b. Interrelationship of Political Capacity Elements

b. Interrelationship of Political Capacity Elements

The elements of political capacity, which constitute the political capacity of a political actor, including armed capacity, economic capacity, and ideological capacity, are also interrelated. Among them, the most important relationship in political terms can be modeled by the following differential equation.

         [Fmla.3.1.3]           \( \frac {d} {dt}\vec{L(t)} = \begin{bmatrix}\frac {d} {dt}A \\  \\ \frac {d} {dt} E\\  \\ \frac {d} {dt} \vec{I} \end{bmatrix} \)

 

                                                                  \( = \) \( \begin{bmatrix}
\rho_{1}\frac{\begin{vmatrix} \sum \vec{I_{[-]}}
\end{vmatrix}}{average.of. \begin{vmatrix} \sum \vec{I_{[-]}}
\end{vmatrix}}A \\
 \\
\rho_{2}\frac{A}{average.of.A}E  \\
 \\
\rho_{3}\frac{E}{average.of.E}\left| \vec{I} \right|
\end{bmatrix} \)

 

In this equation  \( dL/dt \)  represents the degree of change in political capacity over time ( \( t \) ), as do  \( dA/dt \),  \( dE/dt \), and  \( d| \vec{I} | /dt \) . It is easier to understand this equation by starting with the simplest second equation, as all three equations express the same concept in the same way.

 

Rate of change in economic capacity

To intuitively explain what the second differential equation means, it states that the stronger the armed capacity of a political actor, the greater the increase in their economic capacity over time. In other words, assuming other conditions are equal, if the armed capacity of a political actor( \( A \) )  is greater than that of another political actor, it can be said that the rate of increase in their economic capacity ( \( dE/dt \) )  is also higher. Here, 'greater' means that the armed capacity of one political actor (individual or group) is 'relatively' greater than that of another political actor. On the contrary, if Goguryeo's armed capacity was smaller, its economic capacity would diminish over time compared to Silla, which had greater armed capacity. Therefore, we have the following.

         [Fmla.3.1.4]     Relative size of  \( A = \frac {A} {average-of-(all) As}  \)

For instance, suppose we measure the capacity size of Goguryeo in ancient Korea's Three Kingdoms era, which was competing with Baekje and Silla. Let Goguryeo's armed capacity be 5, Baekje's armed capacity be 3, and Silla's armed capacity be 2. Then, from Goguryeo's perspective, the relative size of armed capacity is 5/((5+3+2)/3) = 5/3. Accordingly, the unit time increase rate of Goguryeo's economic capacity is calculated to be 5/3 times, which is higher than the economic capacity increase rates of Baekje and Silla.

However, this relative size of armed capacity, i.e., the armed capacity ratio, may only partially reflect the increase rate of economic capacity. The armed capacity ratio of Goguryeo is 5/3, but the actual ratio of Goguryeo's economic capacity increase per unit time can be measured as 4/3. Therefore, we can reduce the error by multiplying it by a number \( \rho_{2} \) , and \( \rho_{2} = \) 4/5 in this case.

Then the following equation would be considered plausible.

          [Fmla.3.1.5]     \( \frac{d}{dt} E = \rho_{2} (relative.size.of.A) E \)

                                                        \( = \rho_{2} \left ( \frac {A} {average.of.(all)A } \right ) E  \)
       
[Fmla.3.1.5] implies that the rate of change in economic capacity is proportional to the relative size of armed capacity. Armed capacity is concretized into legal rights within society, and when black Americans were legally freed after the American Civil War (increase in armed capacity), their income per person increased by 46% between 1857 and 1879 while working fewer hours (increase in economic capacity). In Cote d'Ivoire, after the death of President Félix Houphouët-Boigny in December 1993, crime and violence increased dramatically, and the economy faltered after the mid-1980s, which is a case of national scale. Thus, the relative size of armed capacity affects the rate of change in economic capacity, and when each capacity is appropriately quantified, [Fmla.3.1.5] can be established.

\( \rho_{2} \) is determined by the political environment in which each political actor is situated. In other words, \( \rho_{2} \)  is a variable determined by the political environment. For example, in a military competition environment such as the Three Kingdoms period, military superiority can lead to rapid economic growth, and thus \( \rho_{2} \)  is likely to be large. On the other hand, in an ideological struggle environment after World War II, such as the Cold War, the economic growth induced by military superiority will be relatively small, and therefore the value of \( \rho_{2} \)  will also be small. 

Assuming the initial value as \( E_{0} \), the following Formula holds for \( E \)  when solving the differential equation.

         [Fmla.3.1.6]         \( E= E_{0}^{ \left( \rho_{2} \left ( \frac {A} {average.of.As } \right ) t \right) } \)

The equation implies that the economic capacity of a political actor increases over time in exponential to the relative size of its armed capacity. Thus, the economic capacity of a political actor with a larger armed capacity increases exponentially faster than that of a political actor with a smaller armed capacity.

 

The Rate of Change in Armed capacity

When understanding the equation for \( dA/dt \) , which is the first equation, one can replace armed capacity \( A \) with the more complex concept and symbol \( \sum \vec{I_{[-]}} \) while leaving everything else the same. Here, \( \vec{I_{[-]}} \) refers to the ideological capacity of the members of a political actor. In this book,  \( \vec{I} \)    generally represents the ideological capacity of a political actor, but the reason for adding a  enclosed in square brackets as an exponent is that this ideological capacity represents the ideological capacity of the members of the political actor. "[-]" refers to political actors at one level smaller scale. For example, if A represents the armed capacity of South Korea, \( \vec{I_{[-]}} \)  represents the ideological capacity of South Korean citizens, and if  represents the armed capacity of the European Union (EU), \( \vec{I_{[-]}} \)    represents the ideological capacity of member states of the EU.

On the other hand,  \( \sum \vec{I_{[-]}} \)  is a value that sums up all ideological capacities. Here, the absolute value symbol is added again to make it   \( | \sum \vec{I_{[-]}} | \) , because it is necessary to calculate only the magnitude of ideological capacity by eliminating its direction.  \( \vec{I} \)  is a vector and therefore has both magnitude and direction.

For example, let's say that South Korea is composed of two individuals, Peter and Kevin, and Peter is a conservative capitalist ideologically while Kevin is a modified capitalist. When their ideological resources are  combined (  \( | \sum \vec{I_{[-]}} | \)  ), a slightly more conservative modified capitalist voice will be heard on various issues. In contrast, Peter and Kevin will cooperate to present their arguments by using their knowledge and reasoning to oppose the socialist direction of ideology. At this point, 'conservative modified capitalism' is the direction in which the two people's ideological capacities are summed up, and the knowledge and logic obtained by the cooperation of the two individuals are the magnitude of their ideological capacities.

The absolute value symbol   \( | \sum \vec{I_{[-]}} | \) , which includes all ideological capacities, is used to calculate the magnitude of capacities by removing the directionality from ideological capacities. Therefore,   \( | \sum \vec{I_{[-]}} | \)   can be understood as the 'mentally unified internal capacity' (simply referred to as 'mental capacity') of a political actor. 

Similar to the calculation of the growth rate of economic capacity, a factor  is multiplied to adjust the proportion of mental capacity reflected in the growth rate of armed capacity. Thus, the following formula holds.
 
         [Fmla.3.1.7]          \( \frac{ d }{ dt } A =  \rho_{1} \left ( \frac{ \begin{vmatrix} \sum \vec{ I_{[-]} }
\end{vmatrix} } {average.of. \begin{vmatrix} \sum \vec{ I_{[-]} }
\end{vmatrix} } \right ) A \)

Assuming an initial value of , the differential equation can be solved for  to obtain the following equation.

         [Fmla.3.1.8]         \( M= A_{0} \cdot e^{ \left ( \rho_{1} \left ( \frac{ \begin{vmatrix} \sum \vec{I_{[-]}}
\end{vmatrix} } {average.of. \begin{vmatrix} \sum \vec{ I_{[-]} }
\end{vmatrix} } \right ) t \right ) } \)

 

Rate of change of ideological capacity

The third equation, \( d| \vec{I} |/dt \), only needs to be considered in light of the fact that ideological capacity is a vector. Therefore, this equation means that as economic capacity increases, i.e. when society is more materially abundant, ideological capacity also grows. When scholars can focus on their research due to economic prosperity, they can strengthen their values through knowledge and ideology. In England during the 12th and 13th centuries, when the population grew and prosperity increased, Oxford and Cambridge, the first universities in England, were developed. The flourishing of Enlightenment thinkers such as Montesquieu, Voltaire, and Diderot in 17th-century France was also due to the development of economic conditions such as economic growth and social mobility. The increase in people's literacy rates due to economic development was the basis for the development of Enlightenment thought. Conversely, in the reign of Catherine the Great of Russia in the 18th century, the low economic status of priests resulted in their limited authority (ideological capacity).

However, the size of economic capacity does not affect the direction (content) of ideology. In the Joseon Dynasty, as their economic capacity increased, scholars were able to conduct more research, which enabled them to further develop the logic and knowledge necessary to justify their values. However, the content (neo-confucianism) of their values was led by scholars such as Toegye and Yulgok, and there is little reason to attribute it to the influence of economic abundance. Looking at the development of Christianity in Europe, it does not seem that the level of economic poverty or prosperity has had a significant impact on changing the values (i.e. directionality) included in ideological capacity.

Therefore, assuming that economic capacity only enhances the size of ideological capacity, the following differential equation holds.

          [Fmla.3.1.9]                \( \frac{d}{dt}\left|\vec{I} \right| = \rho_{3}\left ( \frac{E}{average.of.Es} \right ) \left| \vec{I}\right| \)

If we assume the initial value to be \( \vec{I_{0}} \), we can solve this differential equation to obtain the following equation.

          [Fmla.3.1.10]            \( \left| \vec{I}\right| = \left| \vec{I_{0}}\right| e^{\left ( \rho_{3}\left ( \frac{E}{average.of.Es} \right )t  \right )} \)

To summarize the above content, the following equation holds for political capacity ( \( \vec{L_{t}} \) ) as a function of time (t).

         [Fmla.3.1.11]            \( \vec{L(t)} = \begin{bmatrix}A(t) \\  \\ E(t)\\  \\ \vec{I(t)} \end{bmatrix} \)

                                                                  \( = \) \( \begin{bmatrix}
A_{0} e^ { \left (  \rho_{1} \left (   \frac{\begin{vmatrix} \sum \vec{I_{[-]}}
\end{vmatrix}}{average.of. \begin{vmatrix} \sum \vec{I_{[-]}}
\end{vmatrix} } \right )  t \right ) } \\
 \\
E_{0} e^ { \left (  \rho_{2} \left (   \frac{ A }{average.of. As } \right )  t \right ) } \\
 \\
\begin{vmatrix} \vec{I_{0}}
\end{vmatrix}  e^ { \left (  \rho_{3} \left (   \frac{ E }{average.of. Es } \right )  t \right ) } 
\end{bmatrix} \)

 

Since the exponentials of e are too complicated, they can be expressed using the  function, and the phrase "the average of..." can be mathematically represented, resulting in a more complicated but practically equivalent equation. 

Understanding various aspects of the 1st law

 

Here, we will explain the paradoxical aspects of the independence of Samjae capacities and their interrelationships^[Fmla.3.1.3]. This is because ⓐ each component of Samjae capacities is independent of each other (for example, the strength of armed capacity does not affect economic capacity) and ⓑ as stated in [Fmla.3.1.3], Samjae capacities are sequentially strengthened (for example, if armed capacity is strong, economic capacity gradually increases), which seems contradictory.

However, there is no contradiction in the 1st law of Samjae Capacities. Its key idea is that the total quantity of each Samjae capacity is not affected by other Samjae capacities (independence), and the size of a political actor's Samjae capacity within that total quantity is influenced by his or her other Samjae capacities (interaction). For example, suppose that the total quantity of economic capacity within a political regime is 100 and that member a has 10. If a's armed capacity is greater than that of other members, a's economic capacity increases more. The increased economic capacity of a only increases the proportion that a holds within the 100, rather than expanding the total economic capacity within the political regime.

Here, we can find an apparent contradiction between the fact that [Fmla 3.1.11] implies that each of the Samjae capacities grows infinitely over time, and the practical intuition that the growth of each Samjae capacity cannot be infinite. The basic solution to this problem is to introduce the logistic model, which is a fundamental model in differential equation models. In this way, mathematical models developed in other fields can be incorporated into political science's mathematical modeling. This book is limited to providing examples of such possibilities and presenting basic concepts for such applications. Further discussions on more advanced and precise mathematical models will be attempted in other research. For example, in the subsequent discussions in this book, we will only consider the basic geometric growth model [Fmla.3.1.3], without considering the logistic model [Fmla.3.1.13].

On the other hand, let us consider examples of the  \( \rho \)  values that affect the magnitude of each component of Samjae capacities. For the element  \( \rho_{1} \)   that affects the size of armed capacity, it may include advancements in science and technology for creating new weapons, developments in fortification techniques, or political culture. Meanwhile,  that affects the size of economic capacity would include the state of land in agricultural economy or the invention of production machines. These values of \( \rho_{n} \) ( n=1, 2, 3) are independent of the size of other Samjae capacities. That is, the size of armed capacity (A) does not affect the value of  \( \rho_{2} \)  that affects the size of economic capacity, and the size of economic capacity (E) does not affect the value of  \( \rho_{3} \)  that affects the size of ideological capacity.