**6. Conclusions**

In the paradigm of deterministic models inspired as models of historical construction, these constitute a reflection of the previous behavior at the time of its construction without any possibility to consider sudden changes in the values of the variables due to events not reflected in the historical series. This method of using historical data to make estimates from probability distributions often meets with contradictions in economic reality, which undergoes continual change given the common appearance of new factors not reflected in past behaviors. Therefore, it is convenient, as shown in the present paper, to be able to incorporate in the models future unknown values which can be evaluated by experts with a certain degree of possibility, so that, in this way, models can take into account the continuously changing nature of the economic, social and financial environment. This paper has shown that fuzzy modelling allows introducing inaccuracy by incorporating the variability of model parameters in a reasonable manner. This allows us to obtain a model that includes more information and can therefore produce better future predictions. In this sense, the proposed model allows us to ascertain the impact on the prediction process of income behavior being jointly studied for all possible values of the uncertain parameters considered, each with their own degree of possibility.

Summing up, this paper shows how, by starting with some distributions of possibility for the uncertain magnitudes that are taken as inputs, we can determine the corresponding distribution for any equilibrium values that potentially represent income. Assigning possibility distributions to variables that are not precisely known allows us to provide a proper interpretation of economic reality, thus obtaining forecasts more suited to the real world, since more information is taken into account when applying the model.

On the other hand, from the example based on the classic Harrod's model of economic growth in the context of uncertainty, we highlight the following points:

1. Under the established fuzzy conditions, that is, considering the marginal propensity to save *a* and the accelerator *b* as fuzzy numbers, the results obtained with the application of the model have an interpretation within the context of uncertainty in which they are presented, and, in this sense, they give a more generality to the application of the model, which implies, in this context of uncertainty, a closer approach to reality than the still image that results from the classical application. This study in a fuzzy context allows us to incorporate within the model what would be a broad analysis of sensitivity. The study is a logical generalization because if the parameters that we have considered as fuzzy numbers are reduced to crisp numbers, the results obtained coincide with those resulting from the classical application;

2. When analyzing the behavior of the model under the considered hypotheses of uncertainty, we observe that, as in the context of certainty, income and investment must grow at the same rate to be in equilibrium under the conditions imposed by the model. This rate is given by the growth index, which in this study is expressed as a fuzzy number with the corresponding membership function being expressed from the membership functions *μa* and *μb*;

3. The direct application of the arithmetic of the intervals is not always suitable for the calculation of the fuzzy growth index, since, depending on the form of the expression of this index, the expression of the intervals that determine the α-cuts with the direct application of arithmetic has much more entropy than the calculation of the α-cuts obtained by using the interpretation proposed by Buckley and Qu, which is appropriate in the application of the model presented;

4. If we use *G*- = 1 + *a*/ *b* − *a* as the expression for the growth index, we have found that the applications of the rules of fuzzy arithmetic lead to a correct and interpretable result of the α-cuts of the growth rate. Therefore, by using this expression the usual arithmetic can be applied;

5. As is to be expected, the time factor increases the uncertainty in determining future income and investment values, which is translated into the increase in entropy that characterizes the resulting fuzzy numbers as we move forward in time;

6. The initial value of income *Y*0, which we consider as a known and therefore crisp, is a value that influences the results only as a scale factor, as in the classical model;

7. From the knowledge of the membership functions *μa* and *μb*, it is possible to determine the membership function for income in any period, and this function adopts a simple and fully operational expression in the case that *a* and *b* are triangular fuzzy numbers. On the other hand, for investment values the expression obtained for the membership function is not operational and it is essential in practice to calculate the possible values and their degree of possibility through the expression of their *α*-cuts;

8. The purpose of the application presented in this paper is how models based on fuzzy numbers work, as well as to analyze how the model behaves when the parameters linked to the variables are fuzzy numbers and need to be operated according to the arithmetic of uncertainty. In further studies it will be necessary to go beyond the simple Harrod model of economic growth and study and analyze models of exogenous and endogenous growth with the aim of building new models based on uncertainty, more suited to the complex economic reality. It should be noted that governmen<sup>t</sup> expenditure and taxes are not considered in our representation of Harrod's model. Introducing these variables would prove a fairly straightforward task and would not affect the results produced by the model. An approach that involves exogenous and endogenous growth models would be more interesting, since it would allow us to take into account the role of, for example, technology, education, property rights or human capital, relevant variables missing from the Harrod approach;

9. It should also be noted that our representation of Harrod's model considers a closed economy and does not take into account the financial aspect of the economy. Nowadays, money flows around the world as countries trade with one another. Indeed, finance was at the heart of the economic crisis before the world was hit by that of the Covid pandemic. Therefore, an interesting approach would be to consider fuzzy logic in a model aimed at the open economy (with imports, exports and financial flows). This would represent a valuable addition to the Mundell–Fleming setting, which takes into account deterministic values for exchange rates, interest rates and output;

10. Beyond the long-term considerations of economic growth, our approach is also useful for analyzing the economy in the short term. By incorporating fuzzy logic, it is possible to study economic fluctuations related to the business cycle. Currently, dynamic stochastic general equilibrium models introduce fluctuations way means of random shocks into the economy. Fuzzy numbers could be used to represent these shocks so that the model takes into account the uncertainty pervading the real world;

11. Finally, considering macroeconomic variables as fuzzy numbers may also represent a way of improving the current measurements used by traditional economic models. It is well known that real world measures of national income, consumption, saving, investment, exports or imports are subject to uncertainty. Indeed, national accounts data are subject to frequent revisions as new information emerges. Presenting macroeconomic variables as fuzzy numbers would make the uncertainty inherent in such magnitudes apparent and result in policymakers being able to make more informed decisions.

Consequently, the modeling presented in this article expands the possibilities offered by the current Harrod's growth model [29–31], and continues along the path first followed with the proposal to use fuzzy numbers [11,12].

**Author Contributions:** This paper is the result of the joint work by all the authors. J.C.F.-C.; (conceptualization, methodology, conclusions); S.L.-M. (literature review, introduction, validation); R.R.-T. (literature review, validation, conclusions). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Spanish Ministry of Science, Innovation and Universities and FEDER, gran<sup>t</sup> number RTI2018-095518-B-C21.

**Data Availability Statement:** The data and methods used in the research have been presented in sufficient detail to the work so that other researchers can replicate the work. Numerical data has not been copied from another source.

**Conflicts of Interest:** The authors declare no conflict of interest.
