A Foundation for Logarithmic Utility Function of Money

Round 1

Reviewer 1 Report

The paper needs relevant improvements as far as clarity of text and precise statement of assumptions are concerned.

Here are some specific comments and questions

• page 1, lines 9-12:
  this period sets the goal of the paper, but is not clear. The period should be broken in parts to be understood.

• page 1, line 13:
  “. . . is additionally separable . . . ” should be perhaps “. . . is additively separable . . . ”

• page 1, lines 15-16:
  what is the financial meaning of assumption 3?

• page 1, line 26:
  “... Walrassian ...” should be “... Walrasian ...”

• page 1, lines 42:
  what values can take the parameter r?

• page 1, lines 42-43 and page 2 line 44:
  why the utility function in line 42 takes the form in line 44 when r = 1? Anyway, perhaps the authors write r = 1 to mean lim r→1?

• page 2, line 57:
  “... of the i+1 interval ...” should be “... of the (i + 1)-th interval ...”

• page 2, line 61:
  “... deprecation ...” should be “... depreciation ...”

  4

• page 3, line 72:
  “Here M represents. . . ” should be “Here M^i represents”

• page 3, line 79:
  “... no sense consider ...” should be “... no sense to consider ...”

• page 3, equation (13):
  the authors should be more precise and say that this is to represent the problem of maximizing U(τ,x) subject to the constraint τ+px = m

• page 4, line 97 and equation (14):
  does equation (14) provide necessary and sufficient conditions for an optimal solution, or only necessary conditions? What features of util- ities W and V are you implicitly assuming?

• page 4, lines 98 to equation (21):
  authors should say that this is sensitivity (comparative static) analysis of the solution

• page 4, line 103:
  “... matrix Ω i:” should be “... matrix Ω is:”

• page 5, line 115:
  “... change in m whit no ...” should be “... change in m with no ...”

• page 5, line 129:
  it is not clear what “conditions 2) and 3)” refer here

Author Response

-->  page 1, lines 9-12:    this period sets the goal of the paper, but is not clear. The period should be broken in parts to be understood.

A: The abstract has been rewritten in an attempt to explain in more detail both the problem studied and the results obtained from this study. The wording has been made clearer and more concise.

--> page 1, line 13:    “. . . is additionally separable . . . ” should be perhaps “. . . is additively separable . . . ”

A: Of course, you are right. It has been corrected.

-->  page 1, lines 15-16:    what is the financial meaning of assumption 3?

A: In lines 133-137 an explanation has been added about the relation of the matrix to the Hessian of the optimization problem and therefore how it is related to the second-order conditions for the existence of maximum.

-->  page 1, line 26:    “... Walrassian ...” should be “... Walrasian ...”

A: It has been fixed.

-->  page 1, lines 42:    what values can take the parameter r?

-->  page 1, lines 42-43 and page 2 line 44:    why the utility function in line 42 takes the form in line 44 when r = 1? Anyway, perhaps the authors write r = 1 to mean lim r→1?

A: The meaning of the parameter r has been introduced, as well as the redefinition of the function U in the case r=1, where the original expression is not defined.

-->  page 2, line 57:     “... of the i+1 interval ...” should be “... of the (i + 1)-th interval ...”

A: It has been modified.

-->  page 2, line 61:    “... deprecation ...” should be “... depreciation ...”

A: This error has been fixed.

-->   page 3, line 72: “Here M represents. . . ” should be “Here M^i represents”

A: The index has been added   

-->  page 3, line 79:   “... no sense consider ...” should be “... no sense to consider ...”

A: This error has been fixed.

-->   page 3, equation (13):    the authors should be more precise and say that this is to represent the problem of maximizing U(τ,x) subject to the constraint τ+px = m

A: The text has been modified to indicate more clearly that the equation represents the optimization problem under study.

-->  page 4, line 97 and equation (14):    does equation (14) provide necessary and sufficient conditions for an optimal solution, or only necessary conditions? What features of util- ities W and V are you implicitly assuming?

A: Since we have not explicitly considered the quasi-concavity characteristics of the utility functions, we have limited ourselves to consider the problem as a case of conditional optimization, so that the equation is only a necessary condition. These details and those concerning the conditions on the functions have been added/rewritten at this point in an attempt to clarify it (lines 127-128, 133-136 and 204-206).

-->  page 4, lines 98 to equation (21):    authors should say that this is sensitivity (comparative static) analysis of the solution

A: A sentence has been included to point this out.

-->  page 4, line 103:    “... matrix Ω i:” should be “... matrix Ω is:”

A: This error has been fixed.

-->   page 5, line 115:    “... change in m whit no ...” should be “... change in m with no ...”

A: This error has been fixed.

-->  page 5, line 129:    it is not clear what “conditions 2) and 3)” refer here

A: After equation (25) we study the three cases in which this equation is verified. These are the cases referred to in the expression (the sentence has been rewritten for clarification)

Reviewer 2 Report

I have read the article and there are a couple of points that I would like to receive some clarification.

First of all, the novelty of this work should be stated more clearly and in more detail in the paper. I don't think it is stated clearly(or at all) in the current text.

Second, I would also like to see some clarification regarding the solution of equation (13) that is given as equation (14). I do not understand were the differential equation in (14) came from, and I could not find anything relevant elsewhere in the text.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 3 Report

Dear authors,

Your topic is interesting but I have major suggestions you will need to consider: They are here:

1) You have formulated a new way of presenting a utility function. What properties does it have that make it recommended for readers? You consider some assumptions behind it, yes, but what are its properties?

2) In the introduction, you need to justify the problem you are solving and why it is important for readers. Otherwise, readers will continue to use existing forms of utility functions. This needs to be brought and why your work stands out. 

3) Lines 9 to 12 : Writing a paragraph with one sentence is discouraged. The sentence is so long and loses the reader. Please rewrite.

4)Line 20: Start the sentence with "The nature" and not "Nature"

5) Line 20: say "has been subjected to" and not the "object of"

6) Lines 25 to 27 : Writing a paragraph with one sentence is discouraged. The sentence is so long and loses the reader. Please rewrite.

7)Line 115: Italicize m since the m in the equation is italicized. Did you intend to write with or whit?

8) Line 145-147: Writing a paragraph with one sentence is discouraged. The sentence is so long and loses the reader. Please rewrite.

 

 

 

Author Response

1) You have formulated a new way of presenting a utility function. What properties does it have that make it recommended for readers? You consider some assumptions behind it, yes, but what are its properties?

A: Actually our interest has been to show how the logarithmic expression for the utility function of money can be derived as a consequence of a set of basic assumptions. We have modified the abstract to explain this point more clearly. In addition, we have modified the discussion to better explain what results we think this paper can provide. Also, we have included some comments to try to discuss with honesty on the advantages and problems of the final proposal we make for the function W().

We understand that this is what the reviewer is referring to.

2) In the introduction, you need to justify the problem you are solving and why it is important for readers. Otherwise, readers will continue to use existing forms of utility functions. This needs to be brought and why your work stands out.

A: As mentioned in the previous section, section 3 has been modified to explain the main result of the research (the demonstration of the need for linear and logarithmic expressions, especially the latter). It is not the authors' intention to propose the use of only the utility functions studied. Rather, the idea would be to reflect on whether it is possible to derive the expressions of the various utility functions from basic conditions as we have tried to do in this paper.

It is also explained how the proposal for the function presents problems that should be studied later and that it is just a way of linking the results obtained and comparing them with previously published results.

In any case, understanding from the indication that it would be necessary to develop the introduction in greater detail, we have proceeded to include a paragraph             on the need to address the problem studied.

3) Lines 9 to 12: Writing a paragraph with one sentence is discouraged. The sentence is so long and loses the reader. Please rewrite.

A: The abstract has been rewritten in an attempt to explain in more detail both the problem studied and the results obtained from this study. The wording has been made clearer and more concise.

4)Line 20: Start the sentence with "The nature" and not "Nature"

5) Line 20: say "has been subjected to" and not the "object of"

A: Errors 4 and 5 have been fixed.

6) Lines 25 to 27 : Writing a paragraph with one sentence is discouraged. The sentence is so long and loses the reader. Please rewrite.

A: The paragraph has been rewritten by dividing it into sentences to improve the wording.

7)Line 115: Italicize m since the m in the equation is italicized. Did you intend to write with or whit?

A: This error has been fixed.

8) Line 145-147: Writing a paragraph with one sentence is discouraged. The sentence is so long and loses the reader. Please rewrite.

A: The paragraph has been rewritten.

Round 2

Reviewer 1 Report

I have checked the revised version of the paper and read the answers by the authors.
I think the paper is suitable for publication in its present form. 

Reviewer 2 Report

The authors have addressed my concerns sufficiently.

Reviewer 3 Report

Dear Authors,

I am glad you were able to consider my suggestions. 

Sincerely,

Reviewer.