# 7.2 A Closer Look at Rates and Accelerations Due to Innovation

**7.****2 A Closer Look at Rates and Accelerations Due to Innovation**

The example in section 7.1 is about wheat. But, of course, as use of the heavy plough spread across Northern Europe, other crop yield rates also increased. In order to make more ploughs, there were increases in activity in supporting trades such as blacksmithing and carpentry, leatherworks, and much else. As mentioned in section 7.0, partly because of there being more food, there were also increases in populations and in urbanization. For the moment, however, we keep a focus on plough production rates and crop yield rates.

Farms were only so big. There were limits to how many acres a farmer and their family could manage, even with the new type of plough. Eventually, the total number of heavy ploughs in use levelled off as did food production.^{1}See Section 7.0, note 2.

In other words, for an extended period of time, an approximately constant plough production rate led to accelerating crop yield rates. Eventually, however, the need for new ploughs (above and beyond maintenance and replacement) decreased. New ploughs were still needed but emphasis moved to maintenance and replacement. As a result, crop yield rates also levelled off. This is also indicated by equation (16) from the end of section 7.1. When production rates of ploughs moved toward maintenance and replacement, the difference-term in square brackets in equation (16) moved toward zero. Then net accelerations of crop yield rates also moved close to zero. Accelerations near zero mean that crop yield rates were leveling off.

We have drawn attention to rates and accelerations in plough production, crop yield and the numbers of heavy ploughs made and available for use. To understand more about the whole process, we now look to sketches of graphs of general trends in pairs of rates and pairs of accelerations^{2}For readers who have not yet studied calculus, at a point on the graph of a rate, the slope of the graph approximates acceleration at that time..

**Figure 7.2**: Plough production rates and crop yield (production) rates (per year). Time scale is common. There is no vertical scale in the graph.

The plough production rates graph shows that plough production rates increase for a time, level out for a time, and eventually drop off toward whatever rates are needed for maintenance and replacement. Crop yield rates begin to climb somewhat after plough production rates begin to increase. They climb toward a more or less sustained new level.

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**Figure 7.3**: Plough production accelerations and crop yield accelerations (per year per year). Time scale is common. There is no vertical scale in the graph.

Initially, plough production rates increase, which means positive acceleration. As production rates approach being constant, accelerations return toward being near zero. Eventually, plough production rates settle toward maintenance and replacement rates. Downwardly adjusting production rates means that, for a time, acceleration is negative. Before too long, a new level is reached and accelerations return to being near zero. The fact that acceleration goes negative for a time does not reflect any negative consequence for farmers and community. It is a matter of slowing plough production rates to them bring downward toward maintenance and replacement rates, less than initial surge rates. (Otherwise, there would be an accumulation of unneeded ploughs.)

**Figure 7.4**: Ploughs available and used rates, and crop production rates. Time scale is common. There is no vertical scale in the graph.

The numbers of ploughs available for use increases and after a time approaches “carrying capacity.” Somewhat later, crop production rates also increase and eventually they too approach their own carrying capacity. The graph for plough usage rates is also area under the graph of the plough production rates.

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**Figure 7.5**: Acceleration in the numbers of ploughs made and available for use, and crop production accelerations (per year per year). Time scale is common. There is no vertical scale in the graph.

**Regarding the graphs**:

If we look at yield rates, notice that, as pointed to by figure 7.4, it is not a constant production rate of ploughs that directly correlates with yield rates of wheat. It is, rather, the total number of ploughs made and available for use that correlates with yield rates. Since ploughs are reused, the total number of ploughs made and available for use increases over time. That total is a sum (or “integral”), as mentioned in Section 7.1.1.

Figure 7.5 is the graph of slopes of Figure 7.4.

Figure 7.4 represents an initial positive acceleration in the numbers of ploughs available for use. As that number approaches a carrying capacity, acceleration settles back toward zero. Again, this has no negative implications. It is a matter of accelerating to increase the numbers of ploughs available for use and then easing off on (a still positive) acceleration, as the numbers of ploughs available for use plateau. At that stage, positive acceleration approaches zero. With a lag, crop yield rates also accelerate. As crop yield rates reach their carrying capacity, their acceleration also moves back toward zero.

Figure 7.4 represents a *surge* in the numbers of ploughs available for use. With a lag, this is followed by a *surge* in basic production.

Figure 7.5 shows corresponding accelerations. There is a positive acceleration in the numbers of ploughs (surplus products) available for use. With a lag, there is a non-negative acceleration in basic production rates. The long-term effects of this pair of accelerations grounds the possibility of raising the standard of living.^{3}In section 8, we discuss the ‘business cycle.’ Notice that, unlike in the business cycle (a consequence of applying contemporary models), there are no necessary negative implications.Both of these acceleration functions are *pulse functions* in the sense that accelerations are non-negative and both eventually return to near zero.

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