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We propound the economic idea in terms of fractional derivatives, which involves the modified Caputo’s fractional derivative operator. The suggested economic interpretation is based on a generalization of average count and marginal value of economic indicators. We use the concepts of

Fractional calculus is the field of Mathematical study that grows out of the traditional definition of calculus integral and derivative operators as the same way of fractional exponents is an outgrowth of exponents of integer value. The birth of fractional calculus occurred in a letter from G. F. A. de L’Hospital to G. W Leibniz in 1695 posing a possible question “what if the order of derivative

Riemann defined the fractional integral of order

Riemann-Liouville defined the fractional derivative of order

Abel’s integral equation of order

It is notable that Abel was the first one to use fractional calculus for

Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time, taken from [

Before elaborating the economic meaning of fractional derivatives, we put up the economic concept of the standard derivative of the first order. In terms of economy the first-order derivatives show the rapidness in alteration of economic indicators with respect to the investigated factor by assuming that other factors remain constant. The first-order derivative of the function of an indicator defines the marginal value of this indicator. The marginal value is the incremental value through the corresponding indicator per unit increase of the determining factor. In economic study, the principal marginal values of indicators are marginal product, margin utility, marginal profit, marginal cost, marginal revenue, marginal tendency to save and consume, marginal tax rate, marginal demand, and some others.

Mathematically if the dependence

Graphical analysis from a sample graph given in Figure

When the average curve is rising, then

When the average curve is at its minimum, then

When the average curve is falling, then

Three cost curves (average total cost, marginal cost, and marginal revenue) overlap.

In [

Graph in Figure

The graph in figure defined by equations (

So, the formulas enlisted in (

The expressions (

In the study of differential calculus, (

It is worth mentioning that the violation of single valued property of the indicator function

The Caputo’s fractional derivative is one of the most used definitions of a fractional derivative along with other fractional calculus such as

Let

The Caputo approach appears often while modeling applied problems by means of integrodifferential equations. Hence subsequent to our discussion in previous sections, the economic agents can react against identical changes in economic analysis and this is due to the use of integer order derivative. In continuation to this, the concept of noninteger derivative order [

Action of derivative on three stages of functions.

“The left-hand side Caputo’s fractional derivative of order

Caputo’s derivative of order

Let

Note that

“Special Case of modified Caputo’s derivative operator” is defined as follows: let

Denoting the modified Caputo’s fractional derivative operator by

Note that the above power form of modified Caputo’s fractional derivative corresponds to power function studied by Rubin & Wenzel (1996); see [

For

Considering the effects of memory in economic process the generalized concept of marginal and average values of indicator in terms of Caputo’s Left-hand side derivative with

Economic research studies show that memory effect leads to abnormality in economic growth. The memory effect with declining order

For instance to count on changes in results of marginal values, we introduce the modified Caputo’s fractional derivative of order

Note for

It is also usable to replace

For the sake of numerical analysis of marginal values we apply the integer order derivative, Caputo’s fractional order derivative, and modified Caputo’s fractional order derivative, respectively, shown in forthcoming equations and tabulation. Manipulating (

Marginal values for

Depressive value | Marginal Value |
---|---|

| |

| |

0 | 20.0284 |

33/64 | 20.0288 |

17/32 | 20.0292 |

9/16 | 20.0305 |

5/8 | 20.0336 |

11/16 | 20.0382 |

3/4 | 20.0455 |

13/16 | 20.0579 |

7/8 | 20.0832 |

15/16 | 20.1616 |

31/32 | 20.3265 |

63/64 | 20.6906 |

127/128 | 21.5932 |

255/256 | 24.6707 |

1 | 15.0403 |

The sharpening behavior of the fractional values can be seen from (Table

Some of the key notes are as follows:

Equation (

Equation (

The proposed economic indicator in (

The supplementary parameter

We used a modified Caputo’s fractional operator towards the rectification of the indicator function

All data generated or analyzed during the study are included in the article.

The earlier version of this research article was presented in the “First Innovation Conference” on 21 December 2017, At A’Sharqiyah University Ibra, Sultanate of Oman.

The authors declare that there are no conflicts of interest. All the authors agreed with the content of the manuscript.

The work here is supported by UKM grant GUP-2017-064.