Saturday, March 24, 2018

The Curious Union of Chess, Money, and Economic Production

\(c(t) + g(t) = f_w(t)n(t)(1+r(t)) + k_p(t)\)
\(c(t) + g(t) = k_b(t)(1+r(t)) + k_p(t)\)
In the game of Chess, players own the moves they make. Each move advances the game in a continuum from start to finish.

A productive firm works the same way. When a fractional part of a complex product is made or purchased, the firm owns another fractional part of a continuum that evolves into a priced product.
In a curious way, I'm frustrated by Brian Romanchuk's article
"The Curious Profit Accounting of DSGE Models." I can't convince myself that his beginning equation 16.2.3 is correct.[1] [2] It certainly doesn't fit with the real-world economic framework that I observe!

Unable to make sense of the equation but agreeing that an equation linking three sectors of the productive economy is a worthy goal, I would like to remove the wrinkles. A task harder to do than I expected.

The Problem

While my initial criticism of the equation was the number of sectors represented, I eventually realized that the basic flaw stemmed from a misuse of the capital tool. Financial capital was conceptually used as means of production in the same manner as labor so that it became one of the final products. That made no sense to me. In the real world, financial capital is directly exchanged for fractional contributions thereby transferring ownership in both directions.

We can use the example of fractional labor as an illustration to make the point. Using monetary exchange, one current hour of labor is traded for an hour of labor-that-has-been-previously-completed. In other words, when a person works one hour, that effort is effectively traded monetarily for one past hour that another person has sweated through.

The Solution

We can symbolically show past laboring effort expended, then represented by financial capital, and then returned to labor by writing  \(n \mapsto k \mapsto n, \) where \( \mapsto \) is the maps-to symbol, \(k\) is the capital received from previously expended effort and \(n\) is hours expended either past or currently.

This symbolic description using money [5] does not relate hours directly to a rate of payment. It has the limited task of indicating that a block of labor has been traded for a block of money has been traded for a block of labor. The concept of trading blocks for blocks will be woven into this article.

Only by accident does perfect mapping occur in real world exchanges that occur over a time period. Instead, there is always a difference in values that we will describe symbolically by writing  \(n \mapsto k \mapsto (n+p), \) where \(p\) represents a value difference perceived by the owner of the second \(n.\) In words, we say that the second provider of labor must perceive a benefit (or Profit) that encourages a decision to proceed. The benefit may be visible in the form of money or invisible in the form of product preference.

Continue  Defining Sectors

Returning to the subject of sectors, my initial criticism of Brian's equation 16.2.3 was that four sectors were described, not three as he suggested. Now that I am in version 6 (or more!) of developing a 'better equation', I believe that FIVE sectors are required to fully describe the productive economic continuum.

How do I define 'sectors'? If an economic contributor can make a decision, there is a need to represent that ability by assignment to an economic sector. The ability to make a decision implies ownership of some part of the productive amalgam with entitlement to financial reward. The five sectors that I see are consumption, government, productive firm, labor, and financial capital.

The productive firm sector will be assigned a reward coming from profit (if any).

The need for a financial capital sector is not immediately obvious because each of the other sectors have within them members who own financial capital,  The need for a separate sector stems from the ownership impairment decision that must be made when property is rented away. 

Our capital sector will have a undefined stock of preexisting capital. This will be a necessary assumption when we later assume that productive firms borrow and repay all financial capital used to pay expenses during a time period. The capital sector will be assigned a reward in the form of interest payment.

Describe Model Conditions in a Nutshell

The task of a productive firm is to reorder the capabilities of our five sectors into a product exchange evolution. The assembly process will be measured and modeled as a capital expense using three sectors: 1) a stylized all-inclusive labor sector, 2) a sector describing the cost of rented capital, and 3) an unpredictable remainder (Profit?) assumed to be owned by the productive firm sector. The selling process (of the product for consumption by the household and government sectors) provides income. The entire process is assumed to start and complete in one year.

Capital (owned by the capital sector) is existent and adequate. Financial capital is traded fractionally for fractional product parts as the product is assembled and finally sold.

Now Write a Productive Continuum Equality

We will describe the productive process by completing an annual income statement in terms of our five economic sectors. The result is surprisingly satisfying.

Our income statement will be written directly from the description given.[4]  Label the sector terms as labor \(n,\)  borrowed capital \(k_b,\) profit \(k_p,\) household \(c,\) and government \(g.\) With income on the left side, we write the general concept mapped to the real-world as \[c+g \mapsto k_b+n+k_p.\] where profit is the incentive  term.

Before writing an equality, we need to adjust the data from each sector into a place on the monetary scale, noticing that only the labor sector is not already reported in monetary terms. We will adjust labor by using a translation factor \(f_w.\)  The cost of borrowed capital will be calculated using cost factor \(r.\)

To do all of this, we will use a function format[3] with a time notation to indicate that the equality is related to a unique time period. We will write our equality as

(1)       \(c(t)+g(t)=r(t)k_b(t)+f_w(t)n(t)+k_p(t).\)

Equation 1 bridges the continuum of the productive process using five sectors.

Readers may be interested in comparing equation 1 to Brian's equation 16.2.3. In 16.2.3,  terms \(k_b(t)\) and \(f_w(t)n(t)\) seem to have been equated which enables us to simplify the equation. This is allowable if capital is borrowed at the same time and amount that labor cost are paid (albeit an inefficient process). With stylized labor cost clearly paid with borrowed capital, we have  \(k_b(t) = f_w(t)n(t).\) Substitute into equation 1 using the labor term to get \[c(t) + g(t) = r(t) f_w(t)n(t) + f_w(t)n(t) + k_p(t).\] Simplify to read

(2)          \(c(t) + g(t) = f_w(t)n(t)(1+r(t)) + k_p(t). \)

We can also substitute using the borrowed capital term to get

(3)         \(c(t) + g(t) = k_b(t)(1+r(t)) + k_p(t).\)

The difference between expressions (2) and (3) is the emphasis on financial capital or labor. The two expressions should evaluate to the same monetary number.

Equations 1, 2 and 3 are all final versions of an equality that I think describes the productive continuum adjusted to the real-world using five sectors. Perhaps my readers will have a better definition of exactly what the equation describes.

Unfortunately, Brian's 16.2.3 looks like a distant cousin of the equations developed here. I still don't understand 16.2.3.

Making the Unknown Known

In the assumptions (in "model conditions"), we clearly stated that profit \(k_b(t)\) was unknown, yet, we also clearly assumed that the entire equation completed in one time period. If completed, we can learn the value of all terms by examining the accumulated data.

Equation 1 (and equivalent 2 and 3) describes a relationship between economic sectors. It would be correct to attach a historical number to each of the sectors. It would also be correct to predict future numbers for each sector based on changes envisioned.

DSGE models supposedly try to find points of optimization for the sector of interest. The equations developed here would suggest that optimum for one sector would not be optimum for a second sector.

Using the Equation

It is important to remember that the profit term \(k_p(t)\) is a balancing term that maps theory to the real-world. Hence, we should NEVER expect to find numerical values by beginning with a known profit result. Yes, the profit term is real in the real-world, but it represents motivation (or disincentive) for decision makers at the theoretical level.

This model of money and it's use in productive effort seems (to me) very robust. As a robust description, it can become a common reference for departure into other, hopefully improved, models. Three model excursions follow to illustrate this possibility:

1)  This productive continuum equation was built based on micro economic principals. It can be smoothly expanded to the macro economic scale by adding productive firms. When ALL of the productive firms are included, we can begin consideration of the catastrophic transfer problem [If productive firms are always profitable, eventually all of the money available in an economy will be in the ownership of firm management.]. This undesirable conclusion could materialize using this model if financial capital ownership was not widely distributed in the economy. Would government ownership of capital prevent the potential problem?

2)  We can easily model a government that uses two revenue sources, bonds \(g_b(t)\) and taxes \(g_t(t)\). We could write the equation as \(c(t) + g_b(t)b_k(t)+g_t(t) = f_w(t)n(t)(1+r(t)) + k_p(t) \) where the term \(b_k(t)\) converts the bonds to financial capital. This model would need an explanation for how that conversion mechanism worked in the real economy.

3) Still using the model of example 2, which government would be more interested in imposing tariffs and why?

Hmmm. How would we fit foreign sourced production into an equality like we have here? I don't have that question resolved.


A DSGE equation that seems to not fit smoothly with other frameworks caused enough irritation to initiate an effort to find a better fitting equation. The better fitting equality found requires a base framework of assumptions that clearly included a capital owning sector that had decision making authority.

The simplicity of the equation found makes it seem almost trivial. Yet, we needed to assume a robust continuum of sequential money transfers before we could logically connect fractional construction to finished priced product. We also needed to divide the economy into sectors, each with decision making capability and with the ability to contribute to the productive or consumptive processes. The simplicity of the equation hides the rigidity of the required assumptions.

The management role of decision makers has barely been considered here. The role of a lender supplying initial capital is particularly important. This productive continuum model is discontinuous at the starting point. If a lender fails to allow initial production, nothing happens! In a similar fashion, labor, whether organized or individually represented, has a decision making role. Labor is supplied hour by hour. A discontinuity is reached if labor decides to not perform.

In the real economy, consumers can borrow money to buy goods. It is easy to see in this model that production would be stimulated by customer borrowing. Government borrowing may be sustained sequentially over time, potentially forming a mechanism for hyperinflation.

Thanks to Brian Romanchuk for his efforts to present this series of articles. [Brian's final article in the series can be found here.]


[Note 1] One of the nice things about beginning a study of economics when you are older is that you have a lot of experience from which to draw. One of the frustrations is that you may have little or no economic schooling to guide you. This places you in the position of learning everything about economics from the position of being a well educated beginner. I resist accepting economic theories until each element can fit seamlessly with the mating theories, like in assembling a jigsaw puzzle.

[Note 2] This article contains an expanded explanation of the equations first presented (by me) in a comment in Brian Romanchuk's article at

[Note 3] Function notation is a method of mapping inputs into a repeatable pattern. The time term \(t\) as in \(f(t)\) indicates that the data is accumulated over a time interval.

[Note 4] Term \(n\) represents hours of labor. We will use hours of labor as a proxy for all cost of production whether purchased directly as labor or hidden in a bid price for a part or service. The actual expense of a widget is not of concern here. We are only interested in learning the interactions between borrowed money and other sectors. If we later are interested in other interactions, we may need to refine our definitions. Hours of stylized labor must be converted to a financial capital equivalent using term \(f_w\).

[Note 5] The terms 'money', 'financial capital', and (occasionally) simply 'capital' are used somewhat interchangeably in this article. 'Capital' is never used as a reference to fixed capital such as buildings or bonds.

(c) Roger Sparks 2018

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