Mathematical Models in Mechanics and Economics

by Kenneth R. Driessel, Mathematics Department, Iowa State University

Here is some basic information about this web page. (This web page is under construction - last update: December 6, 2016.)

I created this web page in the spring of 2013 for a seminar at Iowa State University. You can look at the seminar announcement: Announcement. This seminar should be of interest to students and researchers in mathematics, engineering and economics.

I posted some notes and references for the seminar. You can look at them: Notes.

During the fall the seminar normally meets in Heady 272 at 3:30 on Wednesday.

Here is a list of the seminar talks.

********************************* 2016 Talks *************************************
  • The seminar will not meet during the rest of the fall.

  • Wednesday, December 7, 2016 Speaker: Sittisak Leelahanon (Kob) Title: The Probability Approach in Econometrics Abstract: We shall discuss the article by T. Haalvelmo with the same title. This article appeared in Econometrica in 1944. You can find a copy of the article on the seminar web page. (It is a searchable file.)

  • Wednesday, November 30, 2016: Speaker: Sittisak Leelahanon (Kob) Title: A Historical Introduction to Econometrics (continued) Abstract: We shall continue to discuss the following book: 'The history of econometric ideas' by M. S. Morgan. In particular, we shall discuss the following parts of the book: Part III. Formal models in econometrics. Introduction to formal models. Chapter 7 Errors-in-variables and errors-in-equations models. Chapter 8 Haavelmo's probability model.

  • Wednesday, November 23: The seminar will not meet this week. (This week the University is on Thanksgiving break.)

  • Wednesday, November 16, 2016: Speaker: Sittisak Leelahanon (Kob) Title: A Historical Introduction to Econometrics Abstract: We shall discuss the following book: 'The history of econometric ideas' by M. S. Morgan. (This book was first published by Cambridge University Press in 1990.) In particular, we shall discuss the following parts of the book: Part I. Business cycles. Part III. Formal models in econometrics.

  • Wednesday, August 17: The seminar will not meet during the rest of the summer.

  • Wednesday, August 10: Speaker: Jun Xiang, Title: Business Cycles in the Real World

  • Wednesday, August 3: Speaker: Jun Xiang, Title: A Modified Cycle-Theory Model

  • Wednesday, July 27: Speaker: Jun Xiang, Title: Model fitting comparison

  • Wednesday, July 20: Speaker: Jun Xiang, Title: Time Series for Schwartz's Model of the Business Cycle

  • Wednesday, July 13: Speaker: Jun Xiang, Title: A Test of Schwartz's Model

  • Wednesday, July 13: Speaker: Jun Xiang, Title: A Test of Schwartz's Model

  • Wednesday, July 6: Speaker: Jun Xiang, Title: The Inverse Problem for J.T. Schwartz's Model of the Business Cycle

  • Wednesday, June 29: Speaker: Jun Xiang, Title: An Empirical Test of J.T. Schwartz's Model of the Business Cycle

  • Wednesday, June 22: Jun Xiang, J.T. Schwartz's Theory of Business Cycles (continued)

  • Wednesday, June 15, Jun Xiang, J.T. Schwartz's Theory of Business Cycles (continued)

  • Wednesday, June 8, 2:10: Jun Xiang, J.T. Schwartz's Theory of Business Cycles

  • Wednesday, June 1, 2:10: Jun Xiang, An Alternative to Aggregation in Input-Output Analysis.

  • Wednesday, May 25, 2:10: Jun Xiang, A review of the Leontief Model.

    ********************************* 2015 Talks *************************************
  • Tuesday, November 30: The seminar will not meet during the rest of the semester.

  • Tuesday, November 23, 2:10: Sittisak Leelahanon (Kob), The Leontief Model and Neoclassical Equilibrium Theory (continued).

  • Tuesday, November 17, 2:10: Sittisak Leelahanon (Kob), The Leontief Model and Neoclassical Equilibrium Theory (continued).

  • Tuesday, November 10, 2:10: Sittisak Leelahanon (Kob) and Ken Driessel, The Leontief Model and Neoclassical Equilibrium Theory.

    Abstract: We shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. In Part A of the book, Schwartz presents the Leontief model. We shall review this model. In Part C of the book, Schwartz presents neoclassical equilibrium theory. In particular, he compares the price theory he presented in Part A with the price theory of neoclassical equilibrium theory.

    Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of Part A and Part C there.

  • Tuesday, November 3, 2:10: Sittisak Leelahanon (Kob), Schwartz's theory of the business cycle (continued).

  • Tuesday, October 27, 2:10: Sittisak Leelahanon (Kob), Schwartz's theory of the business cycle (continued).

    Abstract: We shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. In Lecture 7, Schwartz revises his model of the business cycle. We shall discuss this revised model.

    Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of Lecture 7 there.

  • Tuesday, October 20, 2:10: Ken Driessel, Linear Aggregation of Input-Output Models (continued)

  • Tuesday, October 13, 2:10: Ken Driessel, Linear Aggregation of Input-Output Models

    Abstract: We shall continue to discuss aggregation. In particular, we shall discuss the paper "Linear Aggregation of Input-Output Models" by Erick Howe and Charles Johnson. The authors consider input-output aggregation generally. In particular, they discover what can be deduced axiomatically about the about functional forms of aggregation. They also prove several new results about input-output aggregation and link these results to the existing literature.

    Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find copies of the paper on that page.

  • Tuesday, October 6, 2:10: Sittisak Leelahanon (Kob), Aggregation in Leontief and Schwartz (continued)

  • Tuesday, September 29, 2:10: Sittisak Leelahanon (Kob), Aggregation in Leontief and Schwartz

    Abstract: We shall discuss aggregation in Leontief and Schwartz. In particular, we shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Schwartz. In Lecture 6 Schwartz presents his "aggregative" model of the business cycle. We shall continue to discuss this model. We shall also continue to discuss the following paper: "Quantitive input and output relations in the economic system of the United States" by Leontief. In this paper Leontief presents his version of aggregation.

    Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find copies of the Lecture and the paper on that page.

  • Tuesday, September 22, 2:10: Ken Driessel, Some dynamical systems that model business cycles (continued).

  • Tuesday, September 15, 2:10: Ken Driessel, Some dynamical systems that model business cycles (continued).

    Abstract: We shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. In Lectures 5 thru 12 Schwartz presents various versions of his model of the business cycle. We shall review the versions that Schwartz presents in Lectures 5 thru 9. Then we shall discuss Lecture 10 "Additional discussion of cycle theory". (Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of that Lecture on that page.)

  • Tuesday, September 8, 2:10: Ken Driessel, Some dynamical systems that model business cycles.

    Abstract: We shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. In Lectures 5 thru 12 Schwartz presents various versions of his model of the business cycle. We shall review the versions that Schwartz presents in Lectures 5 thru 8. Then we shall discuss Lecture 9 "A modified cycle-theory model". (Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of that Lecture on that page.)

  • Friday, August 21, Black 1026, 2:10: Ken Driessel, Reflections on Keynesian Economics

    Abstract: We shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. In particular, we shall discuss Lecture 8 "General Reflections on Keynesian Economics. The Numerical Value of the Multiplier". Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of that Lecture on that page.

  • Friday, August 14, Carver 004, 2:10: Ken Driessel, Consumption in the Cycle-Theory Model

    Abstract: We shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. In particular, we shall discuss Lecture 7 "Consumption in the Cycle-Theory Model. Say's Law". Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of that Lecture on that page.

  • Friday, August 7, No Meeting

  • Friday, July 31, Carver 004, 2:10: Ken Driessel, Mathematical Analysis of a Cycle-Theory Model

    Abstract: We shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. In particular, we shall discuss Lecture 6 "Mathematical Analysis of a Cycle-Theory Model. Expansive and Depressive Cases." Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of that Lecture on that page.

  • Friday, July 24, No Meeting

  • Friday, July 17, Carver 004, 2:10: Ken Driessel, A Dynamic Model of an Economy

    Abstract: We shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. In particular, we shall discuss Lecture 5 "Business Cycles. Introductory Considerations" with emphasis on section 4 "A Dynamic Model of an Economy". Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of that section on that page.

  • Friday, July 10, Carver 004, 2:10: Ken Driessel, Input and Output Relations of the U.S.

    Abstract: We shall discuss the following paper: "Quantitive input and output relations in the economic system of the United States" by Wassily Leontief. Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of the paper on that page.

  • Friday, July 3, No Meeting

    Abstract: University Holiday

  • Friday, June 26, Carver 004, 2:10: Ken Driessel, Mathematics as a Tool for Economic Understanding

    Abstract: We shall continue discussing the model of economics developed by Jacob Schwartz. In particular we shall discuss the following 1978 paper of Schwartz: "Mathematics as a Tool for Economic Understanding". Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of the paper on that page.

  • Friday, June 19, Carver 004, 2:10: Ken Driessel, Input-Output (Leontief) Analysis

    Abstract: We shall continue discussing the input-output (Leontief) model of economics. In particular, we shall discuss the chapter 9 "Input-Output Analysis in Economics" in the book "Nonnegative Matrices in the Mathematical Sciences" by Berman and Plemmons. We shall compare this input-output analysis to that of Schwartz. Recall that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find a copy of chapter 9 on that page.

  • Friday, June 12, Carver 004, 2:10: Ken Driessel, The Theory of Prices in the Leontief Model

    Abstract: We shall continue discussing the book "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. The Leontief model leads to an eigenvalue-vector problem. We shall discuss the connection between this problem and prices.

  • Friday, June 5, Carver 004, 2:10: Ken Driessel, Analytical Economics from the Viewpoint of Jacob Schwartz

    Abstract: We shall begin discussing the following book: "Lectures on the Mathematical Method in Analytical Economics" by Jacob Schwartz. Note that at the beginning of this web page there is paragraph concerning notes and references together with a link to a page called "Notes". You can find copies of the first few chapters of the book on that page.

  • Friday, May 29, Carver 004, 2:10: Ken Driessel, Quotients of Topological Spaces and Structural Stability

  • Friday, May 22, Carver 004, 2:10: Ken Driessel, Examples of structural stability

    Abstract: In his book about dynamical systems Steven Wiggins writes: "The mathematical models we devise to make sense of the world around and within us can only be approximations. Therefore, it seems reasonable that if they are to accurately reflect reality, the models themselves must be somewhat insensitive the perturbations and have properties that are 'not atypical', in a sense that is not easy to characterize in a way that is useful in applications. The attempts to give mathematical substance to these rather vague ideas have led to the concept of 'structural stability' and 'genericity' ..." We shall discuss the definitions of these two concepts and a number of elementary examples that illustrate these concepts.

    Here is the definition of 'generic' (from the Wikipedia article "Generic property"): In topology, a 'generic' property is one that holds on a dense open set. Here is the definition of 'structurally stable' (from the Scholarpedia article "Structural stability" by Pugh and Peixoto): If a set is equipped with a topology and an equivalence relation then its 'structurally stable' elements are those interior to the equivalence classes. The 'structure' is whatever is preserved by the equivalence relation; its structure remains the same when a structurally stable element is perturbed.

  • Friday, May 15, Carver 004, 2:10: A.M. Fink, Almost Periodic Functions

    Abstract: Fink writes: "I will give several definitions of these functions and some of the main theorems about the structure of the class of all of them and their uses. [I also will make remarks] about the reason that they are relevant to differential equations and some ways in which one might find them. (This is about the 100th anniversary of their discovery by Harald Bohr.)"

  • Tuesday, May 12, Carver 401, 2:10: Jim Murdock, What mathematical models are worth studying?

    Abstract: Mathematicians occasionally hear the claim that "one should not study" a particular mathematical model because it does not meet someone's criterion for a worthwhile topic, usually because it is "not generic" or "not structurally stable. (This idea is associated with Steve Smale, among others, and is sometimes called the "stability dogma.") A quick counterargument would say that it is not intellectually respectable to deny anyone the right to study whatever they please. A better suggestion is that one should "unfold" the model by adding parameters until it does become structurally stable, and study the unfolded model. A counterargument to that is that many systems have "infinite codimension" and cannot be unfolded. All of these ideas (genericity, structural stability, and codimension) are quite technical. I recently noticed a sentence in the 1983 book Guckenheimer and Holmes, in which they turn the argument upside down by saying that "the only properties of a [mathematical model] that are relevant [to the real world] are those which are preserved under perturbations of [the model]." This approach allows all models to be studied, but provides a practical note of caution about applications of the results, and reflects the actual practice of applied mathematicians. In 1998, and in an improved form in 2009 (with David Malonza), I proposed an approach to unfolding that fits well with Guckenheimer and Holmes' approach (which I had not noticed at the time). My approach gives every (local) dynamical system an unfolding with finite codimension that is valid for a subset of the system's properties. I will discuss these ideas in a nontechnical way, and leave you free to study whatever models you like.

  • Tuesday, May 5: Ken Driessel, Economics and the Lambert W Function

    Abstract: Recall that Tinbergen and Kalecki discussed economic models that involve the following linear delay differential equation: f'(t) = - a f(t-\theta) where a and \theta are constant parameters. The characteristic equation for this equation is z exp(z) = -a * \theta where z is a complex variable. The characteristic roots are important for the analysis of the differential equation. The multi-valued inverse of the function z -> z exp(z) is called the Lambert W function. (In Mathematica this function is called the ProductLog function.) We shall discuss this function.

  • Tuesday, April 28: Ken Driessel, Economics and Almost Periodic Functions

    Abstract: Recall that Tinbergen and Kalecki discussed economic models that involve delay differential equations. The bounded solutions of such equations involve almost periodic functions. We shall discuss the relation between almost periodic functions and solutions of delay differential equations.

  • Tuesday, April 21: Ken Driessel, Tinbergen's Shipbuilding Cycle

    Abstract: In 1931 Jan Tinbergen wrote a paper with title "A Shipping Cycle?". The paper includes the following model: f'(t) = - a f(t-\theta) where where a and \theta are constant parameters. The quantity f(t) represents shipping tonnage. This is an example of a linear delay differential equation. In particular, the parameter \theta is a "lag" or "delay". This equation is a special case of Kalecki's investment equation. Kalicki reduces his equation to Tinbergen's equation. We shall discuss Tinbergen's equation. In particular, we shall describe the character roots of this equation.

  • Tuesday, April 14: Ken Driessel, Kalecki's Theory of the Business Cycle (continued)

    Abstract: Kalecki's 1935 paper includes the following "Investment Equation'': I'(t) = a I(t) + b I(t-v) + c where where a, b, c and v are constant parameters. This is an example of a linear delay differential equation. In particular, the parameter v is a "lag" or "delay". We shall discuss such equations and their relation to Kalecki's theory of the business cycle.

  • Tuesday, April 7: Ken Driessel, Kalecki's Theory of the Business Cycle

    Abstract: Kalecki wrote papers in 1933 and 1935 that presented a model of the business cycle. I suggest that we study this model. In his recent 2009 book "Economics Dynamics", Gandolfo presents an introduction to Kalecki's theory. Gandolfo says that "This model . . . deserves a detailed treatment because it continues to be the source of inspiration for research . . .". We shall see that this model provides an example of a delay differential equation.

    Kenneth R. Driessel, driessel@iastate.edu