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9 wealth (or little wealth), and the estate multiplier e t =W t /B t is infinite (or very large, say 100 or more). It is interesting to note that this theory was formulated precisely at the time when inheritance was at its historical nadir. According to our series, inheritance flows were about 4% of national income in the 1950s-1960s, vs. as much as 20%-25% at the time of estate multiplier economists. Presumably, economists were in both cases very much influenced by the wealth accumulation and inheritance patterns prevailing at the time they wrote. Our advantage over both estate-multiplier and life-cycle economists is that we have more years of data. Our two-century-long perspective allows us to clarify these issues and to reconcile the various approaches into a unified framework (or so we hope). The lifecycle motive for saving is logically plausible. But it clearly cohabits with many other motives for wealth accumulation (bequest, security, prestige and social status, etc.). Most importantly, we show that with low growth rates and high rates of return, past wealth naturally tends to dominate new wealth, and inheritance flows naturally tend to converge towards levels that are not too far from those posited by the estate multiplier formula, whatever the exact combination of these saving motives might be. 3. Data sources and methodology The two main data sources used in this paper are national income and wealth accounts on the one hand, and estate tax data on the other hand. Before we present these two data sources in a more detailed way, it is useful to describe the basic accounting equation that we will be using throughout the paper in order to relate national accounts and inheritance flows. 3.1. Basic accounting equation: B t /Y t = µ t m t W t /Y t If there was no inter vivos gift, i.e. if all wealth transmission occurred at death, then in principle one would not need any estate tax data in order to compute the inheritance flow. One would simply need to apply the following equation: B t /Y t = µ t m t W t /Y t I.e. b yt = µ t m t β t (3.1) With: B t = aggregate inheritance flow Y t = aggregate national income W t = aggregate private wealth m t = mortality rate = (total number of decedents)/(total living population) µ t = ratio between average wealth of the deceased and average wealth of the living b yt = B t /Y t = aggregate inheritance flow-national income ratio β t = W t /Y t = aggregate private wealth-national income ratio Alternatively, equation (3.1) can be written in per capita terms:

t /y t = µ t w t /y t = µ t β t (3.2) 10 With: b t = average inheritance per decedent y t = average national income per living individual w t = average private wealth per living individual Equation (3.1) is a pure accounting equation: it does not make any assumption about behaviour or about anything. For instance, if the aggregate wealth-income ratio β t is equal to 600%, if the annual mortality rate m t is equal to 2%, and if people who die have the same average wealth as the living (µ t =100%), then the annual inheritance flow b yt has to be equal to 12% of national income. In case old-age individuals massively dissave in order to finance retirement consumption, or annuitize their assets so as to die with zero wealth, as predicted by the pure life-cycle model, then µ t =0% and b yt =0%. I.e. there is no inheritance at all, no matter how large β t and m t might be. Conversely, in case people who die are on average twice as rich as the living (µ t =200%), then for β t =600% and m t =2%, the annual inheritance flow has to be equal to 24% of national income. If we express the inheritance flow B t as a fraction of aggregate private wealth W t , rather than as a fraction of national income Y t , then the formula is even simpler: b wt = B t /W t = µ t m t (3.3) I.e. the inheritance-wealth ratio b wt is equal to the mortality rate multiplied by the µ t ratio. In case µ t =100%, e.g. if the age-wealth profile is flat, then b wt is equal to the mortality rate. The estate multiplier e t =W t /B t is simply the inverse of b wt . We will return to the evolution of the inheritance-wealth ratio b wt later in this paper. But for the most part we choose to focus attention upon the inheritance-income ratio b yt and accounting equation (3.1), first because the evolution of the wealth-income ratio β t =W t /Y t involves economic processes that are interesting per se (and interact with the inheritance process); and next because national wealth data is missing in a number of countries, so that for future comparison purposes we **find** it useful to emphasize b yt ratios, which are easier to compute (if one has fiscal data). Also, b yt has arguably greater intuitive economic appeal than b wt . E.g. it can easily be compared to other flow ratios such as the capital share α t or the saving rate s t . An example with real numbers might be useful here. In 2008, per adult national income was about 35,000€ in France. Per adult private wealth was about 200,000€. That is, β t =W t /Y t =w t /y t =560%. The mortality rate m t was equal to 1.2%, and we estimate that µ t was approximately 220%. 22 It follows from equations (3.1) and (3.3) that the inheritance-income ratio b yt was 14.5% and that the inheritance-wealth ratio b yt was 2.6%. It also follows from equation (3.2) that average inheritance per decedent b t was about 450,000€, i.e. about 12.5 22 In 2008, French national income Y t was about 1,700 billions €, aggregate private wealth W t was about 9,500 billions €, adult population was about 47 millions, so y t ≈35,000€ and w t ≈200,000€. The number of adult decedents was about 540,000, so the mortality rate m t ≈1.2%. For exact values, see Appendix A, Tables A2-A4.

- Page 1 and 2: On the Long-Run Evolution of Inheri
- Page 3 and 4: 2 These are truly enormous historic
- Page 5 and 6: 4 observers tend to imagine. E.g. a
- Page 7 and 8: 6 One important limitation of this
- Page 9: 8 was obvious that most wealth deri
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- Page 15 and 16: 14 personal sector. W t is estimate
- Page 17 and 18: 16 important point is that our stoc
- Page 19 and 20: 18 of the tax is that the successor
- Page 21 and 22: 20 implies that we probably under-e
- Page 23 and 24: 22 The 1913-1949 capital shocks cle
- Page 25 and 26: 24 Our purpose here is more modest.
- Page 27 and 28: 26 income ratio β* also tends to b
- Page 29 and 30: 28 First-order approximations again
- Page 31 and 32: 30 and social status conferred by w
- Page 33 and 34: 32 This might possibly entail syste
- Page 35 and 36: 34 steeply increasing age-wealth pr
- Page 37 and 38: 36 19 th century growth rate (1.0%)
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- Page 41 and 42: 40 aggregate wealth. But the middle
- Page 43 and 44: 42 be that the moral value attached
- Page 45 and 46: References 1 A. Ando & F. Modiglian
- Page 47 and 48: R. Giffen, “Recent Accumulations