Modeling and Application of Fractional-Order Economic Growth Model with Time Delay
Round 1
Reviewer 1 Report
The proposed manuscript can be accepted for publications.
However, the manuscript should be greatly improved.
1) The manuscript is devoted to economic dynamics with memory.
Many works have been done in this area, which are practically
not reflected or discussed in the manuscript.
2) Figure 6 with predicted value contains only a short section of predictions,
which does not allow making significant predictions in a nonlinear model.
The predictive part must describe several decades for the trend to be evident.
The past interval is 1980-2020, then t>2020 should be 2020-2060 or 2020-2050 at least.
3) Note the definition of the Caputo fractional derivative is defined for the order q from (n-1,n],
and this derivative gives the integer-order derivative for q=n.
The predictions should be compared to the memoryless model (when q=1).
For example,
the comparison of q=1 and q=0.7 should be considered on Figure 1;
the comparison of q=1 and q=0.85 should be considered on Figure 2;
the comparison of q=1 and q=0.8 should be considered on Figure 3; ...
4) Since the value of the memory parameter "q" is currently practically
not studied in econometrics, then a wide range of parameters 0< q <1
should be described in the manuscript and the corresponding graphs should be built.
Figure 6 should be considered for q=1 and wide range of 0<q<1.
5) The model should be also considered for q>1. For example,
at least values of q close to one (q =1.1, q=1.2) should be described
in the manuscript and corresponded Figures.
Author Response
Reviewer ♯1
Comment 1.1: The manuscript is devoted to economic dynamics with memory. Many works have been
done in this area, which are practically not reflected or discussed in the manuscript.
Response: The authors sincerely thank the reviewer for the time, effort and recognition given to the
manuscript as well as the positive and constructive comments raised. Accordingly, we have extensively
collected and summarized the research contribution and progress in the field of economic dynamics with
memory. The Introduction section has been modified involving the history and recent development in this
field. The corresponding literatures have been cited and added into the revision. For details, please see
Introduction and References in the revised manuscript.
Comment 1.2: Figure 6 with predicted value contains only a short section of predictions, which does
not allow making significant predictions in a nonlinear model. The predictive part must describe several
decades for the trend to be evident. The past interval is 1980-2020, then t > 2020 should be 2020-2060 or
2020-2050 at least.
Response: Thanks for your constructive comments. According to your suggestion, we have carefully
considered and performed a longer section of predictions. The prediction curves in the interval 2020-2050
have been added in Figure 7, 8, 9, 10. With the thirty years of prediction, the trend of working population
(Lt), capital stock (Kt) and TFP (At) turns out to be more significant. For details, please see the Economic
Prediction section in the revised manuscript.
Comment 1.3: Note the definition of the Caputo fractional derivative is defined for the order q from
(n-1,n], and this derivative gives the integer-order derivative for q=n. The predictions should be compared
to the memoryless model (when q=1). For example,
the comparison of q=1 and q=0.7 should be considered on Figure 1;
the comparison of q=1 and q=0.85 should be considered on Figure 2;
the comparison of q=1 and q=0.8 should be considered on Figure 3; ...
Response: Thanks for your insightful comment and pointing out the improving direction. Based on
your suggestion, the definition interval of Caputo fractional derivative is re-examined and the integer-order
derivative of q = 1 is added in the Numerical Simulation part. In particular, we compared the time evolution
curve of q = 1 and q = 0.7 in Figure 1, q = 1 and q = 0.85 in Figure 2, q = 1 and q = 0.8 in Figure 3. For
details, please see the Numerical Simulation section in the revised manuscript.
Comment 1.4: Since the value of the memory parameter “q” is currently practically not studied in
econometrics, then a wide range of parameters 0 < q < 1 should be described in the manuscript and the
2corresponding graphs should be built. Figure 6 should be considered for q = 1 and wide range of 0 < q < 1.
Response: Thanks for your constructive comment. According to your suggestion, the memory parameter
q plays a significant role in the proposed model while is currently practically not studied in econometrics. So
we would like to explore its implication in economics and its influence on economic forecasting. A wide range
of 0 < q < 1 (q = 0.2, 0.5, 0.8) as well as the memoryless model of q = 1 Have been considered in Figure 7
and Figure 8. As can be seen, the model of q = 1 always overestimates the working population and TFP
level. While the appropriate fractional derivative model can effectively offset this overestimation and bring
the predicted value back to the true level. In addition, we find through numerical prediction that within
the interval of 0 < q < 1, as the fractional order decreases, the offsetting effect becomes more significant. It
indicates that the fractional order may imply the growth rate of economic variables. For details, please see
the Economic Prediction section in the revised manuscript.
Comment 1.5: The model should be also considered for q > 1. For example, at least values of q close
to one (q = 1.1, q = 1.2) should be described in the manuscript and corresponded Figures.
Response: The authors sincerely thank the reviewer for the time and effort given to our manuscript
and raising such a constructive comment. In our proposed model, only 0 < q 6 1 is considered and
the corresponding stability conditions are derived in the three theorems in the Main Result section. The
numerical simulation and economic prediction are also completed under the framework of 0 < q 6 1.
If q > 1 are considered, the stability zone will change and the theorems in Section 3 will no longer hold,
resulting in the stability conditions to be deduced entirely again. In the case of 0 < q 6 1, we have made
full discussion from numerical simulation to economic implications, which forms a systematic framework.
Therefore, after our careful consideration, we decided not to include the model of q > 1 in our manuscript.
But the authors sincerely thank you for providing us with such a creative and pioneering suggestion and
idea. We are very eager to further explore the entire framework under the condition of q > 1 in our follow-up
research.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
1) I think that the authors should explicitly indicate that in the Theorems 1-3 the system is locally stable for all delays \tau.
2) In point 5 "Economic Prediction", it is not clear why the author use the time lag value \tau=2, and not some other.
2) 3) On page 7, Equation (28) must end with a comma, not a dot.
Author Response
Reviewer ♯2
Comment 2.1: I think that the authors should explicitly indicate that in the Theorems 1-3 the system
is locally stable for all delays τ .
Response: Thanks a lot for your kind words and helpful suggestions on our paper. According to your
suggestion, we have clarified in the Theorems 1-3 that the system is Lyapunov locally asymptotically stable
for all delays τ > 0. The corresponding modification are highlighted in blue color the Theorems 1-3 in the
revised manuscript.
Comment 2.2: In point 5 ”Economic Prediction”, it is not clear why the author use the time lag value
τ = 2, and not some other.
Response: Thank you very much for your constructive advice to improve this manuscript. Considering
your suggestion, the method to select the suitable delay τ has been carefully discussed in Section 5 and the
corresponding Figure 9 (a) depicts the impacts of different delays on the predicted Kt. As can be seen from
3the figure, τ = 2 best matches with the original data because it captures the delay feature of the capital
accumulation process and offsets the overestimation compared with the no delay case. Additionally, the case
of τ = 4 slightly underestimate the Kt level which is considered to delay for too long. Therefore, τ = 2 is a
more appropriate value for the delay parameter. For details, please see the Economic Prediction section in
the revised manuscript.
Comment 2.3: On page 7, Equation (28) must end with a comma, not a dot.
Response: Thank you for your detailed comment. The comma after Equation (28) has been changed
to a comma. And we have looked over our paper again and again to make sure that all punctuation and
mathematical symbols are precise in the revised manuscript.
Author Response File: Author Response.pdf
Reviewer 3 Report
I recommend the acceptance of the paper. However I have some recommendation that should be addressed.
1) The authors state "our model that technological progress is endogenous and related to both population and capital". They need to exhibit this point in their study.
2)The authors should emphasize the meaningfulness of adding delay and fractional derivatives. What are the stylistic facts you observe from the addition of these features? What additional observations do you get with these features?
3) How do you choose the delay and fractional derivative parameters?
4)What are the impacts of delay and fractional derivatives on the predictions?
Author Response
Reviewer ♯3
Comment 3.1: The authors state “our model that technological progress is endogenous and related to
both population and capital”. They need to exhibit this point in their study.
Response: Thanks a lot for your kind words and insightful comment on improving direction of our
paper. According to your suggestion, we have reclarified why technological progress is endogenous in the 2.2
Model Description part in our revised manuscript. The key step is shown as Equation 6. The technological
progress rate g is assumed to be a linear function with lt and kt as variables, and lt, kt represent the
working population per TFP and the capital stock per TFP respectively. That means technological progress
is endogenous and related to both population and capital. For details, please see the 2.2 Model Description
section in the revised manuscript.
Comment 3.2: The authors should emphasize the meaningfulness of adding delay and fractional derivatives. What are the stylistic facts you observe from the addition of these features? What additional observations do you get with these features?
Response: Thank you very much for your constructive advice to improve this manuscript. Considering
your suggestion, we have carefully explored the meaningfulness and economic implication of adding delay
and fractional derivatives.
As for the fractional derivatives, the memoryless model of q = 1 is considered in Figure 7 and Figure
8. As can be seen, the model of q = 1 always overestimates the working population and TFP level. While
adding fractional derivatives (0 < q < 1) effectively offsets the overestimation. In other words, the fractional
derivatives control the growth rate of economic variables.
Similarly, for the delay parameter, the no delay model (τ = 0) and the delayed one (τ = 2, 4) are
compared in Figure 8. The delayed model captures the delay feature in the capital accumulation process
and bring the predicted value back to the true level. For details, please see the Economic Prediction section
in the revised manuscript.
Comment 3.3: How do you choose the delay and fractional derivative parameters?
Response: Thanks for your insightful comment and pointing out the improving direction. The methods
to select the suitable delay and fractional derivative parameters have been carefully discussed in Section 5
4and corresponding graphs have been given in Figure 7, 8, 9.
For the fractional derivative parameters, to enhance the adaptability of the model, we allow the fractional orders q1, q2, q3 to change dynamically with time t, denoted as q1(t), q2(t), q3(t). Then the minimum
prediction error principle is adopted to solve the optimal q∗(t) as Equation (36) does. In Figure 7 and Figure 8, the selected q∗(t) is proven to outperform the other fixed q on predictions.
For the delay parameters, Figure 9 (a) depicts the impacts of delays on the predicted Kt. As can be seen
from the figure, τ = 2 best matches with the original data, while τ = 0 shows overestimation and τ = 4
shows underestimation. Therefore, τ = 2 is a more appropriate value for the delay parameter. For details,
please see the Economic Prediction section in the revised manuscript.
Comment 3.4: What are the impacts of delay and fractional derivatives on the predictions?
Response: Thanks for your constructive comment. Based on your suggestion, the impacts of delay
and fractional derivatives on the predictions are demonstrated in Figure 7, Figure 8 and Figure 9. We have
considered a wide range of fractional derivatives and delay parameters and carry out economic predictions
separately. For details, please see Figure 7, Figure 8 and Figure 9 and the corresponding descriptive text in
the revised manuscript
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The manuscript can be published in the journal.
