2.1 Data

Population data come primarily from the Human Mortality Database (Reference Database2019), (HMD). I use their data on population by age to construct various population shares across the age distribution. In addition, I use US Census National Intercensal Tables to extend the US demographic series to 1900. I consider two methods for controlling for the population age structure. The first is to use the working age population, defined as the number of individuals aged 20-64 as a percentage of the overall population. The second relies on demographic variables that cover population shares over the full age distribution as in Fair and Dominguez (Reference Fair and Dominguez1991).

In Figure 2a, I plot the working age population for the full panel of countries used in my estimations. This highlights an advantage of such a long sample. While a great deal of work is limited to either a single country, or to panels that include only the period since the baby boomer generation entered the workforce, I can take advantage of a large amount of variation both across countries, as well as in the period before the boomer cohort. While all of the countries in this sample have some sort of baby boom, the severity and timing are different.

Figure 2. Working age populations and excess money growth.

I merge this population data with the macrohistory dataset of Jordà et al. (Reference Jordà, Schularick and Taylor2017) (JST) which provides information on price levels and a wide range of macroeconomic variables of interest. With the addition of Jordà et al. (Reference Jordà, Knoll, Kuvshinov, Schularick and Taylor2019), this also contains useful information on asset returns and credit market conditions. It will be crucial to control for a broad set of macroeconomic and financial controls from these data, which have been shown to be important. While the dataset contains 18 developed countries, my preferred specification drops Ireland and Canada as they lack some of the financial variables used as controls. For the remaining 16 countries,Footnote ⁷ data go back as far as 1870 with information on population age, macroeconomic controls, and financial variables. The shortest panel is 32 years (Belgium) due to lack of data on broad money, but all other series are more than 50 years, with the average in the panel spanning 85 years of data.

A simple quantity theory of money states the growth rate of inflation should be the growth rate of money less growth in output and velocity. I take as my measure of excess money growth the difference between broad money growth rates and growth of output. Assuming a constant velocity, the quantity theory then suggests one-to-one relationship. Figure 2b shows my measure of excess money growth. It is important to note that part of the weakening of money is due to mismeasurement issues around official aggregates. Belongia and Ireland (Reference Belongia and Ireland2016) show that by using the Divisia monetary aggregates of Barnett (Reference Barnett1980), there has been a lengthening of the lag time for money to affect output and prices since the early 1980s, but that the relationship remains strong. This is in line with Barnett et al. (Reference Barnett, Offenbacher and Spindt1984), and many others who show that these Divisia aggregates generally outperform official measures. While these measures are not available for the long run panel I study, it seems plausible that they might reflect a faster convergence to the quantity theory than I find using official measures of broad money from the JST data. I will show for a subsample of my data (the USA from 1967) that my age-dependent estimates are similar when using these more appropriate measures money, though consistent with this literature such money growth measures will have a much stronger estimated impact on inflation.

2.2 Estimation strategy

I wish to study the impact of excess money growth rates on cumulative inflation conditional on the state of demographics. As such, I estimate cumulative impulse response functions (IRFs) of the change in price level to the excess money growth rate. I use the reduced form local projections method of Jordà (Reference Jordà2005), which has been shown in Plagborg-Møller and Wolf (Reference Plagborg-Møller and Wolf2021) to estimate the same IRFs as an equivalently specified VAR, with flexibility regarding a wide range of structural assumptions. In my case, the local projection is one with a structural recursive identification with money growth in prior period affecting contemporaneous values of my outcome and controls. This standard impulse response, not taking into account state dependence, is shown in equation (1):

(1) \begin{equation} \pi _{i,t+h}- \pi _{i,t-1} = \mu _{i}^{h} + m_{i,t-1}\beta ^{h} + (x_{i,t}-\bar{x})\gamma ^{h} + D_{i,t}\gamma _{d}^{h} + \epsilon _{i,t+h}\textit{ ; for } h = 0,\ldots,H. \end{equation}

The outcome, $\pi _{i,t+h}- \pi _{i,t-1}$ , is the cumulative growth rate of price levels from the current period to the $h$ annual horizon. The key independent variable, $m_{i,t-1}$ is excess money growth, defined as growth in M2 money less growth in output. Estimates of horizon $\beta ^{h=0}$ are therefore 1-year relationship between changes in excess money growth rate the year prior and inflation in the current year. Estimates at this horizon might be considered the short run impacts of these policies, while later horizons reflect a medium- to long-term relationship. These are correlations with money and inflation, similar to those estimated in Belongia and Ireland (Reference Belongia and Ireland2016), who study the inflation and output response to growth of Divisia money aggregates using a recursive VAR identification. This is far from a causally identified shock, and a narrative instrument would be preferred and easily incorporated into this framework. However, such an instrument is not readily available for the panel of countries and time period studied. Although these results suggest a strong empirical relationship, I caution against any causal interpretation.

I include in the vector $x_{i,t}$ for a broad set of macroeconomic and financial controls. In order to have comparability with estimates using the Kitagawa-Blinder-Oaxaca (KBO) decomposition described below, all controls are taken as deviations from their sample mean. In the baseline specification, I populate $x_{i,t}$ with growth rate of real output, growth of real consumption per capita, the change in the ratio of investment to GDP, change in the current account-to-GDP ratio, change in government debt-to-GDP ratio, real equity prices, growth in total loans to the household sector, population growth rates, and the change in the short-term bill rate. Two lags, along with contemporaneous values, are included in the regression specification for each control. Additionally, three lags of current inflation are included. I include dummy variables for both world wars, which are interacted with demographics to avoid clouding estimates with large swings in some age groups related to wartime periods. I also include a measure of global GDP growth, here taken as the aggregate GDP growth of the countries in my sample, as a means of controlling for global time varying trends. This captures much of the variation of time fixed effects in the same way as similar long run studiesFootnote ⁸ on this data, without the need to add an additional 143 controls.

Inclusion of time fixed effects gives qualitatively identical results for estimated demographic changes to policy transmission. They have larger error bands and a smaller, though still sizable, response of inflation to money growth over reported horizons. Time fixed effects can bias estimates in the presence of unobserved time trends that are not uniform across countries. Such trends are easy to imagine in this context. For example, structural changes to monetary policy framework, which are likely occur across this sample, with differential implementation and timing. I opt to consider my baseline estimates without such effects. They are included in Appendix D for reference. I include country fixed effects with $\mu _{i}^{h}$ . The term $D_{i,t}$ represents the set of controls for population age structure. This is either the working age population (WAP) or constructed as in Fair and Dominguez (Reference Fair and Dominguez1991) as polynomial controls for the entire population age distribution. I describe construction of the latter below.

2.3 Kitagawa-Blinder-Oaxaca heterogeneous estimates

In Cloyne et al. (Reference Cloyne, Jordà and Taylor2020), the standard local projection framework is extended to incorporate an estimation strategy well known in the microeconomic literature and first widely used in work by Kitagawa (Reference Kitagawa1955), Blinder (Reference Blinder1973), and Oaxaca (Reference Oaxaca1973). They propose this method as a way of parsimoniously dealing with heterogeneity of treatment impacts of policy implementation. Introducing all control variables as demeaned relative to the sample population, and including them in the local projection directly (as above) and also as interactions with the policy variable allows for decomposition of estimates into three terms: average estimates for the policy on the outcome, the average estimates for controls on the outcome, and the change in the policy estimates when controls move away from their sample means.

I note, as they emphasize in their paper, that without a clear identification of independent variation along various dimensions of heterogeneity, it is not possible to give a causal interpretation of this inherently partial equilibrium approach. The estimates coming out of the interaction between a given control and the policy shock require identification of not only the policy, but also the control. However, such concerns are widely present in the large existing literature on state-dependent policy, which typically split samples across discrete states for estimation without clearly dealing with the additional identification issues arising from potential selection across states. Further, the identification challenge outlined for the policy shock above remains, and while estimates are comparable to similar recursively identified VARs, the KBO approach used here does not provide any further identification, but rather a decomposition of the empirical relationship between the policy and inflation across all states of the control set.

The KBO estimates of IRFs by local projections are shown in equation (2). I split out my vector of demographic variables $D_{i,t}$ from other controls $x_{i,t}$ for ease of reference, but they are treated identically.

(2) \begin{equation} \begin{split} \pi _{i,t+h}- \pi _{i,t-1} &= \mu _{i}^{h} + m_{i,t-1}\beta ^{h} + (x_{i,t}-\bar{x})\gamma ^{h} + m_{i,t-1}(x_{i,t}-\bar{x})\theta ^{h} \\[5pt] &+ D_{i,t}\gamma _{D}^{h} + m_{i,t-1}D_{i,t}\theta _{D}^{h} + \epsilon _{i,t+h}\textit{ ; for } h = 0,\ldots,H. \end{split} \end{equation}

Cloyne et al. (Reference Cloyne, Jordà and Taylor2020) show that under fairly standard assumptions the local projections estimate from equation (2) can be estimated by any usual method. Here I use OLS, as is conventional for local projections. All residuals from the estimations below have covariance stationary error processes. While all additional controls used are stationary I(0) processes, the demographic variables (and cumulative inflation itself) are I(1). I discuss in Subsection 2.5 that population age variables are cointegrated with inflation, consistent with prior evidence in the literature. This is important for my results as it implies that my estimates of $\gamma ^{D}$ and $\theta ^{D}$ should be consistent, and differencing demographics to create a stationary I(0) process would be inappropriate in this context.

Equation (2) decomposes estimates into three types of results. A direct relationship between the policy variable and inflation, an indirect relationship between policy (conditional on the state of underlying controls), and composition effects of the changing control set on the outcome itself. Estimates from changing composition of the control set on outcomes directly are given by $\gamma ^{h}$ . For instance, older populations may have an impact on the level of inflation directly, as has been shown in recent work by Juselius and Takáts (Reference Juselius and Takáts2021) and is discussed broadly in Goodhart and Pradhan (Reference Goodhart and Pradhan2020). The direct relationship of the policy variable is given by $\beta ^{h}$ . The KBO approach is such that this is the average effect in sample and is comparable to an equivalently specified estimate without state dependence. The indirect relationships are estimated via $\theta ^{h}$ and are by construction zero when the policy variable is zero or when controls are at their mean $\bar{x}_{i,t}$ . A change in any control from its mean in the population has a $\gamma ^{h}$ estimated impact on inflation itself, while also affecting the size of the estimated policy transmission through $\theta ^{D}$ .

The advantage of this specification is that it provides straightforward way to quantify heterogeneity of policy, while providing clarity in the underlying assumptions embedded in existing state-dependent literature. Much of the literature estimating such effects will opt to estimate coefficients on split samples.Footnote ⁹ While it might be possible to find a somewhat compelling instrument for monetary policy interest rate shocks in some contexts, such as the well known Romer and Romer (Reference Romer and Romer2004) narrative series, there is not, to my knowledge, a readily available instrument for exogenous change in monetary aggregates suitable for this larger panel. Further, identification of these shocks would still require causal variation in demographics as mentioned above, which is likely a more challenging task. For this reason, although I will work hard to show that these results are fairly robust, I reiterate my caution against a causal interpretation.

Given the importance of the question at hand I believe that exploring this empirical relationship is worthwhile. Much of the conventional wisdom around the structural shift in money’s effect on prices also generally lacks clean empirical causal identification. Because of this, the recent empirical advancements in the literature have focused more on identification of interest rate policy shocks. Ignoring money growth may seem reasonable if the underlying structural shift is permanent, as one might expect in a story of policy regime change. I show that demographic channels provide another plausible, and not mutually exclusive, explanation of the decline in importance of monetary aggregates during the great moderation, with different implications going forward as populations continue to evolve.

2.4 Controlling of population structure: An augmented Fair and Dominguez (1991) methodology

Before moving to results, I must first describe how I will control for demographics in equation (2) using the entire population age structure. One of my two main demographic specifications will be to use a methodology similar to that of Fair and Dominguez (Reference Fair and Dominguez1991), who fit the estimated effects for population age shares across the entire age distribution with a low-order polynomial. A naive regression might split the population into $J$ bins, and add all of these population age shares to the estimating equation. This is problematic for two reasons. The first problem is that all of the population shares sum to one, making it impossible to jointly estimate all of the coefficients along with a regression constant. Additionally, as the number of age bins increases, the population shares become increasingly colinear. This latter point often results in significant parameter instability as one splits the age structure into finer and finer groups. The work of Fair and Dominguez (Reference Fair and Dominguez1991) suggests two assumptions:

1. 1. Letting $\alpha _{j}$ be the regression coefficient on population share $p_{j,i,t}$ of age group $j$ in country $i$ and time $t$ . Assume that all of these coefficients across the age distribution sum to zero. In other words:

   (3) \begin{equation} \sum _{j}^{J}\alpha _{j} = 0. \end{equation}

2. 2. Assume that the estimated age coefficients $\alpha _{j}$ can be fitted with a $K$ order polynomial. In other words:

   (4) \begin{equation} \alpha _{j} = \sum _{k}^{K}\gamma _{k}j^{k}. \end{equation}

The first point makes the coefficients on $J$ perfectly colinear population shares estimable without dropping the constant from the equation. The second forces that the estimated coefficients across the age distribution to be relatively smooth and reduces the number of parameters that need to be estimated. How much variation is allowed across the life cycle depends on the order of the polynomial, as increasing the degree of the polynomial allows for more turning points in the estimated coefficients across age. A notable, but not substantial, difference in my approach is that instead of using population age shares to construct these variables, I use these shares less their population mean, consistent with the KBO methodology. By putting these deviations of population age shares from long run means into equation (1) and applying the two assumptions above, I construct demographic variables that estimate the $\gamma _{k}$ parameters of the fitted polynomial. These are given by:

(5) \begin{equation} D_{k,i,t} = \left [\sum _{j}^{J}(p_{j,i,t}-\bar{p}_{i,t})j^{k} \right ] \end{equation}

I use a fourth-order polynomial in my baseline specification and so create $D_{1}-D_{4}$ to include in my analysis. I provide evidence that this choice is the best fit in Appendix C. For a detailed derivation of equation (5), as well as a discussion of the Fair and Dominguez (Reference Fair and Dominguez1991) controls see Appendix A.

2.5 Use of non-stationary demographic controls

Most of the controls used above are growth rates or first differenced variables, and all non-demographics controls pass an augmented Dickey-Fuller test for stationarity. These variables are generally included with the intention of accounting for the shorter run factors that may be important for determining inflation. However, my demographic controls described above are non-stationary, $I(1)$ series. There may be some concern that results below represent a spurious relationship with cumulative inflation, which though stationary in changes at very short horizons is itself $I(1)$ . Montiel Olea and Plagborg-Møller (Reference Montiel Olea and Plagborg-Møller2021) show that lag-augmented local projections provide valid inference over long horizons with both stationary and non-stationary data. Thus as a first protection against spurious relationships, all estimations will include three lags of inflation as controls. The number of lags used here, or indeed specifications with no lagged inflation, has little impact on my coefficients of interest.

Additionally, I find that the working age population as well as the set four of demographic controls described in Section 2.4 are cointegrated with cumulative inflation, my outcome of interest. I show these test results in Appendix B. This is consistent with existing work (Bobeica et al. Reference Bobeica, Nickel, Lis and Sun2017) who study the cointegrated relationship between inflation and age structure. In the presence of a cointegrating relationship, the estimations performed in equation (2) should be super-consistent, with coefficients of interest converging faster than they would if the series were stationary. This suggests that these demeaned levels are more appropriate than a stationary $I(0)$ difference of the demographic controls. As a final check, for all estimated models in the paper I run an augmented Dickey-Fuller test on predicted residuals post-estimation with a unit root rejected in all cases with extremely strong ( $\lt 0.0001$ ) significance.

4.1 Working age population

I first present results for working age population (WAP). While this is a crude estimate of the age structure, and evidence from Figure 4 might suggest that it misses important turning points, it allows for quite simple interpretation and therefore provides useful introduction to population estimates and their relationship with monetary transmission. In Table 2, I show the impulse response function at up to 5-year horizons for these estimates. The direct coefficient of lagged excess money growth accumulated over this whole horizon (0.51) is of similar magnitude, though smaller than the equivalent estimate in Table 1 (0.63), after adding interactions of money growth with WAP, as well as with all other covariates.

Table 2. Response of inflation to excess money growth: working age population

Table reports estimations of IRFs for money growth under working age population specification of population age. Heteroscedastic standard errors are in parenthesis with $ ^{\ast}p \lt 0.10,\; ^{\ast\ast}p \lt 0.05,\; ^{\ast\ast\ast}p \lt 0.01$ . $X_{i,t}$ significance. Regressions include three lags of the dependent variable as well two lags and contemporaneous values of all additional controls and their interactions with lagged money growth.

The direct coefficients on the WAP now tell a different story. The impact of age composition is now positive and quite large. This seems contradictory, as theories connecting WAP to inflation through labor market tightness, such as those put forward by Goodhart and Pradhan (Reference Goodhart and Pradhan2020), suggest a negative effect. A potential explanation could be that hump shaped life cycle consumption, peaking late in working life might explain some inflationary relationship of the WAP. It also seems plausible that some of the estimated effects of age that many have found could work through indirect channels such as those I find here. As above, any implications for inflation that vary throughout working life, as in Figure 4, could make this relationship difficult to cleanly estimate as it cannot distinguish between a large WAP with many 20–40 years old and one with high concentration in late career. In Appendix C Table 7, I also show that while the signs of these coefficients are fairly consistent, their significance depends on inclusion of controls (especially for the direct relationship), suggesting that one should be wary of explanations of aging and inflation that use WAP alone. In the following section, where I use the entire age distribution, my results for the composition effect of age structure on inflation directly will be more or less in line with the inflationary aging hypothesis in the literature while also being strongly robust to choice of control set.

How large are these transmission mechanisms? Figure 5 shows the effect of increasing/decreasing the demeaned WAP by one-half and one standard deviations. The thickest line/markers represents an increase of working age population to 3.93 percentage points above its long run mean, while the smallest line/markers shows a symmetric decrease. The point estimates suggest that a high working age population (thicker curves/markers) brings transmission of money growth down substantially. Taking the USA as an example, the WAP has been persistently above its long run mean starting from 1977 and remains so today (though notably not persistently increasing over that period). In the latest year of the sample, 2017, the working age population in the USA is about 3.06 percentage points larger than the historical average of 56.7% in the sample, but has begun falling somewhat rapidly off a peak that occurred in 2011.

Figure 5. Heterogeneous effect of excess money growth and inflation: WAP.

These are better understood in context by plotting similar variation to Figure 5, but instead feeding empirical values for a given country rather than standard deviations. In Figure 6, I present this age decomposition for three countries: the USA, France, and the United Kingdom. Each left-hand panel contains impulse responses of inflation to money growth over a 5-year window for one of these countries. For reference, the direct money growth relationship, which is an average estimate across the sample, is plotted, along with conditional estimates for the age distribution of that country in three historical periods: 1970, 1990, and 2017, and two future projections: 2030 and 2050. Data on future demographics come from UN (2019) projections. I also plot the evolution of the demeaned working age population in each country in the right-hand panels.

Figure 6. Response of inflation to excess money growth, KBO decomposition: Working age population. Notes: Direct estimates are the average estimate of excess money growth on inflation when population is at long run sample mean. Conditional IRFs reflect this relationship conditional on age distribution of each country in a given year.

In Figure 6b, I show the demeaned value of the WAP for the USA to provide a clear idea of what drives the conditional results in Figure 6a. In particular, the effect of the baby boomer cohort, which represented an unprecedented (in this sample) increase of young dependents in the 1950s and 60s, pushed WAP down in the post-war period with a dramatic reversal over the course of the great moderation. In the decades to come, a return toward the mean WAP will imply gradual return to average estimated money pass through to inflation. One thing worth noting is that a persistent decrease in young dependents in the years following the boomer cohort both exaggerated the rise of WAP during the great moderation and slows the speed of decline as boomers enter retirement.

Similar historical evidence can be seen in France and the UK, though in France working age population has already reached the mean value in the sample, and both countries will see declines much more rapidly than the USA. Increasing longevity will provide negative pressure on the working age population for all of these countries going forward, further increasing estimated money pass through to inflation implied by these estimates.

These results give suggestive evidence that demographic channels increased transmission of money growth to prices in the 1970s, weakened it over subsequent years, and may increase it again as boomers fully leave the WAP. However, the WAP is an imprecise measurement of aging. Using just one statistic for population age structure assumes a number of things. First, it forces any transmission effects to be constant across the working life. We know from work such as Wong (Reference Wong2019) that this is unlikely to be the case with interest rate policy, and Figure 1 suggests it is unlikely the case for life cycle variation in money demand. Additionally, using WAP forces old and young dependents to have symmetric effects. This seems unlikely, especially in the face of lengthening of time-span in retirement, which itself might alter the consumption behavior of older retirees.

4.2 Estimates over the full age distribution

I now present estimates that control for the entire age distribution. The goal is to allow for a more flexible approach than is allowed by using a single variable to capture population age structure dynamics. I use a fourth-order ( $k=4$ ) polynomial to fit 16 population age shares in my baseline model. In Appendix C, I show that this selection likely provides the best trade-off between improved in-sample fit (based on AIC) as well as issues associated with overfitting age coefficients which is common among Fair and Dominguez (Reference Fair and Dominguez1991) controls. The estimates of impulse response functions controlling for this specification are in Table 3. As before, I present the estimation including all covariates, country fixed effects, and interactive effects for the four demographic variables and all other controls.

Table 3. Response of inflation to excess money growth: full age distribution

Table reports estimations of IRFs for money growth under a fourth-order polynomial specification. Clustered standard errors are in parenthesis with ${\dagger} p \lt 0.15,\; ^{\ast}p \lt 0.10,\; ^{\ast\ast}p \lt 0.05,\; ^{\ast\ast\ast}p \lt 0.01$ . $X_{i,t}$ significance. Regressions include three lags of the dependent variable as well two lags and contemporaneous values of all additional controls. F-tests are for joint significance of demographic controls (one for direct controls and another for interactions with lagged money growth).

The cumulative excess money growth coefficient at $h=4$ , a 5-year horizon, is nearly the same as the equivalent estimate Table 1, with average estimates of $59$ % correlation between excess money growth and inflation. While the regression coefficients in Table 3 on both the demographic variables directly and their interactions are not easily interpretable economically (as before), it is clear from reported F-tests, that both the direct estimates and indirect interactions are jointly significant at all horizons. The significance of all four interaction terms with demographics suggests that the model found useful variation in each term of the polynomial in improving fit of the prediction across the age distribution. I show in Appendix C that four terms is ideal for this setup, but results are similar with fewer/more.

The compositional relationships of population age structure on inflation directly are weaker at short horizons. This is in contrast to strong significance of all demographic terms in the specification of Table 1 Column 7. Short horizons are omitted from that table for exposition, but all four demographic variables were significant at the 1% level at all fifteen horizons pictured in Figure 3. While it is perhaps reasonable that aging takes time to influence inflation directly, much of the variation picked up by population age structure coefficients in Table 1 comes through these indirect channels of the population relationship with excess money growth. The sensitivity of inflation to money, conditional on age structure, is strongly significant at all horizons for each of these variables. Moreover, unlike estimates in Table 2, these parameters are fairly insensitive to model specification as shown in Appendix C, Table 7.

As in the baseline average estimates from Table 1, I can plot the compositional relation between age and inflation across the life cycle by backing out the implied coefficients as in Figure 4. More important for my work are the indirect impact of age structure coming through changes in excess money growth. These are estimated in the same way, but using the interaction term and as such are zero when the policy variable is zero. In Figure 7a, the solid blue line represents the composition terms: the average relationship between an age group and inflation unrelated to excess money growth. Though some significance is lost, the overall coefficient estimates by age group are roughly the same as in Figure 4. These once again match well with results from Juselius and Takáts (Reference Juselius and Takáts2021) whose work studies this relationship directly. In addition to these, I show the net effect of adding indirect impact of age: those which come through changes in money growth. These are constructed from the composition coefficients by age group and adding/subtracting a half standard deviation change in excess money growth times the age-specific term implied by the interaction coefficients. Figure 7b simply shows these indirect coefficients alone for a positive money growth shock, which are the difference between the dashed green and solid blue lines in Figure 7a.

Figure 7. Age-specific estimates at H = 4: Composition + Indirect age coefficients.

The total impact of aging, including indirect channels, shows that the indirect effects more or less reinforce the age-specific coefficients. This is somewhat easy to see from the fact that each interaction in Table 3 has the same sign as its non-interactive counterpart. These indirect coefficients are positive for young dependents and very early career workers, negative for early-to-mid career workers and then strongly positive again for late career ( $\gt 55$ ) workers and retirees. Unlike the direct (composition) age coefficients, these do not sum to zero, and what matters is not whether a given coefficient is positive or negative, but their relative size. Moving from a population structure with large weight in young or old populations, to one with weight in the early working ages population will reduce transmission, even leading to a negative net effect, as will be the case below. This could be true even if all of the coefficients in Figure 7b were positive, as the deviations from population sample mean always sum to zero. As such an increase in one group necessarily reflects a decrease in another, and what matters for the net effect of a change in population age structure are how relative weights from population deviations interact with coefficients from Figure 7b.

In Figure 7b, the indirect effect drops substantially for young-to-middle aged workers and then begins to sharply rise for workers in mid-to-late career, with a peak in the 5-year groups on either side of retirement. This quite similar to the stylized picture of money holdings by age from the SCF in Figure 1. Here though I also see the effect of young dependents, who might exert pressure on money demand through spending needs of their parents. A theory of money, needed to finance consumption, may explain these dynamics. Fernández-Villaverde and Krueger (Reference Fernández-Villaverde and Krueger2007) show that consumption has a dramatic hump-shape, peaking somewhere in the mid-late fifties and remaining high throughout early retirement before falling during later retirement years. This is more or less the same picture I see for the indirect effect in the second half of life here. My empirical findings suggest that more theoretical work should be done to investigate how late career workers and early retirees might be an important conduit for monetary transmission to inflation, by generating these life cycle profiles. It seems likely based on these pictures that some wealth and consumption dynamics, which match these shapes could play an important role.

There may be other mechanisms at work. If more experienced workers are more important for the wage pressures through bargaining, this might act as a push factor for inflation. This is a main hypothesis regarding inflationary aging, but may be important for priming the monetary transmission if these wages affect money demand. Other explanations that have been put forth, focused on interest rate policy, suggest that households with large accumulation of wealth become more sensitive to changes in interest rates and therefore more likely to strengthen the effect of policy. However, this is the opposite of the effect expected in work such as Wong (Reference Wong2019),Footnote ¹³ where young households are more sensitive via a different, borrowing constraint, channel. One could imagine such mechanisms for safe interest rate transmission to potentially operate similarly to money if it was modeled as a competing asset as in Aoki et al. (Reference Aoki, Michaelides and Nikolov2019).

The visualizations in Figure 7 are useful, but do not yet provide a clear idea of how recent demographic trends have contributed to monetary transmission. I conduct the same KBO decomposition as with my WAP estimates above, showing the impulse response functions for a 1 percentage point increase in excess money growth conditional on the age distribution of the USA, France and the United Kingdom. These results are plotted in Figure 8.

Figure 8. Response of inflation to excess money growth, KBO decomposition: Full age controls. Notes: Direct estimates are the average estimate of money growth on inflation when population is at long run sample mean. Conditional IRFs are effects of money growth rates conditional on age distribution of each country in a given year.

In Figure 8a, each line represents estimates conditional on a particular realization of the population age distribution for the USA. These then correspond to four values for the constructed demographic variables, which capture movements in deviations of population age shares from their long run mean. As such, each IRF represents the direct estimate of the inflation response to excess money growth (plotted by the blue line) plus the contribution from each of these four indirect demographic channels. The first 20 years of this exercise offer the same, though more dramatic, picture of the change in this relationship than those seen in the previous section. The great inflation of the 1970s is a time when the estimated impact of aging on the money-inflation relationship is strongly positive, with changes in excess money growth rate accommodated by aging such that a one-to-one relationship materialized within 3 years. During the great moderation, represented here by the estimated response in 1990 (and similar though not plotted 2000), this relationship completely disappears, as demographic headwinds create an almost puzzlingly strong negative money-inflation relationship.

What drives the dramatic shift from 1970 to 1990? To understand more clearly I show the demeaned population shares for each of these years in the right-hand panels of Figure 8. Looking at the USA in Figure 8b there are a few large swings from the 1970s (green diamonds) to the 1990s (gold triangles). The first is that the population moves from one with a large share of young dependents, by historical standards, to one with relatively few. These swing from higher than historical sample averages (here the zero line) when the boomers make up this group in the 1970s (aged between 6 and 24 in 1970) to lower for all but the youngest group.Footnote ¹⁴ Looking back on the indirect coefficients across age groups plotted in Figure 7b, these are ages with quite a strong positive impact on the money-inflation relationship so this reduction shrinks the effect. Additionally, there is an increase on the younger working aged group as the boomer generation has moved into this category. By 1990 boomers are between 26 and 44 years old, occupying the early-mid working life trough in Figure 7b. Meanwhile, population aging has not yet begun to take off in the USA, with the relatively small silent generation making up late career workers who have the strongest positive impact on the money-inflation relationship. The great moderation, it would seem, was the perfect demographic storm for a movement from strong to weak money pass through to prices, largely due to the size of the boomers.

By 2017, the boomer cohort is at the peak of the money growth relationship aged 53–71. In the USA, this would suggest a resurgence of money transmission. This effect is limited by the massive increase in the oldest age cohorts, who contribute negatively, due to continued increases in life expectancy over this period, as well as the still small groups of young dependents. According to the OECD, life expectancy at age 65 for men in the USA increased by 2.9 years from 1990 and 2017 from 15.1 to 18 years. This increase is even more dramatic in the United Kingdom (14 to 18.8 years) and France (15.7 to 19.6 years). The combination of having relatively fewer young dependents and relatively more old age retirees leads the effects in these two countries to remain quite low in the most recent data. Projecting forward it is indeed the increasing importance of the eldest age cohort and continued decreasing share of young dependents that drive further downward pressure on money-inflation estimates.

It is clear from the right-hand panels of Figure 8 that the 75 age group has an outsized effect on future projections. As a result, I wish to caution putting too much weight on point estimates here. A known issue of using demographic controls is that over-fitting of the polynomial over the age distribution can tend to lead to extreme values for the youngest and oldest age groups. Further, these out-of-sample projections feed in unprecedented increases in this age group, with no comparable historical data. While decreasing money demand in old age is consistent with the stylized picture in Figure 1, the reversal for the oldest group is dramatic. It is true that the extreme shift of the estimates of future projections in Figure 7b may be subject to such over-fitting. I explain my choice of polynomial order in Appendix C, showing that the fourth degree balances an improved model fit coming from the increase in parameters (measured by AIC), with a worsened out-of-sample fit measured through a method developed in Bilger and Manning (Reference Bilger and Manning2015). Moreover, I show that the implied age coefficients of the indirect estimates by age groups for higher-order look broadly similar, with more extreme values for old age groups, suggesting that $k=4$ captures the relevant age variation without allowing for these swings to become overly extreme. In Appendix C.3, I present estimates and projections using lower-order versions of this polynomial. These give a broadly consistent story of the changing transmission from 1970 to present, with less extreme negative impacts from old age groups in future projections.

4.3 Where is the missing inflation?

Where is the missing inflation? My two separate demographic controls paint similar pictures over a long run view, but give conflicting answers in the years since the global financial crisis. If one thinks that the estimates in Section 4.2 are potentially over-fitting the age data, then they might be content with the idea that working age population suggests that demographic headwinds have still been in place over the last decade, perhaps reconciling seemingly weak inflation in response to relatively loose monetary policy. However, it is possible that the age distribution has already become accommodating and that other forces have been working in the other direction.

Coibion et al. (Reference Coibion, Gorodnichenko and Ulate2019) suggest that an expectation-augmented Phillips curve can account for inflation weakness post global financial crisis in a wide range of countries. With the deflationary pressures of the Great Recession lasting until about 2018. Part of the inability of standard Phillips curves to show this is that economic slack is not well accounted for by traditional measures such as the unemployment rate. Falling participation, for example, might signify more slack in the economy that would be missed otherwise. My present analysis is not able to account for such expectations, nor even for dynamics in employment. As a result, if I were to try to forecast inflation using my model I would, like the traditional Phillips curve, significantly overestimate inflation.

To get a sense of this, I plot the predicted current year inflation rate using my specification from Table 3 for a subsample of countries in Figure 9. To be clear, I expect this to be a poor estimate of inflation. This paper is not making any attempts at forecasting and has explicitly focused on attempting to estimate medium- to long-term policy interactions. However, many of my macroeconomic and financial controls are quite important in determining inflation at shorter horizons and can give some sense as to whether my estimated age-structure transmission mechanism, which is quite strong by 2017 in the analysis of Subsection 4.2, would have counter-factually predicted a large inflation takeoff. It is clear from Figure 9 that in spite of my sharp predicted increase in the transmission of money growth to inflation over this period, the model has not predicted any such takeoff from inflation levels from what might have been expected during the great moderation.

Figure 9. Prediction of current inflation.

These results are reassuring in that they provide some context such that the dramatic transmission mechanisms plotted for the USA in Figure 8a do not necessarily imply a takeoff of inflation in their own right. Rather, they suggest that age structure has shifted toward accommodation of money transmission ceteris paribus. Of course many factors might be working against this in the short run, and those discussed in Coibion et al. (Reference Coibion, Gorodnichenko and Ulate2019) suggest that an overhang from the global financial crisis is plausible. However, with inflation response to the COVID crisis appearing more robust than many expected my results suggest that policy makers should be careful to disregard monetary aggregates as a potential driver, particularly in the USA where demographics suggest a robust relationship at present and where Fed interventions have led to incredible spikes in M2.