4.1.1 Axioms
First, we formulate
axioms by using the concepts introduced in Sect.
2.1. In our paper, the axioms represent all the assumptions that are not empirically founded. Although we try to minimize the use of such assumptions, our models, like all other models of empirical sciences, cannot be formulated without using a minimum of not empirically founded assumptions.
Axiom 1 is a standard for modeling structural change in the three-sector framework (see Stijepic (
2015) for a discussion and the papers listed in Sect.
4.1.3 for some examples of models based on Axiom 1). In Sect.
4.2, we use Axiom 1 in all our models, while the following two axioms are only used in some of our models. Without loss of generality, Axiom 1 extends the modeling horizon
d
+ to infinity such that we can exploit the maximum prediction range of our models. However, most of the following discussion does not require this extension.
The continuity axiom (i.e., Axiom 2) is a typical (long-run) modeling convention in development and growth theory. The models presented by Kongsamut et al. (
2001), Ngai and Pissarides (
2007), Acemoglu and Guerrieri (
2008), Foellmi and Zweimüller (
2008), and Boppart (
2014) are typical examples of multi sector models satisfying Axiom 2.
Axiom 3 is not necessary for formulating our assumptions, describing the empirical evidence, deriving the predictions in Sect.
4.2, or any of our other results. That is, we could write an alternative version of our paper that does
not rely on differentiable functions or the notion of the derivative and generates the same results. However, the use of derivatives abbreviates, among others, the formulation of Properties 1–3, Definition 5, as well as Laws 3 and 4, significantly. Therefore, we rely on Axiom 3.
4.1.2 Laws
In this section, we translate the verbal statements of Regularities 1–6 into mathematical statements, which we name laws and, in particular, Laws 1–6. As we will see in Sect.
4.1.3, Laws 1–6 are supported by the theoretical literature. Thus, their naming (‘laws’) is consistent with the definition of the term ‘law’ discussed in Sect.
1.
In general, a law is defined as a regularity that is valid across
time and
space.
6 The fact that laws are valid across
time (cf.
D) means, among others, that they are valid in future to some extent; thus, we can use them for prediction; as we will see later, there are different ways to extend laws across time (i.e., across the time interval (
D) to which our models refer). ‘
Space’ refers here to countries, where we can distinguish between general laws (i.e., laws that are valid across all countries) and ceteris paribus laws, which are valid only for some countries (see Stijepic
2016, p. 20ff. for a discussion). This distinction is, however, not important in our paper, since our mathematical/logical derivations are the same irrespective of the type of law to which they refer (general vs. ceteris paribus law). Therefore, we assume, henceforth, that the laws are valid for the country set
C. This set may represent all countries of the world or only a subset of them. The readers may decide whether they consider the laws discussed in our paper as general laws or as ceteris paribus laws and, thus, whether our results/predictions are valid for all countries or only for some subset of countries. Overall, Laws 1–6 are statements that are valid across time (
D) and space (
C). We let the readers decide to which countries (‘space’) the laws apply and focus now on the discussion of the period
D.
In the following, we rely on Axiom 1 (and Axioms 2 and 3, in part) and the concepts of elementary calculus and set theory to translate the verbal statements of Regularities 1–6 into Laws 1–6. Recall that Axiom 1 refers to the long run; thus, all the statements of Laws 1–6 are statements about the long-run dynamics. As we will see, while Laws 1 and 2 describe the state of a country \(c \in C\) (at the time points a
c
and b
c
), Laws 3–6 describe the (transitional) dynamics of the country \(c \in C\). The notation used in Laws 1–6 and the verbal statements of Regularities 1–6 jointly imply the following interpretation of the time points a
c
, b
c
, and z
c
: ‘a
c
’ is an ‘early point in development’ of country \(c \in C\); ‘b
c
’ is a ‘later point in development’ of country \(c \in C\); and ‘z
c
’ is the turning point in manufacturing sector dynamics (from the industrialization period to the de-industrialization period) of country \(c \in C\). We start with the translation of Regularity 1 into Law 1.
Law 1 states that each country belonging to the group
C is an agricultural economy (i.e., is characterized by
x
1 > 0.5) over the period of time (
d,
a
c
]. As we can see, Law 1 extends to the lower limit (
d) of the time period considered (
D) (cf. Axiom 1). Thus, Law 1 states that primitive economies are agricultural economies. This fact may also be relevant for long-run predictions where the backward extension of the trajectory, i.e., {
x
c
(
t)
\(\in\)
S:
d ≤
t <
a
c
}, is relevant (cf. Stijepic (
2015, p. 81)).
Note that the period (
d,
a
c
] represents (a part of) the ‘early development phase’ (cf. Regularity 1 and Sect.
4.2.1) of country
c, where
a
c
is indexed by
\(c \in C\). In other words, Law 1 allows for differences in the duration of the phase (
d,
a
c
] across countries
\(c \in C\). This makes sense, since different countries overcome the early development phase at different points of time (see the references listed in Sect.
3 for empirical evidence).
Law 1 is formulated by using the expression \(\forall c \in C\). This reflects our discussion of the fact that (our) laws are valid for a group of countries (C) and not only for one country (c). We adhere to this view when formulating Laws 2–6.
Law 2 states that each country c belonging to the country group C is a services economy (i.e., is characterized by x
3 > 0.5) at the time point b
c
> a
c
. Thus, b
c
represents a point in the later phases of development of country c (cf. Regularity 2).
In Law 2, we do not explicitly define the point
a
c
, since we use Law 2 only in conjunction with Law 1 when modeling structural change in Sect.
4.2 (and Law 1 has already defined the point
a
c
). Laws 1 and 2 jointly state that each country from the country group
C is, first, an agricultural economy (at time
a
c
) and, later, a services economy (at time
b
c
).
Note that Law 2 allows that the point
b
c
differs across countries
\(c \in C\), since
b
c
is indexed by
c. This is consistent with the empirical evidence, which shows that some countries reach the status of a services economy earlier than others (see the references listed in Sect.
3 for empirical evidence).
The difference between Laws 2a and 2b is simple: Law 2b states that country
c is a services economy at time
b
c
and continues to be a services economy for the rest of the time period
D; in contrast, Law 2a does not state what happens after the time point
b
c
(i.e., economy
c may be a services economy or not for
t >
b
c
). This fact is of importance for the predictions of the limit dynamics of labor allocation, as we will see in Sect.
4.2.
Law 3 states that in each country \(c \in C\), there exists a period [p
c
, d
+) of monotonously decreasing agricultural share (\(\dot{x}\)
1c
≤ 0), where d
+ is defined by Axiom 1. In particular, Law 3 extends to the upper limit (d
+) of the time period considered (D) and over all countries belonging to the group C. Moreover, Law 3 allows for cross-country differences in the starting point of the period [p
c
, d
+), since p
c
is indexed by the country index \(c \in C\).
The discussion of Law 4 is analogous to the discussion of Law 3. Law 4 states that each of the countries belonging to the group C is characterized by a monotonously growing services share over the period [q
c
, d
+). The starting point of the period [q
c
, d
+) may differ across countries.
Law 5 states that the dynamics of each country c are characterized by a ‘turning point’ z
c
, where \(c \in C\). Per Law 5, this ‘turning point’ partitions the period [a
c
, b
c
] into a phase of monotonously increasing manufacturing share and a phase of monotonously decreasing manufacturing share. z
c
may differ across countries \(c \in C\), since z
c
is indexed by the country index c. Law 5 refers to the points a
c
and b
c
without defining them explicitly, since we use Law 5 only in conjunction with Laws 1 and 2, which define the points a
c
and b
c
.
Law 6 refers to the time period D and states that in this period, each country \(c \in C\) does not have a point of self-intersection (cf. Definition 3). Law 6 extends over all the countries belonging to the group C and over the whole period D.