**Definition 3.** *Fuzzy equation*.



*A fuzzy equation is an equation in which coefficients or variables are expressed through fuzzy numbers. Without loss of generality, we will write a fuzzy equation:*

$$F\left(\widetilde{A} \; , \, \widetilde{X}\right) = \widetilde{B} \tag{10}$$


*where A and B are fuzzy numbers, which are usually called fuzzy parameters and X is the unknown, which we call fuzzy variable.*

We call the crisp equation associated with the fuzzy Equation (10) the equation:

$$F(a, \, \mathbf{x}) = b \tag{11}$$

where we consider that parameters *a* and *b* of Equation (2) represent uncertain values that can accept several different values expressed through respective distributions of possibility taken from the fuzzy numbers *A* - and *B* - .

Buckley and Qu [27] propose a way of interpreting the fuzzy Equation (10) which is fully consistent with the theory of the possibility. 

Let us consider Equation (10): *FA*- , *X*-= *B*-. Buckley and Qu's idea is to interpret Equation (10) as a family of crisp equations:

$$F(a, \,\,\mathbf{x}) = b \quad a \in \operatorname{Supp}\left(\tilde{A}\right) \quad \text{and} \quad b \in \operatorname{Supp}\left(\tilde{\mathcal{B}}\right) \tag{12}$$

where we assume that *a* accepts all possible values given by the fuzzy number *A* = *a* , *μA*-(*a*)/*a* ∈ *R* and *b* all possible values given by the fuzzy number *B*- = *b* , *μB*-(*b*)/*b* ∈ *<sup>R</sup>* . We then need to find all possible values of *x*, each with their own degrees of possibility, that satisfy any of Equation (12).

To this end, if it is assumed that *F* verifies the hypotheses of the theorem of the implicit function, then from *<sup>F</sup>*(*a* , *x*) = *b* we can determine *x*, so that:

$$\mathfrak{x} = f(a \,, b) \tag{13}$$


Thus, we obtain a new fuzzy equation:


$$
\widetilde{X} = f\left(\widetilde{A} \; , \, \widetilde{B}\right) \tag{14}
$$

whose solution *X* expresses the solution of Equation (10) in the sense of Buckley and Qu.

Thus, we can understand *f A*- , *B*-as a binary operation between two quantities of uncertain values, resulting in another uncertain quantity represented by *X* - . We must therefore study the possibility of this magnitude taking on a specific value *x*, considering that several combinations of possible values for *a* and *b* will exist so that *f*(*a* , *b*) = *x*. Each of the possible values of *x* fulfills some of the equations of the family (12).

Let us now observe how the equation is resolved in practice.

With the hypothesis of continuity of function *f*, and due of the compatibility of the principle of extension with the α-cuts, the α-cuts *Xα* of *X* - are given by:

$$X\_a = \{ \mathbf{x}/\mathbf{x} = f(a, b), \ a \in A\_a \text{ } \ b \in B\_a \} \tag{15}$$


When considering f continuous and *A* - and *B* fuzzy numbers, then the domain of definition of *f* , that is *Aα* × *B<sup>α</sup>*, is a compact set in *<sup>R</sup>2*, and therefore, by virtue of Weierstrass extreme value theorem ensures the existence of maximum and minimum of *f* , resulting in:

$$\mathcal{X}\_{\mathfrak{a}} = \lfloor \underline{\mathcal{X}}(a), \, \overline{\mathcal{X}}(a) \rfloor$$

with:

$$\underline{X}(\mathbf{a}) = \min \left\{ \mathbf{x} / \mathbf{x} = f(a, b), \ a \in A\_{\mathbf{a}}, \ b \in B\_{\mathbf{d}} \right\} \\ \overline{X}(\mathbf{a}) = \max \left\{ \mathbf{x} / \mathbf{x} = f(a, b), \ a \in A\_{\mathbf{d}}, \ b \in B\_{\mathbf{d}} \right\} \tag{16}$$

Therefore, in this case *Xα* is a closed interval, and by extension convex. This means that *X* - is convex. In addition, like *A* - and *B* - , *X* - is obviously normal. Indeed, since *A* - and *B* - are normal, there exist *a*\* and *b*\* with *μA*-(*a*<sup>∗</sup>) = 1 and *μB*-(*b*∗) = 1. Therefore, value *x*<sup>∗</sup> = *f*(*a*<sup>∗</sup> , *b*∗) meet *μX*- (*x*<sup>∗</sup>) = 1. Therefore, the solution *X* - is a fuzzy number. Furthermore, the membership function of *X* - will be expressed by applying the extension principle:

$$\mu\_{\vec{X}}(\mathbf{x}) = \bigvee\_{\{\mathbf{x} / \mathbf{x} = f(a,b)\}} \left( \mu\_{\vec{A}}(a) \wedge \mu\_{\vec{B}}(b) \right) \tag{17}$$

Keep in mind that when applying the compatibility of the first extension with α-cuts, that the uncertain quantities represented by *A* - and *B* - have no interaction, that is, the value assumed by one of these does not affect that assumed by the other. That being said, in order to compute *Xα* it is not generally possible to do so by directly applying the arithmetic of intervals and directly substituting *Aα* and *Bα* in the expression of the function. That is, if

we have the binary operation *f*(*<sup>a</sup>*, *b*) = *a* ∗ *b*, and directly compute the α-cuts by applying the arithmetic of intervals:

$$V\_a = A\_a(\*)B\_a \tag{18}$$

it is not generally verified that *Xα* = *V<sup>α</sup>*. However, if *f* is monotonous with respect to the inclusion of intervals, then the following condition is fulfilled *Xα* ⊆ *V<sup>α</sup>*. Therefore, if the arithmetic of the intervals is directly applied to calculate the α-cuts of *X* - , then wider intervals containing the solution will be generally obtained. If calculating *Xα* is complicated due to the behavior of the function *f* , then *Vα* can be considered as an approximation of the true result.

All of that being said, two common cases can be highlighted in which α-cuts *Xα* can be calculated easily:



**Table 1.** Extremes of the α-cuts depending of the monotony of *f*.

We will use Buckley and Qu's resolution method in our study into the behavior of Harrod's income growth model in a context of uncertainty, since it agrees with the way in which we interpret the values obtained by income in a fuzzy environment.

#### **3. Study and Solution of the Harrod's Growth Model in Conditions of Uncertainty**

Three years after publication of "The General Theory of Employment, Interest and Money" by famed English economist John Maynard Keynes in 1936 [28], another English economist, Henry Roy Harrod studied, following some of Keynes' ideas, the conditions for the harmonious growth of the economy and the factors of instability that may affect it [29].

With this purpose, Harrod determined, assuming a simplified model, the guaranteed rate of growth that keeps over time the balance in the circular flow of income. Years later, along with the parallel work of Domar [30,31] it gave rise to what is known as Harrod– Domar's model of economic growth. As discussed above, this model was overtaken first by the so-called exogenous growth models and then by endogenous growth models, which have always criticized that part of assumptions made by Harrod–Domar do not conform to reality. In these circumstances, in the present article we study how the contribution of fuzzy logic can change the predictions that derive from the model and adjust them more to reality.

The version of the model that we discuss in this article considers that investment constitute a stimulus to aggregate demand (multiplier principle) and, at the same time, cause an increase in productive capacity (accelerator principle). We consider that investment is the result of the growth of productive capacity. For this reason we examine the conditions under which the stimulus of investment towards aggregate demand is exactly offset by the increase in productive capacity that investment entails.

Harrod's index of guaranteed growth indicates a growth path for the economic system according to which the two objectives of investment (stimulating aggregate demand and contributing to the expansion of productive capacity) are in equilibrium, so that greater demand justifies greater productive capacity, as well as greater productive capacity allows meeting greater demand.

It is therefore a question of examining the conditions under which this equilibrium is established. The initial model proposed by Harrod considers a simplified system, without public sector and without relations with other countries. In this case, logically, the equilibrium condition between aggregate demand and productive capacity occurs when saving equals investment.

With respect to saving, as with consumption, the Keynesian condition that establishes income dependence is considered. On the other hand, with respect to investment, due to the complexity of the factors that may influence them, the accelerator principle is used, considering only the induced component that refers to the reaction of investment to changes in the level of income, and thus reflecting the fact that investment constitute an increase in productive capacity. The problem is trying to find the growth rate (according to Harrod's terminology) of income and investment to be in equilibrium.

If we consider a period *t,* the macromagnitudes *National Income* (*Yt*), *Consumption* (*Ct*), *Saving* (*St*) and *Investment* (*It*), we have that the growth model of Harrod will be formed by three conditions expressed through three equations:

(a) A Keynesian-type saving and a consumption equation:

$$S\_t = a \cdot Y\_t \qquad \mathbb{C}\_t = (1 - a) \cdot Y\_t \quad 0 < a < 1 \qquad t = 0, 1, 2, \dots \tag{19}$$

(b) An investment equation based on the accelerator principle, i.e., the investment is proportional to the rate of change in national income over time:

$$I\_t = b \cdot (Y\_t - \chi\_{t-1}) \qquad \qquad b > 1 \qquad \qquad t = 1, 2, 3, \dots \tag{20}$$

where *b* is a constant (the accelerator) that represents the average ratio between the increase in capital and the increase in production, and is therefore generally considered to have a value greater than unity.

(c) The equilibrium condition based on equality between savings and investment, since:

> *St*

$$\mathbf{Y}\_{l} = \mathbf{C}\_{l} + \mathbf{S}\_{l} \qquad AD\_{l} = \mathbf{C}\_{l} + l\_{l} \quad \text{( $AD$  = aggregate demand)}\tag{21}$$

So that equilibrium is given by:

$$= I\_t \tag{22}$$

And we ge<sup>t</sup> that with the equilibrium condition:

$$a \cdot \mathbf{Y}\_t = b \cdot (\mathbf{Y}\_t - \mathbf{Y}\_{t-1}) \tag{23}$$

that is:

$$(b - a) \cdot \mathbf{y}\_t = b \cdot \mathbf{y}\_{t-1} \tag{24}$$

Dividing by *b* − *a*, which is a value strictly greater than zero, gives the relation:

$$
\Upsilon\_t = \left(\frac{b}{b-a}\right) \cdot \Upsilon\_{t-1} \tag{25}
$$

It is an equation in differences of first order that presents immediate solution applying the recurrence, and leads us to determine the expression for income in period *t* given the initial value *Y*0, which results from:

$$\mathbf{Y}\_{l} = \left(\frac{b}{b-a}\right)^{t} \mathbf{Y}\_{0} = \left(1 + \frac{a}{b-a}\right)^{t} \mathbf{Y}\_{0} \tag{26}$$

Thus, the stability of the time trajectory depends on *b*/(*b* − *<sup>a</sup>*). Since *b* represents the capital/production ratio, which, as stated previously, is usually greater than 1, and since a represents the marginal propensity to saving, which is greater than zero and smaller than 1, the base *b*/(*b* − *a*) will be greater than 0 and, generally, greater than 1. Therefore, the trajectory of income *Yt* is explosive, but not oscillating. Thus, according to the relationships given by the model, income grows indefinitely.

We observe that the guaranteed growth rate is the relative percentage of growth between two consecutive periods, which is obviously:

$$G = \frac{a}{b-a} \tag{27}$$

Finally, the value of the investment in period *t* is determined from the equality:

$$S\_t - S\_{t-1} = a \cdot Y\_t - a \cdot Y\_{t-1} = a \cdot (Y\_t - Y\_{t-1}) = \frac{a}{b} \cdot I\_t \qquad t = 1, 2, 3, \dots \tag{28}$$

Given the equilibrium condition (*St = It*) we have:

$$I\_t - I\_{t-1} = \frac{a}{b} \cdot I\_t \implies \left(1 - \frac{a}{b}\right) \cdot I\_t = I\_{t-1} \implies I\_t = \frac{b}{b-a} \cdot I\_{t-1} \tag{29}$$

from which the following relationship is deduced:

$$I\_t = \left(\frac{b}{b-a}\right)^t \cdot I\_0 = \left(1 + \frac{a}{b-a}\right)^t \cdot I\_0 \qquad\qquad\qquad t = 1, 2, 3, \dots \tag{30}$$

which indicates the type of investment growth required to sustain equilibrium according to the model assumptions, which implies full employment. We observe that income and investment growth take the same form, so they must grow at the same required rate to sustain the equilibrium situation.

In this model, we see that if we know the true values of the marginal propensity to save, the accelerator, and the value of income at the initial time, we can determine the subsequent values of income and investment as well as the value of the growth rate to maintain equilibrium.

As we have argued for the models of previous works [11,12], the use of fuzzy numbers to express the marginal propensity to save and the accelerator, makes subsequent calculations difficult, but it allows the possibility of studying the behavior of the model when we operate with uncertain values, thus obtaining more information and giving a wider range of applications of the model. By operating with fuzzy numbers, we will determine the values of income, investment and growth rate as fuzzy numbers under conditions of uncertainty defined through their α-cuts, as well as their function of theoretical membership from the approach and solution of the equation given by the model under conditions of fuzziness.

When the values of *a* and *b* are expected to be of a similar amount in all periods, but uncertain, they can be considered as fuzzy numbers *a* and - *b* and equal for each period; or, in other words, having the same membership function for each value of *t*. They can then be expressed generically as fuzzy subsets and as α-cuts, thus:

$$\begin{aligned} \widetilde{a} &= \left\{ (\mathbf{x}, \,\mu\_{\widetilde{a}}(\mathbf{x})) \right\} = \left\{ a\_{\mathfrak{a}} = \left[ \underline{\mathfrak{a}}(\mathbf{a}), \,\overline{\mathfrak{a}}(\mathbf{a}) \right] \,\, 0 \le a \le 1 \right\} \\ \widetilde{b} &= \left\{ (\mathbf{x}, \,\mu\_{\widetilde{b}}(\mathbf{x})) \right\} = \left\{ b\_{\mathfrak{a}} = \left[ \underline{\mathfrak{b}}(\mathbf{a}), \,\overline{\mathfrak{b}}(\mathbf{a}) \right] \,\, 0 \le a \le 1 \right\} \end{aligned} \tag{31}$$

where *μa* and *μb* represent their respective membership functions, and *aα* and *bα* the respective α-cuts at the level α. According to the assumptions of the model, the following conditions are established:

$$0 < \underline{a}(a) \le \overline{a}(a) < 1 \qquad \text{and} \qquad \overline{b}(a) \ge \underline{b}(a) > 1 \tag{32}$$

The value of income is determined from the fuzzy equation, thus:

$$
\widetilde{Y}\_t = \left(\frac{\widetilde{b}}{\widetilde{b} - \widetilde{a}}\right)^t \cdot Y\_0 \tag{33}
$$

To solve it we first determine the fuzzy number:

$$
\check{M} = \frac{\check{b}}{\tilde{b} - \tilde{a}}\tag{34}
$$

Given the crisp equality: *M* = *b*/(*b* − *<sup>a</sup>*), we can consider *M* as the result of the binary operation *f*(*<sup>a</sup>*, *b*) = *a* ∗ *b* = *b*/(*b* − *<sup>a</sup>*). From the principle of extension, we know that if we have a binary operation *f*(*<sup>a</sup>*, *b*) between two quantities, we can extend the operation to the case where the quantities are uncertain and apply the principle of extension to obtain: *f a* , *b* = *M*-. In this case, if we consider *μa* and *μb* to be continuous membership functions, then, by applying the extension principle to the binary operation *f*(*<sup>a</sup>*, *b*), the membership function of the fuzzy number *M* - will be given by:

$$\mu\_{\vec{M}}(z) = \bigvee\_{\{z/z = f(x,y)\}\bigvee} \left(\mu\_{\vec{a}}(x) \wedge \mu\_{\vec{b}}(y)\right) \tag{35}$$

Since *f* is a continuous function, by virtue of Buckley's theorem [25], we have that the α-cuts of *M* - are:

$$M\_{\mathfrak{a}} = \left[ \underline{M}(\mathfrak{a}), \,\, \overline{M}(\mathfrak{a}) \right] = \{ z = f(\mathfrak{x}, y) / \,\, \mathbf{x} \in a\_{\mathfrak{a}} \,\, \, y \in b\_{\mathfrak{a}} \}\tag{36}$$

where logically:

$$\underline{M}(\boldsymbol{\alpha}) = \min \{ z = f(\mathbf{x}, \boldsymbol{y}) / \, \, \mathbf{x} \in a\_{\alpha} \, \, \, \, \, \, \, \underline{y} \in b\_{\alpha} \} $$
 
$$\overline{M}(\boldsymbol{\alpha}) = \max \{ z = f(\mathbf{x}, \boldsymbol{y}) / \, \, \, \mathbf{x} \in a\_{\alpha} \, \, \, \, \, \, \underline{y} \in b\_{\alpha} \} $$

Since in our case *f*(*<sup>x</sup>*, *y*) is a continuous function whose domain of definition is *aα* × *b<sup>α</sup>*, which is a compact set in *R*2, the existence of *<sup>M</sup>*(*α*) and *<sup>M</sup>*(*α*) is assured and, thus, *Mα* is a closed interval and consequently *M* - is convex. The condition of normality of *M* - is immediately deduced from the fact that *a* and - *b* are normal, meaning *M* - is a fuzzy number.

However, since we are not dealing with the hypotheses from Moore's theorem [26], if the arithmetic of the intervals is directly applied to calculate the *α*-cuts *Mα* by calculating *f*(*<sup>a</sup>α*, *bα*), an unwanted result can be obtained in the sense that the range is wider than the true *M<sup>α</sup>*. -

Indeed, if we calculate the *α*-cuts of *M* from the fuzzy equation:

$$
\check{M} = \frac{\stackrel{\curvearrowright}{b}}{\stackrel{\curvearrowright}{b} - \stackrel{\curvearrowright}{a}} \tag{37}
$$

We have that since the function *f*(*<sup>a</sup>*, *b*) = *b*/(*b* − *a*) is continuous in its domain and increasing with respect to *∂ f ∂a* > 0) and decreasing with respect to *∂ f ∂b* < 0, when applying the results given by Buckley and Qu [27] we get:

$$\begin{aligned} \underline{M}(a) &= \min \{ z = f(\mathbf{x}, y) / \, \mathbf{x} \in a\_{\mathfrak{a}} \, , \, y \in b\_{\mathfrak{a}} \} = f\left(\underline{a}(a), \, \overline{b}(a)\right) = \frac{\underline{\mathfrak{z}}(a)}{\underline{\mathfrak{z}}(a) - \underline{\mathfrak{z}}(a)} \\\ \overline{M}(a) &= \max \{ z = f(\mathbf{x}, y) / \, \mathbf{x} \in a\_{\mathfrak{a}} \, , \, y \in b\_{\mathfrak{a}} \} = f(\overline{a}(a), \, \underline{\mathfrak{z}}(a)) = \frac{\underline{\mathfrak{z}}(a)}{\underline{\mathfrak{z}}(a) - \overline{\mathfrak{z}}(a)} \end{aligned} \tag{38}$$

Instead, with the application of the arithmetic of the intervals, we would obtain:

$$\begin{split} \mathcal{M}\_a^\* &= b\_a(:)(b\_a(-)a\_a) = \left[\underline{b}(a), \,\,\overline{b}(a)\right] \left(:\right) \left( \left[\underline{b}(a), \,\,\overline{b}(a)\right] \left(-\right) \left[\underline{a}(a), \,\,\overline{a}(a)\right] \right) \\ &= \left[\underline{b}(a), \,\,\overline{b}(a)\right] \left(:\right) \left[\underline{b}(a) - \overline{a}(a), \,\,\overline{b}(a) - \underline{a}(a)\right] = \begin{bmatrix} \frac{\underline{b}(a)}{\overline{b}(a) - \underline{a}(a)} \, \, \, \frac{\overline{b}(a)}{\underline{b}(a) - \overline{a}(a)} \end{bmatrix} \end{split} \tag{39}$$

From this expression we ge<sup>t</sup> what Moore [26] the interval extension of *α*-cuts of *M* , since it holds:

$$M\_{\alpha} \subseteq M\_{\alpha}^\* \tag{40}$$


If we start instead from the crisp equality:

$$N = 1 + \frac{a}{b - a} \tag{41}$$

Then, we set the fuzzy equation:

$$
\widetilde{N} = 1 + \frac{\widetilde{a}}{\widetilde{b} - \widetilde{a}} \tag{42}
$$

it turns out that in this case the *α*-cuts of *N* coincide with the extension by intervals obtained by directly applying the arithmetic.


Indeed, for the calculation of the α-cuts of *N* , we consider the function *g*(*<sup>a</sup>*, *b*) = 1 + *a b*−*<sup>a</sup>* , easily checking that this function is increasing in *a* and decreasing in *b*, and by application of monotony in Table 1:

$$N\_a = \lfloor \underline{N}(a) \rfloor \,\overline{N}(a) \rfloor$$

where:

$$\begin{aligned} \underline{N}(a) &= \min \{ z = \emptyset(x, y) / \ge x \in a\_a \, , \, y \in b\_a \} = 1 + \frac{\underline{q}(a)}{\underline{b}(a) - \underline{q}(a)} \\ \overline{N}(a) &= \max \{ z = \emptyset(x, y) / \ge x \in a\_a \, , \, y \in b\_a \} = 1 + \frac{\underline{q}(a)}{\underline{b}(a) - \overline{q}(a)} \end{aligned} \tag{43}$$


On the other hand, with the direct application of the arithmetic of the intervals, we obtain in this case:

$$N\_a^\* = 1(+) \left[ a\_a(:)(b\_a(-)a\_a) \right] = [1,1](+) \left[ \left[ \underline{a}(a), \overline{a}(a) \right](:) \left[ \left[ \underline{b}(a), \overline{b}(a) \right](-) \left[ \underline{a}(a), \overline{a}(a) \right] \right] \right]$$

$$= [1,1](+) \left[ \left[ \underline{a}(a), \overline{a}(a) \right](:) \left[ \underline{b}(a) - \overline{a}(a), \overline{b}(a) - \underline{a}(a) \right] \right] \tag{44}$$

$$= [1,1](+) \left[ \frac{\underline{a}(a)}{\overline{b}(a) - \underline{a}(a)}, \frac{\overline{a}(a)}{\underline{b}(a) - \overline{a}(a)} \right] = \left[ 1 + \frac{\underline{a}(a)}{\overline{b}(a) - \underline{a}(a)}, \ 1 + \frac{\overline{a}(a)}{\underline{b}(a) - \overline{a}(a)} \right]$$

which, as we see, matches the α-cut *Nα* computed in (43).

We see, therefore, with this particular application, the importance of applying the method of solving fuzzy equations proposed by Buckley and Qu [27], since, otherwise, if we start from Equation (37) and calculate the α-cuts of *M* - by directly applying the arithmetic of the confidence intervals, we could not ensure that all values of the α-cut are greater than 1, since *b*(*α*) does not need to be smaller than *b*(*α*) − *<sup>a</sup>*(*α*). On the other hand, calculating the α-cuts from Equation (38), it is ensured that the lower end of each α-cut, that is *<sup>b</sup>*(*α*)/*b*(*α*) − *a*(*α*) always has a value greater than 1, and thus, it turns out that the base of the exponential function which gives the income growth, despite being an uncertain value, is greater than 1 and gives rise to an increasing trajectory for income.

In addition, another very important question is that, because the hypotheses of Moore's theorem [26] are not fulfilled, from the equality *M* = *N* the fuzzy equality *M* - = *N* - would not be deduced if we applied the arithmetic of the intervals in the calculation of

α-cuts. However, if we determine the α-cuts from Equations (38) and (43), respectively, we can write that *M*- = *N*-, because for each level α ∈ [0,1] we have *Mα* = *N<sup>α</sup>*. Indeed:

$$\begin{split} \mathcal{N}\_a &= \begin{bmatrix} \underline{\mathsf{N}}(a), \overline{\mathsf{N}}(a) \end{bmatrix} = \begin{bmatrix} 1 + \frac{\underline{\mathsf{g}}(a)}{\mathsf{k}(a) - \underline{\mathsf{g}}(a)} \; \; 1 + \frac{\overline{\mathsf{z}}(a)}{\underline{\mathsf{k}}(a) - \overline{\mathsf{a}}(a)} \end{bmatrix} \\ &= \begin{bmatrix} \frac{\overline{\mathsf{b}}(a)}{\mathsf{b}(a) - \underline{\mathsf{g}}(a)} \; \; \; \frac{\underline{\mathsf{k}}(a)}{\underline{\mathsf{k}}(a) - \overline{\mathsf{a}}(a)} \end{bmatrix} = \mathcal{M}\_a \end{split} \tag{45}$$

To determine the membership function of the fuzzy number *M*-, we apply the extension principle, and obtain the expression:

$$\mu\_{\bar{M}}(z) = \bigvee\_{\{z/z = y/(y-x)\}} \left(\mu\_{\bar{a}}(x) \wedge \mu\_{\bar{b}}(y)\right) \tag{46}$$

If we use the partial order relation of the confidence intervals defined by:

$$[a\_1, b\_1] \le [a\_2, b\_2] \iff a\_1 \le b\_1 \quad \text{and} \quad a\_2 \le b\_2$$

note that for each level *α* we have *M*α > [1,1]. Therefore, because the base exponential function greater than 1 is always increasing; it will turn out that of trajectory given by the equality:

$$
\tilde{Y}\_t = \tilde{M}^t \cdot Y\_0 \qquad \qquad t = 0, 1, 2, \dots \tag{47}
$$

we deduce the income membership function in period *t*:

$$
\mu\_{\bar{Y}\_l}(\mathbf{x}) = \mu\_{\bar{M}^l} \left( \frac{\mathbf{x}}{Y\_0} \right) = \mu\_{\bar{M}} \left[ \left( \frac{\mathbf{x}}{Y\_0} \right)^{\frac{1}{7}} \right] \tag{48}
$$

As well as its *α*-cuts:

$$\mathbf{Y}\_{t}\left(\mathbf{Y}\_{t}\right)\_{a} = \begin{bmatrix} \underline{Y}\_{t}(a), \ \overline{Y}\_{t}(a) \end{bmatrix} = \begin{bmatrix} (\underline{\mathbf{M}}(a))^{t} \cdot \mathbf{Y}\_{0} \ \ \left(\overline{\mathbf{M}}(a)\right)^{t} \cdot \mathbf{Y}\_{0} \end{bmatrix} = \begin{bmatrix} \left(\overline{\mathbf{b}}(a)\right)^{t} \cdot \mathbf{Y}\_{0} & \left(\underline{\mathbf{b}}(a)\right)^{t} \cdot \mathbf{Y}\_{0} \\\\ \left(\overline{\mathbf{b}}(a) - \underline{\mathbf{a}}(a)\right)^{t} & \left(\underline{\mathbf{b}}(a) - \overline{\mathbf{a}}(a)\right)^{t} \end{bmatrix} \tag{49}$$

On the other hand, the income growth factor is given by the fuzzy number:

$$
\widetilde{G} = \frac{\widetilde{a}}{\widetilde{b} - \widetilde{a}}\tag{50}
$$

which has membership function:

$$
\mu\_{\tilde{G}}(\mathbf{x}) = \mu\_{\tilde{M}}(\mathbf{x} - 1) \tag{51}
$$

where its α-cuts are:

$$\mathbf{G}\_{\mathfrak{a}} = \left[ \underline{\mathbf{G}}(a), \overline{\mathbf{G}}(a) \right] = \left[ \frac{\underline{\mathfrak{a}}(a)}{\overline{b}(a) - \underline{\mathfrak{a}}(a)}, \frac{\overline{\mathfrak{a}}(a)}{\underline{\mathfrak{b}}(a) - \overline{\mathfrak{a}}(a)} \right] \tag{52}$$

Finally, in an analogous way to income, we determine the membership function of investment from the equation:

$$
\vec{I}\_t = \vec{M}^t \cdot \vec{I}\_0 \qquad \qquad t = 1, 2, 3, \dots \tag{53}
$$

considering that *I*0 = *<sup>a</sup>*·*Y*<sup>0</sup> and, therefore:


$$
\mu\_{\overline{I}\_0}(\mathbf{x}) = \mu\_{\overline{a}}\left(\frac{\mathbf{x}}{\mathbf{Y}\_0}\right) \tag{54}
$$

Further, applying the principle of extension, we obtain the expression for the membership function:

$$\mu\_{\bar{l}\_l}(z) = \bigvee\_{\{z/z = x \cdot y\}} \left(\mu\_{\bar{M}^t}(x) \wedge \mu\_{\bar{l}\_0}(y)\right) = \bigvee\_{\{z/z = x \cdot y\}} \left(\mu\_{\bar{M}}\left(x^{\frac{1}{t}}\right) \wedge \mu\_{\bar{a}}\left(\frac{y}{\bar{\chi}\_0}\right)\right) \tag{55}$$

The corresponding *α*-cuts for investment in period *t* are given by:

$$\begin{split} \left(I\_{l}\right)\_{\mathfrak{a}} &= \begin{bmatrix} \underline{I}\_{l}(\mathfrak{a}), \ \overline{I}\_{l}(\mathfrak{a}) \end{bmatrix} = \begin{bmatrix} \left(\underline{M}(\mathfrak{a})\right)^{\mathfrak{t}} \cdot \underline{\mathfrak{a}}(\mathfrak{a}) \cdot \mathcal{Y}\_{\mathbb{O}} \ \left(\overline{\mathcal{M}}(\mathfrak{a})\right)^{\mathfrak{t}} \cdot \overline{\mathfrak{a}}(\mathfrak{a}) \cdot \mathcal{Y}\_{\mathbb{O}} \end{bmatrix} \\ &= \begin{bmatrix} \frac{\left(\overline{\mathfrak{F}}(\mathfrak{a})\right)^{\mathfrak{t}} \cdot \underline{\mathfrak{a}}(\mathfrak{a}) \cdot \mathcal{Y}\_{\mathbb{O}}}{\left(\underline{\mathfrak{F}}(\mathfrak{a}) - \underline{\mathfrak{a}}(\mathfrak{a})\right)^{\mathfrak{t}}}, \ \frac{\left(\underline{\mathfrak{F}}(\mathfrak{a})\right)^{\mathfrak{t}} \cdot \overline{\mathfrak{a}}(\mathfrak{a}) \cdot \mathcal{Y}\_{\mathbb{O}}}{\left(\underline{\mathfrak{F}}(\mathfrak{a}) - \overline{\mathfrak{a}}(\mathfrak{a})\right)^{\mathfrak{t}}} \end{bmatrix} \end{split} \tag{56}$$

#### **4. Analysis of the Particular Case in Which the Parameters Are Expressed through Triangular Fuzzy Numbers (TFN)**

We now study the particular case for which *a* and *b* are triangular fuzzy numbers (TFN). As it is well known considering *a* and *b* as TFN is the result of assuming in the process of estimation that for the marginal propensity to save in a given period we can indicate two values *a*1 and *a*3 which correspond to the minimum and maximum estimations, respectively, and a value *a*2 that we believe the most likely. Likewise, we interpret parameter *b* as a TFN. Thus, we consider:

$$
\widetilde{a} = (a\_1 \ , \ a\_2 \ , \ a\_3) \qquad \text{and} \qquad \bar{b} = (b\_1 \ , \ b\_2 \ , \ b\_3) \tag{57}
$$


With membership functions *μa* and *μb* which are obviously know, as well as their expressions through the α-cuts *aα* and *b<sup>α</sup>*.

From *a* and *b* we can determine the fuzzy number *M*- that we later use for determining income. Following the general methodology set out in the previous section, we obtain the α-cuts of the fuzzy number *M*- which, logically, will no longer be triangular due of the quotient that defines it:

$$M\_{\mathfrak{a}} = \left[ \underline{M}(a), \,\overline{M}(a) \right] = \left[ \frac{b\_3 - (b\_3 - b\_2) \cdot a}{b\_3 - a\_1 - (b\_3 - b\_2 + a\_2 - a\_1) \cdot a}, \,\frac{b\_1 + (b\_2 - b\_1) \cdot a}{b\_1 - a\_3 + (b\_2 - b\_1 + a\_3 - a\_2) \cdot a} \right] \tag{58}$$

From this expression we ge<sup>t</sup> the membership function:


$$\mu\_{\widetilde{M}}(\mathbf{x}) = \begin{cases} 0 & \text{if } \qquad \mathbf{x} < \frac{b\_{3}}{b\_{3} - a\_{1}}\\ \frac{(b\_{3} - a\_{1}) \cdot \mathbf{x} - b\_{3}}{(b\_{3} - b\_{2} + a\_{2} - a\_{1}) \cdot \mathbf{x} - (b\_{3} - b\_{2})} & \text{if } \qquad \frac{b\_{3}}{b\_{3} - a\_{1}} \le \mathbf{x} \le \frac{b\_{2}}{b\_{2} - a\_{2}}\\ \frac{(b\_{1} - a\_{3}) \cdot \mathbf{x} - b\_{1}}{(b\_{2} - b\_{1}) - (b\_{2} - b\_{1} + a\_{3} - a\_{2}) \cdot \mathbf{x}} & \text{if } \qquad \frac{b\_{3}}{b\_{2} - a\_{2}} \le \mathbf{x} \le \frac{b\_{1}}{b\_{1} - a\_{3}}\\ 0 & \text{if } \qquad \qquad \qquad \mathbf{x} > \frac{b\_{1}}{b\_{1} - a\_{3}} \end{cases} \tag{59}$$

From which we find the membership function of income in period t by using (48):

$$
\mu\_{\bar{Y}\_t}(\mathbf{x}) = \mu\_{\bar{M}} \left[ \left( \frac{\mathbf{x}}{Y\_0} \right)^{\frac{1}{t}} \right]
$$

So that the expression for *Yt* by means of α-cuts is:

$$(\Upsilon\_t)\_a = \left[ \left( \frac{b\_3 - (b\_3 - b\_2) \cdot a}{b\_3 - a\_1 - (b\_3 - b\_2 + a\_2 - a\_1) \cdot a} \right)^t \cdot \Upsilon\_0 \,, \left( \frac{b\_1 + (b\_2 - b\_1) \cdot a}{b\_1 - a\_3 + (b\_2 - b\_1 + a\_3 - a\_2) \cdot a} \right)^t \cdot \Upsilon\_0 \right] \tag{60}$$

Following again the general methodology for this particular case, we determine the α-cuts for the growth factor:

$$G\_{\mathfrak{a}} = \left[ \frac{a\_1 + (a\_2 - a\_1) \cdot a}{(b\_3 - a\_1) - (b\_3 - b\_2 + a\_2 - a\_1) \cdot a}, \frac{a\_3 - (a\_3 - a\_2) \cdot a}{(b\_1 - a\_3) + (b\_2 - b\_1 + a\_3 - a\_2) \cdot a} \right] \tag{61}$$

from which we compute the membership function:

$$\mu\_{\mathcal{G}}(\mathbf{x}) = \begin{cases} 0 & \text{if } \qquad \qquad \qquad \qquad \ge \frac{a\_1}{b\_3 - a\_1} \\ \frac{(b\_3 - a\_1) \cdot \mathbf{x} - a\_1}{(b\_3 - b\_2 + a\_2 - a\_1) \cdot \mathbf{x} + (a\_2 - a\_1)} & \text{if } \qquad \frac{a\_1}{b\_3 - a\_1} \le \mathbf{x} \le \frac{a\_2}{b\_2 - a\_2} \\ \frac{a\_3 - (b\_1 - a\_3) \cdot \mathbf{x}}{(b\_2 - b\_1 + a\_3 - a\_2) \cdot \mathbf{x} + (a\_3 - a\_2)} & \text{if } \qquad \frac{a\_2}{b\_2 - a\_2} \le \mathbf{x} \le \frac{a\_3}{b\_1 - a\_3} \\ 0 & \text{if } \qquad \qquad \qquad \qquad \qquad \times > \frac{a\_3}{b\_1 - a\_3} \end{cases} \tag{62}$$


as can be clearly seen from the membership function *G* , it does not maintain the triangular fuzzy number structure.

Finally, if we replace the values in this particular case in the general Equation (56), we would ge<sup>t</sup> the expression for investment in each period through its *α*-cuts. For the function of membership of investment we refer to the general case as it is not a simple operational expression.

## **5. Example of Application**

As an illustrative example of the model that we have just developed, we complement the explanation with a specific numerical case, considering *a* and - *b* as triangular fuzzy numbers with the following values:

$$
\tilde{a} = (0.15, \, 0.17, \, 0.20) \qquad \qquad \bar{b} = (3.5, \, 4, \, 5)
$$

Additionally, considering the initial value for income *Y*0 = 100, we will determine the fuzzy values of the guaranteed growth index *G* - , as well as income *Y* - *t* and investment *I* - *t* for period *t* = 4.


In this case, the membership functions of *a* (marginal propension to save) and *b* (accelerator) are formed by linear sections, as seen in Figures 1 and 2, respectively.

The analytical expression for the membership function of the fuzzy number *a* is:

**Figure 1.** Membership function of marginal propension to save (mps).

**Figure 2.** Membership function for the accelerator.

The expression of its α-cuts is:

$$a\_{\mathfrak{a}} = [0.15 + 0.02 \cdot \mathfrak{a} \text{ , } 0.2 - 0.03 \cdot \mathfrak{a}] \qquad \qquad 0 \le \mathfrak{a} \le 1.$$


On the other hand, for the fuzzy number *b* we have:

$$\mu\_{\overline{b}}(x) = \begin{cases} 0 & \text{if } & x < 3.5\\ \frac{x - 3.5}{0.5} & \text{if } & 3.5 \le x \le 4\\ 5 - x & \text{if } & 4 \le x \le 5\\ 0 & \text{if } & x > 5 \end{cases}$$

For the expression of its α-cuts it is:

$$b\_{\mathfrak{a}} = [3.5 + 0.5 \cdot \mathfrak{a} \text{ , } 5 - \mathfrak{a}]$$

From the fuzzy equation:

$$
\widetilde{M} = \frac{\widetilde{b}}{\widetilde{b} - \widetilde{a}}
$$

Given the membership functions for *μa* and *μb*, we ge<sup>t</sup> the membership function of the fuzzy number *M* - which, as shown in Figure 3, does not correspond to a triangular fuzzy number, because, as we see, this membership function is made up of nonlinear sections. Its expression is given by:

$$\mu\_{\bar{M}}(\mathbf{x}) = \begin{cases} 0 & \text{if } & \mathbf{x} < 1.030 \\ \frac{4.85\mathbf{x} - 5}{1.02\mathbf{x} - 1} & \text{if } & 1.030 \le \mathbf{x} \le 1.044 \\ \frac{3.5 - 3.3\mathbf{x}}{0.53\mathbf{x} - 0.5} & \text{if } & 1.044 \le \mathbf{x} \le 1.060 \\ 0 & \text{if } & \mathbf{x} > 1.060 \end{cases}$$


**Figure 3.** Membership function for *M* .

From the fuzzy number *M* - we determine the guaranteed growth rate, expressed by means of the fuzzy number *G* - , with α-cuts:

$$G\_{\mathfrak{A}} = \left[ \frac{0.15 + 0.02 \cdot \alpha}{4.85 - 1.02 \cdot \alpha}, \; \frac{0.2 - 0.03 \cdot \alpha}{3.3 + 0.53 \cdot \alpha} \right] \qquad 0 \le \mathfrak{a} \le 1$$

The membership function *μG*- takes a similar form to *μM*- since as we have seen in Equation (51), they are closely related. Its expression is:

$$\mu\_{\widehat{G}}(\mathbf{x}) = \begin{cases} 0 & \text{if } & \mathbf{x} < 0.030 \\ \frac{4.85\mathbf{x} - 0.15}{1.02\mathbf{x} + 0.02} & \text{if } & 0.030 \le \mathbf{x} \le 0.044 \\ \frac{0.2 - 3.3\mathbf{x}}{0.53\mathbf{x} + 0.03} & \text{if } & 0.044 \le \mathbf{x} \le 0.060 \\ 0 & \text{if } & \mathbf{x} > 0.060 \end{cases}$$

As shown in Figure 4, it is not a triangular fuzzy number because it is made up of rational sections.

**Figure 4.** Membership function for the growth index.

Finally, we determine the fuzzy expression for income for *t* = 4, whose α-cuts are:

$$Y\_{4\kappa} = \left[ \left( \frac{5 - \alpha}{4.85 - 1.02 \cdot \alpha} \right)^4 \cdot 100 \,, \left( \frac{3.5 + 0.5 \cdot \alpha}{3.3 + 0.53 \cdot \alpha} \right)^4 \cdot 100 \right]$$

as well as its membership function from Equation (48):

$$\mu\_{\overline{Y}\_4}(\mathbf{x}) = \begin{cases} 0 & \text{if } & \mathbf{x} < 112.55 \\ \frac{1.5337 \cdot \sqrt[4]{\pi} - 5}{0.3225 \cdot \sqrt[4]{\pi} - 1} & \text{if } & 112.55 \le \mathbf{x} \le 118.79 \\\\ \frac{3.5 - 1.0435 \cdot \sqrt[4]{\pi}}{0.1676 \cdot \sqrt[4]{\pi} - 0.5} & \text{if } & 118.79 \le \mathbf{x} \le 126.24 \\ 0 & \text{if } & \mathbf{x} > 126.24 \end{cases}$$

As in all non-triangular membership functions in the example, the coefficients are approximate, but with the degree of accuracy tolerable by the uncertainty situation of the problem.

Finally, to determine the fuzzy expression for investment in period *t* = 4, we take the uncertain value of the initial investment given by the triangular fuzzy number:

$$\tilde{I}\_0 = (15, 17, 20) \qquad \qquad l\_{0\pi} = [15 + 2 \cdot \pi \text{ , } 20 - 3 \cdot \pi]$$

with which, by application of Equation (56), we determine the *α*-cuts for investment in period *t* = 4 which are given by:

$$I\_{4a} = \left[ \left( \frac{5 - a}{4.85 - 1.02 \cdot a} \right)^4 \cdot (15 + 2 \cdot a) \,, \left( \frac{3.5 + 0.5 \cdot a}{3.3 + 0.53 \cdot a} \right)^4 \cdot (20 - 3 \cdot a) \right]$$

If we consider the *α*-cut at level *α* = 0, we ge<sup>t</sup> the support for investment in the considered period:

$$I\_{4,0} = \begin{bmatrix} I\_{\underline{4}}(0), \ \overline{I\_4}(0) \end{bmatrix} = \begin{bmatrix} 16.94, \ 25.30 \end{bmatrix}$$

from which we ge<sup>t</sup> the minimum (16.94) and maximum (25.30) estimates for investment in that period.
