Trouble with the equation of exchange.

[Widdekind]

economies consist of three kinds of economic entities

1. individuals (I)
2. corporations (C)
3. Governments (G)

their respective total aggregate expenditures, in domestic markets, are accounted as I/C/G, to which X must be added (expenditures from foreigners), and from which M must be subtracted (expenditures, in domestic markets, on foreign products) [note: i have reversed the traditional notation C <---> I, typically interpreted as (individual) Consumption and (corporate) Investment]

Chart 1 - Primary Money Flows in the American Economy
the solid-line outer circuit represents the nonfinancial expenditure-income payments, in exchange for "real" goods and services. The dash-line inner circuit represents money advanced in exchange for credit claims (or securities). The counterclockwise money flows shown in the diagram are thus conceptually matched by (unseen) equal and opposite flows of real goods, services, and securities.

The "credit market" is an analytical abstraction for conceptually "pooling" all credit flows between economic units...

The money balance of each sector is represented in the diagram by a small circle...

The "monetary agency" is an analytical abstraction representing the source of all new money. New money created in connection with bank loans is shown as entering the circuit via the credit market. Conceptually, the banks borrow the new money...from the monetary agency...

I + U + M = E + A + C
I = nonfinancial net income
U = primary uses of credit
M = new money (increase in currency and demand deposits)
E = nonfinancial expenditure (consumption + capital investment)
A = primary credit advances
C = change in cash balance​

...The basic money-flow equation is equally appropriate to the economy as a whole, or to individual economic units. Its three subsidiary equations are true only for the economy as a whole: I = E, U = A, M = C

employing my own notation, for a given currency's economy:

Commodities (including Labor) Market
Incomes = Outflows
Revenues + Borrowings = Expenses + Loans + change-in-Reserves

Credit Market
Borrowings = Loans + new-Money​

subtracting the second equation, from the first equation, we have that (w.h.t.):

Revenues = Expenses + [change-in-Reserves - change-in-Money-supply]
I = E + (d/dt)[R - M]​

that is the "Income/Expenditure" equation, albeit without distinguishing domestic from foreign economic entities:

Revenues = domestic + foreign = Y + M
Expenses = domestic + foreign = I+C+G + X​

For simplicity, Reserves & Money-supply need not be accounted separately, for domestics & foreigners.

Y + M = I+C+G + X + (d/dt)[R - M]
GDP = Y = I+C+G + NX + (d/dt)[R - M]​

where the last term does not differentiate domestic from foreign:

estimating the correction term

First, the total supply, of all conceivable forms of pseudo-Money in USD (M4):

M4 = M3 + Money-Market-Funds
~= (M2-M1) + (MZM-Currency)​

Second, Domestic Reserves (MB-M0) of USD (DR); and Foreign Reserves of USD (FR)*. The latter increased steadily, from $0.6T in 1995, to $3.5T in 2011.

* ignoring I/C/G Reserves (individual, corporate, & Government cash-on-hand)​

From 1995 through 2011, M-R increased by +$6T over 17 years. Ergo, since 1995, US GDP estimates may have been overstated, by $350B / year, mis-accounting net Money creation ([d/dt][M-R]) as earned income (GDP).

M4
M4 - DR
M4 - DR - FR (estimated)

 

[Widdekind]

another take, which i find more confusing:

"From a macroeconomic perspective, households, government and firms (corporations) are the 3 fundamental players of money and resources. This flowchart diagram below illustrates how they interact and exchange resources" (econ-guru)​

 

[DSGE]

another take, which i find more confusing:

"From a macroeconomic perspective, households, government and firms (corporations) are the 3 fundamental players of money and resources. This flowchart diagram below illustrates how they interact and exchange resources" (econ-guru)​

What's confusing?


By the way, you can't get any actual information out of circular flow. It's not a theory or anything. It's just... some of the exchanges that happen in an economy.

 

[itfitzme]

another take, which i find more confusing:

"From a macroeconomic perspective, households, government and firms (corporations) are the 3 fundamental players of money and resources. This flowchart diagram below illustrates how they interact and exchange resources" (econ-guru)​

What's confusing?


By the way, you can't get any actual information out of circular flow. It's not a theory or anything. It's just... some of the exchanges that happen in an economy.

I am curious what you mean by you cannot get any information out of circular flow diagram?

Do you mean this particular one or any circular flow diagram?

This one presents both GDP = C + T + S and GDP = C + I + G + X - M.

 

[DSGE]

another take, which i find more confusing:

"From a macroeconomic perspective, households, government and firms (corporations) are the 3 fundamental players of money and resources. This flowchart diagram below illustrates how they interact and exchange resources" (econ-guru)​

What's confusing?


By the way, you can't get any actual information out of circular flow. It's not a theory or anything. It's just... some of the exchanges that happen in an economy.

I am curious what you mean by you cannot get any information out of circular flow diagram?

Do you mean this particular one or any circular flow diagram?

This one presents both GDP = C + T + S and GDP = C + I + G + X - M.

Yes. It's not a theory, it's not anything. It just shows you some streams of spending that happen.

Y = C+I+G+X-M isn't shown or proven by circular flow. All you need for that is:

Income necessarily equals expenditure.
Put income on the left hand side.
Put expenditure on the right hand side.
Disaggregate either as much as you like (say separate expenditure into consumption expenditure, investment expenditure, etc)

Circular flow does nothing.

 

[itfitzme]

What's confusing?


By the way, you can't get any actual information out of circular flow. It's not a theory or anything. It's just... some of the exchanges that happen in an economy.

I am curious what you mean by you cannot get any information out of circular flow diagram?

Do you mean this particular one or any circular flow diagram?

This one presents both GDP = C + T + S and GDP = C + I + G + X - M.

Yes. It's not a theory, it's not anything. It just shows you some streams of spending that happen.

Y = C+I+G+X-M isn't shown or proven by circular flow. All you need for that is:

Income necessarily equals expenditure.
Put income on the left hand side.
Put expenditure on the right hand side.
Disaggregate either as much as you like (say separate expenditure into consumption expenditure, investment expenditure, etc)

Circular flow does nothing.

It's a model.

That's a curious statement.

Why then, do you suppose, it is presented then?

How do you suppose one gets to MV = PQ, GDP = C + T + S, GDP = C + I + G + X - M without some presentation that the economy is a closed loop without sources or sink and some demonstration that, in fact, all elements have been accounted for?

 

[DSGE]

I am curious what you mean by you cannot get any information out of circular flow diagram?

Do you mean this particular one or any circular flow diagram?

This one presents both GDP = C + T + S and GDP = C + I + G + X - M.

Yes. It's not a theory, it's not anything. It just shows you some streams of spending that happen.

Y = C+I+G+X-M isn't shown or proven by circular flow. All you need for that is:

Income necessarily equals expenditure.
Put income on the left hand side.
Put expenditure on the right hand side.
Disaggregate either as much as you like (say separate expenditure into consumption expenditure, investment expenditure, etc)

Circular flow does nothing.

It's a model.

What are its assumptions? What interesting things does it do? What questions can it answer?

That's a curious statement.

Why then, do you suppose, it is presented then?

Because it's easier for intro to econ students to understand that income = expenditure if it's shown on a diagram. It's just like venn diagrams. You can't do any proofs or models with a venn diagram. They're just good if you're a visual learner.

How do you suppose one gets to MV = PQ, GDP = C + T + S, GDP = C + I + G + X - M without some presentation that the economy is a closed loop without sources or sink and some demonstration that

Exactly like I did above. You just have to notice that all expenditure in necessarily income and vice versa. Then you can disaggregate it as much as you like, as long as your aggregates form a partition.

Y = E

So I can partition E into Private expenditure and Public expenditure

Y = Priv + Pub.

(Assume a closed economy) I can disaggregate expenditure further into consumption and investment:

Y = C_priv + I_priv + C_pub + I_pub

Maybe let's group all public expenditure into the one term, G:

Y = C_priv + I_priv + G
Y = C + I + G (drop subscripts because all pub stuff is in G)

Maybe I want to disaggregate private consumption into expenditure on durable goods and non-durables:

Y = C_durable + C_non-durable + I + G

You can do anything you want. As long as the left hand side partitions income and the right hand side partitions expenditure. You don't get anything actually useful from looking at the diagram because you've made no assumptions.

, in fact, all elements have been accounted for?

The diagram doesn't do that. In order to put down all elements you have to think of them first and verify that they're a partition. But if you've done that, just write it down. No need to draw them as nodes on a graph.

 

[itfitzme]

M2 is increased by my savings while M1 is decreased. Yet, someone else then borrows against it to increase M1 again.

according to Wikipedia, M2 includes all forms of "Money" included in M1, plus longer-term savings Deposits. If so, then any transfer of "Money" affecting M1 automatically affects M2 in the same way. You may be employing the word "M1" to mean "checking Deposits", and "M2" to mean "saving Deposits". Technically, one would need to define "dM1 = M1-M0 = checking Deposits", and "dM2 = M2-M1 = savings Deposits".

Yeah, whatever what I thinking? Der.... You are right, of course and I was thinking M1-M0 and M2 - M1 but misstated it.

I was more interest in just expressing that I ran into some issues in trying to just sum things up definitively, given all the borrowing, saving, borrowing against that, and saving again.

I could only hope that the total of all reserves and savings is equal to the total of all credit (accounting for the reserve rate). I am, though unwilling to assume that what seems reasonably apparent is in fact so unless I have actually detailed an accurate and precise, comprehensive model.

I was interested, originally, in determining if, in fact, it all just summed to zero. That is if, should all the debt be paid off, there would be no remaining supply of money in the system.

I got distracted from that. And in later learning that the existence of money is expected to be dependent upon the continuous recirculation of debt, fundamentally beginning with the fractional reserve process, it seemed reasonable to assume that the volume of funds created by the fractions reserve process is of such a great magnitude that the question was a mute point.

 

[itfitzme]

Yes. It's not a theory, it's not anything. It just shows you some streams of spending that happen.

Y = C+I+G+X-M isn't shown or proven by circular flow. All you need for that is:

Income necessarily equals expenditure.
Put income on the left hand side.
Put expenditure on the right hand side.
Disaggregate either as much as you like (say separate expenditure into consumption expenditure, investment expenditure, etc)

Circular flow does nothing.

It's a model.

What are its assumptions? What interesting things does it do? What questions can it answer?



Because it's easier for intro to econ students to understand that income = expenditure if it's shown on a diagram. It's just like venn diagrams. You can't do any proofs or models with a venn diagram. They're just good if you're a visual learner.

How do you suppose one gets to MV = PQ, GDP = C + T + S, GDP = C + I + G + X - M without some presentation that the economy is a closed loop without sources or sink and some demonstration that

Exactly like I did above. You just have to notice that all expenditure in necessarily income and vice versa. Then you can disaggregate it as much as you like, as long as your aggregates form a partition.

Y = E

So I can partition E into Private expenditure and Public expenditure

Y = Priv + Pub.

(Assume a closed economy) I can disaggregate expenditure further into consumption and investment:

Y = C_priv + I_priv + C_pub + I_pub

Maybe let's group all public expenditure into the one term, G:

Y = C_priv + I_priv + G
Y = C + I + G (drop subscripts because all pub stuff is in G)

Maybe I want to disaggregate private consumption into expenditure on durable goods and non-durables:

Y = C_durable + C_non-durable + I + G

You can do anything you want. As long as the left hand side partitions income and the right hand side partitions expenditure. You don't get anything actually useful from looking at the diagram because you've made no assumptions.

, in fact, all elements have been accounted for?

The diagram doesn't do that. In order to put down all elements you have to think of them first and verify that they're a partition. But if you've done that, just write it down. No need to draw them as nodes on a graph.

Might we suppose that a substantial portion of people have a different style of thinking and might find greater utility in visualizing it in a graphical format rather then a list?

 

[itfitzme]

Yes. It's not a theory, it's not anything. It just shows you some streams of spending that happen.

Y = C+I+G+X-M isn't shown or proven by circular flow. All you need for that is:

Income necessarily equals expenditure.
Put income on the left hand side.
Put expenditure on the right hand side.
Disaggregate either as much as you like (say separate expenditure into consumption expenditure, investment expenditure, etc)

Circular flow does nothing.

It's a model.

What are its assumptions? What interesting things does it do? What questions can it answer?



Because it's easier for intro to econ students to understand that income = expenditure if it's shown on a diagram. It's just like venn diagrams. You can't do any proofs or models with a venn diagram. They're just good if you're a visual learner.

How do you suppose one gets to MV = PQ, GDP = C + T + S, GDP = C + I + G + X - M without some presentation that the economy is a closed loop without sources or sink and some demonstration that

Exactly like I did above. You just have to notice that all expenditure in necessarily income and vice versa. Then you can disaggregate it as much as you like, as long as your aggregates form a partition.

Y = E

So I can partition E into Private expenditure and Public expenditure

Y = Priv + Pub.

(Assume a closed economy) I can disaggregate expenditure further into consumption and investment:

Y = C_priv + I_priv + C_pub + I_pub

Maybe let's group all public expenditure into the one term, G:

Y = C_priv + I_priv + G
Y = C + I + G (drop subscripts because all pub stuff is in G)

Maybe I want to disaggregate private consumption into expenditure on durable goods and non-durables:

Y = C_durable + C_non-durable + I + G

You can do anything you want. As long as the left hand side partitions income and the right hand side partitions expenditure. You don't get anything actually useful from looking at the diagram because you've made no assumptions.

, in fact, all elements have been accounted for?

The diagram doesn't do that. In order to put down all elements you have to think of them first and verify that they're a partition. But if you've done that, just write it down. No need to draw them as nodes on a graph.

Curiously, the entire body of circuit it analysis in electronic engineering, which is structurally identical to the circular flow in economics, is dependent upon the skill of modeling the circuit in just this manner, the significant difference being the details of how the element react. Though, admittedly, circuit analysis is infinitely more complicated in the combinations. It does have the advantage that the elements are more well behaved.

 

[DSGE]

Might we suppose that a substantial portion of people have a different style of thinking and might find greater utility in visualizing it in a graphical format rather then a list?

Yes.

They're just good if you're a visual learner.

 

[itfitzme]

Might we suppose that a substantial portion of people have a different style of thinking and might find greater utility in visualizing it in a graphical format rather then a list?

Yes.

They're just good if you're a visual learner.

How about if you are visual thinker, a process which includes teaching oneself?

 

[DSGE]

Might we suppose that a substantial portion of people have a different style of thinking and might find greater utility in visualizing it in a graphical format rather then a list?

Yes.

They're just good if you're a visual learner.

How about if you are visual thinker, a process which includes teaching oneself?

Teaching yourself that income equals expenditure, sure. Other than that, what information do you get out of circular flow? Circular flow just demonstrates an income-expenditure identity. It has no real content. What would somebody be trying to learn from it?

 

[itfitzme]

Yes.

How about if you are visual thinker, a process which includes teaching oneself?

Teaching yourself that income equals expenditure, sure. Other than that, what information do you get out of circular flow? Circular flow just demonstrates an income-expenditure identity. It has no real content. What would somebody be trying to learn from it?

Well, given that it exists suggests that there has been considerable demand for it. And given that there has been considerable demand, then it clearly has utility for a large number of individuals.

Do I really need to know what utility another person gets from the product that they demand? Or it sufficient for me to just know that they find utility in it?

Does it have to be a proof to have utility? For that matter, what constitutes a proof?

Might we suppose that visual processing in the brain invokes distinctly different and important mental processes that are not available from the verbal or symbolic processes?

Faraday found no purpose for using mathematical notation and yet he proved the existence of and natural laws of the electro-dynamic forces. Apparently mathematical notation had no utility for him.

Einstein is considered to have done all of his thinking in a visual form, only committing it to a mathematical format after he had realized the answer. The math was, for him, not a proof but simply a method of presenting the information. He later, in working on his theory of general relativity "struggled" with using mathematical notation.

The fundamental question that I have for you is why do you consider a proof to be dependent on mathematical notation or any particular method of communication and representation? Cannot a proof be presented in purely graphical format?

Or even more significantly, isn't a proof really a real world, physical demonstration of the actual cause and effect?

Isn't mathematics, graphical representations, schematics, and common language nothing more then a method of communication of ideas?

 

[DSGE]

Well, given that it exists suggests that there has been considerable demand for it. And given that there has been considerable demand, then it clearly has utility for a large number of individuals.

I don't know about you, but my experience is that circular flow is covered once, briefly, at the beginning of an Introduction to Macroeconomics course. It just provides the intuition, for visual learners, as to why Y = C+I+G+X-M. After that it's never seen or heard from again.

Do I really need to know what utility another person gets from the product that they demand?

Does it have to be a proof to have utility? For that matter, what constitutes a proof?

Might we suppose that visual processing in the brain invokes distinctly different and important mental processes that are not available from the verbal or symbolic processes?

Faraday found no purpose for using mathematical notation and yet he proved the existence of and natural laws of the electro-dynamic forces. Apparently mathematical notation had no utility for him.

Einstein is considered to have done all of his thinking in a visual form, only committing it to a mathematical format after he had realized the answer. The math was, for him, not a proof but simply a method of presenting the information. He later, in working on his theory of general relativity "struggled" with using mathematical notation.

The fundamental question that I have for you is why do you consider a proof to be dependent on mathematical notation or any particular method of communication and representation? Cannot a proof be presented in purely graphical format?

Or even more significantly, isn't a proof really a real world, physical demonstration of the actual cause and effect?

Isn't mathematics, graphical representations, schematics, and common language nothing more then a method of communication of ideas?

Recall that I come from a pure maths background. Most of the proofs I've ever done have been in words in english sentences. But that's incidental, since I only brought up "proof" in regard to how venn diagrams don't constitute one, despite being a helpful visual aid.

So I don't care how it's done. Maths, no maths, words, graphs. If you disagree with me and think it's useful for anything other than the intuition behind Y = C+I+G, tell me about some interesting things I can learn from circular flow? What are some conclusions I can come to with it?

 

[itfitzme]

Well, given that it exists suggests that there has been considerable demand for it. And given that there has been considerable demand, then it clearly has utility for a large number of individuals.

I don't know about you, but my experience is that circular flow is covered once, briefly, at the beginning of an Introduction to Macroeconomics course. It just provides the intuition, for visual learners, as to why Y = C+I+G+X-M. After that it's never seen or heard from again.

Do I really need to know what utility another person gets from the product that they demand?

Does it have to be a proof to have utility? For that matter, what constitutes a proof?

Might we suppose that visual processing in the brain invokes distinctly different and important mental processes that are not available from the verbal or symbolic processes?

Faraday found no purpose for using mathematical notation and yet he proved the existence of and natural laws of the electro-dynamic forces. Apparently mathematical notation had no utility for him.

Einstein is considered to have done all of his thinking in a visual form, only committing it to a mathematical format after he had realized the answer. The math was, for him, not a proof but simply a method of presenting the information. He later, in working on his theory of general relativity "struggled" with using mathematical notation.

The fundamental question that I have for you is why do you consider a proof to be dependent on mathematical notation or any particular method of communication and representation? Cannot a proof be presented in purely graphical format?

Or even more significantly, isn't a proof really a real world, physical demonstration of the actual cause and effect?

Isn't mathematics, graphical representations, schematics, and common language nothing more then a method of communication of ideas?

Recall that I come from a pure maths background. Most of the proofs I've ever done have been in words in english sentences. But that's incidental, since I only brought up "proof" in regard to how venn diagrams don't constitute one, despite being a helpful visual aid.

So I don't care how it's done. Maths, no maths, words, graphs. If you disagree with me and think it's useful for anything other than the intuition behind Y = C+I+G, tell me about some interesting things I can learn from circular flow? What are some conclusions I can come to with it?

Well, me personally, having had my progress cut short by the recession, I find it very useful, if for no other reason as to maintain a sense of the foudation that I am trying to build from.

I think that, given any particular individual may have not ever seen it before, should they present it, it is useful. If they have never seen it, telling them it is useful is bit dismissive and suggests to them that it is useless in all manners of understanding and should be summarily ignored, even dismissed.

I have this feeling of wanting to defend their learning process and validate their need for what you now take as obviously apparent and otherwise meaningless.

I cannot so easily dismiss it, especially given someone that hasn't seen it and says, "it confuses me." Clearly, when understood, there isn't anything confusing about it. So the very fact that they should suggest confusion tells me that it is new to them.

I'm just saying.

 

[DSGE]

Well, given that it exists suggests that there has been considerable demand for it. And given that there has been considerable demand, then it clearly has utility for a large number of individuals.

I don't know about you, but my experience is that circular flow is covered once, briefly, at the beginning of an Introduction to Macroeconomics course. It just provides the intuition, for visual learners, as to why Y = C+I+G+X-M. After that it's never seen or heard from again.

Do I really need to know what utility another person gets from the product that they demand?

Does it have to be a proof to have utility? For that matter, what constitutes a proof?

Might we suppose that visual processing in the brain invokes distinctly different and important mental processes that are not available from the verbal or symbolic processes?

Faraday found no purpose for using mathematical notation and yet he proved the existence of and natural laws of the electro-dynamic forces. Apparently mathematical notation had no utility for him.

Einstein is considered to have done all of his thinking in a visual form, only committing it to a mathematical format after he had realized the answer. The math was, for him, not a proof but simply a method of presenting the information. He later, in working on his theory of general relativity "struggled" with using mathematical notation.

The fundamental question that I have for you is why do you consider a proof to be dependent on mathematical notation or any particular method of communication and representation? Cannot a proof be presented in purely graphical format?

Or even more significantly, isn't a proof really a real world, physical demonstration of the actual cause and effect?

Isn't mathematics, graphical representations, schematics, and common language nothing more then a method of communication of ideas?

Recall that I come from a pure maths background. Most of the proofs I've ever done have been in words in english sentences. But that's incidental, since I only brought up "proof" in regard to how venn diagrams don't constitute one, despite being a helpful visual aid.

So I don't care how it's done. Maths, no maths, words, graphs. If you disagree with me and think it's useful for anything other than the intuition behind Y = C+I+G, tell me about some interesting things I can learn from circular flow? What are some conclusions I can come to with it?

Well, me personally, having had my progress cut short by the recession, I find it very useful, if for no other reason as to maintain a sense of the foudation that I am trying to build from.

I think that, given any particular individual may have not ever seen it before, should they present it, it is useful. If they have never seen it, telling them it is useful is bit dismissive and suggests to them that it is useless in all manners of understanding and should be summarily ignored, even dismissed.

I have this feeling of wanting to defend their learning process and validate their need for what you now take as obviously apparent and otherwise meaningless.

I cannot so easily dismiss it, especially given someone that hasn't seen it and says, "it confuses me." Clearly, when understood, there isn't anything confusing about it. So the very fact that they should suggest confusion tells me that it is new to them.

I'm just saying.

Perhaps it's confusing because because somebody looks at it and sees a whole bunch of nodes and lines going everywhere. Maybe it's less confusing to know "all it says is that income equals expenditure"?

 

[Widdekind]

"starting from first principles" per DSGE...

An economy is composed of "economic entities" (individuals, corporations, Governments). Over the course of one "time period" (dt, e.g. quarter, year), account the "net profits (np)" of every "entity"; np is a rate, namely, the net amount of Money gained or lost by that "entity", during that period (dm), divided by the amount of time (dt) during which those "gains/losses" were accumulated. Thus, np = dm/dt is a Money-flow, measured in [$/time].

The sum of all np is the aggregate NP of the economy, as a whole (NP = sum(np)). If the Money-supply is constant, then NP=0; otherwise, NP=dM/dt - dR/dt, the rate of expansion/contraction of the Money-supply (largely determined by central banks) dM/dt, less the rate at which Money is sequestered into Reserves (largely by central banks) -dR/dt. For the following, i ignore other Money "sinks", e.g. physical destruction of currency, unaccounted Reserves.

Income - Outflow = (d/dt)[M-R]
(Y+Im) - (I+C+G+Ex) = (d/dt)[M-R]
Y = I+C+G + (Ex-Im) + (d/dt)[M-R]

d[M4 - domestic Reserves - foreign Reserves] (per year, foreign Reserves estimated from data for China)

Naively, when the "correction term" is positive, then US Gross Income was higher than reported, because of a build up, of net new Money, within the economy, which was received as Income, without being subsequently spent ("unspent income", "got it kept it"). Conversely, when the "correction term" was negative, then US Gross Income was lower than reported, because some expenditures (I+C+G + [Ex-Im]) were actually drained out of the economy, into Reserves, without returning into the "stream of spending", as somebody else's Income ("check lost in the mail", "spent it 'twas lost", "spent it stayed spent"). E.g. after 2008, massive "hoarding" of bailout Money, as "excess Reserves", by bailed-out banks, drained trillions of new dollars, straight out of the US economy; with new Money already sequestered, by domestic banks, further Reserves increases, by foreign banks, represented a net "dollar drain" from the (active) US economy.

As a fraction of GDP, the "correction term" computes to nearly 0.1 (10%) in some years:

Reportedly, USD are a "semi-global" currency, utilized in economic transactions, world-wide, e.g. two-thirds of the $1T of physical cash resides abroad. Private use of USD "beyond the US economy" would only enter the "correction term" as de facto "R", i.e. "foreigners are swapping dollars around inside central bank Reserve vaults". Subtraction of half of the "Currency" component generates graphs visually indistinguishable from the plots provided.

 

[DSGE]

An economy is composed of "economic entities" (individuals, corporations, Governments). Over the course of one "time period" (dt, e.g. quarter, year), account the "net profits (np)" of every "entity";

What are we defining "net profits" as?

The sum of all np is the aggregate NP of the economy, as a whole (NP = sum(np)). If the Money-supply is constant, then NP=0; otherwise, NP=dM/dt - dR/dt, the rate of expansion/contraction of the Money-supply

What?

Income - Outflow = (d/dt)[M-R]
(Y+Im) - (I+C+G+Ex) = (d/dt)[M-R]

Well that's obviously not true because Income necessarily equals expenditure. That's an identity. It always holds. That's the entire point of the circular flow diagram. Income = Expenditure. Not, income = expenditure + some other stuff. You've made a mistake somewhere (I can't quite tell because this is hard to follow), because you end up concluding that 0 = 1.

 

[Widdekind]

What are we defining "net profits" as?

net gain/loss of Money (dm) per time period (dt), i.e. np = dm/dt [$/time], the difference between Incomes (Revenues + Borrows) - Outflows (Expenses + Loans)

that's obviously not true because Income necessarily equals expenditure. That's an identity. It always holds.

what about "cash on hand" ?

Outflows can derive, from Inflows; or from draw-down of current Stock:

dS/dt = +I -O​

for an entire economy, Money "Inflows" by Money creation (anciently, by mining & minting, or by piracy & plunder) +dM/dt, and Money "Outflows" into Reserves (anciently, by being pillaged, paying tributes) -dR/dt:

Incomes - Outflows = +dM/dt -dR/dt = (d/dt)[M-R]
(Y+Im) - (C+I+G+Ex) = d/dt (M-R)​

E.g. imagine nobody buys anything (Expenditures = zero); everybody could still have Income, if new Money entered the economy ("victorious general showers the masses with Money from campaign"):

Y-0 = dM/dt​

or, imagine that everybody is spending allot, but then they get conquered, and lots of Money is extracted into (foreign) "Reserves"; then net Income could be zero

0 = CIGNX - dR/dt​

i.e. "there lots of spending, and foreigners wind up with most of it" ('twas spent & stayed spent). i suspect you are implicitly assuming, that nobody carries any cash-on-hand (in any macro-economically significant extent); such seems to be true, for everybody besides banks.