The Kondratiev cycle and the population
stress model
The Kondratiev cycle in the
pre-industrial era.
The Kondratiev cycle was introduced in the discussion on American historical cycles. It refers to a fifty-year cycle in prices, interest rates and other economic variables and is most easily seen as a cycle in producer prices from the late 18th century through the early 20th. Figure 1 shows a price index for South England over six centuries.1 Price cycles like those seen in 19th century are not visible. Further analysis will be required to test for cycles. A polynomial spline was employed to provide a trend for the index. The price data over 1162-1550 was fit to the fourth degree polynomial while data over 1500-1800 was fit to a fifth degree polynomial. Finally data over 1450-1525 was fit to a second degree polynomial. The trend employed the first equation for the 1162-1477 period and the second equation for the 1521-1800 period. The last polynomial was used in between the others. As shown in the figure the fit was excellent. The ratio of the raw price to the trend was calculated and the minimum value of this ratio subtracted.
The
result was a data series (gray columns in Figure 1) I call price
distress. Price distress is the value of the price index relative to the
long-term trend for a given year. Higher values should reflect shorter-term
price movements most likely due to food scarcity. For example, the highest
value of price distress occurred in 1316, in the midst of the Great Famine of
1315-1317. Heavy rains led to a bad harvest in 1315. In 1316 they happened
again and Europe experienced the worst famine in its history.7 The
historian David Fischer describes the horror:8
When other sources of food ran out, people began to
eat one another. Peasant families consumed the bodies of the dead. Corpses were
dug up from their burial grounds and eaten. In jails the convicts ceased to be
fed; we are told that starving inmates ferociously attacked new prisoners and
devoured them half alive. Condemned criminals were cut down from the gallows,
butchered, and eaten. Parents killed their children for food, and children
murdered their parents.
The
famine extended in 1317, and the 7th highest price distress value occurred
then. The third highest price distress was in 1557 and the eighth-highest in
1556. A well-known historical famine occurred over 1556-7.9
But nature seemed to be against Mary too, for the
harvests also failed two years in succession. In 1556 people were scrabbling
like pigs for acorns and dying of starvation.
The fifth highest value of price distress occurred in 1597, and was associated with another major famine, the Great Dearth:10
On Christmas Day 1596, in the bleak midwinter of a
second consecutive year of harvest failure, the privy council
reminded the archbishops of Canterbury and York of Queen Elizabeth's 'great and
princely care' at this 'tyme of scarsety' to 'provide for the relief of the
poorer sort of people'. After reiterating the provisions for market regulation
specified by the book of orders, last issued in October 1595, the council
condemned the continuation of 'want and dearth'.
A discrete Fourier transform (Figure 2) was run to look for evidence of characteristic cycles in the data. Peaks were seen at 37 years and over 43-53 years, providing evidence for the existence of cycles with lengths around these values. In an attempt to identify these price cycles, price distress was analyzed by calculating average values over periods of approximately 25 years, which would roughly correspond to one-half of a 43-53 year cycle. The periodization was selected in order to produce discernible periods of alternating high and low average values. Eighteen periods of average length 25 years were found over the 1162-1650 period and are given in Table 1. The average price distress for the high-value periods was 0.4 compared to 0.31 for the low. This difference was very statistically significant (p < 0.0001). This demonstration provides a high degree of confidence that fairly regular cycles can be identified in the price distress data by using a moving average. Figure 1 shows a 10-year moving average to help visualize the cycles, presenting a graphical representation of the embedded cycles in the data. Caution is required when using moving averages to ascertain cyclical behaviors in time series data because of what is known as the Slutsky-Yule effect. The Slutsky-Yule effect refers to the observation, first made by Eugen Slutsky, that moving averages of random data can produce apparently periodic fluctuations as an artifact.11 This means the price oscillations shown in Table 1 could be a statistical artifact. One way to deal with this issue would be to find another variable for which a causal link between it and price distress can be hypothesized.
Table 1. Average values of normalized price, unrest and building activity over 1176-1650
|
Period |
Price Distress |
Bldg. Index (normalized) |
Unrest (normalized) |
Duration |
|
1176-1202 |
0.22 |
0.89 |
0.79 |
26 |
|
1203-1230 |
0.49 |
0.93 |
1.66 |
27 |
|
1231-1254 |
0.34 |
0.96 |
0.53 |
23 |
|
1255-1282 |
0.39 |
1.12 |
0.74 |
27 |
|
1283-1305 |
0.30 |
0.67 |
1.08 |
22 |
|
1308-1328 |
0.48 |
0.52 |
0.84 |
20 |
|
1329-1352 |
0.21 |
1.88 |
1.07 |
23 |
|
1353-1381 |
0.45 |
0.53 |
1.54 |
28 |
|
1382-1405 |
0.32 |
1.39 |
0.65 |
23 |
|
1406-1435 |
0.41 |
0.55 |
0.56 |
29 |
|
1436-1459 |
0.41 |
1.33 |
1.15 |
23 |
|
1460-1487 |
0.42 |
0.59 |
0.76 |
27 |
|
1488-1520 |
0.33 |
1.68 |
0.83 |
32 |
|
1521-1557 |
0.43 |
-- |
1.10 |
36 |
|
1558-1579 |
0.31 |
-- |
0.60 |
21 |
|
1580-1602 |
0.37 |
-- |
1.08 |
22 |
|
1602-1626 |
0.35 |
-- |
0.40 |
24 |
|
1627-1650 |
0.40 |
-- |
1.79 |
23 |
|
Average Low (std.
dev.) |
0.31 (0.06) |
1.26 (0.44) |
0.80 (0.27) |
Avg: 25 (4) |
|
Average High (std.
dev.) |
0.43 (0.04) |
0.71 (0.25) |
1.12 (0.44) |
-- |
|
Significance: p < |
0.1% |
1.2% |
3.9% |
-- |
The iron law of wages holds that in pre-industrial times real wages will tend towards subsistence levels at equilibrium. Nominal wages in Medieval and early-modern times were set by custom and tended to change slowly. In the short run, a rise in price level will decrease real wages and living standards to below equilibrium levels; workers will be dissatisfied. Dissatisfaction arising from high prices might find expression as an increased incidence of social unrest.
This same variable had been examined in the paper on modern American cycles. As previously done, events from Appendix A were used to produce a time series of unrest event frequencies. A trend line for this series was calculated using a centered 100 year moving linear regression. The event frequency data was divided by the trend to obtain a normalized unrest frequency. Average values of the normalized event frequencies were calculated for the price-defined periods and appear in Table 1.
Although a moving average analysis of unrest would likely produce its own Slutsky cycle, it will not be the same cycle as the Slutsky cycle obtained from prices. That is, periods of high and low unrest are not likely to align with periods of high and low price distress unless there really is a correlation between the two. In the absence of such a correlation the average level of unrest during the high price periods should not be significantly different from the average unrest during low price periods. As shown in the table, a higher frequency of unrest events was associated with the periods of high price distress. Average unrest during the nine high-price periods was 1.12 compared to 0.80 for the low price periods. This difference was statistically significant (p < 0.04).
Another phenomenon which could be affected by price distress/unrest would be construction projects. It is reasonable that periods of low distress/unrest should be associated with economic prosperity as indicated by a higher frequency of construction starts. I compiled a list of construction start dates for medieval monasteries and churches and recorded the number of construction starts in each year from the mid-9th century down to the early 16th (see Appendix B) As with unrest, I normalized the data using a trend calculated with a running 100 year regression. Average values of the normalized construction starts were calculated for the price-defined periods and appear in Table 1. As expected, the average value during periods of high distress was 0.71 compared to 1.26 for periods of low distress. This difference was statistically significant (p < 0.02). The finding of two independent measures that show statistically-significant alignment with periods of high or low price distress provides strong evidence in favor of approximately regular price cycles of a length consistent with 50-year Kondratiev cycles pre-modern times.
Table 2 presents a continuation of Table 1. After 1790 American reduced price was used in place of British price distress for normalized price. Eight periods of average length 26 years were found over the 1650-1872 period and are given in Table 2. The variability in length of the post-1650 cycles is much greater that the pre-1650 cycles, as shown by the larger standard deviation (9 versus 4). These observations suggested that the price cycles shown in Table 2 might be Slutsky-Yule cycles. The correspondence between high levels of unrest and high price distress disappeared after 1650. A new correspondence arose: unrest was higher during the periods of low price distress. This result was statistically significant (p < 0.03). Thus, fifty-year price cycles continued on after 1650, but they showed different social dynamics.
Table 2. Average values of normalized price and unrest over 1650-1872
|
Period |
Price |
Unrest |
Duration |
Kondratiev Cycle* |
|
1627-1650 |
0.40 |
1.79 |
23 |
|
|
1651-1692 |
0.34 |
1.02 |
42 |
1650-1689 (D) |
|
1693-1711 |
0.44 |
0.12 |
19 |
1689-1720 (U) |
|
1712-1746 |
0.35 |
1.35 |
35 |
1720-1747 (D) |
|
1747-1773 |
0.39 |
0.97 |
27 |
1747-1762 (U) |
|
1774-1793 |
0.34 |
0.84 |
20 |
1762-1790 (D) |
|
1794-1818 |
0.52 |
0.36 |
25 |
1790-1814 (U) |
|
1819-1844 |
0.43 |
1.64 |
26 |
1814-1843 (D) |
|
1845-1872 |
0.46 |
0.42 |
28 |
1843-1864 (U) |
|
Avg Lo (std. dev) |
0.36 (0.05) |
1.21 (0.36) |
Avg: 28 (8) |
-- |
|
Avg Hi (std. dev.) |
0.45 (0.06) |
0.47 (0.36) |
-- |
-- |
|
p < |
4.6% |
2.6% |
-- |
-- |
*Dates over 1650-1790
were obtained from Goldstein12 Values after 1790 from US prices.
In the paper on American cycles evidence was provided supporting the idea that a Kondratiev cycle has been operative since the early 18th century. Dates for this cycle are given in Table 2. Kondratiev cycles are typically defined in terms of price movements as opposed to price levels. Periods of rising prices are called upwaves (U in Table 2) and periods of falling prices are downwaves (D in Table 2). Kondratiev cycles defined in this way were obtained from an American producer price index and from Joshua Goldstein's base dating scheme presented in his excellent book on long cycles.12 The two measures align with each other showing that the different methods used in the two papers track the same cycle.
In summary, there is a reasonable amount of evidence supporting the idea that a Kondratiev cycle has been operative in the Anglo-American political economy from the 12th century right down to the present. Three kinds of dynamics have been seen. Up until the mid-17th century, socioeconomic bad times were associated with periods with a relatively high price level. This pattern reversed over the next two centuries: socioeconomic bad times were associated with periods when the price level was low and/or falling. In still more recent times, socioeconomic conditions followed a composite political/economic cycle (the PE cycle). In the next sections a possible mechanism for observed patterns during the pre-1650 era is developed.
Evidence for a relation between price and population
Figure 3 shows the price index plotted along with English population estimates from 1541 onward that were obtained from parish registers.13 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. Two independent rising trends can show apparent correlation simply because both series are correlated with a common variable (time) and so appear correlated with each other. This problem can be circumvented by examining the 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 than 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 the response is delayed correlation may be between current population growth and inflation next year or the year after. Figure 4 shows a plot of inflation versus population growth in in the previous 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 0.214 for 209 points, and was statistically significant (p < 0.002).
This finding supports the idea that population affects price levels, most likely through the impact of population on food demand. Changes in population at a time of relatively fixed agricultural output will produce high prices. The correspondence of extremely high price distress with historical famines described earlier is one example of this. The correlation of annual price inflation with population growth rate supports the idea that periods of elevated price and socio-economic hard times reflect periods of elevated population and vice-versa. That is, the price cycles and positive relation between price and unrest up to the 17th century should reflect population cycles over the same time. . As long as price and population are related, this dynamic should persist. Figure 5 shows population and price plotted together for the 1600-1850 period. The correlation between the two seen in Figure 3 is evident in the data up to about 1700. After 1700, population begins to become uncoupled from price. Thus, we should expect the positive relation between price and unrest to break down in the late 17th century, which is exactly what was observed. In the next section a negative feedback model is proposed to describe the dynamics of the pre-1650 cycles.
A mathematical model for
Kondratiev cycles in the early period
For a static or very slowly changing agricultural technology, agricultural output will be a function of the quantity and quality of the available farmland. When there is adequate arable land, the less desirable land will not be cultivated and agricultural productivity per farmer will be high, giving a large food supply relative to demand. Prices will be low. If the population rises faster than new farmland can be cleared or already cleared lands improved, farming productivity will fall and so will food supply relative to demand. Prices will be high. As population growth continues, the margin of surplus production in good years over deficits in bad years falls, increasing the severity or frequency of famines. A malnourished population becomes more susceptible to disease, magnifying the severity of plagues. Fertility declines as well. Both of these Malthusian checks serve to reduce the rate of natural increase, which, over time, will bring population into line with the amount of arable land.
These ideas can be expressed in terms of the logistics model:
1. Pi = Pi -1 + r Pi -1 (1 - Pi -1 / PMAX)
Here Pi is the population in year i and Pi-1 is the population in the previous year. PMAX is the carrying capacity of the environment, expressed as the maximum population that can be supported with the current amount of arable land. The parameter r is the natural rate of increase, the rate of population growth that would be seen under conditions of excess food supply. Logistic growth starts out exponential, but then starts to slow, eventually stopping as the population reaches PMAX (Figure 1).
The population stress model uses a lagged version of the logistics model:
2. Pi = Pi -1 + r P i-D t (1 - Pi-D t / PMAX)
Here P is the population of physical adults; it does not include infants and small children. Pi-D t is the value of P Dt years in the past. The model says growth rate in P is based on the value of P in the past, not the present. Changes in P are assumed to reflect changes in flux of entrants into physical adulthood, not increases in the death rate of adults. Changes in the flux individuals surviving infancy will show up as changes in P Dt years down the road, where Dt is the time needed for a survivor of infancy to reach physical adulthood. Food availability is assumed to be determined by P. Thus infant survival is governed by P/PMAX which affects the growth rate of P Dt years later.
Since it takes about 12 years for a baby to grow into a physical adult, (in terms of caloric requirements) Dt is set at 12. Figure 6 shows plots of equations 1 and 2 with PMAX =100, r = 9%/year and D t = 12 years. The unlagged model shows a rising P gradually approaches PMAX where it remains Growth in the unlagged model over and undershoots PMAX (Figure 6). These oscillations gradually die out and P reaches a steady-state at PMAX.. The cycle length is about 48 years for Dt =12. The damped cycles are a problem, the only reason the cycles were even seen in the figure is because the impulse given by the initial growth was so large. In reality PMAX is never going to be so much higher than P (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 can provide this forcing function. Famines can be modeled as a temporary reduction in PMAX due to random, mostly weather-related factors. This was modeled by subtracting 10 from PMAX in famine years, defined as years in which a random number between 0 and 1 fell below 0.10. Also, 10 was subtracted for any year that followed a famine year and for which a random number fell below 0.5. This definition of famine produces clusters of reduced PMAX roughly every decade. Figure 6 also shows the same lagged plot but with famines added. Sporadic famines exert a perturbation to the lagged population cycle that helps keep the cycle going. Assuming a direct relation between food availability (represented by P/PMAX) and price, such a mechanism could give rise to the price cycles described above.
The lagged logistics model is not very physically realistic. Its advantage is simplicity. Nevertheless it shows that a lagged Malthusian-type mechanism is capable of producing cycles of the observed length.
Summary
This paper and the one on American cycles propose that long cycles have been operative in Anglo-American history since the 12th century. A sudden change in cycle length suggested that two mechanisms were operative in these cycles, one for the period since the early 19th century and one before then. In this paper, it was shown that the pre-19th century cycle operated by at least two mechanisms. Before the late 17th century, the cycles were consistent with a lagged Malthusian feedback of population on its growth rate. Changes in population directly affect prices and adverse socioeconomic conditions. This mechanism is replaced by another one that was operative from the late 17th century through the early 19th. The operation of this mechanism is characterized by an inverse relation between price and adverse socioeconomic conditions.
References
1. The index is a composite of two indices. The first is the Phelps-Brown index of consumables in South England that begins in 1264.2 The other is an index constructed from English grain and livestock prices3-6 that ran from 1162 to 1325.
2. Phelps Brown E. H. and Sheila V. Hopkins, Builders' Wage-rates, Prices and Population: Some Further Evidence, Economica (February 1959): 18-37
3. Farmer, D.L. (1956-57) Some Price Fluctuations in Angevin England, Economic History Review 9: 34-43.
4. Farmer, D.L. (1959) Some Grain Price Movements in Thirteenth Century England, Economic History Review 11: 412-17.
5. Farmer, D.L. (1957) Some Livestock Price Movements in Thirteenth Century England, Economic History Review 10: 207-20.
6. Farmer, D. L., Prices and Wages, in H. E. Hallam ed., The Agrarian History of England and Wales, Volume II, 1042-1350, Cambridge: Oxford University Press, 1988.
7. The Great Famine of 1315, webpage: www.ku.edu/kansas/medieval/108/lectures/black_death.html
8. Fischer, David Hackett, The Great Wave, Price Revolutions and the Rhythm of History, New York: Oxford University Press, 1996, p. 37.
9. Excerpt from Dunn, Jane, Elizabeth and Mary, Cousins, Rivals, Queens, New York: Random House, 2004: (www.randomhouse.com/knopf/catalog/display.pperl?isbn=9780375408984&view=excerpt).
10. Hindle, Steven, Dearth, Fasting and Alms: The Campaign for General Hospitality in Late Elizabethean England, Past and Present, 172, 44. (www2.warwick.ac.uk/fac/arts/history/staff/shindle/publications/dearth.pdf)
11. Slutsky, Eugen, (1937) The summation of random causes as the source of cyclic processes, Econometrica 5: 107.
12. Goldstein, Joshua S. Long Cycles, New Haven: Yale University Press, 1988
13. Mitchell, B. R. British Historical Statistics, Cambridge University Press, 1988.
