Primer: The Kalecki Profit Equation (Part I)

The Kalecki profit equation -- named after the economist Michal Kalecki -- describes how aggregated profits are determined by national accounting identities. (Note that Jerome Levy came up with a similar approach earlier; the equation is sometimes referred to as the Kalecki-Levy profit equation.) The results are perhaps not obvious if we look at profits from a bottom up perspective. From the perspective of business cycle analysis, the key point to note is that net investment is a source of profits. Meanwhile, since firms invest in order to grow profits, we get a self-reinforcing feedback loop. From a policy perspective, we see that governmental deficits also add to profits, which implies that increasing deficits add to profits in a recession, helping put a floor under activity.

Given the length of this discussion, it will be broken into a multi-part article. (Link to part 2.) This first article will discuss the role of savings and the distribution of income on profits. My objective is to explain the practical importance of the profit equation, and I make a number of assertions about business cycle behaviour to do so. Other articles will expand upon this analysis, and provide a stronger justification of these assertions. I also want to add a disclaimer that I am not particularly concerned about the details of national accounting; I am writing about a simplified accounting system that would reflect the outcomes of economic models. There may be some details around various definitions that I am glossing over.

If we wanted to apply the profit equation to real world national accounts data, the final equation contains a great number of terms. This complexity distracts from the basic principles of the equation. Instead, this treatment will start off with the equation for very simple model economies, and then add terms as we add complexity to the models. The usefulness of the Kalecki profit equation is for understanding the model dynamics that transfer to real world behaviour, rather than playing with national accounts identities. We can build up the equation term-by-term, and so we can have an intuition of the role of each term.

This treatment is largely based on the one found in Section 5.3.6 of Marc Lavoie's Post-Keynesian Economics: New Foundations (link to review). I would also note the historical discussion found in "Profits: The Views of Jerome Levy and Michal Kalecki" by S. Jay Levy; Levy's tended to be more focused on the details of the national accounts.

Given my aversion to the theoreteical concept of "money," I use "cash" in discussion. Cash can refer to any instruments that are used exchange, which may or may not be formal monetary aggregates (although all the instruments in monetary aggregates are undoubtedly cash).

Model 1: Simplest Two-Sector Model

We will start with the simplest case: an economic model with just two sectors (the business sector and workers), and no investment. Furthermore, we will assume that the business sector does not pay dividends. In order to eliminate investment, we will assume that the business either provides services or highly perishable goods; there are no inventories. We will refer to this is model #1, and it has an associated profit equation.

We will treat the business sector as if it were a single employer. If there are multiple employers, we might see that some are profitable and others run at a loss; what we are interested in is aggregate profit, so that distribution does not matter.

The business sector's profits are equal to revenue minus expenses.

• Revenue equals purchases from households.
• Expenses are equal to wages paid to workers.

We can immediately see that if households spend 100% of wages (and have no other source of spending), revenue equals expenses and profits are equal to zero. There is a circular flow of cash out of the firm to workers which then returns as revenue, and no cash drops out of the loop. However, if workers save some of their income, revenue will be less than expenses; cash has dropped out of the loop. This gives us a version of the profit equation with just one term:

(Model 1 Profits) = - (Household savings).

Even this simplest version of the profit equation has a couple interesting theoretical properties.

The first thing to note that since household savings subtract from profits, rising savings "all else equal" has a negative cyclical effect. Variants of this concept show up throughout discussions of simple monetary models; for example, the desire to hoard money could allegedly lead to a recession. However, there does not seem to be any evidence of such an effect in market economic analysis. From my perspective, the emphasis on household savings as a cyclical factor by academic economists caused me to view their work as being unrealistic. The problem is the overly simplified near-barter economies academics use to illustrate their models -- the issue is not changes in what households view as "saving," rather the willingness to incur debt.

If the household sector borrows in aggregate during the accounting period, it could consume more than it earns from wages. That is, if household saving is negative, profits are positive.

Within this two sector model, there are two avenues to allow this borrowing.

1. The business sector can lend to households; for example, by offering financing on purchases. The business sector could end up with an unchanged cash balance, but it will gain financial assets -- the debt-like claims emitted by household sector.
2. Individual households can lend to other households. However, this can only be sustained for as long as lending households hold cash instruments.*

Household consumption patterns appear to be fairly stable across the cycle (during the expansion, at least). However, the willingness to incur debts is more cyclical, particularly with respects to housing and car purchases.** We need to drop the parables about Farmer Bob buying apples from Farmer Alicia, or Bob's desire to hold gold coins, rather the issue is whether their kids Cynthia and Doug are buying a condo in the big city.

The next thing to note is that one could view the equation as implying that household frugality is a negative. This is in stark contrast to the conventional story that we need to increase household savings to boost growth. ("Farmer Bob needs to hold back some of his corn to plant next year!") This model stacks the deck against frugality by assuming that investment is zero; I will return to this when investment is added into the model.

Model 2: Dividends

The next addition to the model is to add dividends into the mix. Neoclassical models have a hard time with dividends, for reasons that will be discussed below. However, the inclusion of dividends into the equation has some major theoretical and political economy effects.

A dividend is a payment by a firm to its owners, and dividend income is normally the objective for capitalist firms. The normal assumption is that dividends are paid out of profits (current or historical) -- although the private equity industry has mucked that belief up.

Dividends and wages are the two main cash flows from the business sector to the household sector in economic models. (In the real world, some individuals either are self-employed or part of private partnerships that end up being lumped in the household sector in the national accounts. As a result, real world data could see more types of interactions than the tidy world of economic models.) Although the cash flows may be fungible, they have very different implications for profits: dividends are not an expense.

If we assume that the household sector has zero savings, all business cash outflows return in revenue. So if 20% of outflows are dividends, only 80% of outflows are wages. This means that profits (revenues minus expenses) are 20% of revenue -- or equal to the dividend payments. However, the household savings is generally non-zero, and we end up with the profit equation #2, with an added term:

(Model 2 Profits) = - (Household savings) + (Dividend payments).

(In other words, we added another term to the equation.)

One typical way to present Kalecki's model is to add some behavioural information that acts to constrain outcomes. The assumption is that workers and capitalists are distinct sets of households, and that workers spend 100% of their wages (and do not dis-save). This has the implication that that household sector saving is exactly equal to the savings of the households that receive dividends.

This implies a different profit equation:

(Kalecki Assumption Profits) = (Dividend payments) - (capitalist savings),

which by definition of savings implies:

(Kalecki Assumption Profits) = (Capitalist consumption).

 This explains Kaldor's aphorism***:

Capitalists earn what they spend, and workers spend what they earn.

Once again, this version of the equation only holds in models with the constraint that workers' aggregate savings are exactly equal to zero; this will not be true in real world data (except as a statistical fluke). Furthermore, since workers and salary owners can also own equities (courtesy of the move to self-directed retirement schemes), we have a blurring in the distinction between "workers" and "capitalists." Nevertheless, it does seem safe to argue that dividend income is largely received by the upper income quantiles, and those quantiles have a greater propensity to save. Therefore, the behavioural constraint might be a reasonable approximation of reality.

The addition of the dividend term to the profit equation has limited real world significance in terms of cyclical analysis. Over short spans of time, dividends are stable, and so this term is not too significant a source of volatility when compared to the other terms to be added. Over multi-decade periods, the changes in dividend policy would presumably show up in secular profit trends, and so assertions about the long-term would need to take dividend changes into account. However, the addition of dividends to the profit equation is very important from a theoretical perspective, as well as for political economy.

(The current popularity of stock buybacks might muddy the theoretical waters. In a stock buyback, the business sector outflows are in exchange for financial assets, and the seller will not count the sales proceeds as normal income. At most, there will be capital gains. Trying to analyse this properly would require something like a stock-flow consistent model that takes into account equity market valuations and capital gains. Those models are complex, and raise a great number of thorny questions. I may return to this question, but I do not see an obvious answer as to the macro effects of a switch from dividends to stock buybacks.)

The theoretical problem with dividends being a source of profits is that is creates a self-reinforcing feedback loop between profits and dividends, as greater profits allows for greater dividends. For simpler models like stock-flow consistent models, this is not a particular issue. However, if one believes that model outcomes are in some sense the result of optimising behaviour, this creates difficulty. If the business sector (or capitalists) were truly aiming for optimal outcomes, the solution would likely be pathological, with extremely high dividends that are not saved. Neoclassical models avoid these problems by avoiding actual optimisations, instead sectors follow heuristics that are sub-optimal relative to choices that are implied by taking into account the full macro model. (See this article for a longer explanation of that assertion.) This accords with my previous experience in applied mathematics: for practical problems, optimisations almost always result in pathological outcomes.

Even if we accept that economic outcomes are not in any sense optimal, there are important implications of the role of dividends. The implication is that the level of profits in an economic system are essentially arbitrary; for any level of output, we can split the output between workers and capital and have a system that respects cash flow constraints (accounting identities). An economic model needs to pin down the distribution of income using some assumptions about behaviour in order to have a single solution.

Neoclassical models hide distributional questions under the carpet by assuming that workers and capital receive their just desserts: marginal contributions determine the levels of wages and profits, and hence the income shares. As a result, it is possible to ignore the politics of income distribution. By contrast, the post-Keynesian tradition explicitly notes the arbitrary nature of income distributions. To be fair, "mainstream" economists are now more willing to discuss the effects of income distribution (at least the leftward end of the mainstream).

From a short-term perspective, distributional questions are a second order effect. I have seen arguments to the effect that the structural sluggishness of recent decades is due to a lower wage share of national income, but I am agnostic on the validity of that view. Investment trends are much more important for profit determination, as I discuss next.

Model 3: Investment

I will conclude this part of the article with the addition of investment, which is arguably the most important cyclical component during an expansion. For those who are new to economics, one needs to keep in mind that the definition of "investment" in economics does not match the way the word is often commonly used. For example, people will often use "investing in the stock market" as a way of describing the purchase of shares; this does not fit the definition used in economics. Instead, "investment" here refers to spending by firms that creates non-financial assets that presumably will generate future profits. The trick is that this can either be fixed investment, or investment in inventories. The issues around inventory investment will be discussed later.

(The usual convention in the national accounts is that investment is largely an activity of the business sector; household spending is usually classified as consumption. This is an artefact of the nature of the way that national accounts are measured. For example, national statisticians cannot tell whether my purchase of a computer is to support consulting activities -- an investment -- or to play the latest video games -- which is a form of consumption. The discussion here follows the convention that only the business sector invests.)

Investment is another cash outflow by businesses that is not an expense. As a result, circular flows result in revenue that is not matched by a wage expense -- profits.

The Model 3 equation now reads:

(Model 3 Profits) = (Net Investment) - (Household savings) + (Dividend payments).

The transactions for investment are more complicated than the previous models. We can imagine three basic channels for investment in this simplified framework. (If we add more sectors, there are more possibilities.)

1. The firm pays workers $100 to create a capital good. The $100 payment is not treated as an expense, it is instead "capitalised." From the workers' perspective, it does not make a difference whether the $100 is expensed or not; it is a household income-producing cash flow that can be spent. The household cash flow then recirculates through the system as before. Profits are higher since we are now no longer deducting some of the cash flows from revenue.
2. The firm pays workers $100 to produce products that are not sold. The unsold products will end up in inventory. The inventory will be held on the balance sheet at the production cost (which is equal to to wage payments need to produce the goods). The hope is that the goods in inventory will be sold in a future accounting period, and the profits on the sale are equal to the sales price less the cost of the goods as valued in inventory. (Under most circumstances firms cannot mark the value of inventories as equal to their final selling price, as firms could generate "profits" by just producing goods that cannot be sold and dumping them into inventory.) This then leads to another observation -- selling the goods out of inventory represents disinvestment (negative investment), and this amount would be subtracted from gross investment in the next accounting period to get net investment. (Note that the profit equation specifies net investment, not gross investment.) 
3. Firm A purchases a capital good for $100 from Firm B. Firm B held the good in inventory, with a value of $80. The $100 cash outflow for Firm A is not an expense; and so the transaction is profit neutral. For Firm B, the profit on the transaction is equal to the selling price ($100) less the cost of goods sold ($80) -- which implies a profit of $20. The net investment for the aggregate business sector is also $20. Firm A has a new fixed investment of $100, while Firm B has an inventory disinvestment of $80. It is easy to see that cash flows can get quite complicated once we allow for intra-sector cash flows (e.g., business to business flows). For example, an investment project could have a mixture of purchased inputs as well as worker pay. The key is that investment creates outflows that are not matched to an expense. 

Depreciation adds another wrinkle to the concept of net investment. Most capital goods have a finite lifespan, and their value is written down over time. The decrease in asset value is known as depreciation, and it shows up as an expense (the cost of capital). Depreciation is subtracted from other investment in order to get net investment. That is,

Net Investment = (Gross fixed investment) - (Depreciation) + (New goods added to inventory) - (Value of goods sold from inventory).

If we had a static economy with fixed nominal prices, depreciation expense would converge to equal gross investment. For example, assume that all capital goods depreciate by 10% of their initial value every year for 10 years. If firms invest $100 per year, that would create an expense of $10 per year for 10 years. After doing this for 10 years, the level of depreciation expenses would equal the new investment. However, the usual condition for modern economies (outside Japan) is that prices and volumes of investment are rising over time, so the new investment is larger than the depreciation of earlier investments (which have a lower nominal value).

The net investment component of the profit equation is much more important from a cyclical perspective than household saving and dividends. The reason is that investment is pro-cyclical: firms invest if they expect higher future demand -- and profits. Since investment itself generates profits, expectations of higher profits can be viewed as self-fulfilling. This means that expansion is the natural state for capitalism; the problem is that other factors can derail profits and investment, and then the self-reinforcing feedback depresses activity.

I would summarise the post-Keynesian view of how firms operate during an expansion as follows. (I will fill in references in later.) It is a mistake to believe that firms attempt to solve some optimisation problem in order to plan their activities; they are missing too much useful information to fill in the optimisation parameters. Instead, they need to use rules of thumb (heuristics), such as extrapolating past growth trends. Obviously, they do a great deal of analysis of their market, but there is an obvious great amount of uncertainty about the direction of the overall economy. They will then come up with some baseline forecast of demand for their products.

• They will plan production to meet demand, and to grow their inventories in line with sales. That is, they typically want to keep the ratio of inventory to sales at some target ratio, such as holding one month's sales in inventory. (Since inventory is a stock and sales are a flow, the ratio has units of time.)
• They will launch fixed investments if they believe that they need to add capacity to meet projected demand.

The implication is that if firms are projecting demand growth over their planning horizon, they will generally plan on both increasing inventories as well as increasing fixed investment. 

The future does not always meet past projections. Two competing firms may have ramped up capacity beyond the demand for their products, and so sales end up below their projections. If they are producing goods that are held in inventory, they will end up with a higher inventory-to-sales ratio than desired. By itself, the higher inventory levels do not cause a loss. However, the unplanned inventory build represents a hit to cash flow -- they paid cash to produce the inventory. Their balance sheet weightings shift from cash to inventories -- and they need cash to meet expenses and repay debts. The inventory build has to be reversed -- which implies a reduction in investment.

This means that "investment" is not an unalloyed positive, as it is sometimes portrayed in economic parables. One very often encounters stories about Robinson Crusoe economies, or economies that consist of barter among various proprietors (fisherman, shoemakers, etc.). In such an exchange economy, investment is represented as frugally abstaining from consuming output to add to productive capacity. However, in a monetary economy, "investment" may just be the piling up of unsold goods at firms -- and those firms will go bankrupt if that inventory build is not reversed. This explains why "Keynesian" economists often emphasise the role of demand within the economy.

Concluding Remarks

The discussion of the Kalecki Profit equation will conclude with the addition of two extra terms, which take into account the government and external sectors.

Link to the second part.

Footnotes:

* These sorts of complicated intra-sector transactions can break overly simplistic analysis that wants to develop mechanistic models of cash flows in the economy. As a result of things like intra-sector borrowing, measured gross debt levels can move around without resulting in changes in measured GDP. Conversely, if one assumes that debt is only issued to purchase goods and services, one can incorrectly believe that there is an iron law relating debt changes to GDP changes.

** The purchase of an existing house does not itself add to GDP, whereas it will inflate household debt (in the typical case where the buyer will end up with a larger mortgage than the seller). This makes sense, as we do not consider the proceeds of a house sale to be income. That said, there are fairly large income effects in a typical housing transaction. Things like realtor fees, mortgage insurance, legal fees, welcome tax, and moving expenses represent income for the providers of services, and are often rolled into the mortgage. As a result, a portion of the increase in mortgage debt does directly raise GDP. This matches up with the experience of housing booms: they buoy economic activity, but the effect is smaller than the change in mortgage debt would imply.

*** N. Kaldor, "Alternative Theories of Distribution," Review of Economic Studies, 23 (2), pages 83-100. The quote itself is in page 96. I got the reference from Marc Lavoie's Post-Keynesian Economics: New Foundations. He notes that this quotation is often mis-attributed to Kalecki.

(c) Brian Romanchuk 2018