According to Eric Naevdal in his 2001 article “Optimal Regulation of Eutrophying Lakes, Fjordes, and Rivers in the Presence of Threshold Effects,” the use of threshold effects is widespread in natural and social sciences alike. A threshold represents a gateway from one description of the world into another, which may be quite different. More precisely a threshold effect describes a process by which the magnitude of the response variable changes significantly as the triggering stimulus exceeds some critical value. One can think of threshold effects as devices to explain nonlinear phenomena.
A way of understanding how threshold effects work is by means of an example. In economic growth, one of the main theories developed to explain the process of per-capita income growth in an economy is the one that allows for human capital to affect the production of output overtime. Human capital refers to the stock of knowledge that is accumulated through education and technological progress. These resources make workers more productive and able to adapt to new technologies. Most of the theoretical models that deal with the relationship between human capital and growth assume that this relationship is linear: Increasing the level of human capital should yield a higher rate of economic growth irrespective of the level of human capital, something, according to Pantelis Kalaitzidakis, Theofanis Mamuneas, Andreas Savvides, and Thanasis Stengos in their 2001 study, not borne by empirical evidence.
Costas Azariadis and Alan Drazen observe in their 1990 article “Threshold Externalities in Economic Development” that long-run growth rates exhibit persistent differences between more and less developed economies. Historically this derives from the “Big-Push” ideas of P. Rosenstein-Roden (1943) and Ragnar Nurkse (1953) who have noted that an economy can remain stuck in an underdevelopment state unless there is a substantial investment effort. Some low-income economies appear to be caught in a low-growth environment and lag behind countries similar in terms of endowments. The standard growth model cannot account for such persistent differences. While one can seek explanations for this stylized fact in non-economic factors, there is an economic explanation based on the existence of technological externalities with a threshold property that allows for economies with very similar structures to exhibit different growth experiences. Specifically, once human capital attains a certain threshold level aggregate production possibilities may expand especially rapidly. Economies that have not attained this threshold level may languish in a self-perpetuating state of persistent underdevelopment.
Nonlinearities due to the presence of threshold externalities in the production of human capital can lead to different growth experiences, including possible low-growth (or low-development) “traps,” where low investment in human capital (education) will discourage further human capital accumulation (acquisition of additional skills) and, hence, result in a low-growth trap. Consequently, a group of countries with unequal initial educational endowments may never catch up to each other; the growth rate of ones with higher endowments will diverge from those with lower initial endowments of human capital. This suggests a role of government intervention in the educational sector to introduce policies (such as educational subsidies) that support skill acquisition and higher educational attainment.
As explained above, threshold effects are the main cause for the presence of nonlinearities in the process that underlies economic growth. From a modeling point of view, threshold effects only produce a specific type of non-linearity that is quite restrictive. The process that is induced by thresholds can be described by piece-wise linear segments that are individually defined by thresholds. Thresholds act as the “knots” that connect the different segments together. In that case a threshold effect is an “all or nothing” effect as values below and above the threshold result in different regimes. Alternatively, such regime change could occur gradually and not as abruptly by means of a smooth transition. The amount of smoothing done to the knots that connect the linear segments will result in different types of nonlinear effects. Assuming that smoothing is arbitrary there is a large class of such smooth transition nonlinear effects that includes threshold effects as a special case. Yet despite their limitations, threshold effects have proven to be powerful devices to introduce nonlinear-ity in economic modeling, especially growth theory.
Azariadis, Costas, and Alan Drazen. 1990. Threshold Externalities in Economic Development. Quarterly Journal of Economics 105: 501–526.
Kalaitzidakis, Pantelis, Theofanis Mamuneas, Andreas Savvides, and Thanasis Stengos. 2001. Measures of Human Capital and Nonlinearities in Economic Growth. Journal of Economic Growth 6 (3): 229–254.
Naevdal, Eric. 2001. Optimal Regulation of Eutrophying Lakes, Fjordes, and Rivers in the Presence of Threshold Effects. American Journal of Agricultural Economics 83 (4): 972–984.
Nurkse, Ragnar. 1953. Problems of Capital Formation in Underdeveloped Countries. Oxford: Basil Blackwell.
Rosenstein-Rodan, P. 1943. The Problem of Industrialization of Eastern and South-Eastern Europe. Economic Journal 53: 202–211.