**1. Introduction**

As is well known, macroeconomics studies how economic systems work from an aggregate point of view as a result of the interactions that take place between different economic agents. Given this consideration, we can state that this field of economic science has the following two aims:


All economic models rely on a theoretical basis that tries to simplify the complexity of economic systems by stating a set of relationships between variables linked by parameters that are a priori unknown (but in certain cases can be estimated econometrically). The dependence of the model on several parameters implies that when it is necessary to determine the adequacy of the model to reality, the exact values of most of the parameters embedded in the model are not known. This involves taking uncertain quantities in the models as defined, and thus, when the model is applied, a result close to the empirical reality is already expected, but in practice it can often be wrong. Solow [1] already warns of this fact with the statements "all theory depends on assumptions which are not quite true", and "the art of successful theorizing is to make the inevitable simplifying assumptions in such a way that the final results are not very sensitive". The aim of this article is to present a

complementary and novel point of view, introducing a fuzzy logic model in order to obtain results based on an infinity of possible inputs. This new modelling based on fuzzy logic [2] allows that empirical reality can be one of the expected results of the model and therefore the actual result is not expected to be outside the range of possible expected values. To sum up, the idea is not to check how much we have deviated from the actual solution with the model, but to confirm that this actual solution is one of the possible solutions derived in the model.

In order to limit the set of possible solutions yielding the best forecast, it is common to resort to certain constructs of the theory of fuzzy sets, known as fuzzy numbers, and which can be associated with numerical values with a degree of possibility [3]. The idea of considering the future in a pessimistic, optimistic and highly plausible scenario is the basis of these types of numbers that collect all values with some degree of possibility greater than zero. It is the ease of use of these mathematical structures [4–7] that allows uncertainty to be incorporated efficiently into any economic model regarding the behavior of economic, social and financial scenarios. This is especially necessary when sudden changes in the values of the variables are expected due to events not reflected in the historical series.

The incorporation of fuzzy logic into economic models has been addressed in a multitude of papers [8–10]. Following this line of research, in recent years the fuzzy logic perspective has been proposed for various growth models [11,12]. The present work, included in this current line of research, proposes the creation of a dynamic model of income behavior based on the classic Harrod model. This model is seminal in the economic growth literature. It is well known that this model has several shortcomings [13,14]. For example, the model assumes that productive capacity is proportional to capital stock, an assumption that has been shown to be false. In fact, the model was overtaken first by the so-called "exogenous growth" models of Solow [1] and Swan [15] and then by the "endogenous growth" models [16]. Essentially, exogenous growth models explain longterm growth using a variable which exogenous to the model: technology. Endogenous growth models try to explain the way in which the evolution of technology allows longterm growth. This article recovers Harrod's original approach in order to show how the contribution of fuzzy logic to a very simple model can change the predictions that stem from the model and bring them closer to reality. This paper must be understood, then, as a first step in a process that will have to continue with the application of fuzzy logic to exogenous growth models, first, and endogenous, later.

In order to meet our primary objective, the main body of the work has been divided into three different blocks. Section 2 presents a brief summary of the fuzzy logic tools used by the new model, specifically the method used to solve equations with fuzzy parameters. Section 3 details the initial Harrod model for determining income through the use of fuzzy parameters. The results are simplified in Section 4 to make them more operational when we analyze the particular case in which uncertain parameters are expressed through triangular fuzzy numbers (TFN). Section 5 provides a numerical example with the proposed new model. Finally, the paper ends with the conclusions and references.
