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.

Introduction

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