Periodic shortages of food can arise though a lagged negative feedback relation between demand for food (reflecting population) and fertility. A large adult population (relative to food supply) causes high food prices and more frequent or more severe famines. This is turn causes reduced birth rate and, after a lag, smaller adult population. A change in the number of newborns will have little immediate impact on food demand. As the babies grow older, they consume more food and so will affect food demand. Increased birth rate will, after a lag, produce increased food demand, which causes high food prices and increased famine, which feeds back to reduce birth rate. The introduction of a lag into a negative-feedback effect process like this will cause oscillations. These oscillations should show up as alternating periods of rising and falling prices. For these oscillations to be a cause of the saeculum they must define a cycle two generations in length. For the observed generation length of 26 years, we are looking for evidence for a price cycle with a typical length of 52 years. Such price cycles have been reported and are known as Kondratiev cycles.

I can express these ideas quantitatively in terms of a simple mathematical model. Start with the logistics model for population growth:

D-1. P_i = P_{i -1} + r P_{i -1} (1 - P_{i -1} / P_MAX)

Here P_i is the population in the present year i and P_i-1 is the population in the previous year, i - 1. P_MAX is the "carrying capacity" of the environment, expressed as the maximum population that can be supported. The parameter r is the natural rate of increase, the rate of population growth that would be seen under conditions of excess food supply.

Figure D-1. Plot of equations A.1-2 with P_MAX =100, r = 9%/year and D t = 14 years

Figure D-1 shows the sigmoid or "S-shaped" population growth curve obtained using equation D-1. Population initially shows exponential growth, which slows and eventually stops as population approaches P_MAX. The period of exponential growth is observed when P is small compared to P_MAX and the limits to growth represented by P_MAX are not much of a factor. As P comes closer to P_MAX, the limits to growth exert more of an effect, slowing growth and eventually halting it as P reaches P_MAX.

The population model for the saeculum holds that changes in fertility result in a change in adult population after a lag period. Let us denote this lag period as D t. The cause of the birth rate D t years ago was the conditions of that time, which according to the logistics model are determined by the population D t years ago, which we will denote as P_{i-D t}. Substituting P_{i-D t} into the growth term in equation 17 yields

D-2. P_i = P_{i -1} + r P _{i-D t} (1 - P_{i-D t} / P_MAX)

Equation D-2 is the lagged version of the logistics model. It says the growth rate of the adult population today reflects the birth rate D t years ago, which was affected by the size of the adult population then. Since it takes about 14 years for a baby to grow into a physical adult a reasonable choice for D t would be 14 years. Figure D-1 also shows a plot of equation 18 with P_MAX =100, r = 9%/year and D t = 14 years. The two curves show the same initial exponential growth. The unlagged model shows slowing growth before the lagged one does. This occurs because current growth in the lagged model reflects the population of the past, which is smaller than the current population; growth is slowed less and so is faster. Current growth in the unlagged model reflects current population which is larger and so the growth rate is smaller.

This difference becomes more pronounced as P approaches P_MAX. Growth in the unlagged model slows to zero as P reaches P_MAX, while growth continues in the lagged model, reflecting P_{i-D t} (the population of D t years ago). As a result, P rises well above P_MAX, only stopping when P_{i-D t} reaches P_MAX, that is, D t years after P reached P_MAX. At this point P is at the top of its cycle. It then accelerates downward for D t years as P_{i-D t} rises from P_MAX to the cycle top, at which point P reaches its maximum downward velocity. It then decelerates for D t years (still moving downward) as P_{i-D t} falls from the cycle top to P_MAX. When P_{i-D t} reaches P_MAX, P stops falling; it has reached its cycle bottom. The time from the cycle top to cycle bottom is thus 2 D t years. This means the time for a complete cycle is 4 D t years.

Figure D-1 shows the damped oscillations around P_MAX that are obtained from the lagged model using D t of 14 years. The cycle length is 56 years, which is consistent with L =28, which is close to the observed value for L. Although the lagged population growth model shows how this mechanism can produce population cycles of the correct length using a plausible assumption for D t, the cycles are damped. The only reason the cycles are even seen is because the impulse given by the initial growth was so large. But in reality P_MAX is never going to be so much higher than P in an agricultural population (there is no need to clear so much excess land). Thus for this mechanism to work, some sort of forcing function is needed that provides a periodic push to keep the cycle going.

Sporadic famines provide this forcing function. Famines can be modeled as a temporary reduction in P_MAX due to random, mostly weather-related factors. This was modeled by subtracting 5 from P_MAX in famine years, defined as years in which a random number between 0 and 1 fell below 0.05 (5%). A famine year was also any year that followed a famine year and for which a random number fell below 0.5. This definition of famine produces clusters of famine years roughly every twenty years.

The factor of interest that we can track is food price. Food prices will reflect the size of the population (P) relative to the carrying capacity of the land (P_MAX). When P / P_MAX is low price will be at some base level. As population approaches and exceeds P_MAX, price will rise at an increasing rate above its basal level. Equation 19 shows an expression for price that shows this behavior.

D-3. Price - Basal Price = (P / P_MAX)ⁿ

Figure D-2 shows the same lagged plot as shown in Figure D-1, but with famines added. Sporadic famines exert a perturbation to the lagged population cycle that helps keep the cycle going. Hence the oscillations do not completely die out as they did in Figure D-1. Also shown is a plot of equation D-3 with n = 5. Price simply follows the same damped oscillations as population, only it is amplified by the factor n. Famines show up as spikes in the price plot due to the sudden drop in P_MAX, which is magnified by the large value of n.

Figure D-2. Effect of sporadic famines on the lagged logistics model

Figure D-3 shows the same model as in Figure D-2, but with random noise added. The model itself is trendless, but I wished to compare it to data which shows a trend. The model output was added to an external trends designed to be broadly similar in shape to the population and price trends that occurred in England during 12^th to 17^th centuries. The purpose of adding the trend was to facilitate visual comparison of the model to actual data (Figure D-4). Figure D-3 doesn't show obvious cycles just as the real data (Figure D-4) does not, despite the fact that cycles are explicitly built into the model. Yet when subjected to cycle analysis (see Appendix B) the cycles can be seen (red lines in Figures D-3 and D-4). This is very important. Kondratiev cycles are usually hard to see (like those in Figure D-4) and only reveal their cycles upon further analysis. Any realistic model for the Kondratiev cycle would have to produce a price series in which the cycles are similarly hard to see by inspection, yet readily revealed by analysis.

Figure D-3. Population model with superimposed rising population and price trend

Figure D-4. Price data and estimated population for England over six centuries

This exercise shows that a lagged feedback model can produce price behavior that looks like an actual price series. That is, the population model is a plausible explanation for how a Kondratiev price cycle could be created. But to assert that population dynamics causes the Kondratiev requires some evidence. A necessary (but not sufficient) requirement for the validity of population model is a direct relation between population and price.

Figure D-5 shows support for the idea that population is related to price. Here a consumer price index from South England³⁹ is plotted along with English population estimates from 1541 onward that were obtained from parish registers.⁶¹ The price and population plots line up well, implying close correlation. Indeed, a linear regression analysis of price as a function of population over the 1541-1750 period shows a correlation coefficient (r) of 0.88, which is very high. Successive values in time series are often correlated with one another, however. This persistence is known as serial correlation and needs to be taken into account when testing correlation significance between two time series. Serial correlation can severely reduce the effective number of degrees of freedom in a time series, meaning that the significance of correlation between two series may be greatly exaggerated.

Figure D-5 English price index and population 1500-1700

Two independent rising trends usually show serial correlation simply because they are both rising. This problem can be circumvented by examining correlation between the rates of change of the variables. That is, are annual price inflation and population growth correlated? If price movements truly follow population movements, when population moves up or down in a given year, price should move in the same direction more often that not. A regression analysis of annual price inflation against annual population growth should show a statistically significant correlation if population and price are truly related to each other.

The correlation can involve a lag, depending on the response time of the presumed connection between price and population. If population changes show up in price quickly, the correlation should be between population growth and inflation in the same year. If it is slower the correlation may be between population growth and inflation in the next year, or the year after that.

Figure D-6. Inflation versus population growth in the previous year (1542-1750)

Figure D-6 shows a plot of inflation next year versus population growth in the current year. Regression against inflation in the same year or two years later gave a poorer correlation than the one shown in the figure. The correlation coefficient obtained was 0.215 for 208 points, and is statistically significant at the 99.8% level. This pretty much establishes that price and population are related. What has not been established is whether population is responsible for producing Kondratiev-type cycles in price. I have demonstrated the existence of price cycles, but have not shown that such cycles also occur in population in a manner that correlates with price cycles.

Direct cycle analysis of population data does not reveal convincing cycles of Kondratiev length. This is not surprising as the cycles expected by the population model are very shallow and I would expect them to be swamped out by measurement errors. Thus, it will not be possible to show a direct correlation between Kondratiev price cycles and the proposed corresponding cycles in population. An indirect approach might be useful here.

The population model says that when population is low (P < P_MAX) food prices are low on average, reflecting surplus food production. During these times large food stores will accumulate. Conversely, when P > P_MAX food prices are high on average, reflecting food production shortages. During these times food stores will be very low. Consider the impact of a drought when P < P_MAX and food stores are large. Although the harvest in the drought year may be quite poor, the availability of food will not be severely affected because of the abundant food stores that can be tapped. Price will rise, of course, but not tremendously. In contrast, a drought during a time when P > P_MAX will result in dearth or even famine and prices will skyrocket.

Thus, there are two regimes for the response of price to presumably random weather-related short harvests. During the "low" portion of the population/Kondratiev cycle, price does not respond greatly to a short harvest, because food stores are large. Conversely, during the "high" portion of the cycle price responds greatly to random short harvests. If the population model is valid, random adverse weather should produce large impacts on price leading to high price volatility during one part of the cycle and not at others. This means that the Kondratiev cycle should appear as a price volatility cycle if the population model is correct.

Price volatility is most directly measured using relative standard deviation (RSD). RSD is the standard deviation divided by the average of a set of data. Figure D-7 shows RSD values calculated from the price data in Figure 17 over 5, 10 and 15 year moving periods. A regular series of high-volatility episodes appears in the figure.

Figure D-7. Running relative standard deviations of price in England over six centuries

The length of the moving period is important, because of what is known as the Slutsky effect. The Slutsky effect refers to the observation, first made by Eugen Slutsky, that moving averages of random data can produce apparently periodic fluctuations as an artifact.³⁸ As I showed experimentally in The Kondratiev Cycle, Slutsky cycles typically run about three times the averaging period, with wide variations.³⁹ The 15-year plot shows eight clear peaks in volatility that stand out from the background and a possible ninth one around 1375. This is about as many as one would expect if these cycles were Slutsky cycles. The 10-year plot shows the same eight peaks plus another one around 1375 that matches with the one from the 15-year plot. There are about half as many peaks in the 10-year plot as would be expected if price volatility were random. The 5-year plot identifies standout peaks with asterisks. Each of these match up with peaks at the longer time frames. The peaks do not shift positions, nor do the number of standout peaks change with averaging period, as would be expected for Slutsky cycles. These observations support the conclusion that periods of high price volatility do not occur randomly. The volatility peaks are spaced 55± 13 years apart. Thus, a real volatility cycle does exist in the price data that is of the correct length for a Kondratiev cycle, that is, about two generations in length.

Before I conclude that the population mechanism causes the Kondratiev cycle, it would be instructive to consider an alternate explanation for cyclical prices and famine. For example, suppose weather is cyclic? Cyclical drought, for example, would cause cyclical poor harvests and cyclical high prices. They is no way to establish whether or not cyclical weather was responsible for cyclical prices in England during late Medieval and early Modern times. However, one would think if weather were cyclical, one could find evidence for it from in any weather records.

Figure D-8. Moving averages of the Palmer Drought Index over three centuries in Iowa

Figure D-8 shows average values for the Palmer drought severity index for Iowa over the last three centuries.⁴⁴ The Palmer index measures drought conditions with negative values indicating drought. Annual June-July index values were subjected to 5, 10, and 15 year moving averages in an attempt to reveal drought cycles. The 15-year moving average reveals five distinct troughs indicative of periods of unusually intense or persistent drought. The 10-year moving average reveals nine similarly distinct troughs. The 5-year moving average also shows nine distinct troughs, not all of which line up with the 10-year troughs. These variations suggest that the cycles shown the figure are Slutsky cycles. Compare how the number of troughs and their positions shift with moving average period in this figure, with consistency of peaks in Figure D-7. Figure D-8 can serve as a control for what a Slutsky cycle looks like, helping to establish both that the volatility cycles are real and are probably not caused by cyclical weather.

In summary, cycles of price volatility likely reflect cycles of food shortage that are probably not due to corresponding weather cycles. This finding, and the fact that population and price are correlated, are strong evidence in support for the population model for the Kondratiev cycle. By model I mean the general idea of the Kondratiev cycle as a lagged negative feedback of population on population growth, not the specific mathematical model I presented. The model I developed is grossly simplified and not very physically realistic. I did not explicitly consider deaths, simply lumping them in with births as a net population increase factor. A structured model dealing with individual population cohorts would be more realistic, but it would also be far more complex and require many arbitrary parameters whose values would have to be guessed. My goal here is not to provide a quantitative model, but rather to show the power of a single simple mechanism to explain a number of Kondratiev features such as its length and why it shows up in price. Given the existence of a price cycle generated independently of the saeculum, the economic and social aspects of the Kondratiev cycle that help generate the saeculum follow.