¯)/e¯, where e¯ is the average employment rate of adults in the sample. t t t 17

in the upper-left panel) mimics movements in the equilibrium real rate. The real rental rate of capital increases as the baby-boom generation enters the labor force and capital is relatively scarce. The net saving rate follows a similar pattern as the rental rate of capital in that it peaks around 1980 and declines sharply after the year 2000. Changes in the net saving rate reflect the desire of the baby-boom generation to save for retirement. After these workers leave the labor force, they draw down their capital, depressing net saving rates. When we include dependent children in our model, the contour of the net saving rate is somewhat changed from 1980 through 2010, reflecting the resources that the baby-boom generation allocates toward its dependent children. However, these differences are small relative to the effect of retirement saving and spending by baby-boom generation. During the period in which the proportion of adults active in the labor market is at its highest, the capital-labor ratio is relatively low, even though net savings is high (as shown in the lower-left panel). The capital-labor ratio rises even as the net saving rate falls because the retirement of workers from the baby-boom generation puts downward pressure on the growth rate of the labor supply. The real wage (measured in efficiency units, and shown in the bottom-right panel) remained low and falls through most of the 1960s and 1970s, reaching a trough in the 1980s, before steadily rising. The movements in the real wage are much smaller than the movements in the capital-labor ratioandreflecttherelativescarcityoffactorsofproduction; thatis,laborisrelativelyscarceinour model after workers in the baby-boom generation begin to retire. Even though the net saving rate is declining after the year 2000, the model predicts that the equilibrium real interest rate is falling because the capital-labor ratio is dramatically increasing. This observation cautions that simple dynamic analyses that ignore the rich variation in demographic factors and composition of the population might draw incorrect conclusions. Moreover, it suggests that demographic factors offer an explanation for puzzlingly weak business investment in the wake of the Great Recession, which some authors, such as Banerjee et al. (2015), have instead traced back to persistent uncertainty and limited investment opportunities. 4.3 Parsing the effects of fertility, mortality, and employment rates To assess the respective contributions of changes in fertility, mortality, and employment rates to macroeconomic developments since 1980, we perform alternative simulations in which we freeze 18

demographic variables from 1960 onward.14 We use the model with dependent children as our baselinemodelandwekeepβ fixedacrosssimulationssoastoisolatetheeffectsofthedemographic data. We present the counter-factual paths of the equilibrium real rate in the top panel of Figure 12. When we keep fertility, mortality, and employment rates all fixed at their 1960 values, we see that there would have been no net decline in the equilibrium real interest rate since 1980. Put differently, the entirety of the decline in the equilibrium real interest rate that our model finds for the recent decades is a direct consequence of the demographic changes that happened from 1960 onward. To analyze the respective contribution of each demographic factor, we consider the effects of freezingeachdemographicvariableinturn.15 Thetwomostconsequentialdemographicchangesare thedropinfertilityratesandtheincreaseinemploymentrates,whicheachcontributes½percentage point to the 1¼–percentage-point decline in the equilibrium real rate between 1980 and the present. Intuitively, the demographic dividend stemming from the drop in fertility rates in the 1960s and 1970s facilitated the accumulation of capital in aggregate. Moreover, the retirement of a relatively large cohort of baby boomers has further put upward pressure on the aggregate capital-labor ratio of late, leading to a fall in the equilibrium real interest rate. Similarly, employment rates were relatively low in 1960, as we showed in Figure 7. Since that time, the number of households with two adults who participate in the labor market has increased dramatically. As such, the aggregatelaborsupplyhasincreasedthroughthelatterpartofthe20thcentury, inturnfacilitating aggregate capital formation and exacerbating the deceleration in the aggregate labor supply once baby boomers began to retire. The contribution of lower mortality rates to the fall in the equilibrium real rate since 1980 is only half as large, at ¼ percentage point, than those of the fall in fertility rates and rise in employment rates. That changes in life expectancy alone have had only a small effect on the equilibrium real interest rate over this period is perhaps unsurprising. To put these changes in 14Thatis,weassumethatapersonagedainanyperiodafter1960:Q1facesthesamemortalityrate,fertilityrate, or employment rate (or combinations of these three variables, depending on the simulation) as someone aged a did in 1960:Q1 under our baseline calibration. The net migration flows by age and period are kept the same as under our baseline calibration. 15The contributions of demographic factors to the fall in the equilibrium real rate since 1980 are roughly additive in the model, a fact that we verified by solving the model under all possible combinations of frozen variables. 19

mortality rates in perspective, we note that the mean life expectancy at birth conditional on the mortality rates observed at the start of 1980, at 79.6 years, was only three and half years less than the corresponding statistic based on the mortality rates prevailing in 2015. By contrast, the total gain in life expectancy over the 1900–2100 period is nearly three and a half decades. Moreover, the gains in life expectancy over the past few decades reflect primarily a fall in mortality rates during an adult’s years in retirement, thus having almost no effect on the aggregate supply of labor. And withyoungercohortsdiscountingtheconsumptionoftheirolderselves,theincentivestoaccumulate more capital during the productive years are little changed either. On net, the aggregate stock of capital rises only modestly, which is consistent with a small fall in the equilibrium real rate at the margin.16 Tonarrowinonthetimingofthedemographicchangesthatweremostimportantforthedecline in the equilibrium real interest rate, we do similar experiments in which we freeze demographic variables starting in 1980. These new counter-factual paths for the equilibrium real interest rate are shown in the bottom panel of Figure 12. It is clear that holding demographic variables fixed at their 1980 levels, either independently or jointly, has essentially no effect on the subsequent path of interest rates predicted by the model. The reason for this result is that, by 1980, fertility rates were close to levels that prevail even today (which can be seen in Figure 6), employment rates had increased markedly but then stabilized (which can be seen in Figure 7), and, as mentioned above, mortality rates have fallen only somewhat modestly. In sum, the subsequent decline in real interest rates due to demographic factors was mostly due to demographic changes that took place before 1980. The subsequent dynamics reflect the life cycle of those demographic changes, meaning the contours of the path of interest rates largely reflects the impact of the baby-boom generation. Figure 13 shows that keeping employment rates or mortality rates fixed at either their 1960 or 1980 levels has little effect on the growth rate of real GDP overall. Instead, the bulk of the fall in real GDP growth of late is accounted for by declines in the aggregate labor supply related to the gradual retirement of the baby-boom generation. As can be seen from Figure 13, fixing fertility rates at their 1960 levels would have kept real GDP growth from falling. However, because 16Ourestimateofthecontributionoflongerlifeexpectancytothefallintheequilibriumrealinterestrateismuch smallerthan foundbyCarvalhoet al.(2016). Thedifference likely reflects thefact that, intheseauthors’ model, all retirees face the same mortality rates independently of their age or asset holdings. By contrast, in our model, the mortality rates are calibrated to the age and cohort of each adult; an increase in longevity primarily affects older adultswhosecapitalholdingshavealreadydeclined,sothattheeffectontheaggregatecapitalstockismorelimited. 20

fertility rates fell, our model predicts that real GDP growth initially increased as members of the baby-boom generation reached working age and then started falling in the 1980s. The model also predictsthatthemostdramaticdeclinesinrealGDPgrowthduetodemographicchangeshappened shortly after the year 2000, again around the time that the baby-boom generation began to retire. 4.4 The effects of productivity growth We add productivity growth to the model to assess its effects on the transition to the new normal for the U.S. economy. Authors such as Summers (2014), Gordon (2016), and Fernald (2016) argue that the rate of potential GDP growth in the future will be modest relative to its average over the post-war period. Changes in TFP have two direct effects in theory. First, because productivity enters the production function, GDP growth should slow if productivity growth slows. Second, real interest rates are proportional to TFP growth in a broad class of models, meaning interest rates should fall if productivity growth falls. In the standard RBC model, the real interest rate is given by (1+∆z)ν r = −1, β where 1+∆z ≡ Z /Z . t t−1 We perform a dynamic simulation of our model in which technology growth matches prominent estimates of its historical and projected trend. Our series captures relatively rapid productivity growth before the early 1970s, a slow down from the 1970s through the early 1990s, a temporary pickup associated with the information technology boom, and finally modest rates of technology growth from the early 2000s onward of the kind envisioned by Gordon (2016). See our appendix for details on the construction of our series. Consistent with our earlier approach, we assume that these developments are anticipated and calibrate the household’s discount factor, β, such that the average real rate in the 1980s corresponds to the average point estimate of Johannsen and Mertens (2016b). Absent this adjustment, the introduction of positive technology growth in the model would raise the path of the equilibrium real rate compared with our baseline specification with no technology growth.17 17The calibration requires a value of β that is slightly above 1. In a standard RBC framework, values of β above 1 would be inconsistent with the existence of an equilibrium with balanced growth because the incentives to save would be so strong that households would want to accumulate an ever increasing amount of (technology-adjusted) capital. Bycontrast,inourOGmodel,themortalityrisklowerstheeffectivediscountfactorsofhouseholdstolevels that ensure the existence of an equilibrium. 21

Figure10showsthatproductivitygrowthshiftsupthemodel’spredictedpathforGDPgrowth. Even though we do not use data on aggregate output, the model’s predictions for real GDP growth are remarkably similar to the growth rate of the Hodrick–Prescott filtered trend in actual output, also shown in Figure 10.18 It is noteworthy how our model captures the size of the decline in real GDP growth from the 1980s to the present. The reason is that, aside from some years of very high productivity growth in the 1990s, data from Fernald (2014) indicate that TFP growth was low even in the 1980s. By contrast, high levels of TFP growth in the 1960s correspond to a larger difference between our baseline calibration and our calibration with TFP growth in terms of real GDP growth. Thus, even with historical movements in TFP, our model predicts that much of the decline in real GDP growth since the 1980s is associated with demographic change. Relative to the baseline case, the boom in productivity growth initiated in the 1990s keeps the equilibrium real rate elevated through much of the first decade of the 2000s. The equilibrium real rate only begins falling in earnest around the onset of the Great Recession, proceeding to drop almost one percentage point in the span of about 10 years to a level similar to that in our baseline simulation. Importantly, the fall in the equilibrium real rate under this scenario is largely concomitantwiththeglobalfinancialcrisis. Assuch, factoringinthetemporarilyhighproductivity gainsofthe1990sandearly2000ssuggestsamaterialriskthatobserverscouldconfusethepersistent but ultimately temporary consequences of the financial crisis with other influences originating from the demographic transition and technology growth cycles. Looking at the whole period shown, Figure 14 points to a rather weak connection between the equilibrium real rate and the pace of technology growth over long horizons. Notably, even though technology growth was elevated by historical standards prior to the early 1970s, and should thus have been associated with a high equilibrium real rate, realized real interest rates were quite low at the time. Accordingly, the figure features a persistently large gap between the equilibrium real rate predicted by the model and the estimates of Johannsen and Mertens (2016b) in the 1960s and 1970s. 18Atabusinesscyclefrequency,onemightexpectthatcalibratingthemodeltomatchdataontheaggregatelabor supplywouldmostlycapturemovementsinaggregateoutputgiventhatthecapitalstockisrelativelyfixedinthenear term. However, at low frequencies, the endogenous dynamics of capital accumulation is also potentially important for explaining movements in the trend of real GDP growth. 22

5 Discussion and robustness Judging the empirical success of our model requires taking a stand on the low-frequency paths of real interest rates and the growth rate of real GDP that the model should seek to capture. We take the change in our Hodrick–Prescott filtered output series as a good estimate of the decline in GDP growth; other low-frequency filters suggest similar contours.19 By contrast, estimates of permanent changes in interest rates are much more uncertain, a point demonstrated in Kiley (2015). Rachel and Smith (2015) seek to explain a 450-basis-point fall in the neutral rate of interest over the past 30 years, a movement that is in line with the fall in real long-term rates over the period. We judge their benchmark to be much too large to reflect the influence of secular factors alone. As we showed in Figure 1, the 1980s were preceded by relatively low but positive real interest rates in the 1960s and early 1970s, and then negative real interest rates in the remainder of the 1970s as inflation picked up. The historically high real interest rates of the 1980s occurred as the Federal Reserve tightened the policy stance to rein in inflation. Unsurprisingly, the estimates of Johannsen and Mertens (2016b), which net out cyclical influences, suggest a significant but much smaller drop since the 1980s. Another reason for our discomfort with the size of the drop in the neutral rate reported by Rachel and Smith (2015) is that some of the explanatory factors in their analysis were alreadyatplaypriortothe1980s; oneimplicationofextendingthedatafartherbackintimeisthat the drop in the neutral rate since, say, the 1960s would be implausibly large to us. Additionally, workby Hamilton etal.(2015)presentsevidencethatiftheequilibriumrealratehasdeclined, then that decline is likely smaller than the magnitude reported by Rachel and Smith (2015). Therefore, we assess that a decline in the equilibrium real interest rate of less than 2 percentage points since 1980 is most consistent with the new normal for the U.S. economy.20 Our results are subject to a number of potential caveats. A particularly important assumption 19That said, we view the typical value of the Hodrick–Prescott filter Lagrange multiplier for quarterly data, at 1,600, as much too low to eliminate all cyclical fluctuations, especially in light of the unusually prolonged recovery from the global financial crisis. 20Alternative estimates of the long-term real interest rate that are popular in economic analysis include the timeseriesmodelof Laubach andWilliams (2003) andthe relatedmodelof Holstonet al.(2016). We seethese estimates as less preferable for our purposes than those of Johannsen and Mertens (2016b) in light of their apparent cyclical sensitivityandthefactthattheyimplyrealequilibriumratesinthe1960sand1970sthatarewellabovetherealized real yields on Treasuries and other risk-free instruments during those decades. The initially elevated estimates in those papers appear to be due to the assumption of a direct relationship between movements in trend productivity growth,whichwashighatthetime,andtheequilibriumrealrate. SeeKiley(2015)foranexplorationofidentification in the Laubach and Williams (2003) model. 23

we have made is that adults have no intentional bequest motive, an assumption that is standard in the OG literature. Relaxing this assumption would likely weaken the size of the decline in the equilibrium real rate since 1980 by facilitating the extent to which economic agents can engage in intertemporal substitution across generations and thus smooth the effects of the baby boom and other demographic elements. By comparison, the effects on real GDP growth would likely be modest because demographics affect the pace of output growth primarily through changes in the labor supply, which is exogenous in our environment. From an empirical perspective, the extent to which intentional bequests matter for the determination of the equilibrium level of the aggregate capital-labor ratio—and thus for equilibrium interest rates—is unclear. Intentional bequests are certainly observed in practice but often come in the form of housing, which Yang (2009) argues has different properties than capital over the life cycle due notably to large transactions costs, the impossibility of fractionating most capital held in the form of housing, and the utility that housing provides to owners. The consideration of housing is beyond the scope of our model. Below, we consider a number of alternative scenarios to assess the sensitivity of our results to other model features. We first consider the possibility that the elasticity of substitution between factorsofproductionislowerthanwehaveassumed. Thisconsiderationismotivatedbythecentral role played by the aggregate capital-labor ratio in the model for explaining movements in the equilibrium real rate, as well as by empirical evidence that the long-run elasticity of substitution is lower than implied by Cobb-Douglas production functions. We next consider an alternative specification of preferences that some authors have argued is especially successful at matching consumption/saving decisions over the life cycle. We conclude with a comparison of the U.S. and foreign demographic transitions. 5.1 Elasticity of substitution in production So far, our analysis has focused on how demographic characteristics affect the supply of labor and capital by households in equilibrium. We now provide a sensitivity analysis of the elasticity of substitution between capital and labor in production, 1/(1−ρ), which affects the demand for factors of production in equilibrium, and thus also plays a role in determining equilibrium factor prices and quantities. In particular, the dynamic of the equilibrium real rate in the model is tightly connected to that of the (technology-adjusted) capital-labor ratio. Using Equations (8) and (10), 24

we can write the equilibrium real rate as r = RK −δ = α (cid:18) α+(1−α) (cid:18) Z t L t (cid:19)ρ(cid:19)1− ρ ρ −δ. t t K t In our baseline calibration, we assume that the representative firm’s production function is of the Cobb-Douglas form, so that the elasticity of substitution between factors of production is one (that is, ρ = 0 and r = α(Z L /K )α−1 −δ). The empirical relevance of this standard assumption is t t t t questionable, withthe surveyofChirinko (2008)suggestinga long-run elasticitythat isappreciably lower than one.21 Inspection of the above equation indicates that, for a given capital-labor ratio, both the level of r and its sensitivity to movements in the capital-labor ratio fall as we lower the t elasticity of substitution from 1. In this robustness check, we decrease the elasticity to 0.75 while still imposing that the parameter α is consistent with a share of capital in aggregate income of 35 percent. We also re-calibrate the value of β such that the model’s average equilibrium real rate in the 1980s matches the point estimate of Johannsen and Mertens (2016b). Figure 14 shows that lowering the elasticity of substitution accentuates the extent of the fall in the equilibrium real rate since 1980 that can be attributed to demographics. The current difference with respect to the baseline case is somewhat small, however, at about 2 tenths of a percentage point. 5.2 Adult-equivalent preferences The predicted peak in household consumption in our baseline model occurs somewhat earlier in the life cycle than it does in the data. As an alternative, we consider a variant of Browning and Ejrnæs (2009) preferences in which total household consumption increases in the age of each of the children, thus postponing the peak in household consumption to when dependent children approach adulthood. These authors’ specification of the utility function is of additional interest 21The evidence reviewed in Chirinko (2008) is consistent with a long-run elasticity somewhere between 0.4 and 0.6, with the preferred value of the more recent study by Chirinko and Mallick (2015) being at the bottom of that range. UnderastandardCESproductionfunctionlikeEquation(8),onecannotlowertheelasticityparameterdown tothatrangewhilesimultaneouslyimposingthattheshareofcapitalinaggregateincomeisalignedwiththatinthe data. In particular, for a capital share of aggregate income s and a balanced-growth real return on capital (net of K depreciation) of r, Equation (8) requires setting α equal to rρ/(s )ρ−1. If we set r such that it conforms with the K estimates of Johannsen and Mertens (2016b) in the 1980s, then the value of α consistent with s =0.35 exceeds one K whenever the elasticity of substitution between capital and labor is assumed to be below 0.72. 25

to us because they show that it can explain the magnitude and timing of the rise and peak in life-cycle consumption without recourse to other causal mechanisms, such as liquidity constraints, precautionary motives, and non-separabilities between consumption and labor supply—features that are also absent in our environment. Household utility is given by A (cid:88) −a βsΓ (cid:0) C a h +s,t+s e−fa+s,t+s (cid:1)1−ν efa+s,t+s. (15) a,t,t+s 1−ν s=0 The function f = δ log(m ) allows for economies of scale in consumption based on the number a,t h a,t of adult-equivalent members of the household, denoted m .22 In simulations not shown, we obtain a,t results similar to our benchmark model when we use this specification of preferences with the same parametrization as Browning and Ejrnæs (2009). 5.3 Open-economy considerations One limitation of our framework is that it does not incorporate international influences on the U.S. equilibrium real rate. Or, there are reasons to expect foreign developments to contribute to the determination of domestic interest rates. Notably, there is a strong correlation across real interest rates worldwide, indicating either common driving factors, strong international spillovers, or both.23 In a similar vein, the global saving glut hypothesis of Bernanke (2005) posits that U.S. interest rates have felt downward pressure from an excess of global savings emanating from Asian and oil-producing economies; this hypothesis challenged the idea that the dynamics of the U.S. current account since the mid-1990s reflected primarily domestic developments. More generally, movements in U.S. financial variables are sometimes traced back to foreign economic and political developments, as has been evident during bouts of shifting global risk appetite in the wake of the financial crisis. While we do not aim to explain the effects of these foreign elements on U.S. real interest rates, it is nonetheless legitimate to ask whether open-economy considerations might 22Although the utility function is specified in terms of total household consumption, we retain the dependence of thehousehold’seffectivediscountfactorintermsofthesurvivalrateofadultsonly. Currentandprojectedmortality rates are rather low for the adult’s child-rearing years, so that Γ remains close to one until late in life when a,t,t+1 household consumption coincides with adult consumption. 23On the correlation in real interest rates across countries, see, among many contributors, Desroches and Francis (2006) and King and Low (2014). 26

affect our estimates of the contribution of demographic factors. A number of elements give us confidence that our finding that demographic factors are an economically important driver of the U.S. equilibrium real interest rate is robust to open-economy considerations. One such element is that the United States and the world’s other advanced economies are undergoing demographic transitions that are similar in terms of magnitude and timing. In the top panel of Figure 15, we show that the step down in U.S. working-age population growth since the 1970s echoes a similar pattern in OECD economies and in the world more broadly. The 2000– 2030 period features an especially sharp slowdown in population growth. In the bottom panel, we show that the U.S. mean life expectancy at birth has been rising in tandem with that in OECD economies. Similar gains in life expectancy at birth are observed when we include both developed and developing economies, although the level is pulled down by still markedly higher death rates in the less-developed countries. Another element is the size and direction of international capital flows. In our model, demographic factors put downward pressure on the return on domestic capital. In an open-economy setting with international capital mobility, a fall in the rate of return on domestic capital would, all else equal, induce domestic investors to seek investment opportunities abroad, resulting in a net capital outflow. To the extent that, in aggregate, U.S. investors have succeeded in finding investment opportunities abroad, our closed-economy estimates of the effect of demographics on the U.S. equilibrium real rate could be overestimated. However, in contrast with the prediction of a capital outflow, the U.S. current account balance has been solidly negative since the early 1980s. A third piece of evidence is that attempts at incorporating demographics in open-economy settings do not suggest economically large differences between closed-economy and open-economy predictions for U.S. real interest rates. In a closely-related study, Krueger and Ludwig (2007) build a multi-country OG model to explore the effects of demographics factors on the rate of return on capital. For their 2005–2080 simulation period, their model projects net declines in the U.S. real interest rate of 86 basispoints under the multi-country version of theirmodel and of 79basis points under the closed-economy version. The difference in point estimates between versions is arguably small and, taken at face value, suggests that foreign demographic developments tend to amplify rather than to dampen the extent to which the U.S. real rate is being pressured downward by demographic developments. Our model without dependent children, which is close to the model 27

consideredbyKruegerandLudwig(2007),predictsthattheequilibriumrealratewillfallby54basis points over their sample period. By comparison, our benchmark model with dependent children predicts a net fall in the equilibrium real rate of 79 basis points. The results of Krueger and Ludwig (2007) present a number of limitations with respect to our questions of interest. One such limitation is that Krueger and Ludwig (2007) focus on projected developments over the course of this century, in contrast with our goal of explaining movements in the equilibrium real rate and other aggregate variables over the past several decades. The initial period in their simulation period—2005—coincides with a period of historically rapid fall in the equilibrium real rate in our model. Importantly for our ability to expand our framework to a multicountry setting, foreign demographic data are generally not as detailed as the U.S. demographic data we use to calibrate our model. Moreover, the time coverage of the foreign demographic data tends to be limited, in turn restricting one’s ability to study phenomena unfolding over several generations. For example, their model does not incorporate dependent children and variation in the labor supply across birth cohorts, which we found to be important for explaining the timing and magnitude of the fall in the equilibrium real rate since 1980. Similarly, their exercise leaves aside non-OECD economies for which there is a dearth of demographic data. Nonetheless, these results and the other evidence presented above, taken together, give us reasonable confidence that demographicdevelopmentshavebeenaneconomicallyimportantdriveroflow-frequencymovements in U.S. real interest rates over the past few decades. 6 Conclusion In this paper, we used a calibrated OG model with a rich demographic structure to investigate the extent to which past and projected changes in U.S. family composition, life expectancy, and the labor supply have had an influence on real interest rates, GDP growth, and other variables. We found that these demographic factors alone account for a 11–percentage-point decline in the 4 natural rate of interest and real GDP growth since 1980—a magnitude that we argue is in line with empiricalestimates. Moreover,themodelisconsistentwiththerateofaggregateinvestmenthaving slowed appreciably over that period. Looking forward, the model suggests that low interest rates, low output growth, and low investment rates are here to stay, suggesting that the U.S. economy 28

has entered a new normal. We believe that our results will be of interest to academics and policymakers alike. The model implies that the transition to the new normal has been especially rapid over the past decade or so because of demographic factors most-directly associated with the post-war baby boom—a phenomenonthatwasfullypredictable—andthepassingeffectsoftheinformationtechnologyboom on productivity growth. The rapidity at which our model economy has been transitioning to a new normal suggests a risk that the permanent effects of demographic factors could be misinterpreted as persistent but ultimately transitory downward pressure on the natural rate of interest and net savings stemming from the global financial crisis. Going forward, our results have implications for assessing fiscal sustainability. Moreover, the persistence of a low equilibrium real interest rate means that the scope to use conventional monetary policy to stimulate the economy during typical cyclical downturns will be more limited than it has been the case in the past for a given inflation target. 29

References Aaronson, S., T. Cajner, B. Fallick, F. Galbis-Reig, C. Smith, and W. Wascher (2014). Labor ForceParticipation: RecentDevelopmentsandFutureProspects. Brookings Papers on Economic Activity 49(2), 197–275. Banerjee, R., J. Kearns, and M. J. Lombardi (2015). (Why) Is Investment Weak? BIS Quarterly Review March. Bell, F. C. and M. L. Miller (2005). Life Tables for the United States Social Security, 1900–2100. Social Security Administration–Office of the Chief Actuary, Actuarial Study (120). Bernanke, B. (2005). The Global Saving Glut and the U.S. Current Account Deficit. Speech delivered at the Sandridge Lecture of the Virginia Association of Economists, Richmond, VA, March 10. Browning, M. and M. Ejrnæs (2009). Consumption and Children. The Review of Economics and Statistics 91(1), 93–111. Carvalho, C., A. Ferrero, and F. Nechio (2016). Demographics and Real Interest Rates: Inspecting the Mechanism. European Economic Review. Chirinko, R. S. (2008). σ: The Long and Short of It. Journal of Macroeconomics 30(2), 671–686. Chirinko, R. S. and D. Mallick (2015). The Substitution Elasticity, Factor Shares, Long-Run Growth, and the Low-Frequency Panel Model. CESifo Working Paper Series (4895). Coile, C., K. S. Milligan, and D. A. Wise (2016). Social Security and Retirement Programs around the World: The Capacity to Work at Older Ages – Introduction and Summary. NBER Working Paper Series No. 21939. Colby, S. L. and J. M. Ortma (2015). Projections of the Size and Composition of the U.S. Population: 2014 to 2060. United States Census: Current Population Reports P25-1143. Curtis, C. C., S. Lugauer, and N. C. Mark (2015a). Demographic Patterns and Household Saving in China. American Economic Journal: Macroeconomics 7(2), 58–94. 30

Curtis,C.C.,S.Lugauer,andN.C.Mark(2015b). DemographicsandAggregateHouseholdSaving in Japan, China, and India. National Bureau of Economic Research Working Paper No. 21555. Desroches, B. and M. Francis (2006). Global Savings, Investment, and World Real Interest Rates. Bank of Canada Review 2006(Winter), 3–17. Eggertsson, G. B. and N. R. Mehrotra (2014). A Model of Secular Stagnation. National Bureau of Economic Research Working Paper No. 20574. El-Erian, M.A.(2010). NavigatingtheNewNormalinIndustrialCountries. Speech, PerJacobsson Foundation Lecture, International Monetary Fund. Fehr, H., G. Halder, S. Jokisch, and L. J. Kotlikoff (2003). A Simulation Model for the Demographic Transition in the OECD: Data Requirements, Model Structure and Calibration. W.E.P. - Wurzburg Economic Papers 45, University of Wurzburg, Chair for Monetary Policy and International Economics. Fernald, J. (2014). A Quarterly, Utilization-Adjusted Series on Total Factor Productivity. Federal Reserve Bank of San Francisco working paper 19. Fernald, J. G. (2015). Productivity and Potential Output before, during, and after the Great Recession. NBER Macroeconomics Annual 29(1), 1–51. Fernald, J. G. (2016). Reassessing Longer-Run US Growth: How Low? SF-FRB Working Papers Series. Gordon, R. J. (2015). Secular Stagnation on the Supply Side: U.S. Productivity Growth in the Long Run. Digiworld Economic Journal (100). Gordon, R. J. (2016). The Rise and Fall of American Growth: The US Standard of Living since the Civil War. Princeton University Press. Hamilton, B. E. and C. M. Cosgrove (2010). Central Birth Rates by Live-Birth Order, Current Age, and Race of Women in Each Cohort from 1911 through 1991: United States, 1960–2005. Table 1. Hyattsville, MD: National Center for Health Statistics. Released: June 30, 2010. 31

Hamilton, J. D., E. S. Harris, J. Hatzius, and K. D. West (2015). The Equilibrium Real Funds Rate: Past, Present and Future. National Bureau of Economic Research Working Paper No. 21476. Hogan, H., D. Perez, and W. R. Bell (2008). Who (Really) Are the First Baby Boomers? Joint Statistical Meetings Proceedings, Social Statistics Section, 1009–1016. Holston, K., T. Laubach, and J. Williams (2016). Measuring the Natural Rate of Interest: International Trends and Determinants. Board of Governors of the Federal Reserve System. Ikeda, D. and M. Saito (2014). The Effects of Demographic Changes on the Real Interest Rate in Japan. Japan and the World Economy 32, 37–48. Johannsen,B.K.andE.Mertens(2016a). ATimeSeriesModelofInterestRatesWiththeEffective Lower Bound. FEDS Working Paper No. 2016–033. Johannsen,B.K.andE.Mertens(2016b). TheExpectedRealInterestRateintheLongRun: Time Series Evidence with the Effective Lower Bound. FEDS Notes. Washington: Board of Governors of the Federal Reserve System, February 9, 2015–077. Kiley, M. T. (2015). What Can the Data Tell Us About the Equilibrium Real Interest Rate? Finance and Economics Discussion Series 2015–77, Board of Governors of the Federal Reserve System (U.S.). King, M. and D. Low (2014). Measuring the ‘World’ Real Interest Rate. National Bureau of Economic Research: Working Paper No. 19887. Krueger, D. and A. Ludwig (2007). On the Consequences of Demographic Change for Rates of Returns to Capital, and the Distribution of Wealth and Welfare. Journal of monetary Economics 54(1), 49–87. Laubach, T. and J. C. Williams (2003). Measuring the Natural Rate of Interest. The Review of Economics and Statistics 85(4), 1063–1070. Rachel, L. and T. Smith (2015). Secular Drivers of the Global Real Interest Rate. Bank of England Working Paper. 32

Reichling, F. and K. Smetters (2015). Optimal Annuitization with Stochastic Mortality and Correlated Medical Costs. American Economic Review 105(11), 3273–3320. Rogoff, K. (2015). Debt Supercycle, Not Secular Stagnation. VOX: CEPR’s Policy Portal, April 22., http://www.voxeu.org/article/debt-supercycle-not-secular-stagnation. Shackleton, R. (2013). Total Factor Productivity Growth in Historical Perspective. Congressional Budget Office, Working paper 1. Summers,L.H.(2014). Reflectionsonthe’NewSecularStagnationHypothesis’. Secularstagnation: Facts, causes and cures, 27–40. Yaari, M. E. (1965). Uncertain Lifetime, Life Insurance, and the Theory of the Consumer. The Review of Economic Studies 32(2), 137–150. Yang, F. (2009). Consumption over the Life Cycle: How Different Is Housing? Review of Economic Dynamics 12(3), 423–443. 33

Figure 1: Expected real return on 10-year Treasury security 15 10 l l l l l 5 ll llll ll l llll l lll ll l l l l l ll lll ll llllll lllll lllll lllllllllllllllll llllllllllllll l l llllllll ll lllllllll lll l llllllllllllllllll 0 1955 1965 1975 1985 1995 2005 2015 Date dezilaunna ,tnecreP 10−year Treasury 10−year realized inflation l MSC expected inflation Source: Authors’ calculation using data from FRED. Notes: The “10-year Treasury” rate is the series GS10 in FRED. The “10-year realized inflation” rate is the annual average inflation rate of core PCE inflation. The “MSC expected inflation” rate is the answer to the question about the expected change in prices in general over the next 5 to 10 years in the Michigan Survey of Consumers. 34

Figure 2: Estimated/Projected survival function by year of birth 100 75 50 25 0 0 20 40 60 80 100 120 Age (in years) tnecreP 1900 1960 1980 2000 2020 Source: Authors’ interpolation of decennial mortality tables by Bell and Miller (2005). 35

Figure 3: Annual U.S. live birth estimates and projections: 1900 to 2030 4 l l lll ll l l l l ll l l l lllll l l l lllll llllllllllllllllll l l lll l l llll l l l l ll ll l l ll l l l ll l l l ll l l lll 3 l l llll l ll l l l l llllllllllll l l l ll l l l l l 2 ll l l l ll ll l ll 1900 1920 1940 1960 1980 2000 2020 Date )snoillim ,launna( shtriB lll Census−consistent NCHS United Nations Interpolated births Source: Authors’ calculations using data from the National Center for Health Statistics (NCHS), the United Nations’ World Population Prospects: Revision 2015 (WPP), and the U.S. Census Bureau. Notes: We construct a historical birth series by stitching yearly estimates from the U.S. Census Bureau for the 1900–1971 period, yearly estimates from NCHS for the 1972–2014 period, and five-year forecasts from the WPP estimates. The solid line shows our interpolation. 36

Figure 4: Median age difference between men and women at time of first marriage l 4.0 l 3.5 l l l 3.0 l l l l l ll l l ll 2.5 ll ll l l l l lll l llllll l l l l ll lll l l l l l ll lllll ll l l l l 2.0 l l l l l lllll ll l l 1.5 1890 1910 1930 1950 1970 1990 2010 2030 Date sraeY l Census estimate Fitted polynomial Source: Authors’ calculations using data from the U.S. Census Bureau. Notes: Our interpolation assumes that the median age difference between men and women at the time of first marriage, averaged in 2013 and 2014, corresponds to the long-run value. 37

Figure 5: Share of births by age of mother (in years) 40 30 20 10 0 1920 1940 1960 1980 2000 2020 Date tnecreP Less than 20 20−24 25−29 30−34 35−39 40 and above Source: Authors’ calculations using U.S. data from Hamilton and Cosgrove (2010). 38

Figure 6: Average number of dependent children by age and cohort of the representative parent 1.5 1.0 0.5 0.0 15 25 35 45 55 65 Age (in years) nerdlihc tnednepeD 1900 1960 1980 2000 2020 Source: Authors’calculationsusingdatafromHamiltonandCosgrove(2010), theNationalCenter for Health Statistics, the United Nations’ World Population Prospects: Revision 2015, and the U.S. Census Bureau. 39

Figure 7: Employment rates by age and calendar year 80 60 40 20 0 0 20 40 60 80 100 120 Age (in years) tnecreP 1960 1980 2000 2020 Source: Authors’ interpolation of historical data and projection for 2024 from the U.S. Bureau of Labor Statistics. Notes: The figure shows U.S. employment rates conditional on age. The historical yearly data begin in 1948. The data shown are an interpolation to a quarters of age performed using cubic splines. 40

Figure 8: Dynamic simulation of the model under baseline demographics: equilibrium real rate 2.5 2.0 1.5 1.0 0.5 0.0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 2030 Date dezilaunna ,tnecreP Dependent children No dependent children J−M posterior mean Source: Johannsen and Mertens (2016b) and authors’ model simulations. Notes: The figure shows the short-term equilibrium real rate in dynamic simulations of our model. The figure also shows the posterior mean estimate of the expected real interest rate in the long run from Johannsen and Mertens (2016b), along with 50-percent uncertainty bands. 41

Figure 9: Dynamic simulation under baseline demographics: other variables 63 60 57 54 1960 1970 1980 1990 2000 2010 2020 2030 Date tnecreP Employment−population ratio Model input Data 100 80 60 40 1960 1970 1980 1990 2000 2010 2020 2030 Date tnecreP Dependency ratios Children−to−labor Inactive−adults−to−labor Source: Authors’ calculations based on Bureau of Labor Statistics and Census Bureau data. Notes: The top panel shows the employment-population ratio in the data and in our model after weapplytheHodrick–PrescottfilterwithaLagrangemultiplierof25,000. Thebottompanelshows the number of dependent children in the model scaled by the number of full-time equivalent labor endowments. The bottom panel also shows the number of full-time equivalent adults who are inactive scaled by the number of full-time equivalent labor endowments. 42

Figure 10: Dynamic simulation of real GDP growth 5 4 3 2 1 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 2030 Date dezilaunna ,tnecreP GDP growth, baseline model GDP growth, model with TFP growth Growth of trend of H−P filtered data Source: Authors’ calculations using data from the U.S. Bureau of Economic Analysis and authors’ model simulations. Notes: The figure shows real GDP growth under the baseline simulation, which has no time variation in technology growth, and the alternative scenario in which technology growth varies according to an estimate of its historical and projected trend. The figure also shows the growth rate of the trend in realized real GDP after applying the Hodrick–Prescott filter with a Lagrange multiplier of 25,000. 43

Figure 11: Dynamic simulation under baseline demographics: other variables 11 10 9 8 1960 1980 2000 2020 2040 2060 Date tnecreP Real rental rate, annualized 10.0 Dependent children No dependent children 7.5 5.0 2.5 0.0 1960 1980 2000 2020 2040 2060 Date tnecreP Net saving rate 0.2 0.1 0.0 −0.1 1960 1980 2000 2020 2040 2060 Date 0691 morf ecnereffid goL Capital−labor ratio 0.2 0.1 0.0 −0.1 1960 1980 2000 2020 2040 2060 Date 0691 morf ecnereffid goL Real wage Source: Authors’ model simulations. Notes: The lines labeled “Dependent children” and “No dependent children” show simulation results under the assumption that representative adults receive utility from their children’s consumption and receive no utility through an additive term, respectively. The natural logarithms of the real wage and the capital-labor ratio are normalized so that their levels are zero in 1960. 44

Figure 12: Equilibrium real rate with demographic variables fixed in either 1960 or 1980 onward 2.5 2.0 1.5 1.0 0.5 0.0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 2030 Date dezilaunna ,tnecreP Demographics fixed in 1960 Baseline Constant mortality Constant employment Constant fertility Constant mortality, employment, fertility 2.5 2.0 1.5 1.0 0.5 0.0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 2030 Date dezilaunna ,tnecreP Demographics fixed in 1980 Baseline Constant mortality Constant employment Constant fertility Constant mortality, employment, fertility Source: Johannsen and Mertens (2016b) and authors’ model simulations. Notes: The top panel shows dynamic simulations of the model in which we hold constant, first separately and then jointly, the mortality rates, fertility rates, and employment-population ratios by age from 1960 onward. The bottom panel shows a similar set of alternative scenarios in which the demographic variables are held constant from 1980 onward. In all cases, all other parameters are held at their baseline values. The results are shown against the 50-percent uncertainty bands from Johannsen and Mertens (2016b). 45

Figure 13: Real GDP growth rate with demographic variables fixed in either 1960 or 1980 onward 3 2 1 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 2030 Date dezilaunna ,tnecreP Demographics fixed in 1960 Baseline Constant mortality Constant employment Constant fertility Constant mortality, employment, fertility 3 2 1 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 2030 Date dezilaunna ,tnecreP Demographics fixed in 1980 Baseline Constant mortality Constant employment Constant fertility Constant mortality, employment, fertility Source: Authors’ model simulations. Notes: The top panel shows dynamic simulations of the model in which we hold constant, first separately and then jointly, the mortality rates, fertility rates, and employment-population ratios by age from 1960 onward. The bottom panel shows a similar set of alternative scenarios in which the demographic variables are held constant from 1980 onward. In all cases, all other parameters are held at their baseline values. 46

Figure 14: Real interest rate, alternative scenarios 3 2 1 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 2030 Date dezilaunna ,tnecreP Baseline model Model with historical TFP growth Model with low elastity of substitution Source: Authors’ model simulations. Notes: The figure shows the equilibrium real rate under the baseline simulation, which has no timevariationintechnologygrowth, andthealternativescenarioinwhichtechnologygrowthvaries according to an estimate of its historical and projected trend. The figure also shows an alternative scenarioinwhichtheelasticityofsubstitutionbetweencapitalandlaborintheproductionfunction is set to 0.75, as opposed to 1 under the baseline simulation. The results are shown against the 50-percent uncertainty bands from Johannsen and Mertens (2016b). 47

Figure 15: U.S. versus global demographic changes 2 1 0 1950 1970 1990 2010 2030 2050 2070 2090 Date etar launna ,tnecreP Working−age population growth United States OECD countries World 90 80 70 60 50 1950 1970 1990 2010 2030 2050 2070 2090 Date sraeY Life expectancy at birth United States OECD countries World Source: Authors’ calculations using data from the World Population Prospects: Revision 2015 published by the United Nations. Notes: The working-age population is defined as the set of individuals aged 15 to 64 years. We construct the OECD measure of mean life expectancy at birth as a population-weighted average of that statistic across OECD member countries. 48

Technical Appendix to “Understanding the New Normal: The Role of Demographics” A Introduction This technical appendix provides information on data sources and data filtering as well as on the solution method we follow to obtain dynamic solutions of our overlapping-generation (OG) model. To ensure full reproducibility of our data series and simulation results, we are making available a collection of Jupyter notebooks that contains all steps necessary to create the data series and solve the model, along with further instructions. The data and model files are coded in the Julia programming language. The head replication fileisGJLS_readme.ipynb. ThereplicationfilesalsoincludeasetofRfilestoreproduceallfigures. Jupyter, Julia, and R are all freely available. Please email the authors to obtain the latest set of files. B Data B.1 Employment-population ratios Weconstructyearlyandquarterlyemployment-populationratios(EPRs)seriesbyageandbirthcohortfrom 1900 onward. These EPRs are used to calibrate individual labor endowments in the model. The EPRs are defined as Number of Persons Employed EPR=100× . Civilian Non-Institutional Population The creation of EPRs by age and birth cohort over an extended sample period requires a bit of data manipulation to deal with sparse historical data and to project on-going trends in the labor supply of older workers. The time series must also be filtered to remove business cycle variation, which our model does not seek to explain. 1

B.1.1 Data sources We downloaded the BLS data on August 1, 2016, from http://data.bls.gov/cgi-bin/srgate. We arranged the data into CSV files to make them easy to read by our programs. The mnemonics for civilian non-institutional population by age group can be found in Table 1. The mnemonics for total civilian employment by age group can be found in Table 2. The mnemonics for unemployment rate by age group can be found in Table 3. The sources for the Census Bureau’s total population estimates are described below. B.1.2 Estimating the EPRs before 1967 The historical data underlying the EPRs have been published by the BLS starting in 1948. However, the published agecategories have changedovertime. Startingin1967, we havemonthlydata by5-year agebins from 20 to 74 years old, as well as for 16 to 19 years and 75+ years old. Prior to 1967, we have monthly data for 16 to 19 years old, 20 to 24 years old, and 65+ years old, as well as for 10-year age bins from 25 to 64 years old. Given our interest in low-frequency movements, we initially work with annual data—thus avoiding issues pertaining to seasonal adjustment—and recover quarterly EPR series through cubic spline interpolations of the trend in yearly rates. WeestimatetheEPRsforthemissingfive-yearagebinspre-1967usinginformationfromthe10-yearage bins. After inspecting the data, we note that the level of the 25 to 29 years old series and of the 30 to 34 series move in roughly equal percentage-point steps with the 25 to 34 series. For these series, we impute the missing pre-1967 levels by making the identifying assumption that the percentage point difference between the series for 25 to 29 years old and for 30 to 34 years old is equal to the difference observed in 1967: EPR 30to34−EPR 25to29 =EPR 30to34−EPR 25to29 . t t 1967 1967 We then back out the missing level for 5-year bins from POP25to29 POP30to34 EPR 25to34 = t EPR 25to29 + t EPR 30to34 . t POP25to34 t POP25to34 t t t We adopt the same approach to impute missing statistics by five-year age bins for other prime-age workers, for which the employment-population ratio is fairy constant until 55 years old. We also follow the same approach for workers aged between 55 and 64 years old, although inspection of the data suggests somewhat less stability for that age group. For workers aged 65 years or more, we adopt an imputation procedure that is better suited to capture shifts in labor supply late in life. Upon inspection of the data from 1967 onward, we note that the EPRs for 65 to 69 years old, 70 to 74 years old, and 75+ years old are 2

roughly proportional to each other over time. Our imputation method pre-1967 minimizes a weighted sum of square deviations from the EPRs prevailing in 1967, POP65to69 (cid:18) EPR65to69 EPR65to69(cid:19)2 POP70to74 (cid:18) EPR70to74 EPR70to74(cid:19)2 t × t − 1967 + t × t − 1967 65+ 65+ 65+ 65+ 65+ 65+ POP EPR EPR POP EPR EPR t t 1967 t t 1967 POP 75+ (cid:18) EPR 75+ EPR 75+ (cid:19)2 + t × t − 1967 65+ 65+ 65+ POP EPR EPR t t 1967 in a way that is consistent with the published EPRs for the broader age group, POP65to69 POP70to74 POP 75+ EPR65+ = t EPR 65to69 + t EPR 70to74 + t EPR 75+ . t POP 65+ t POP 65+ t POP 65+ t t t t One slight complication for the pre-1967 data is that we do not have estimates of the non-institutional population for all age bins of interest. We simply replace the missing population ratios in the equations above with the corresponding ratios from the U.S. Census estimates of the total population. B.1.3 Projecting the EPRs through 2024 The EPRs of older workers have been increasing over the past few decades along with life expectancy and health conditions, reversing a pattern of declining EPRs in earlier decades. Our projection of EPRs going forward makes use of the BLS’ “Employment Projections: 2014–24”. The BLS creates these projections to look at employment by industries. Although they do not include projections of EPRs by age in 2024, they do include projections of the labor force participation ratio (LFPR) by age bins. We use that information, along with an assumption about equilibrium employment rates by age, to recover EPRs by age in 2024. In particular, we assume that the labor market in 2015 is in equilibrium and that the unemployment rates by age that prevailed in that year also prevail in 2024. Forindividualsagedlessthan25yearsormorethan54years,theLFPRsinthe“EmploymentProjections: 2014–24” directly line up with the EPR age bins derived above. For prime-age workers between the age of 25 and 54 years, the “Employment Projections: 2014–24” are structured by ten-year age bins as opposed to five-year age bins. For these age categories, we make the identifying assumption that the percentage-point difference between two contiguous five-year bins is constant between 2014 and 2024, then retrieve the fiveyearLFPRsconsistentwiththepublishedten-yearLFPRs. Forexample,forthe25-34yearsoldcategory,we compute LFPR2024 and LFPR2024 using the definition of the LFPR for the 25-34 years old category 25to29 30to34 POP2024 POP2024 LFPR2024 = 25to29×LFPR2024 + 30to34×LFPR2024 , 25to34 POP2024 25to29 POP2024 30to34 25to34 25to34 3

and the restriction that LFPR2024 −LFPR2024 =LFPR2014 −LFPR2014 . 25to29 30to34 25to29 30to34 One minor complication is that the “Employment Projections: 2014–24” do not include detailed noninstitutional population statistics, so the relative noninstitutional population shares in the above equations are unobserved for 2024. In a manner similar to the imputation of pre-1967 data, we approximate these shares using corresponding data from the U.S. Census Bureau’s 2014 National Population Projections for the total U.S. population. See the notebook “Migration.ipynb” for detailed sources. Another minor complication is that the historical labor market data have been revised since the publication of the “Employment Projections: 2014–24.” For this reason, we adjust the level of the 2024 LFPR data such that they imply the same change (as opposed to level) in LFPR by age bins between 2014 and 2024 as reported in the publication. The 2024 LFPRs were downloaded at the link www.bls.gov/emp/ep_table_ 303.htm on August 1, 2016. B.1.4 The EPRs before 1948 and beyond 2024 The imputation of EPRs before 1948 is subject to a great deal of uncertainty for several reasons. Notably, the two world wars altered the size and composition of the U.S. labor force. Moreover, home production was arguably more important in light of a large share of the population living in a rural environment. Both of these elements raise questions that are outside of the purview of our model. For this reason, we simply carry back in time the EPRs by age prevailing in 1948. Similarly, the projection of EPRs beyond 2024 is subject to a great deal of uncertainty. On the one hand, one might expect further improvement in health and life expectancy to continue to raise the LFPR of older persons. On the other hand, the historical correlation between these variables is very weak outside of the recent decades. Moreover, some of the rise in EPRs over the past couple decades might reflect in part increases in the minimum age at which one becomes eligible to draw social security benefits. Our approach is to simply level off the post-2024 EPRs at their 2024 levels. B.1.5 Interpolation and filtering Tomakethedataconsistentwiththetimeperiodinthemodelsimulation,thedatabyagegrouparecarried backward in time before 1948 and forward in time after 2024. Cyclical variation in the data by age group is next removed using an Hodrick–Prescott filter with a Lagrange multiplier of 25,000. We then interpolate the trends in the data by age group across age and over calendar periods using cubic splines. Although our 4

model pools EPRs across genders, we separately compute EPRs for men and women to help us provide a narrative of the changes in EPRs over time. Figures by gender and for the total population are provided in the Jupyter notebook. B.2 Mortality rates Ourestimatesofmortalityratesbyageandbirthcohortarebasedoninformationcontainedinthepublication “Life Tables for the United States Social Security Area 1900-2100” from the Social Security Administration. Wedownloadedthetablesfromhttps://www.ssa.gov/oact/NOTES/as120/LifeTables_Body.htmlonAugust 1, 2016. We use two sets of tables: Tables 6 report estimates of the marginal probabilities of death for a person of a given age (in years) at the turn of a particular decade. Tables 7 report estimates of the annualized probabilities of death by years of age for an individual born at the turn of a particular decade. Bell and Miller (2005) provide mortality tables for each gender but not for the total population. To compute mortality rates for all individuals, we first assume that the ratio of live male births to live female births is constant over time. We then aggregate, for each birth cohort and age, the mortality rates of males and females using their respective shares of the birth cohort that has survived up to that age. The probabilities of surviving to a given age for each cohort and gender are based on the information contained in Tables 6 and Tables 7. The ratio of male births to female births is set to 1.05 based on the analysis of Mathews and Hamilton (2005) at the Center for Disease Control. A value of 1.05 is close to the median yearly value over their 1940-2002 sample period, with the values ranging roughly from 1.046 to 1.059. For Table 7, we simply rescale the male cohort by 5 percent, then take a weighted average across gender based on the number of males/females alive at beginning of period. We apply the same procedure to Table 6. To the extent that mortality rates by age for people alive in 1900 differ from those applying to people born in 1900,theprocedureisinexact. However,anydifferenceitmightcreateontheweighedmortalityratesacross gender should be small and only affect the seeding of the 1900 population in the simulation. Once we have aggregated the data across genders, we use cubic splines to interpolate the information from Tables 6 and Tables 7 across ages and calendar periods from 1900 to 2100. With those interpolations at hand, we assume that the estimated rates conditional on age in 2100 prevail in all subsequent periods. Similarly, we assume that the estimated rates in 1900 prevailed in all previous periods. In theory, we could use the information in Tables 7 to impute mortality rates beyond 2100 for people born in 2100 or earlier. Similarly, we could use the information in Tables 6 to impute mortality rates prior to 1900 for people born before that year. However, the available usable information before 1900 and beyond 2100 is sparse and, in any case, its usage would have little to no effect on the simulation results over the period of interest. For 5

this reason, we find it convenient to estimate the mortality rates by age of people alive between 1900 and 2100, and carry those rates backward and forward in time as necessary in our applications. B.3 Population and net international migration In our model, population size and composition evolves over time as people are born, die, age, and migrate. The construction of our birth rates and mortality rates series was detailed above. This subsection discusses how, given our birth and mortality rates, we create our net migration rates in a way that ensures that the size and composition of the total population closely matches the Census Bureau’s historical estimates and projections of population from 1900 to 2060 under our baseline specification of the model. B.3.1 Population estimates and projections We amalgamate several Census Bureau products to create a consistent resident population series by single yearsofageoverthe1900-2060period. AlldatasourcesmeasuretheresidentpopulationasofJuly1ofeach calendaryear. Upto2010,weuse“intercensal” populationstatistics,forwhichCensusstaffadjustedthedata in ways that reconcile discrepancies between decennial censuses and intermediate estimates. For the 2010- 2014 data, we use the 2015 vintage of “postcensal” population estimates, which are based on adjustments to the 2010 Census figures based on estimates of births, deaths, and migration. Similarly, we use the latest set of population projections from 2014 to 2060 based on the 2010 Census. The data sources are listed below, along with the age coverage available. • [1900 to 1939] We downloaded the “Historical National Population Estimates/National Estimates by Age, Sex, Race: 1900-1979.” The annual data cover years of age 0 through 74 years old and 75+ years old. • [1940 to 1979] The source is the same as above. The annual data cover years of age 0 through 84 years old and 85+ years old. • [1980 to 1989] We downloaded the “Quarterly Intercensal Resident Population” estimates by single years of age and retained estimates as of July 1 of each year. • [1990 to 2000] We downloaded the “Intercensal Estimates of the United States Population by Age and Sex, 1990-2000: All Months.” The data cover years of age 0 through 99 years old and 99+ years old. • [2001 to 2013] We downloaded the “2000-2010 Intercensal Estimates April1. 2000 to July 1, 2010.” The data cover years of age 0 through 84 years old and 85+ years old. 6

• [2014to2060]Wedownloadedthe“2014to2060PopulationsProjectionsbasedonCensus2010(released 2014).” The data cover years of age 0 through 99 years old and 100+ years old. Historicalpopulationestimatesweredownloadedfromhttps://www.census.gov/popest/data/historical/. Populationprojectionsweredownloadedfromhttps://www.census.gov/population/projections/data/ national/2014.html. The links were last accessed on August 1, 2016. B.3.2 Net international migration The derivation of net migration estimates uses information on our interpolation of mortality rates from Bell and Miller (2005). Per our timing assumptions, births, deaths, and migration take place at the beginning of the period whereas population measurement takes place at the end. Thus, we treat a population estimate for July 1 as corresponding to the population at the end of the second quarter of the calendar year. By definition, Population age,t =(Population age−1,t−1 +netmigration age,t )×Γ age,t,t , whereΓ istheprobabilitythatanindividualagedageatthebeginningofperiodtisstillaliveattheend age,t,t of that period. As is apparent, we assume that mortality rates are the same for new immigrants as existing residents. For infants, the term Population age−1,t−1 is replaced by our estimate of the number of live births for the period. Because the net migration statistics are computed as residuals, they may absorb sources of variation in populations statistics other than migration. These sources include year-to-year measurement errors in population estimates/projections and imprecision in the imputation of mortality rates by age and birth cohorts. For the data beyond 2060, we assume that our estimates of net migration linearly converge to those published by the Census for 2060. These Census estimates are roughly consistent with a stabilization in the absolute level of net migration around 2060. Our estimates and those of the Census for 2060 are similar with two minor exceptions. First, our estimated number of the net migration of infants (that is, less than a year old) is mildly negative whereas that of the Census is a bit positive, likely reflecting differences in assumptions between the UN projections that we used for projecting live births through 2100 and the assumptions underlying the Census population projections (the details of these Census assumptions are not publicly available). Second, we have slightly negative net migration for older people, likely capturing differencesinassumptionsaboutmortalitylateinlife. Bothelementsareunlikelytoaffectourmacroeconomic results as they have little if any incidence on the aggregate capital-labor ratio. Due to our interpolation to a quarterly frequency of age and calendar period, our population statistics need not correspond exactly to those published by the Census Bureau. However, the difference accumulated 7

to only about 1/4 percent of the level of the population by 2060, indicating that the cumulative discrepancy is very small. B.4 Live births This subsection deals with our stitching of historical estimates and projections of live births in the United States to produce a consistent series from 1900 onward. It uses information from four data sources: • Yearly intercensal population estimates of the number of kids aged less than 1 year old from the U.S. Census Bureau from 1900 to 2060 (detailed earlier). • Monthly data on the number of live births in the United States produced by the National Center for Health Statistics (NCHS) since January 1972. • Five-year birth projections from the United Nations (U.N.) Population Program. • Mortality rates by age and birth cohort based on Bell and Miller (2005) (detailed earlier). Before 1972, our series is based on the Census Bureau’s intercensal population estimates, which we adjusted forinfantmortalityusingourinterpolationofBellandMiller(2005)mortalityratetables. For1972to2014, we directly use yearly NCHS data, which we see as the most reliable estimates available. For 2015 onward, we use a projection of the U.N. data on the Census and NCHS samples (with five-year averages), which is consistent with NCHS data in recent years. The resulting stitched data series mixes yearly and five-year frequencies. To obtain yearly and quarterly series over the 1900-2100 period and beyond, we interpolate the series using cubic splines. We impose no growth in the total number of births in the last 2.5 years of quarterlyobservations,consistentwiththelevelingoffoftheU.N.birthseries. Thisassumptionhelpsensure the existence of an ergodic distribution of population with no net growth under our baseline demographic assumptions, although this stationarity is not necessary to solve for the balanced growth equilibrium of our model economy. To derive yearly birth estimates based on yearly Census population figures, we initially assume that the numberofquarterlybirthsisconstantbetweenpopulationmeasurements,whichcorrespondtoJuly1ofeach calendar year. This identifying assumption is made to take into account high infant mortality in the initial quarters of life. The steps in the underlying quarterly birth series are then smoothed when we aggregate the data back to a yearly frequency and then use splines to extract a smooth quarterly series. Consistent with these timing assumptions, we treat Census population estimates as of July 1 as if they correspond to the measure of the surviving population at the end of the second quarter of calendar years. Given that the 8

calibrationusesquarterlydata,alternativetimingassumptionswithrespecttobirths,deaths,andpopulation measurement would have negligible consequences for our population and births estimates. We now detail the extraction of estimates of live birth from the yearly population series. Let POP [0,1),t be the number of infants less than a year old estimated by the Census at the end of period t. Let B be the t number of births at the beginning of period t; consistent with the model, births take place before deaths during the period, so that newborns face a nonzero probability of not reaching period t+1. The number of infants less than a year old is the sum of surviving births in the current and previous three quarters, POP =γ B +γ γ B +γ γ γ B +γ γ γ γ B , (1) [0,1),t 0,t t 1,t 0,t−1 t−1 2,t 1,t−1 0,t−2 t−2 3,t 2,t−1 1,t−2 0,t−3 t−3 where γ ≡ Γ are parameters governing the probability of survival in period j of someone age i in i,j i,j,j that period. For simplicity, we temporarily assume that the number of births is constant over four-quarter intervals ending in the second quarter of each calendar year, that is, B = B = B = B . Under t t−1 t−2 t−3 this assumption, we can extract a quarterly sequence {B }1979:Q2 from knowledge of mortality rates and t t=1900:Q3 the yearly Census estimates of the number of infants less than a year old. For the first half of 1900, we are missing estimates of quarterly mortality rates for 1899 that would be used to compute B =B 1900:Q1 1900:Q2 fromthenumberofinfantsinthesmoothed1900Censusestimates. Wesimplypositthatthemortalityrates are the same, conditional on age, as our interpolation for the first quarter of 1900 (that is, γ =γ a,Q(cid:48) a,1900:Q1 for Q(cid:48) < 1900:Q1). Averaging quarterly observations for each calendar year, we finally obtain a series of annual birth estimates from 1900 to 2060. Only this latter annual series is used when we interpolate our various data sources into a quarterly series covering the full sample period. Moreover, the data from that series for 1972 onward are only used to investigate consistency with the NCHS and U.N. data series. Forthe1972–2014period,wegetannualbirthfiguresdirectlyfromNCHS.1 TheannualNCHSestimates trackstheannualestimatesthatwederivedabovefrommortalityratesandinfantpopulationestimatesfrom 1972 to 1979 when the two samples overlap. We thus construct an annual births series from 1900 to 2014 by stitching the yearly figures derived from Census estimates for 1900 to 1971 with those from NCHS for 1972 to 2014. To obtain annual birth estimates after 2014 onward, we note that, in recent decades, yearly observations fromtheNCHSseriesessentiallycoincidewiththecorrespondingfive-yearWPPdatameasuredatthemiddle year (the blue circles). We thus allocate WPP data after 2009 to the mid-point of their five-year intervals through 2100, and stitch those five-year observations with the annual series through 2014. We then use cubic splines to extract quarterly birth estimates from 1900:Q2 to 2099:Q4. Of note, by the end of the 1Wecould,intheory,usethemonthlyNCHSinformationtoconstructthequarterlyestimatesusedinthemodel. Doingso wouldrequireremovingapparentseasonalpatterns. 9

current century, the WPP data project that live births have stabilized. Using this observation, we impose that the number of live births is constant from the last WPP data point (that is, 2097.5) through 2400. This assumption, along with constant mortality rates and no immigration, ensures the existence of a stable population distribution in the model over the long run. Finally, we note that our live birth estimates do not adjust for infants who might have migrated to the UnitedStatesoraforeigncountryintheperiodtheywereborn. Theseriesmayalsobesubjecttoimprecise reporting or minor errors in our imputation. That said, our procedure that uses net international migration to ensure that the size and composition of total population remains consistent with the population figures published by the Census Bureau should correct any discrepancy. B.5 Fertility rates Totakeintoaccounttheeffectsoffamilycompositiononhouseholds’economicdecisions,weneedtoallocate births to their parents. In this subsection, we first construct a time series of live births from a variety of sources. We then allocate the births to their mothers based on the historical shares of births by the age of mothers. Wefinallyadjusttheageofthemotherinawaythatfactorsinthehistoricalagedifferencebetween fathers and mothers, so that adults are representative of both genders. Per our timing assumptions in the model, we posit that births, deaths, and population measurement occur sequentially during the period. B.5.1 Female fertility rates ThedataonthenumberofbirthsbyageofmotheristakenfromtheCentersforDiseaseControlandPrevention’s National Center for Health Statistics (NCHS). The data can be found under “Cohort Fertility Tables” at www.cdc.gov/nchs/cohort_fertility_tables.htm (last accessed on August 8, 2016). We downloaded thedataonAugust8,2016. Weusetwosetsofhistorictables. ThefirstsetisputtogetherbyHamiltonand Cosgrove (2010) and consists of two separate data releases. The first release is titled “Table 1. Central birth rates, by live-birth order, current age, and race of women in each cohort from 1911 through 1991: United States,1960-2005”. Thesecondreleaseistitled“Table1. Centralbirthrates,bylive-birthorder,currentage, and race of women in each cohort from 1957 through 1995: United States, 2006-2009”. The methodology and data sources are consistent across the two releases. We saved the data in separate CSV files that are read in the notebook below. Consistent with the timing of population estimates in the Census, women who belong to, say, the 1960 cohort, are born within a twelve-month period centered on January 1, 1960, and thus running from July 1, 1959, to June 30, 1960. Thesecondsetofdataisbasedonthehistoricalpublication“FertilityTablesforBirthCohortsbyColor: 10

United States, 1917-73”, DHEW Publication No. (HRA) 76-1152, U.S. Department of Health, Education, and Welfare, April 1976, Washington D.C. The link to a PDF of that publication is http://www.cdc.gov. nchs/dad/misc/fertiltbacc.pdf (last accessed on August 8, 2016). We extracted the information from Table 4A, “Central birth rates for all women during each year 1917-1973 by age and life-birth order for each cohort from 1868 to 1959, Total birth group”, and entered that information manually into a CSV file. Thetwosetsoffilesdonotuseidenticalmethodology. Inparticular, therearemethodologicaldifferences injudgmentaladjustmentsforunder-registrationofbirths,under-enumerationofthenumberofwomen,and misstatement of age for all years. That said, the differences appear too small to have a material effect on the key simulation results in our paper. We also note that the shares of births by age of the mother are fairly similar for the overlapping period between 1960 and 1973. For these reasons, we use the 1960-2009 data intheir entirety, and onlythe 1917-1959 datafrom the earlysample. Finally, we assume thatthe share of births attributable to mothers by age were constant at their 1917 level in all prior years. Similarly, we carry forward the shares in 2009 to all subsequent years. B.5.2 Median age difference between men and women parents We adjust the age of the mother by an estimate of the age difference between married men and women to obtain the age of the representative parent. This adjustment is based on our interpolation of historical information on the median age of men and women at time of their first marriage published by the Census Bureau. The data are obtained at www.census.gov/hhes/families/data/marital.html (downloaded on August 8, 2016). The publication is called “Estimated Median Age at First Marriage, by Sex: 1890 to the Present”. These data cover the period 1890 to 2015. Yearly estimates are available staring in 1947; prior to that year, only decennial estimates corresponding to census years are available. Themedianagegapbetweenmenandwomenhasbeensomewhatstableoverthepastseveraldecades,at between1.5and2.5years. Inourapplication,weassumethataverageagedifferenceobservedinrecentyears will prevail beyond 2015, then fit a polynomial to the historical and projected values to obtain a smooth series. B.5.3 Teen pregnancies and dependent children A question arises with teen pregnancies because young parents may still be dependent children themselves. In the absence of detailed information, we follow a somewhat simple procedure. Whenever a child is born to a representative parent whose average age is less than 18 years old, we temporarily allocate the child to her representative grandparent until the representative parent turns 18 years old, at which point the representative parent becomes fully responsible for her child’s consumption. With all the above information 11

at hand, we can allocate children at the end of 1899 to their parents and calculate the average number of dependents entering the parents’ consumption function. We can then iteratively calculate the population andfamilystructureoverthefullsampleperiodbylettingthepopulationevolveaccordingtobirths,deaths, and net migration. Finally, we note that our procedure works only with averages rather than actual family cells. For example, deaths of adults and children do not result in discrete jumps in the number of surviving parents or dependents. B.6 Total factor productivity For our alternative dynamic simulation with historical total factor productivity (TFP) growth, we primarily usehistoricalquarterlyestimatescompiledbyFernald(2014). ThedataweredownloadedononMay25,2016, from the San Francisco Fed’s website at the following link: http://www.frbsf.org/economic-research/ indicators-data/total-factor-productivity-tfp/. Theestimatesrunfrom1947:Q2to2016:Q1. More documentation on the methodology can be found on the website as well as the accompanying Excel spreadsheet. We extend the Fernald series forward in time using projections made by Gordon (2015). This author estimates that total factor productivity will grow at a annual rate of 1.1 percent per year through 2032, a rate that we extend beyond that date. We also extend the Fernald series back in time using an informed guess of average technology growth in earlier decades. The rate chosen—2 percent—is in the middle of the range of estimates surveyed by Shackleton (2013). Oursimulationsdonotseektoexplainhigh-frequencymovementsintheequilibriumrealrate,GDP,and other aggregate variables. For this reason, we feed our model with a smoothed version of the quarterly (log) TFPgrowthseries. Tocreatethissmoothedseries, wefirstconvertourextendedFernaldTFPgrowthseries to levels and then use the Hodrick–Prescott filter on the log of that resulting series to extract a smooth trend; the Lagrange multiplier is set to 25,000. We finally convert the smoothed level series back to growth rates (again measured in log changes). C Model economy The model we consider in this technical appendix is slightly more general than in the paper in that it allows for subsistence levels in the consumption of adults and their dependent children. The introduction of these objects in done in a manner that preserves the balanced growth properties of the model. 12

C.1 Households A representative adult aged a in period t maximizes A (cid:88) −a βsΓ (cid:32)(cid:0) C a+s,t+s −Z t+s C¯(cid:1)1−ν +(cid:15)(n )η (cid:0) C a c +s,t+s −Z t+s C¯c(cid:1)1−ν(cid:33) , (2) a,t,t+s 1−ν a+s,t+s 1−ν s=0 where n is the number of dependent children, C is the adult’s consumption, and Cc is the average a,t a,t a,t consumption of each of its dependent children. We have subsistence levels of consumption, labeled Z C¯ t and Z C¯c, respectively, which grow at the same pace as labor-augmenting technology, Z . For tractability, t t we allow the number of dependent children to be fractional. The parameter (cid:15) ∈ [0,1] controls the weight placed by adults on their children’s consumption whereas η ∈ [0,1] captures the effects of family size on householdsutility. Adultsderiveutilityonlyinperiodsinwhichtheyarealive, andhavenobequestmotive. The probability that an adult aged a in period t survives through the end of the next s periods is given by the cohort-specific function Γ . We assume that no adult survives past the age of A periods by setting a,t,t+s Γ =0 for s>A−a. Households face the following sequence of budget constraints a,t,t+s C +n Cc +K =(K +φΞ ) (cid:0) RK +1−δ (cid:1) +e W , a+s,t+s a+s,t+s a+s,t+s a+s+1,t+s+1 a+s,t+s t+s t+s a+s,t+s t+s where K is the age a adult’s capital holdings at the beginning of the period, e is the age a adult’s labor a,t a,t endowment, RK is the real rental rate of capital, and W is the real wage rate. We assume that only a t t fraction φ ∈ [0,1] of the capital held by adults who die, Ξ , at the beginning of the period is redistributed, t with the remainder being lost. The first-order optimality conditions of the household are (cid:0) C −Z C¯(cid:1)−ν =λ (3) a+s,t+s t+s a+s,t+s (cid:15)(n )η(cid:0) Cc −Z C¯c (cid:1)−ν =n λ (4) a+s,t+s a+s,t+s t+s a,t+s a,t+s λ =βλ (cid:0) RK +1−δ (cid:1) (5) a+s,t+s a+1+s,t+1+s t+1+s Equations (3) and (4) can be written as Cc −Z C¯c (cid:16) (cid:17)1 a+s,t+s t+s = (cid:15)nη−1 ν . C −Z C¯ a+s,t+s a+s,t+s t+s 13

This proportionality allows us to express the household objective function in terms of adult consumption A (cid:88) −a βsΓ    (cid:0) C a+s,t+s −Z t+s C¯(cid:1)1−ν +(cid:15)(n )η (cid:18)(cid:16) (cid:15)nη a+ − s 1 ,t+s (cid:17) ν 1 (cid:0) C a+s,t+s −Z t+s C¯(cid:1) (cid:19)1−ν   a,t,t+s 1−ν a+s,t+s 1−ν    s=0 A (cid:88) −a βsΓ (cid:32)(cid:18) 1+(cid:15)(n )η (cid:16) (cid:15)nη−1 (cid:17)1− ν ν(cid:19)(cid:0)(cid:0) C a+s,t+s −Z t+s C¯(cid:1)(cid:1)1−ν(cid:33) a,t,t+s a+s,t+s a+s,t+s 1−ν s=0 A (cid:88) −a βsΓ a,t,t+s (cid:32)(cid:18) 1+(cid:15)ν 1 (cid:18) n a (η + − s 1 ,t )( + 1 ν − s ν)+ην (cid:19)(cid:19)(cid:0)(cid:0) C a+s,t+s 1 − − Z ν t+s C¯(cid:1)(cid:1)1−ν(cid:33) s=0 A (cid:88) −a βsΓ a,t,t+s (cid:32) (cid:16) 1+(cid:15)ν 1 (cid:16) n a η + − ν s 1+ ,t ν +s (cid:17)(cid:17) (cid:0)(cid:0) C a+s,t+s 1 − − Z ν t+s C¯(cid:1)(cid:1)1−ν(cid:33) s=0 A (cid:88) −a βsΓ a,t,t+s (cid:16) 1+(cid:15)ν 1 (n a+s,t+s )1+η− ν 1(cid:17) (cid:0) C a+s,t+s 1 − − Z ν t+s C¯(cid:1)1−ν (6) s=0 with a sequence of budget constraints given by (cid:16) 1+(cid:15)ν 1 (n a+s,t+s )1+η− ν 1(cid:17) C a+s,t+s −(cid:15)ν 1 (n a+s,t+s )1+η− ν 1 Z t+s C¯+n a+s,t+s Z t+s C¯c +K =(K +φΞ ) (cid:0) RK +1−δ (cid:1) +e W . a+s+1,t+s+1 a+s,t+s t+s t+s a+s,t+s t+s We write the intertemporal Euler equation in terms of adult consumption as (cid:18) C a+1,t+1 −Z t+1 C¯(cid:19)ν =βΓ (cid:0) 1+RK −δ (cid:1) . (7) C −Z C¯ a,t,t+1 t+1 a,t t C.2 Firms Firms maximize (α(K )ρ+(1−α)(Z L )ρ)ρ 1 −L W −RKK , t t t t t t t takingfactorpricesasgiven. Weassumethatthefirm’stechnologygrowsataconstantrate,Z /Z =1+z. t t−1 At the optimal solution, the representative firm pays a rental rate on capital equal to the marginal product of capital, (cid:18) Y (cid:19)1−ρ RK =α t , (8) t K t 14

and a real wage rate equal to the marginal product of labor, W (cid:18) Y (cid:19)1−ρ t =(1−α) t . (9) Z Z L t t t C.3 Aggregation, market clearing, and equilibrium In our OG economy, the size of the population varies from period to period as new individuals are born and others die. Let µ be the number of individuals aged s at the beginning of period t after a new cohort of s,t babies have been delivered but before deaths have occurred. Equilibrium in the labor market requires that A (cid:88) L = µ e , t s,t s,t s=Ac+1 while equilibrium in the capital market requires A A (cid:88) (cid:88) K = µ Γ K +φ µ (1−Γ )K . t s,t s,t,t+1 s,t s,t s,t,t+1 s,t s=Ac+1 s=Ac+1 Thetotalcapitalstockiscomposedoftwoterms. Thefirsttermisthecapitalthatsurvivingadultsbrought intotheperiod. Thesecondtermistheredistributedcapitalfromdeadadults. Finally,goods-marketclearing requires that production be equal to the sum of consumption and gross investment aggregated across the cohort-representative adults, A A (α(K t )ρ+(1−α)(Z t L t )ρ)ρ 1 = (cid:88) µ s,t (cid:16) 1+(cid:15)ν 1 (n s,t )1+η− ν 1(cid:17) C s,t + (cid:88) µ s,t (K t+1,s −(1−δ)(K t,s +φΞ t )). s=Ac+1 s=Ac+1 In a perfect-foresight equilibrium, in each period, (a) goods and factor markets clear, (b) factor prices correspond to their marginal product, (c) the bequests received by surviving adults sum up to the capital held by adults who passed away at the beginning of the period (net of any destroyed capital), and (d) household optimality conditions are satisfied. C.4 Solution method We follow an iterative procedure to compute an equilibrium in which (a) cohort-representative adults and the representative firm optimize conditional aggregate variables and demographic factors and (b) aggregate 15

objects are consistent with the aggregation of individual decisions across adults and firms. This equilibrium isconditionalonthepathoftheparametersofthemodel,thedemographicvariables,andinitialdistributions of the population and capital holdings. The steps in the procedure are as follows: 1. Compute two balanced growth equilibrium to the model. The first such equilibrium is parametrized to demographic data at the start of 1900. It is used to obtain a reasonable distribution of capital holdings by age in the initial period. The second such equilibrium is parametrized to the demographic data assumed to prevail after 2100, and corresponds to the balanced growth path toward which the economy ultimately converges. 2. Compute the exogenous path of the labor supply by aggregating the labor endowments across adults. (cid:110) (cid:111)T 3. GuessthepathR(0) = R(0) andotheraggregateobjects,whereT isaperiodsufficientlyfar t t=1900:Q1 into the future (in our case, 2399:Q4) that the economy has essentially converged back to its long-run balanced growth path. As a simple way to initialize R(0), we use a linear interpolation of the 1900 and 2100balancedgrowthequilibriumvaluescomputedinthestepabove. Wethenprojectthe2100values through the end of the simulation period. We initialize the paths of all other aggregate objects, such as bequests, factor prices, etc., using similar linear interpolations. 4. Given the paths of the aggregate objects and initial distribution of capital holdings, solve the problem of the cohort-representative adults who were alive at the beginning of 1900. Then, solve the problem of each cohort-representative adult entering adulthood in subsequent periods. 5. Aggregate the individual decisions of firms and cohort-representative adults period by period to compute new paths of the aggregate objects, including for R˜(0). (cid:13) (cid:13) 6. Calculate the Euclidian distance between the two interest rate paths. If (cid:13)R˜(0)−R(0)(cid:13) < ε, where (cid:13) (cid:13) ε is positive but sufficiently small to claim numerical convergence, then you have found a numerical (cid:13) (cid:13) solution. If (cid:13)R˜(0)−R(0)(cid:13)≥ε, then update the interest rate path using (cid:13) (cid:13) R(0) =ωR˜(0)+(1−ω)R(0), where ω ∈(0,1]. Similarly update the paths of other aggregate objects, then return to step 4. In our main results, we select the discount factor (that is, β) such the model’s average equilibrium real rate in the 1980s coincides with the corresponding estimate from Johannsen and Mertens (2016). We implement the search for this discount factor using a bisection method. 16

C.5 Solving the problem of the cohort-representative adult Themostcomputingintensiveelementisstep4becauseitrequiressolvingeachcohort-representativeadult’s life-cycle problem. The computation of a solution is made simpler by two sets of assumptions. First, the problemissolvedunderperfectforesightofdemographicandaggregatevariables. Second,asweshowbelow, the problem of the adult is linear in leads and lags of her capital holdings conditional on the paths of the aggregate objects and demographics. This linearity allows us to use linear algebra techniques to quickly derive a solution. Before we do so, we scale the equilibrium objects so that we have a stationary equilibrium along the balanced growth path. Let the generic variable X˜ =X /Z , where Z is labor augmenting technology. We begin by scaling the t t t t representativefirm’soptimalityconditions—Equations(8)and(9)—thatrelatetherealrentalrateofcapital to its marginal product, (cid:32) (cid:33)α−1 K˜ R =α t , t L and the real wage rate to the marginal product of labor, (cid:32) (cid:33)α K˜ W˜ =(1−α) t . t L The intertemporal Euler equation is C˜ −C¯ (cid:18) Z (cid:19) a C + ˜ a 1 , , t t+ − 1 C¯ =(βΓ a,t,t+1 )ν 1 (R t+1 +1−δ)ν 1 Z t+ t 1 , and the scaled budget constraint is (cid:16) 1+(cid:15)ν 1 (n a,t )1+η− ν 1(cid:17) C˜ a,t −(cid:15)ν 1 (n a,t )1+η− ν 1 C¯+n a,t C¯c+ Z Z t+1K˜ a+1,t+1 = (cid:16) K˜ a,t +φΞ˜ t (cid:17) (R t +1−δ)+e a,t W˜ t . t Replacing the consumption terms in the intertemporal Euler equation, we get, (cid:18) (cid:19) Z Z − t+2K˜ + R +1−δ+Ω t+1 K˜ −Ω (R +1−δ)K˜ Z a+2,t+2 t+1 a+1,t+1 Z a+1,t+1 a+1,t+1 t a,t t+1 t =Ω φ(R +1−δ)Ξ˜ −φ(R +1−δ)Ξ˜ a+1,t+1 t t t+1 t+1 +(1−Ω ) (cid:0) C¯+n C¯c(cid:1) +Ω e W˜ −e W˜ , a+1,t+1 a,t a+1,t+1 a,t t a+1,t+1 t+1 where Ω a+1,t+1 ≡(βΓ a,t,t+1 )ν 1 (R t+1 +1−δ)ν 1 (cid:32) 1+ 1 (cid:15) + ν 1 (cid:15) ( ν 1 n a (n + a 1 , , t t+ )1 1 + )1 η + − ν 1 η− ν 1(cid:33)(cid:18) Z Z t+ t 1 (cid:19) . 17

We note that the above expression is linear in the leads and lags of individual capital holdings conditional on aggregate variables and demographic factors. This observation allows us to express the equilibrium conditions of a cohort-representative adult aged a in period t as a system of linear equations, M(a,t)K˜ (a,t) = Ψ(a,t). The vector of length (A−a) containing the scaled capital holdings of the representative adult is   K˜ a+1,t+1     K˜ a+2,t+2     K˜ (a,t)=   ...   .      K˜   A−1,t+A−a−1    K˜ A,t+A−a The entries of the squared matrix M(a,t)=[m (a,t)] are i,j  (cid:16) (cid:17)  R t − + Ω j +1−δ+Ω a (R +j,t+j +1 Z Z t − + t+ j δ − j ) 1 if i j f = j = i+ i 1 a+j+1,t+j+1 t+j m (a,t)= i,j  Z Z t+ t 0 + j+ j 1 i o f t j he = rw i i − se 1 . The elements of the vector Ψ(a,t)=[ψ (a,t)] are j   +Ω a+1,t+ + 1 Ω ( ( 1 a + R + − Ω t 1 + , a Ω t + + a 1 1 1 + , e − t 1 a + , , t 1 δ t + W ) ( ˜ 1 R φ ) t Ξ˜ (cid:0) t − C t + ¯ − e + 1 a+ ( − n R 1 a , t δ t + + + ) 1 1 1 K , ˜ W t + ˜ + a t , 1 t 1 + C¯ 1 − c(cid:1) δ)φΞ˜ t+1 if j =1 ψ (a,t)= j  +Ω a+j,t Ω +j a+ (R j, + t t + + ( j j 1 e − a − 1 + + j Ω − 1 1, − t+j δ − ) 1 φ ) W Ξ (cid:0) ˜ ˜ C t t ¯ + + + j j − − n 1 1 − − ( e R a+ t+ j, C j t ¯ + + c j (cid:1) W 1 ˜ t − +j δ)φΞ˜ t+j if 1<j ≤A−a . a+j,t+j a+1,t+1 C.6 An adult-equivalent specification WeconsideravariantofthepreferencesinBrowningandEjrnæs(2009)inwhichtotalhouseholdconsumption increasesintheagesofthechildren. Themotivationforconsideringthisalternativespecificationistwo-fold. First, the predicted peak in household consumption in our baseline model occurs earlier in the life cycle than it does in the data; the adult-equivalent consideration recognizes that older children tend to require 18

moreresourcesforschooling,entertainment,etc,thuspostponingthepeakinconsumptionlaterinthelifeof the household. Second, this specification has encountered empirical success in terms of generating realistic life-cycle dynamics. Browning and Ejrnæs (2009) show that an adult-equivalent specification can match pre-retirementconsumptiondynamicsofadultcouples—includingthesharpriseintheearlyyears—without recoursetoanyothercontributingfactors(whichincludeliquidityconstraints,rulesofthumbs,precautionary motives, and non-separability between consumption and the supply of labor). C.6.1 Household problem As was the case with the main version of the model, we consider a more general specification of the utility function that features subsistence levels in consumption. This specification is also consistent with balanced growth. One difference with the baseline specification of the utility function is that the consumption of adults and their dependent children can no longer be expressed in an additive manner. Household utility is given by A (cid:88) −a βsΓ (cid:0) C a h +s,t+s e−fa+s,t+s −Z t+s C¯(cid:1)1−ν efa+s,t+s, (10) a,t,t+s 1−ν s=0 where C¯ is a subsistence level of consumption per adult equivalent family members. Households are subject to a sequence of budget constraints Ch +K =(K +φΞ )(R +1−δ)+e W . a+s,t+s a+s+1,t+s+1 a+s,t+s t+s t+s a+s,t+s t+s The function f =δ ln(m ) allows for economies of scale in consumption based on the number of adulta,t h a,t equivalent members of the household, denoted m .2 We compute the number of adult-equivalent members a,t of the household as (cid:88) Ac m =1+ n g(σ), a,t a,σ,t σ=0 where A is the maximum age before a dependent child becomes financially independent in the model and c n is the number of children aged σ who are dependent from a representative adult aged a in period t. a,σ,t The function g captures how the age of the children translates into adult-equivalent household members. 2Althoughtheutilityfunctionisspecifiedintermsoftotalhouseholdconsumption,weretainthedependenceonthesurvival rateofadults. Currentandprojectedmortalityratesareratherlowfortheadult’schild-rearingyears,sothatΓa,t,t+1 remains closetooneuntillateinlifewhenhouseholdconsumptionscoincideswithadultconsumption. Thisassumptionislessinnocuous fortheearlydecadesofthesamplewhenlifeexpectancywasappreciablyshorter. 19

Similar to Browning and Ejrnæs (2009), we assume that (cid:18) σ (cid:19) (cid:18) σ (cid:19)2 (cid:18) σ (cid:19)3 g(σ)=µ +µ +µ +µ . 0 1 A 2 A 3 A c c c This function has g(0) = µ and our calibration will be such that it is increasing in σ. We further require 0 that µ =1−µ −µ −µ so that g(σ)→1 as a dependent child approaches adulthood. 3 0 1 2 C.6.2 Equilibrium conditions under alternative adult-equivalent specification The objective is to maximize Equation (10). We can write the Lagrangian as L= A (cid:88) −a βsΓ (cid:26)(cid:0) C a h +s,t+s e−fa+s,t+s −Z t+s C¯(cid:1)1−ν efa+s,t+s −λ (cid:16) Ch +K a,t,t+s 1−ν t+s a+s,t+s a+s+1,t+s+1 s=0 (cid:17)(cid:27) −(K +φΞ )(R +1−δ)−e W . a+s,t+s t+s t+s a+s,t+s t+s Optimization requires C˜ a h +1 C˜ ,t a h + ,t 1 e e − − f f a a , + t 1 − ,t+ C 1 ¯ −C¯ =(βΓ a,t,t+1 )ν 1 (R t+1 +1−δ)ν 1 (cid:18) Z Z t+ t 1 (cid:19) ≡Ω a+1,t+1 , where we have scaled the variables. Scaling the budget constraint and replacing consumption, we get (cid:18) (cid:19) Z −Ω e−fa,t(R +1−δ)K˜ + (R +1−δ)e−fa+1,t+1 +Ω e−fa,t t+1 K˜ a+1,t+1 t a,t t+1 a+1,t+1 Z a+1,t+1 t Z − t+2e−fa+1,t+1K˜ =Ω (R +1−δ)e−fa,tφΞ˜ −(R +1−δ)e−fa+1,t+1φΞ˜ Z a+2,t+2 a+1,t+1 t t t+1 t+1 t+1 +Ω e−fa,te W˜ −e e−fa+1,t+1W˜ +(1−Ω )C¯. a+1,t+1 a,t t a+1,t+1 t+1 a+1,t+1 The solution to the cohort-representative household continues to be represented by a linear system of equations, M(a,t)K˜ (a,t)=Ψ(a,t). The elements of the (S−a)×(S−a) matrix of coefficients are m j,j (a,t)=(R t+j +1−δ)e−fa+j,t+j +Ω a+j,t+jZ Z t+ t+ j− j 1 e−fa+j−1,t+j−1 if 1≤j ≤S−a m j+1,j (a,t)=−Ω a+j+1,t+j+1 e−fa+j,t+j(R t+j +1−δ) if 1≤j <S−a . m j,j+1 (a,t)=−Z Z t+ t+ j+ j 1e−fa+j,t+j if 1≤j <S−a m (a,t)=0 all other entries j,j(cid:48) 20

The elements of Ψ(a,t) are   + Ω a (1 +1 − ,t+ Ω + − 1 a Ω e ( + − R a 1 f + , t a t + 1 + ,t , 1 1 t e + ) + a 1 , C t ¯ 1 ( W˜ R + − t t Ω − δ + ) a e + 1 e − − 1 − f , f t a + a + δ + 1 ) 1 1 , ( , e t t + R − + 1 1 t f e φ a + a , Ξ t ˜ + φ 1 t 1 + Ξ˜ , − t 1 + t 1 δ W ) ˜ K˜ t+ a 1 ,t if j =1 ψ (a,t)= . j  +Ω a Ω + a j + ,t j + , − t j + ( ( j R R e − a t t + + + e j j a + j − − + + 1 ( 1 j 1 , , + t 1 t + + − − j 1 j e − Ω − δ − 1 ) e f δ e a − ) + − f j e f a , − t a + + + j f j − ) j a W , + 1 t C ˜ ¯ + , j t t − j + + φ 1 j , j − Ξ t ˜ + 1 t W j + ˜ − j 1 t+ φ j Ξ˜ − t 1 +j−1 if 1<j ≤A−a a+j,t+j 21

References Bell, F. C. and M. L. Miller (2005). Life Tables for the United States Social Security, 1900–2100. Social Security Administration–Office of the Chief Actuary, Actuarial Study (120). Browning, M. and M. Ejrnæs (2009). Consumption and Children. The Review of Economics and Statistics 91(1), 93–111. Fernald, J. (2014). A Quarterly, Utilization-Adjusted Series on Total Factor Productivity. Federal Reserve Bank of San Francisco working paper 19. Gordon, R. J. (2015). Secular Stagnation on the Supply Side: U.S. Productivity Growth in the Long Run. Digiworld Economic Journal (100). Hamilton, B. E. and C. M. Cosgrove (2010). Central Birth Rates by Live-Birth Order, Current Age, and Race of Women in Each Cohort from 1911 through 1991: United States, 1960–2005. Table 1. Hyattsville, MD: National Center for Health Statistics. Released: June 30, 2010. Johannsen, B. K. and E. Mertens (2016). The Expected Real Interest Rate in the Long Run: Time Series Evidence with the Effective Lower Bound. FEDS Notes. Washington: Board of Governors of the Federal Reserve System, February 9, 2015–077. Mathews, T. and B. E. Hamilton (2005). Trend Analysis of the Sex Ratio at Birth in the United States. National Vital Statistics Reports 53(20). Shackleton, R. (2013). Total Factor Productivity Growth in Historical Perspective. Congressional Budget Office, Working paper 1. 22

Table 1: Data sources: civilian non-institutional population by age group Age Total Men Women 16 to 19 years LNU00000012 LNU00000013 LNU00000014 20 to 24 years LNU00000036 LNU00000037 LNU00000038 25 to 29 years LNU00024932 LNU00000163 LNU00000326 30 to 34 years LNU00024933 LNU00000171 LNU00000332 35 to 39 years LNU00024934 LNU00000172 LNU00000333 40 to 44 years LNU00024935 LNU00000179 LNU00000338 45 to 49 years LNU00024936 LNU00000181 LNU00000340 50 to 54 years LNU00024937 LNU00000188 LNU00000345 55 to 59 years LNU00000094 LNU00000189 LNU00000346 60 to 64 years LNU00000096 LNU00000197 LNU00000352 65 to 69 years LNU00024938 LNU00000203 LNU00000358 70 to 74 years LNU00024941 LNU00024939 LNU00024940 75+ years LNU00024942 LNU00015346 LNU00015349 25 to 34 years LNU02000089 LNU02000164 LNU02000327 35 to 44 years LNU02000091 LNU02000173 LNU02000334 45 to 54 years LNU02000093 LNU02000182 LNU02000341 55 to 64 years LNU02000095 LNU02000190 LNU02000347 65+ years LNU02000097 LNU02000199 LNU02000354 Source: U.S. Bureau of Labor Statistics. 23

Table 2: Data sources: civilian employment by age group Age Total Men Women 16 to 19 years LNU02000012 LNU02000013 LNU02000014 20 to 24 years LNU02000036 LNU02000037 LNU02000038 25 to 29 years LNU02024932 LNU02000163 LNU02000326 30 to 34 years LNU02024933 LNU02000171 LNU02000332 35 to 39 years LNU02024934 LNU02000172 LNU02000333 40 to 44 years LNU02024935 LNU02000179 LNU02000338 45 to 49 years LNU02024936 LNU02000181 LNU02000340 50 to 54 years LNU02024937 LNU02000188 LNU02000345 55 to 59 years LNU02000094 LNU02000189 LNU02000346 60 to 64 years LNU02000096 LNU02000197 LNU02000352 65 to 69 years LNU02024938 LNU02000203 LNU02000358 70 to 74 years LNU02024941 LNU02024939 LNU02024940 75+ years LNU02024942 LNU02015346 LNU02015349 25 to 34 years LNU00000089 LNU00000164 LNU00000327 35 to 44 years LNU00000091 LNU00000173 LNU00000334 45 to 54 years LNU00000093 LNU00000182 LNU00000341 55 to 64 years LNU00000095 LNU00000190 LNU00000347 65+ years LNU00000097 LNU00000199 LNU00000354 Source: U.S. Bureau of Labor Statistics. 24

Table 3: Data sources: unemployment rate by age group Age Total Men Women 16 to 19 years LNU04000012 LNU04000013 LNU04000014 20 to 24 years LNU04000036 LNU04000037 LNU04000038 25 to 29 years LNU04024932 LNU04000163 LNU04000326 30 to 34 years LNU04024933 LNU04000171 LNU04000332 35 to 39 years LNU04024934 LNU04000172 LNU04000333 40 to 44 years LNU04024935 LNU04000179 LNU04000338 45 to 49 years LNU04024936 LNU04000181 LNU04000340 50 to 54 years LNU04024937 LNU04000188 LNU04000345 55 to 59 years LNU04000094 LNU04000189 LNU04000346 60 to 64 years LNU04000096 LNU04000197 LNU04000352 65 to 69 years LNU04024938 LNU04000203 LNU04000358 70 to 74 years LNU04024941 LNU04024939 LNU04024940 75+ years LNU04024942 LNU04015346 LNU04015349 Source: U.S. Bureau of Labor Statistics. 25