Figure 3 demonstrates the three-dimensional kernel-based transition probability for the Relative Service Value-Added per Capita (RSVAPC) of all countries. Secondly, Figure 4 presents a contour map along with the transition dynamics of the national units. In Figure 3, the relative frequency indicates the probability of transition at the country level from one particular RSVAPC value in year t to another RSVAPC value in year t1. It is noteworthy that the RSVAPC value is calculated based on the global average; hence, one is the average value. Consequently, a value less than one and larger than one marks a below-average and above-average RSVAPC, respectively. Figure 3 indicates that the majority of countries had an RSVAPC value of nearly 0.1, indicating a below-average RSVAPC.

In Figure 4, the contour map shows a top view of the three-dimensional graph. It can be visualized that Figure 4 could be separated into several vertical lines, with each line serving as a probability density function by itself—showing the probability of transitions from period tto period t+1. For instance, when a specific country has the RSVAPC value of 0.1 at period t, a vertical line intersecting the horizontal axis at 0.1 can be imagined. The intersection points of this imaginary vertical line with the contour lines illustrate the probability of climbing from the initial level of 0.1 to other values of RSVAPC plotted on the vertical axis. It can be seen from the graph that the imaginary vertical line peaks at the point where the RSVAPC value on the vertical axis is also 0.1. This indicates that a country with 0.1 as an initial value of RSVAPC will have a relatively high probability to remain at 0.1 in the following year. It can also be seen that the imaginary vertical line will reach the last contour line when the RSVAPC value on the vertical axis is equal to 0.65. This means that there is a low probability to climb from 0.1 to 0.65 and it is even lower when moving up to higher regions where contour lines are not visible.

In conclusion, the persistence is quite high, based on Figures 3, 4. Evidently, as shown in Figure 4, there is a very high probability for the countries to remain in their current states of service output capacity. The probability mass peaks along the diagonal, thereby showing that there is a very high probability for the countries to remain at their current positions. This illustrates very slow progress in the development of the service sector. Moreover, as the variability (the probability of moving upward/downward in the next period) in probability mass is relatively low for countries with RSVAPCs less than 0.4, perseverance is observed to be more severe for countries with lower values of RSVAPC. Thus, Figures 3, 4 clearly show that the development of the service sector in countries with extremely low values of RSVAPC remains sluggish, thereby indicating a disparity in the development patterns of the global service sector. This is the first piece of evidence for the unbalanced and inadequate development of the concurrent service sector. The disparity and rigidity inherent in the transition dynamics in both the figures will subsequently translate into a country-level long-run steady-state distribution of RSVAPC.

Figure 5 shows the long-run steady-state ergodic distribution of the countries. As observed, several countries are likely to converge where the RSVAPC value is 0.08—the highest peak observed from the distribution. As one is the global average; industrial development in numerous countries will be very low if the transition dynamics do not change. It can also be observed that convergence club at a higher level, with a small peak where the value of RSVPAC value is 5.1. It indicates that these countries will congregate at this particular value of RSVAPC. Despite the higher values of RSVAPC as compared to the global average, the situation is unlikely to be impressive, since it suggests that many countries across the globe, with few exceptions, will incline toward low production capacity as a norm.

Considering that the service sector will fuel the future development of the service sector, the ideal overall situation to manage the issues of the low and uneven capacity of production would be a steady convergence to the global average. Eventually, countries with below-average values of RSVAPC will need to climb upward and/or the ones with above-average values to move downward. However, countries with high production capacity should not lower their output. As a consequence, there will a strong plea of attention to countries with below-average RSVAPC values, especially those with low or no capability to move up. This study helps in identifying these countries by observing the MPP.

In Figure 6, the MPP plots the probability of net upward mobility, ranging from -100 to 100, as a percentage against the RSVAPC values. A positive value represents a country with a positive probability of net upward mobility, while a negative value illustrates a country with a negative probability of upward mobility. When moving from a region above the horizontal axis to the one below the horizontal axis, the MPP will intersect the horizontal axis. It is imperative to note that countries on the left-hand side of the intersection point have a positive value while the ones on the right-hand side of the intersection point have a negative value. Therefore, a large number of countries will converge around the point of intersection. Consequently, the transition dynamics underlying the MPP in Figure 6 enable a thorough understanding of the shape of the ergodic distribution in Figure 5. Figure 6 shows that the MPP intersects at the horizontal axis where the value of RSVAPC is 0.16. Subsequently, until the MPP intersects the horizontal axis again at the point where the value of RSVAPC value is 3.9, it remains negative. Therefore, within the distribution, only those countries with extremely low levels of output (RSVAPC values of less than 0.16) and extremely high levels of output (RSVAPC values of more than 3.9) showed a higher likelihood to move upward.

Based on this alarming result, one may think about the reason most, if not all, countries failing to make progress in their service sectors, while a few of them continue to outperform the rest. To examine the service sector growth in detail, the database is divided into the wealthier Global North and the poorer Global South.

In Figure 7, the three-dimensional kernel-based transition probabilities for RSVAPC of the two important regions (the Global North and the Global South) with annual transitions can be observed. Subsequently, Figure 8 shows the corresponding contour maps.

Figure 7A illustrates two peaks, situated at RSVAPC values of 1 and 5, in the three-dimensional graph for the Global North. This suggests that for most countries of the Global North, the development in the service sector was well above the global average with a distinct group of countries experiencing five times above the average production capacity. On the other hand, Figure 7B demonstrates a contrasting scenario. Here, the peak is located at an extremely low value of RSVAPC—0.05, which is also far below the global average. This shows severe underdevelopment in most countries of the Global South, thereby indicating a disparity in the degree of global service development.

A larger variability (i.e., the likelihood of moving upward/downward in the following period) in the Global North as compared to the Global South can be observed in Figures 8A,B. Moreover, a rise in production capacity in both the Global North and the Global South leads to an increase in variability. It demonstrates that rigidity is relatively pronounced in countries with low values of RSVAPC—a key source of unbalanced development.

Figure 9 shows the transition dynamics translating into the long-run steady-state ergodic distribution. From Figure 9A, it can be observed that the two most obvious peaks are situated at the RSVAPC values of 1.35 and 5.18. This suggests an encouraging future development in the service sector of the Global North. Meanwhile, Figure 9B illustrates that the two peaks appear at RSVAPC values of 0.09 and 0.9, respectively. This finding is considered disturbing due to its demonstration that most countries of the Global South will have extremely low levels of RSVAPC relative to future development in the service sector. Moreover, the ergodic distribution is more scattered in the Global North as compared to the Global South. It indicates that countries are mostly tied and focused on the two below-the-average peaks with a few exceptions in the latter.

In Figure 10, the MPP suggests a reason behind the current forms of the ergodic distributions. It can be observed from Figure 10A that the Global North countries with an RSVAPC value of less than 1.13 have a positive net probability of upward mobility. At this intersection, the first peak in the ergodic distribution remains evident. The second intersection point presents an RSVAPC situated at level 3.5, representing a second peak in the ergodic distribution. It is worth noting that these two peaks appear above the global average. Consequently, the MPP shows the influence of transition dynamics on a higher development in the service sector of the Global North. Conversely, in Figure 10B, the entire MPP appeared almost below the horizontal axis which suggests a negative net probability of upward mobility. The first intersection point is situated where the RSVAPC value is 0.09, which is far below the former first intersection point (i.e., RSVAPC value of 1.13). It reflects that the Global South countries have relatively low service output than the global average. It is also important to note that the MPP is positive for the RSVAPC values between 0.2 and 0.9, leading to the emergence of another peak in the ergodic distribution. This is due to the presence of a few countries such as China in the Global South with a high production capacity and upward mobility. However, these exceptions are unlikely to provide any help for the development of the overall service sector in the Global South. Without such sporadic cases of countries with an RSVAPC value of less than 0.09, the entire MPP of the Global South remains below the horizontal axis. It implies that only the Global South countries with extremely low service output (those countries with an RSVAPC value less than 0.09) have a chance for upward mobility. Additionally, countries that have RSVAPCs greater than 0.09 will fail to maintain progress in their service development.

The alarming findings suggest a possibility of increasing disparity between the Global North and the Global South in the future. The development of the service sector in the Global North will continue to make greater strides, while the output capacity in many Global South countries struggles to reach the global average. This prompts the need for a re-examination of the development policy in most countries of the Global South. Moreover, the governments are required to develop pragmatic policies to address and remove a large number of obstacles (e.g., to reduce transaction costs) to the service sector development to bridge the gap between the two regions.

To assess the correlation between income and industrial development without the influence of sporadic cases in the Global South, the ergodic distributions for different income groups of countries are considered. The ergodic distributions and the MPP for four respective income groups are shown in Figures 11, 12.

In Figure 11A, the entire ergodic distribution of low-income countries is compressed to the left of the global average, with convergence clubs at the levels of 0.02, 0.04, and 0.12. Owing to the highly concentrated distribution, the peaks are visibly pronounced, and the convergence clubs are closely situated. In Figure 12A, the MPP for the low-income countries indicates that countries have the chance to move upward, only when the values of RSVAPC are less than 0.02, between 0.03 and 0.05, and between 0.08 and 0.12. When the RSVAPC value reaches 0.17, the net probability of upward mobility is at -100. Accordingly, low-income countries are unable to transit through the threshold where the RSVAPC value is 0.17. Upon reaching this threshold, a level less than half of the global average, the RSVAPC value of the country will decline, thereby moving it downward in the distribution. This suggests the possibility of a service development trap in low-income countries. It is an alarming situation, as development in the service sector is the future growth engine in both developed and developing countries. For low-income countries, development in the service sector is the major pathway to achieve a reasonable living standard in the future. These study findings imply that a country continues to face poverty if it exists unless there is external interference.

Figure 11B shows the ergodic distribution for the lower-middle-income countries. The distribution is more dispersed than that of low-income countries, where all the convergence clubs—0.1, 0.22, and 0.31—are below the global average. This shows that the development of the service sector in lower-middle-income countries is far from encouraging. In Figure 12B, the MPP confirms the implications of the ergodic distribution for the lower-middle-income countries. The first intersection point appears where the level of RSVAPC is very low at 0.1. The second and the third range of positive net upward mobility appear where the levels of RSVAPC are approximately 0.15–0.22 and 0.23–0.31, respectively. When the value of RSVAPC reaches 0.5, the net probability of upward mobility becomes -100. In other words, the lower-middle-income countries with no or relatively low levels of service output have a better chance of making proper progress in service development. However, if the output level is close to one-third of the global average (i.e., an RSVAPC level of 0.31), they struggle to maintain progress in their development. Moreover, when the level of RSVAPC for lower-middle-income countries is 0.5, the net upward mobility reaches -100. It suggests that a decline in an RSVAPC level of a country to 0.5, will bring the mobility downward in the distribution. Although a negative net upward mobility appears at a higher output level in low-income countries, a development trap in lower-middle-income countries can be found. After attaining a slightly higher level of economic growth, these countries fail to continue making further progress in development required for sustainable growth.

In upper-middle-income countries, the situation appears to be better. Figure 11C shows three convergence clubs. The first two peaks are below the global average at 0.37 and 0.7, respectively. However, the third peak appears at level 0.98, which is close to the global average. Despite its similarity with the low-income and lower-middle-income countries, the overall shape of the ergodic distribution for the upper-middle countries is not comparable with the previous two distributions, as it is more dispersed and has a higher average level of RSVAPC. In Figure 12C, the MPP for the upper-middle-income countries shows a positive net upward mobility for RSVAPC values ranging from 0.8 to 0.95. It suggests some upper-middle-income countries with RSVAPC values around the global average can maintain sustainable development. However, the net upward mobility becomes -100 when the value of RSVAPC reaches 1.45. To avoid the development trap, the upper-middle-income countries are required to reach and sustain a high rate of development.

In Figure 11D, the first peak of the ergodic distribution for high-income countries is situated at the RSVAPC level of 1.7, while the second peak is situated at 5.2. Looking at this distribution, many high-income countries will have industrial development above the global average. Moreover, some of the countries even have five times the production capacity of the rest of the world. The MPP in Figure 12D shows that the first intersection point appears when the level RSVAPC level is at 1.5, while the second intersection point appears when the level of RSVAPC level is at 3.5. These two intersection points provide the two convergence clubs situated in the ergodic distribution. Unlike the previous MPPs for the three different income groups, there is no obvious development trap within the high-income countries. The most negative net upward mobility is about -50 when the level of RSVAPC reaches 6.7.

In the conclusion of the comparison based on four different income groups, countries with higher per capita income will perform better in the development of their service sector than those with low per capita income. Based on the distribution dynamic analysis, the growth of the service sector is linked with per capita income, thereby creating demand for services within the market. More importantly, based on the ergodic distribution and the MPP analysis, various underdeveloped countries lack the ability for upward mobility in the distribution without any external interferences. Therefore, it is not a natural and inevitable process for a specific country to evolve from agricultural to industrial and then to a service economy.

Figures 13, 14 illustrate regional ergodic distributions and MPP, respectively. There are similarities between the ergodic distributions for the East Asia and Pacific region, the Europe and Central Asia region, the Latin America and Caribbean region, and the Middle East and North Africa region. From the ergodic distributions for these four regions, it can be observed that numerous countries cluster around the levels of RSVAPC which is less than half of the global average, while a few countries cluster around the levels of RSVAPC above the global average. For these regions, the MPP confirms the implications of their ergodic distribution. The MPP of these regions shows that the first intersection points are situated below one-half of the global average. Additionally, they have some positive net mobility when they reach a higher level of RSVAPC.

The progress of development remains the lowest in two particular regions—the Sub-Saharan Africa region and the South Asia region. For these regions, the ergodic distributions are far below the RSVAPC value of 2. Moreover, the ergodic distribution for the Sub-Saharan Africa region is the most compact, indicating the two pronounced peaks at levels 0.02 and 0.12, respectively. In the South Asia region, the two peaks are slightly higher—0.08 and 0.95—as compared to that of the Sub-Saharan Africa region. However, when the value of RSVAPC value reaches 1.87 in the Sub-Saharan Africa region and 1.17 in the South Asia region, the net upward mobility value becomes -100. It implies that despite slightly better development progress in the South Asia region as compared to the Sub-Saharan Africa region, the former will encounter a development trap with a lower value of RSVAPC.

Compared to all the regions, North America has a relatively unique ergodic distribution and MPP. For this region, the ergodic distribution appears above an RSVAPC value of 4.7 and the first intersection point for the MPP is situated at 5.2. Moreover, it is the only region where the RSVAPC of all countries is above the global average. It is worth noting that the region will face from development trap. The net upward mobility reaches -100 in the North America region when the value of RSVAPC reaches 6.8. While the development traps in the Sub-Saharan Africa region create a hindrance in service development, the development trap in the North American region will delay development breakthroughs of the service sector across the world.

Based on the regional evidence, the primary peaks of the ergodic distributions for most regions are far below the global average, except North America. This indicates that almost all regions will not attain convergence to the global mean. The convergence clubs in all regions suggest that the unequal industrial development at the regional level is universal. The focus must be placed on the removal of upward-moving obstacles for countries that have a low level of RSVAPC and negative net upward mobility.