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Wednesday, March 14, 2018

The Curious Profit Accounting Of DSGE Models

One of the more puzzling aspects of neo-classical economic theory is the assertion that profits are zero in equilibrium under the conditions that are assumed for many models. One should re-interpret this statement as "excess profits" are zero, but there are still some awkward aspects to the treatment of profits in standard macro models. This article works through the theory of profits for an example dynamic stochastic general equilibrium (DSGE) model, and discusses the difficulties with the mathematical formulation.

The example is taken from Chapter 16 ("Optimal Taxation With Commitment") in the textbook Recursive Macroeconomic Theory, by Lars Ljungqvist and Thomas J. Sargent (I have the third edition). For brevity, the text will be abbreviated as [LS2012] herein. If the reader is mathematically trained and wishes to delve into DSGE models, this textbook is the best place to start. The mathematics is closer to the original optimal control theory that DSGE macro is based upon, whereas other treatments follow the mathematical standards of academic economics, the difficulties with which are discussed later in this article.


This article represents some initial notes for a proposed future academic article. The thesis of the article is straightforward: mainstream academic macroeconomics has evolved in a direction that results in a literature that is surprisingly opaque to outsiders. There is no argument that this is a deliberate strategy of obscurantism (although one could suggest that this is a revealed preference). I have discussed this project before, so I do not wish to dwell on this background. However, this article is meant to be blandly academic, although there are some punchier observations near the end.

Post-Keynesian authors have been complaining about the neo-classical treatment of capital for decades. This article makes no attempt to fit itself in with that literature. Having looked at [LS2012], I am not entirely happy with those treatments. The literary approach favoured by post-Keynesian academics is perhaps a hindrance in this case. It is hard to relate the post-Keynesian criticisms of capital in models -- often based on decades-old models that do not appear in the mainstream literature -- to modern DSGE models. In my view, it is possible to see problems with the mathematical formulation of capital and profits by just looking at the the modern models, and accepting most of the assumptions at face value.

Why are Pure Profits Zero?

Having just complained about literary critiques of DSGE macro, I will of course give a literary introduction to explain why pure profits are allegedly zero. I am deliberately doing this before I start referring to the equations in [LS2012], as I expect that I will lose many readers as soon as they reach the first equation. At the minimum, I want readers to take away an introduction to how profits are looked at in DSGE models.

There are a great many DSGE models proposed in the literature, and so generalisations are difficult. The model in Chapter 12 in [LS2012] is interesting because it actually works through the accounting of profits in some detail. Conversely, the business sector is not heavily developed in most representative agent macro DSGE models (which are the class of most interest to outsiders). For example, in Chapter 2 of Monetary Policy, Inflation, and the Business Cycle by Jordi Galí (2008), the business sector is covered by four short paragraphs, and four equations. The condition to maximise profits is the only result of interest; the level of profits is ignored. In particular, there is no discussion whatsoever what happens if profits are non-zero: where does the money go?

I would paraphrase the logic in [LS2012] as follows. (I want to underline that this is my phrasing, weaknesses in the formulation just reflect my choice of wording. However, the discussion in [LS2012] probably relies on the reader being familiar with the literature already.)

The model being discussed in Chapter 12 in [LS2012] is the Ramsey Problem. It is a three-sector model, with a government, household sector, and firms. The production function is simple, and depends upon capital and labour inputs. The way in which the model operates is that the business sector operates with no capital. It rents capital from the household sector each period, and pays a rate of interest on that capital. This rental cost is subtracted from the profit equation, and is a source of income to the household sector. If we wanted to map this to the real world, we would realise that this rental cost probably corresponds to dividend payments, which are not an expense from an accounting perspective. The term pure profits refers to profits less this rental expense (or dividends). When a mainstream model suggests that pure profits are zero, what it really means is that all profits are immediately paid out as dividends.

This assumption greatly simplifies the model. Each time period, businesses essentially pop out of nothingness, borrow capital from the household sector, operate their business to maximise the one-period profits, disburse all cash flows to the household sector, and then disappear. There is no need for an inter-temporal maximisation problem, since the firm effectively disappears at the end of the time period. We are left with only the household sector trying to maximise its utility, subject to the policy rules of the government.

This obviously makes no sense from a real world perspective; no one in their right mind is going to lend to firms that have no equity. One may also object to how the questions of distribution neatly disappear from the framework: since we assume that individual households are homogeneous, they all have equal capital holdings, and so there is no conflict between capital and labour. Conversely, if some households had a monopoly on capital holdings, there would be an obvious conflict of interest between them and households that can only supply labour.

In any event, there are a number of implications that follow from this structure.
  • As noted earlier, the business sector has no inter-temporal maximisation; it just reacts almost mindlessly to marginal productivity and the cost of capital in each period.
  • With business sector financial asset holdings assumed to be zero for all time, the government's fiscal balance is the mirror of the household sector's. This explains the belief in the inter-temporal government budget constraint.
  • The equilibrium assumption implies that there is a simple relationship between the one-period Treasury bill interest rate and realised profits (equation 16.2.12 in [LS2012]; the rate of interest is the after-tax profit rate plus one minus the depreciation rate). Such an assumption appears highly problematic in real-world financial analysis, even if we include an equity risk premium.

Model Discussion

We will now turn to the parts of the model in Chapter 12 of [LS2012] that matter for profit accounting. (Equation numbers from that text are given as 16.x.y.) It should be noted that this model is deterministic, which simplifies the mathematics, but it may not conform to intuition regarding real world firm behaviour, where uncertainty exists.

There is a single good produced in the economy, and one unit of this good can be converted to one unit of capital, denoted $k(t)$. The number of hours worked is denoted $n(t)$. (I prefer to denote time dependence by using parentheses rather than subscripts as in the text, since there are some otherwise subscripted variables.) Furthermore, we need to introduce a variable that denotes the amount of capital that is borrowed at any time period: $k_b(t)$. This variable does not appear in [LS2012]; those authors appear to assume that $k(t)=k_b(t),$ without justifying this assumption. We assume that $0 \leq k_b(t) \leq k(t)$, based on physical arguments. A negative amount of physical capital used in production makes little sense ($k_b(t) < 0$), and it is impossible to borrow more capital than exists ($k_b > k(t)$).

The first component is the production function $F$ We are assuming constant returns to scale, which means that production function has the form (16.2.4):
F(t, k_b, n) = F_k(t) k_b(t) + F_n(t) n(t).
We define household consumption as $c(t)$ and government consumption as $g(t)$. The accounting for real output is given by (16.2.3):
c(t) + g(t) + k(t+1) = F(t, k_b, n) + (1-\delta)k(t),
where $\delta \in (0,1)$ and is the rate of capital depreciation. (Note that this equation implies that one could literally consume capital.) Note that this equation features both borrowed capital ($k_b$) and the total level of capital ($k$).

The accounting in the model is done in terms of the price of the good at time $t$, and not in the currency unit. At the time of writing, I am unsure about some of the implications; in particular, the real value of inherited financial balances depends upon the price level. (Going forward, if we express future financial balances in real terms there is no problem if we use real rates instead of nominal rates.)

The firm's pure profit ($\Pi$) in real terms is given by (16.2.17):
\Pi(t) = F(t, k_b, n) - r(t) k_b(t) - w(t) n(t),
where $w$ is the real wage, and $r$ is the rental cost of capital. Importantly, there is a buried assumption that all output is sold (market clearing assumption).

The authors then assert that the first order conditions of the firm's problem are given by (16.2.18):
r(t) = F_k(t),
w(t) = F_n(t).
These conditions are presumably arrived at by differentiating the expression for $\Pi$ by $k_b$ and $n$ respectively. The interpretation:
In words, inputs should be employed until the marginal product of the last unit is equal to its rental price. [LS2012, p. 619].
Such assertions to find first order conditions are common in the literature. In this case, the linearity of the function with respect to $k_b$ and $r$ does make this operation relatively safe, but in general, we need to look at the constraints on the variables. The maximum that is implied by taking the derivative of the objective function might lie outside the feasible set of solutions. Given the complex relationships that exist between the variables in these models, one should properly be examining all pertinent constraints that exist.

Under the assumption that the first order conditions indeed hold, there are a number of implications. The first is that $\Pi$ equals zero, since all the terms cancel out. The next is that this happens for any feasible choice of $k_b$ or $n$. If we put labour hours to the side, we see that any choice of $k_b(t)$ in the interval $[0, k(t)]$ is optimal. Since firms are not making any money, there is no need to employ any available capital. The authors' decision to assume that $k_b(t)$ always equals $k(t)$ by the expedient tactic of replacing $k_b(t)$ by $k(t)$ in the equations represents an end run around the indeterminacy of the optimisation problem.

This suppression of the variable $k_b$ is arguably inexcusable. If we assume that firms always borrow all capital ($k_b(t) = k(t)$) it drops entirely from the optimisation problem. The value of $k(t)$ is inherited from the previous time period (or the problem initial conditions), and so it is a constant at time $t$. It makes no sense to differentiate with respect to a constant, and so the "first order condition" $r(t) = F_k(t)$ has no mathematical validity.

It might be possible to come up with some reasoning why $k_b$ has to equal $k$ for all time. Intuitively, if there is capital that was not rented, the households owning that capital would end up in a suboptimal position, and so they would offer their unrented capital at a lower rate. This presumably would break some assumption about the nature of equilibrium. However, the mathematical reasoning behind that logic is completely ignored within the description of the solution of the problem. In other words, readers are supposed to use their imagination to fill in missing pieces of the proof. Such faith-based logic is not a normal feature of mathematical publications.

The cancellation of all the terms in the profit equation eliminates more interesting possibilities from consideration. I will quickly sketch out what happens with a nonlinear production function. We will have it ignore labour hours, and instead have the form:
\tilde{F}(k_b(t)) = \sqrt{k_b(t)}.
We can then invoke the "first order condition" assertion, and find that $\tilde{\Pi}$ is maximised when:
\frac{1}{2 \sqrt{k_b(t)}} = r(t),
or (assuming $k_b(t) > 0$, $r(t) > 0$):
k_b(t) = \frac{1}{(2 r(t))^2}.
If we set $r(t) = 1.1$, $k_b(t) = 0.2066$, and the maximum pure profit equals 0.2272 (numbers rounded). This non-zero pure profit then causes difficulties for the DSGE model accounting: where does it go? If it is retained within the firm, the firm has non-zero capital available at the beginning of the next period, and it properly should start examining an inter-temporal optimisation problem. However, that is not an issue for the given model, which we return to.

We can then ask ourselves: what happens if the cost of renting capital differs from $F_k$? If the rental cost was greater than $F_k$, the amount of capital rented would equal zero, which results in a not particularly interesting solution (only labour would be used in the production process). If the rental cost was less than $F_k$, we would end up with the business sector wanting to borrow an infinite amount of capital. Since the amount of capital at the beginning of the period is fixed, this could be dealt with by allocating capital to firms on a lottery basis. However, once again, one could presumably invoke "equilibrium" to eliminate such possible solutions. Once again, there is no mathematical discussion within [SL2012] why such an outcome does not meet the definition of "equilibrium."

However, one could note that the entire logic of renting capital is based on a rather questionable premise: that the price of capital itself is fixed. One way of clearing the market for capital if the rental rate is below the marginal productive value is for the price of capital itself to rise. Admittedly, there is a question of logical coherence of such an assumption in a single goods world, but it is clear that a model that implies that households could start eating railway tracks is not a good fit to reality. All we need to do is to make the more plausible assumption that capital cannot revert to being a consumer good, and we can coherently allow for the price of beginning-of-period capital to diverge from the end-of-period goods price. Such an framework restores freedom to set the policy rate away from the (after-tax) marginal productive capacity of capital.

Finally, under the assumption of a zero pure profit, financial asset holdings should have no effect on household behaviour if the household sector were truly optimising its utility function. No matter what relative prices are, or the level of output, the household sector will receive 100% of the revenue of the business sector. The change in the holdings of financial assets will just be the flip side of the government's budget deficit. Since financial asset holdings cannot be affected by consumption decisions, the optimal solution will always be to choose the real activity values for $n(t)$ and $c(t)$ to optimise the utility function (16.2.1). However, it appears that DSGE models are to be interpreted as each sector solving an optimisation independently, and then these "optimal" strategies are then force-fitted into a single model. It is unclear whether this actually qualifies as a true mathematical optimisation problem, as what we are seeing is what happens when components of the model are following heuristics relative to the true overall mathematical system. That is, if the household sector acts in a way that suggests its financial asset balances matter, it is following a sub-optimal heuristic. Why is this particular heuristic privileged versus other potential heuristics?

Concluding Remarks

This model discussion provides a useful example of how mathematical details are buried within the DSGE literature for even what appear to be extremely simple models. In the absence of such details, it is very difficult to see whether mathematical operations within proofs are indeed legitimate.

Furthermore, we also see that the business sector is typically highly undeveloped in these macro models; only the household sector (and government) undertakes inter-temporal optimisation. The sole role of business sector optimisation is to enforce relationships based on the marginal productivity of input factors in the production function.

(c) Brian Romanchuk 2018


  1. DSGE and profit ― forget it! MMT and profit ― forget it!
    Comment on Brian Romanchuk on ‘The Curious Profit Accounting Of DSGE Models’

    Everybody knows that DSGE as the actual version of the microfoundations approach is dead. From this follows that a paradigm shift is needed: “There is another alternative: to formulate a completely new research program and conceptual approach. As we have seen, this is often spoken of, but there is still no indication of what it might mean.” (Ingrao)

    One of the most conspicuous blunders of DSGE is the profit theory. Brian Romanchuk observes: “One of the more puzzling aspects of neo-classical economic theory is the assertion that profits are zero in equilibrium under the conditions that are assumed for many models. One should re-interpret this statement as ‘excess profits’ are zero, but there are still some awkward aspects to the treatment of profits in standard macro models.”

    Indeed, there is something deeply wrong with the profit theory since Adam Smith. Conventional ‘wisdom’ asserts: “The consensus to date has been that it is mathematically impossible for capitalists in the aggregate to make profits.” (Keen) And: “But, from the macro perspective of Walrasian general equilibrium, the total profits in this case cannot be other than zero (otherwise, we would need a Santa Claus to provide the aggregated positive profit) but this does not preclude the possibility of short-run profits and losses of individual firms canceling each other out.” (Boland)#1

    The curious thing is that macroeconomic profit has been greater than zero for most of the time in most of the known market economies up to the present. This is an empirical fact. Obviously, there is something wrong with conventional profit theory and Brian Romanchuk is not the first to notice it. As the Palgrave Dictionary puts it: “A satisfactory theory of profits is still elusive.” (Desai)

    This, indeed, is the most damning verdict about economics which claims to be a science: after 200+ years, economists still cannot tell what the pivotal magnitude of their subject matter ― profit ― is. This does not only apply to DSGE but to the four main approaches: Walrasianism, Keynesianism, Marxianism, Austrianism are mutually contradictory, axiomatically false, materially/formally inconsistent and all got the pivotal economic concept profit wrong.

    While Brian Romanchuk notes that DSGE profit theory must be false he passes over the fact that MMT, the approach he pushes, is not one iota better.

    To make matters short, the axiomatically correct relationships are given here without further explanation.#2 It holds, with Qm monetary profit/loss, Sm monetary saving/dissaving, I investment expenditures, G government spending, T taxes, X export, M import, Yd distributed profit:

    (i) Qm=−Sm in the elementary production-consumption economy,
    (ii) Qm=I−Sm in the elementary investment economy,
    (iii) Qm=(G−T)+(I−Sm) in the investment economy with government deficit/surplus,
    (iv) Qm=Yd+(X−M)+(G−T)+(I−Sm) in the open economy with distributed profit.

    From (iii) follows that ― given business sector investment I and household sector monetary saving Sm ― Public Deficit = Private Profit. This tells one that the MMT policy of deficit spending/money creation benefits alone the one-percenters.

    It never follows the MMT tripartite balances equation (I−S)+(G−T)+(X−M)=0 (v). The comparison with the axiomatically correct Profit Law (iv) makes it clear that MMT ― just like DSGE ― deals with a zero profit economy, i.e. Qm,Yd =0.

    Because the foundational balances equation of MMT (v) is provably false the whole of MMT is worthless, just like DSGE.

    Egmont Kakarot-Handtke

    #1 Debunking Squared

    #2 MMT is idiocy and fraud

    #3 Macro for retarded economists

    1. Oh yes. I thought Nick Rowe had covered this profit accounting thing?

    2. Egmont's schema is given by his definition of income. Revert to the normal definition of income and his schema reduces to the Keynesian formulation.

  2. Brian Romanchuk, Anonymous

    Take notice that what you call “normal definition of income” is one of the worst methodological idiocies of economics. Needless to emphasize that the “normal economist” in his incurable scientific incompetence does not realize it.

    Here, for intelligent non-economists, the Humpty Dumpty Fallacy in full detail.

    In the elementary investment economy, macroeconomic profit Q is defined as the sum of profit in the consumption goods industry, i.e. Qc≡C−Ywc, and the investment goods industry, i.e. Qi≡I−Ywi, that is, Q≡(C−Ywc)+(I−Ywi) or Q≡C+I−Yw (i). Profit Q is greater than zero if the value of output C+I is greater than total wage income Yw.

    Now, Humpty Dumpty introduces a redundant definition by saying that profit may be called “income of the business sector” and that this “income” can be added up with the wage income of the household sector to “total income” Ψ thus

    (a) Ψ≡Q+Yw  and now (i) is rewritten
    (b) Q+Yw ≡C+I and then, hey presto,
    (c) Ψ≡C+I that is, “total income” is “by definition” identical to “value of output” or in the usual sloppy parlance “income = value of output” which obviously contradicts (i) and ― strangely enough ― makes profit disappear.

    This definitional idiocy can be traced back to Keynes “Income = value of output = consumption + investment. Saving = income − consumption. Therefore saving = investment.” (GT, p. 63)

    Take notice that “income” is NEVER equal to “value of output” and by implication that “saving” is NEVER equal to “investment” because profit is NOT “income”.

    In accounting terms, wage income Yw is a flow from the business to the household sector and consumption expenditures C is a flow in the opposite direction and profit is the difference of the two flows Q≡C−Yw. To add a flow and a balance together is a category mistake. No accountant worth his salt would ever do it but economists are Humpty Dumpties who do not even understand the elementary mathematics that underlies accounting.#1, #2

    The analogous flow to wage income Yw is distributed profit Yd. It is methodologically CORRECT to add the two flows Yw and Yd together to total income but it is INCORRECT to add the flow Yw and the balance Q together.#3

    All this is way above the head of the “normal economist” who misspecifies the foundational economic concepts profit/income/saving/distributed profit from Adam Smith onward to DSGE and MMT.

    To argue that the “normal economist” treats profit since 200+ years without any qualms as “income” is to confirm that the “normal economist” is an incurable idiot, and that, in turn, explains the indisputable fact that economics is a failed/fake science.

    Egmont Kakarot-Handtke

    #1 A tale of three accountants

    #2 The Common Error of Common Sense: An Essential Rectification of the Accounting Approach

    #3 How Keynes got macro wrong and Allais got it right

    1. Whatever.

      As a note to anyone else reading this:

      1) Egmont Kakarot-Handtke (hereafter “EKH”) defines “profit” one way.
      2) Nobody else uses that definition. Once we get to the real world, there are technical disputes about the precise definition of “profits,” so it is incorrect to say that everyone agrees.that said, there is general agreement on how to define profits in simpler cases (such as in a mathematical model).

      As someone with applied mathematics training, I would note that researchers are free to use their own definitions of terms. (One may note how physics has a technical definition of “force” versus usages like “the Visigoths attacked with overwhelming force.”) So long as you remain consistent to your own definition, you’re OK.

      This is heretical to EKH, who insists that there is only one definition of profit (his). He then ignores explanations as to how English and mathematics works in practice.

      And that’s my last comment on this thread.

  3. Brian Romanchuk

    “There are always many different opinions and conventions concerning any one problem or subject-matter …. This shows that they are not all true. For if they conflict, then at best only one of them can be true. Thus it appears that Parmenides … was the first to distinguish clearly between truth or reality on the one hand, and convention or conventional opinion (hearsay, plausible myth) on the other.” (Popper)

    This exactly is the task of the scientist: to figure out which of the conflicting ‘opinions and conventions’ is true. Economists have badly failed at this task.

    Keynes is a case in point. He was entirely clueless: “His Collected Writings show that he wrestled to solve the Profit Puzzle up till the semi-final versions of his GT but in the end he gave up and discarded the draft chapter dealing with it.” (Tómasson et al.). And: “Keynes related his definition of income expressly to ‘the practices of the Income Tax Commissioners.’ He was in grave doubt whether ‘it might be better to employ the term windfalls for what I call profits.’ But he was quite sure that ‘saving and investment are, necessarily and by definition, equal ― which after all, is in full harmony with common sense and the common usage of the world.’” (Coates)#1

    After-Keynesians are no better: Kalecki defined profit as P=Cp+I, Minsky as P=I, and Keen applies the commonsensical but provably false Humpty Dumpty definition total income = wages plus profits.#2

    And so it goes on. Ricardo’s profit theory is false,#3 same with Marx,#4 same with MMT. Your assertion “there is general agreement on how to define profits in simpler cases (such as in a mathematical model)” is laughable.

    Nothing shows better the scientific incompetence of economists than the fact that every half-wit applies his own confused definition of profit.

    Physics has one definition of energy and this magnitude is an element of a consistent set of foundational magnitudes. Economics has a wild variety of inconsistent profit definitions. And this is why economists never get above the level of confused blather.

    The MMT balances equation reads (I−S)+(G−T)+(X−M)=0, the AXEC balances equation reads (I−S)+(G−T)+(X−M)−(Q−Yd)=0. Only one equation can be true. As someone with applied mathematics training you can certainly spontaneously tell which one.#5

    Egmont Kakarot-Handtke

    #1 Marshall and the Cambridge school of plain economic gibberish

    #2 Heterodoxy, too, is scientific junk

    #3 Ricardo, too, got profit theory wrong. Sad!

    #4 Profit for Marxists

    #5 Rectification of MMT macro accounting

  4. Eq. 16.2.3 moves a little fast for my taste. It introduces three sectors, one of which is capital.

    My thinking is that consumption by both households and government consumes capital in the sense that capital is part of the production function. This leaves open the question of how capital is produced.

    I like to think of three kinds of capital: financial, tools, and purchased production inputs. Purchased production inputs would be things like fertilizer (for farmers) and parts (for automakers). Tools would be productive property like factories and wrenches.

    To stay within these divisions, I would avoid combining household and government consumption in the method of 16.2.3.

    While government may contribute financial capital (with acts of money creation), households (maybe acting as cooperative assembles) are required to create factories and tools. The creation of factories and tools is a productive effort in itself, with ultimate consumption of the productive object occurring eventually.

    I am interested in acquiring the reference material but it is certainly not free! The biggest problem for me is that Amazon does not provide the version information. I see that version 4 is now available so maybe I should go new, ensuring the version 4 copy.

    Thanks for the post and inspiration.

    1. It’s literally a single good economy so there can only be one kind of capital. Output has to be allocated to its various uses (consumption, capital), and so all sectors have to show up in the allocation equation, in order for the accoumting to add up.

  5. Hmmm. It seems to me that labor is also capital--a renewable resource but transformed into 'total capital' only to the extent that it is productively utilized.

    Of course, if labor is 'capital', then are borrowed capital and total capital still a single kind of good?

    In my own mind, all the inputs and outputs are translated to currency equivalents. Following that conversion, 'capital' becomes the common element that allows us to unite the various economic components.

    I am still trying to wrap my mind around the definitions.

    Would edition 4 correlate with your work or do you think the changes would be confusing?

    1. Could we think of seeds for the growing of wheat as 'borrowed capital'?

      Hmmm. I'm puzzling why we would have a depreciation term if that analog was used.

    2. Roger Sparks

      DSGE is the most recent actualization of the microfoundations approach which had been kicked-off +140 years ago by Jevons/Walras/Menger. DSGE is based on the neo-Walrasian axiom set: “HC1 economic agents have preferences over outcomes; HC2 agents individually optimize subject to constraints; HC3 agent choice is manifest in interrelated markets; HC4 agents have full relevant knowledge; HC5 observable outcomes are coordinated, and must be discussed with reference to equilibrium states.” (Weintraub)

      The representative economist has not realized it but methodologically these premises are forever unacceptable. It should be pretty obvious that the neo-Walrasian hard core contains THREE NONENTITIES: (i) constrained optimization (HC2), (ii) rational expectations (HC4), (iii) equilibrium (HC5).

      Methodologically, the microfoundations approach has already been dead in the cradle. It was Keynes who realized this and tried to move to macrofoundations. However, in his bottomless incompetence Keynes messed the paradigm shift up. This is why the proto-scientific maximization-and-equilibrium rubbish is still around.

      But Neoclassicals did not only get the axiomatic foundations of economics wrong but also the mathematics. The proof has been given by the mathematician Jonathan Barzilai.#1

      Whoever discusses in our days a DSGE model proves that he has no grasp whatsoever of science and mathematics. DSGE is the economic analogue of the Flat Earth Theory and considered worthy of discussion only by some simpletons.

      Egmont Kakarot-Handtke

      #1 An Open Letter to the President of the American Economic Association and An Open Letter to the President of the Canadian Economics Association

    3. Roger -
      I don’t think the 4th edition much will be different, although they may have added material. I doubt that they will have addressed my concerns...

      Converting to dollar equivalents does not work. You need gasoline to power your car; you cannot fill it up with dollar bills and have it run. A single good economy is obviously simplistic, but it does reflect the reality that we need to distinguish financial assets from real ones.

    4. Roger Sparks

      Congratulations! It seems that you have reanimated Brian Romanchuk’s defunct brain cells. And what revolutionary insights they have produced in the shortest time:

      • “You need gasoline to power your car; you cannot fill it up with dollar bills and have it run.”
      • “… we need to distinguish financial assets from real ones.”

      Who has ever thought such bold thoughts? I am looking forward to the continuation of this mind-boggling dialogue of imbeciles.

      Egmont Kakarot-Handtke

    5. Egmont - there’s plenty of more insights like that to come!

  6. After much additional thought, I am ready to venture another comment.

    As you describe the model in Chapter 12, it seems not to be a three sector model but a four sector model. The fourth sector is the capital sector.

    In Eq. 16.2.3, we find the capital sector described twice (as k(t+1) and as k(t). This would be consistent with capital measured at both the beginning and end of the period. It is also consistent with your suggestion that capital may carry over between periods.

    We can also consider equation 16.2.3 in terms of ownership. Household, government, and the productive sector all have ownership of some part of the productive stream at some time during the measured interval. The capital sector has claim to the term kb resulting from restricted renting of capital to the other sectors during the year. The productive decision made by capital owners is a yes-or-no decision about renting capital (made once each measured period).

    You will ask who owns the capital sector? Ownership would be distributed between the three other sectors. I think our real world experience would agree with that possibility.

    If we adopted my suggestion of four sectors, we would want to change the limits of depreciation to be -1,1 to allow appreciation of capital.

    You are certainly not supportive of DSGE models. I am thinking that investing in the book may not be worth the cost. (Another example of trying to maximize utility.)

  7. Brian Romanchuk, Roger Sparks

    Brian Romanchuk characterizes the DSGE profit equation: “The firm’s pure profit (Π) in real terms is given by (16.2.17): Π(t)=F(t,kb,n)−r(t)kb(t)−w(t)n(t), where w is the real wage, and r is the rental cost of capital.”

    Because both of you have never understood what profit is you do not know that there is NO such thing as “profit in real terms”. Take notice that profit is a feature of the monetary economy and that it cannot be captured by a real model.

    In order to see this, one has to go back to the most elementary economic configuration, that is, the pure production-consumption economy which consists of the household and the business sector.#1

    In this elementary economy, three configurations are logically possible: (i) consumption expenditures are equal to wage income C=Yw, (ii) C is less than Yw, (iii) C is greater than Yw.

    • In case (i) the monetary saving of the household sector Sm≡Yw−C is zero and the monetary profit of the business sector Qm≡C−Yw, too, is zero. The product market is cleared, i.e. X=O in all three cases.
    • In case (ii) monetary saving Sm is positive and the business sector makes a loss, i.e. Qm is negative. The market clearing price is lower than in (i).
    • In case (iii) monetary saving Sm is negative, i.e. the household sector dissaves, and the business sector makes a profit, i.e. Qm is positive. The market clearing price is higher than in (i).

    It always holds Qm+Sm=0 or Qm=−Sm, in other words, at the heart of the monetary economy is an identity: the business sector’s surplus = profit equals the household sector’s deficit = dissaving. And vice versa, the business sector’s deficit = loss equals the household sector’s surplus = saving. This is the most elementary form of the macroeconomic Profit Law.

    Profit is a purely nominal magnitude: NO share of output O corresponds to it. Under the condition of market clearing, output always goes in full to the wage income receivers. The correspondence of profit is an increase of money in the business sector’s cashbox, as every economist could know from Marx’s famous formula M-C-M’.

    That no share of output corresponds to profit is logically obvious because the correspondence of loss would be a negative share of output and that is a nonentity.

    There is NO such thing as a “real” profit. Profit is a nominal variable and the counterpart of dissaving. The DSGE concept of profit is as brain-dead as can be and one has to be a brain-dead economist to take it seriously for more than one second.#2, #3

    Egmont Kakarot-Handtke

    #1 The elementary production-consumption economy is given by three macro axioms: (A1) Yw=WL wage income Yw is equal to wage rate W times working hours. L, (A2) O=RL output O is equal to productivity R times working hours L, (A3) C=PX consumption expenditure C is equal to price P times quantity bought/sold X.

    #2 The Profit Theory is False Since Adam Smith

    #3 How the intelligent non-economist can refute every economist hands down

    1. Egmont, I’m not a fan of DSGE modelling, but if I am going to discuss them, I need to discuss what the models are. I’d be happy to ignore them, but the mainstream of the profession does not agree with me.

      In this case, all that it done is divide through by the price level. As long as prices are non-zero, that is a legitimate mathematical operation. How you interpret the equation is anothr issue, but not in the realm of mathematics.

    2. Egmont,

      To me, profit is real and physical. Let me explain how I come to that conclusion.

      Several steps are required.

      First, money itself is real and physical. Money becomes a physical scale (like a scale measured in feet or meters).

      Next, we measure productive inputs against our money scale. Labor becomes a number. Parts become a number. Interest charges become a number. You can see the series of production components that is being developed.

      Because money is viewed as a physical asset, and then borrowed in the DSGE 16.2.3 equation that Brian uses, it becomes logical to call 'capital' (which is another name for 'money') a sector of it's own. This is the basis if my previous comment to Brian.

      Now turn to profits. When something is built with a goal for future resale, the translation from materials to production to ultimate resale is extremely uncertain. The best we can do is to make an estimate and then translate that estimate into money terms.

      As an estimator, one can estimate either to low or too high. Of the two choices, too high is better because of the physical reality of money. The estimator hopes that the actual final recovery of physical money (which follows actual sales and conversion of product back to money) results in a surplus greater than the original cost of production. This surplus (of physical money) has name--PROFIT.

      It is not clear to me that DSGE modeling treats the production and capital interactions correctly. I do appreciate that Brian is trying to bridge the gaps.

      Unfortunately, I don't think that Brian agrees with me that money is a physical reality. You don't agree that profit exists.

      I think the logical gaps between our three positions are very wide.

    3. Brian Romanchuk

      You say: “In this case, all that it done is divide through by the price level. As long as prices are non-zero, that is a legitimate mathematical operation.”

      This is simply NOT the case. Nobody, in fact, divides anything by the price level. The DSGE profit equation reads Π(t)=F(t,kb,n)−r(t)kb(t)−w(t)n(t) where F(.) is the “real” production function, where w is the “real” wage, and r is the “real” rental cost of capital. There is NO nominal variable in the equation and NO division through P.

      The “real” DSGE profit formula is pure methodological BS and only good for the demonstration of the galactic dimension of economists’s scientific incompetence.#1 Because you cannot even read the profit equation you do not realize that is all in real variables while profit is a nominal variable that has to be determined by National Accounting.

      Nobody in his right mind applies a Cobb-Douglas production function to determine profit.

      Egmont Kakarot-Handtke

      #1 The futile attempt to recycle Sraffa

    4. (1) Dude! I am not a fan of DSGE macro. However, I am trying to avoid making wild assertions. Writing that "Egmont Kakarot-Handtke thinks this is totally bogus!" is not enough to prove a logical point.
      (2) I noted that I had reservations with the accounting of the profit equation. However, I was sticking to what was demonstrably logically inconsistent within the equations, without asserting opinions about accounting *should* be done.
      (3) The profit equation includes the production function since there is a buried assumption about market clearing (which may or may not have been specified in an earlier chapter of the book). That is, they have done the replacement (business sales) -> (output of the production function). Mathematically, it could be legitimate, except that it probably breaks the way that they are writing the optimisation problem. In order for their other manipulations to be valid, they likely need to keep the two variables (production, sales) distinct, and then somehow deal with the possibility that they can be different. This is very much equivalent to the way that they identified total capital and borrowed capital, which messes up their mathematics. Although interesting, I cannot hope to cover every questionable mathematical manipulation they did within a single article. This is going to be at least a three-part series.

    5. Brian Romanchuk

      To take DSGE seriously for longer than one minute and to wonder whether market clearing is assumed (of course, this defines equilibrium in GE) is a sure indicator for substandard intellectual performance.

      Egmont Kakarot-Handtke

  8. Brian,

    I am still thinking about "In Eq. 16.2.3, we find the capital sector described twice (as k(t+1) and as k(t)"

    In Eq. 16.2.3, we find the relationship (1-dk(t)). What is the significance of this relationship?

    1. Roger Sparks

      You say: “I think the logical gaps between our three positions are very wide.”

      Then, we have to determine in earnest who is right and who is wrong. As Popper said, if statements are contradicting, this shows that they are not all true.

      You do not even understand the problem of macroeconomic profit. Marx did: “How can they continually draw 600 p. st. out of circulation, when they continually throw only 500 p. st. into it? From nothing comes nothing. The capitalist class as a whole cannot draw out of circulation what was not previously in it.”

      This led to the conclusion: “The consensus to date has been that it is mathematically impossible for capitalists in the aggregate to make profits.” (Keen) And this zero-profit conclusion is obviously NOT in accordance with the empirical evidence since 200+ years.

      Exactly at this point resides the logical gap = black hole of economists in general and you and Brian Romanchuk in particular.

      The mathematical solution of Marx’s problem of the very existence of macroeconomic profit reads Q=−S, that is, the business sector as a whole can only draw more out of circulation if the household sector throws more into it, in plain English, profit = dissaving.

      This is the core of the life-formula of the economic system we happen to live in. Needless to emphasize that neither you, nor Brian Romanchuk, nor Walrasians, nor Keynesians, nor Marxians, nor Austrians ever understood it.

      Egmont Kakarot-Handtke

  9. Egmont,

    "profit = dissaving"

    Like my grandma used to say "you can't take your money to heaven."

    Profits do you no good unless they can be used to improve your life and (hopefully) the life of others.

    As you intimate, if one sector captures all the capital, no matter what the means of capture, then that sector controls the macro=-economy. That observation remains true whether the controlling sector is household, government or productive.

    With the importance of capital in mind, I think we need to recognize capital owners as a distinct sector. The beauty of the capitalist system is that the owners of capital become widely distributed between the household and productive sectors. The role of government in capitalism is hotly debated and certainly plays a role in the formation of capital and in the ownership allocation of capital.

    Let's not freeze our thinking by looking at profits as a destructive weapon with fatal consequences for the macro-economy. Instead, think of profits as gains awaiting redistribution. If in capitalism we can have gains from profits widely distributed within the household sector, then broad sharing of the fruits of labor would have been achieved.

  10. Roger Sparks

    You repeat the core of the axiomatically correct Profit Law, profit = dissaving, and then you go on saying “Profits do you no good unless they can be used to improve your life and (hopefully) the life of others.”

    You obviously do not grasp the implication of what I called the “life-formula of the economic system we happen to live in”. Since profit is the very condition of the functioning of the economy, the life-formula tells you also when the economy we happen to live in will break down.

    This is something that neither DSGE nor MMT nor any other approach will tell you because economists do not know what profit is since economics was established as a proto-science by the silly blatherer Adam Smith.#1

    The market economy breaks down as soon as macroeconomic profit turns into loss and this is, in the most elementary case, when dissaving stops. The macroeconomic Profit Law for the general case reads Qm=Yd+(X−M)+(G−T)+(I−Sm) and you can figure out for yourself the conditions that turn profit eventually into loss.

    If you do not understand the life-and-death formula of the monetary economy there is no need to stop emanating blatant nonsense ― you can still have an absolutely senseless conversation about the crappy DSGE profit formula with Brian Romanchuk or, what amounts to the same, channel your grandma.

    Egmont Kakarot-Handtke

    #1 Mathematical Proof of the Breakdown of Capitalism

  11. If all economists and politicians were legally obligated to define terms such as profit* 'one' way, and one way only, then all debates would cease. But because such a law is ridiculous, the next best thing is for everyone to try and make a concerted effort to define words and meanings, as the law itself defines them. With this, we can then agree that when it comes time to protect or enforce what we so dearly cling to at least we can explain it to the judge what it is we are enforcing.

    *see Economics as a Philosophical Science by Collingwood

    In this sense, there is a stark difference between flows and stocks. One cannot legally (or physically) transfer a flow, but one can transfer the right or claim to a flow, which is the stock. A surplus, in the physical and nominal sense, implies an increase, and the increase is an increase in claims.

    Profit, is an increase in claims. Profit is property.

    In the real sphere, an increase in claims is an increase in ownership of real resources, which decreases everyone elses claims - put in legal parlance means, an increase in rights as against the rest of the world. If one man owns all the resources in the world, there is nothing left to profit from - there are no more resources to accumulate.

    In the monetary sphere, profits are an increase in claims against particular entities in the form of bank deposits, stocks, bonds, currencies etc.

    In order for financial wealth to exist, there must be an equal number (>0) of claims as there are obligations/liabilities/duties (however you wish to define it).

    Profit is an increase in these legal claims. In reality there is no limit to legal claims in the monetary sphere, provided people are willing to extend obligations out into infinity.

    From a micro-perspective, one mans profits (increase in legal claims) can simply mean a redistribution of legal claims, meaning, someone somewhere else has lost some.

    From a macro-perspective however, redistribution is nonsensical because the question must always be asked, what enabled the very first profit to exist, which translated means, what created the first ever debt?

    As the Bank of England pointed out in 2014 'Introduction to Money', if all debts in society were paid off, there would be no money, just 'stuff'. If monetary profits cannot exist unless money exists, this demonstrates without any doubt, profits = dissaving.

    1. ANC Driver

      To sum up. In their DSGE textbook Recursive Macroeconomic Theory, Lars Ljungqvist and Thomas J. Sargent define macroeconomic profit in real terms. This is methodologically as idiotic as one can get because profit is a nominal variable. The DSGE model suffers from a dimensional inconsistency. Alone for this reason, it is scientifically worthless.

      Neither DSGEers themselves nor their critics have realized that the whole approach is proto-scientific rubbish. On second thought, however, this is not really astounding because the representative economist swallows garbage like utility maximization and supply-demand-equilibrium already since 140+ years as if it were manna.

      The alternative to microfounded DSGE is macrofounded Post-Keynesianism. It is not one iota better. Macro profit theory is provably false since Keynes.#1 This is the false MMT balances equation (X−M)+(G−T)+(I−S)=0, and this is the true equation (X−M)+(G−T)+(I−S)−(Q−Yd)=0 with profit and distributed profit greater zero.#2

      Will Lars Ljungqvist, Thomas Sargent, Brian Romanchuk, Roger Sparks, and the rest of mentally retarded microfounded macroeconomists ever get it? No chance, these folks are lost in vacuous proto-scientific space already for centuries.

      Egmont Kakarot-Handtke

      #1 Forget Keynes

      #2 True macrofoundations: the reset of economics

  12. Brian;

    It dawned on me this morning that equation 16.2.3 might be incorrectly written.

    Eq. 16.2.3 makes a lot more sense to me if it is written

    \[c(t) + g(t) + k(t+1) = (F(t, k_b, n))(1-\delta k(t)).\]

    Written this way, the last term-group is a modifier for inflation.

    Is this correct?

    1. I don’t see how that’s a modifier for inflation. The equation has been divided through by the price level; with all due respect to a certain person, that’s a valid mathematical operation.

      In any event, that’s not Ljungqvist’s and Sargent’s model. I am discussing their model, not hypothetical better models.

    2. Brian Romanchuk

      You say “The equation has been divided through by the price level … that’s a valid mathematical operation.”

      The problem is that you are a substandard mathematician. Because of this, you do not realize that DSGE is materially and formally flawed.

      The analytical superstructure of DSGE is based upon this set of hardcore propositions a.k.a. axioms:
      HC1. There exist economic agents.
      HC2. Agents have preferences over outcomes.
      HC3. Agents independently optimize subject to constraints.
      HC4. Choices are made in interrelated markets.
      HC5. Agents have full relevant knowledge.
      HC6. Observable economic outcomes are coordinated, so they must be discussed with reference to equilibrium states. (Weintraub, p. 109)#1

      HC3 introduces marginalism which is the all-pervasive principle of standard economics. There are two methodological flaws here: (i) HC3 is an idiotic behavioral assumption, and (ii), constrained optimization of an ordinal preference order is an invalid mathematical operation as the mathematician Jonathan Barzilai has proven (link has been given above).

      Marginalism has been proto-scientific garbage from the very start.#2 DSGE is the proof that economists are so stupid that have not realized in 140+ years that their axiomatic foundations HC1/HC6 are invalid on all methodological counts.

      The microfoundations approach is dead. Standard economics is dead. DSGE is dead. Nothing less than a paradigm shift will do. Methodologically it holds: If it isn’t macro-axiomatized it isn’t economics.

      Egmont Kakarot-Handtke

      #1 Weintraub, E. R. (1985). General Equilibrium Analysis. Cambridge, London, New York, NY, etc.: Cambridge University Press.

      #2 Putting the production function back on its feet

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