Identities Do Not Imply Equilibrium

In macroeconomics, it is often useful to start from accounting identities between aggregate magnitudes. Since identities are true by definition, they offer a good way to organize concepts. However, it is important to keep in mind that identities in themselves say nothing about causation. In order to build a theory on the basis of accounting identities, it is necessary to make behavioral assumptions. Unlike the identities themselves, these behavioral assumptions are of course contestable.

Consider, initially, a simple two-sector closed-economy model with no government or external sector. For such an economy, actual saving, S, equals actual investment, I:

S = I

This is an identity, true by definition. However, it does not necessarily imply an equilibrium situation. Macroeconomic equilibrium can be defined as a situation in which plans are realized. In the simple two-sector model, equilibrium requires that desired saving, Sd, be equal to desired investment, Id:

Sd = Id

The significance of desires (or plans) is that when they go unrealized there will be disequilibrium and an impetus for changed behavior. Aggregate demand, AD, will differ from output or aggregate supply, AS, and firms will experience unanticipated variations in inventory levels. These unexpected changes in inventories are counted in national accounting as part of actual investment. However, the changes were not intended, so can be thought of as unintended investment, Iu. In a disequilibrium situation, the excess demand or excess supply will equal unintended investment by definition: Iu = AS – AD.

Summarizing the situation:

Excess supply implies Iu > 0; excess demand implies Iu < 0. Likewise, Iu > 0 implies I > Id; Iu < 0 implies I < Id.

Equilibrium requires AS – AD = 0, which occurs when Iu = 0 and I = Id.

Equilibrium also requires actual saving to equal desired saving (S = Sd). Otherwise, households will be motivated to alter spending and saving behavior.

To theorize further it is necessary to make behavioral assumptions. For example, it is relevant to ask how firms will respond to unanticipated variations in inventories.

Economists influenced by Keynes or Kalecki assume that firms will respond to a situation of excess supply in which they experience an unexpected buildup in inventories mainly by cutting back production. There might also be price reductions but output adjustments are assumed to dominate. Modern day neoclassicals generally take the same view in the short run, although they suppose that the price mechanism would restore full-employment output in the long run if it wasn’t for various so-called ‘rigidities’ in the system such as minimum-wage laws, unions or monopolistic firms.

In the case of excess demand and an unanticipated dwindling of inventory stocks, the response of firms is assumed by Keynes and Kalecki influenced economists to depend on whether there is excess capacity. If there is, firms are thought to expand output to meet the stronger demand. This includes an increase in desired investment to restore inventories to the preferred level. The extra desired investment will cause a multiplied increase in income. As a result, desired saving will also rise. Income will continue to adjust until desired saving is equal to desired investment. As output nears full employment, there may also be some price effects but the output response is assumed to dominate. It is only at full capacity that ‘pure inflation’ could occur, a situation in which only prices could rise in response to further increases in demand. Modern day neoclassicals also generally accept this logic in the short run.

It is behavioral assumptions such as these that are at the heart of disagreements among macroeconomists. When Keynesian and Kaleckian economists hold that investment determines saving through income adjustments, they are not talking about actual investment and saving, but desires. They are referring to the kind of multiplier process described in the previous paragraph in which planned saving adjusts passively to planned investment through income adjustments. Likewise, when neoclassicals maintain that in the long run investment adjusts to saving via interest-rate adjustments, they are referring to desires, not actual saving and investment.

Similar logic applies to more elaborate models. For example, in a three-sector closed economy with government but no external sector the following identity holds:

(G – T) = (S – I)

On the left-hand side, G is government spending and T is tax revenue. In words the identity says:

Budget Deficit = Net Private Saving

This identity refers to actual net private saving and says nothing in itself about desires, equilibrium or causation.

Equilibrium in this model requires that desired net private saving equals the budget deficit. Otherwise, even though the identity holds, there will be impetus for behavioral change.

As in the two-sector model, any disequilibrium is captured in the definition of unintended investment and unintended saving. If desired net private saving, (I – S)d, exceeds the budget deficit, economists assuming Keynesian or Kaleckian causation suppose that there will be negative income adjustments. This is argued to occur because the private sector is attempting to hit a net saving target that is incompatible with the budget deficit. Faced with weak demand conditions, firms will cut back investment, impacting negatively on income and the government’s tax revenues. This will be exacerbated if households attempt to maintain their level of planned saving alongside weak investment. This is possible in a model with more than two sectors because it is now Sd + T that adjusts passively to Id + G rather than simply Sd adjusting to Id. If the private sector fights the government’s fiscal stance by attempting to maintain high net private saving, the budget will endogenously move further into deficit. The result will be unemployment and a “bad deficit” (one caused endogenously by weak economy activity) rather than a “good deficit” (one due to exogenous changes in government spending and taxing to boost income).

Conversely, if the budget deficit exceeds desired net private saving, the effects will depend on whether there is excess capacity. If there is, there will be an increase in output and income in response to stronger private spending. If there isn’t, pure inflation will occur.

Again, most modern day neoclassicals would expect similar in the short run even if they don’t couch the argument in terms of the sectoral balances.

12 thoughts on “Identities Do Not Imply Equilibrium

  1. “(it equals Y – (C + I))”

    You can read that as saying is that income (Y) can only buy what is available for sale in aggregate (in a money system).

    It doesn’t really matter whether it is ‘C’ or ‘I’ and quite frankly the classifications of items into either of those categories have a whopping great grey area in the middle. And that’s before you get into the vexed area of ‘materiality’.

    We could use ‘B’ instead as a moniker for ‘stuff that’s been bought’ and switch the whole thing onto a cash basis.

    You can then classify:

    Net Private Savings = Income less Stuff That’s Been Bought.


    (Y – B)

    which then gives you to ex post identity of:

    (G – T) = (Y – B)

    at which point you can move onto the debate about whether (G – T) implies (Y – B) or (Y – B) implies (G – T).

  2. PeterC

    One question.

    “Neoclassicals subsequently accepted this causation [i.e. the Kalecki/Keynes one, from investment to income to saving] in their ‘short run’ but persisted with a real interest-rate mechanism in their ‘long run’, arguing that real interest-rate movements adjust investment plans to full-employment saving”.

    (I take it long and short runs are defined as in Alfred Marshall’s sense).

    I find it difficult to visualize why should (or indeed how could) the economy shift from working in a Kaleckian/Keynesian mode in the short run, to working in a neoclassical mode (real interest rate adjustment) in the long run.

    Whatever one thinks of the notion of short run, it was defined by Marshall as a **microeconomic** category, applicable only to individual firms.

    Maybe, being flexible with the definition of short run, one can extend it from individual firms to industries, i.e. sets of similar firms (similar sizes, technologies, products, etc), as Marshall himself did; but I find it near impossible to conceive of a meaningful single short run applicable equally to all firms in a economy.

    Let’s think of two extreme cases: nuclear power stations and convenience stores. For a nuclear power station the short run is presumably measured in months or maybe years (and it’s hard to imagine additional labour input would matter); for the next corner convenience store it is conceivably measured in days, maybe even hours.

    (In fact, I’d argue that for many small businesses, the most immediately pressing constrain is not even capital inputs, but labour, but let’s leave this aside)

    So, assuming my doubt is pertinent, this is my question: if it is difficult to determine a single short run, how can the economy shift from the K/K mode to the neoclassical mode after the short run? Do neoclassicals address this difficulty?

  3. Good question, Magpie. The short run in macro models is often taken to be a period for which it is reasonable (or at least not too unreasonable) to assume the productive capacity of the economy remains unchanged. So, in a short-run model, the effects of investment on income are considered, but the effects of investment on productive capacity are ignored. In a long-run model, both effects of investment are taken into account.

    The neoclassical position — post Kalecki and Keynes — acknowledges that causation goes from investment to income and saving but maintains that the productive capacity consistent with full employment and intertemporal preferences will be brought about in the long run by investment adjusting to the level consistent with full capacity savings in response to real interest-rate movements.

    Even in the long run, there is still a causative effect of investment on income and saving, but the level of investment is argued to be determined by the real interest rate.

  4. ” but the level of investment is argued to be determined by the real interest rate.”

    Which I presume is a mystical value that can only be augured by high priests in the temple of bankers. And then only if everybody pays them lots of money.

    Funny that I’ve never met an entrepreneur yet who is that bothered about interest rates. They are generally more interested in whether they can sell anything and what mark up they can get.

    It must be an interesting aggregation function they’ve got to transform people who really don’t care about interest rates into an aggregate that obsesses about it as though nothing else matters.

  5. So, the definition of short run in macro is different to that on micro (just in case I was mistaken, I re-checked with The Penguin Dictionary of Economics).

    That is a bit puzzling.

    What does it mean exactly that “the productive capacity of the economy remains unchanged”?

  6. Magpie: The emphasis can be different in different neoclassical models, but basically the long run refers to a situation where everything relevant to the model is flexible. In some contexts, this means all ‘factors’ of production are flexible, in contrast to the short run in which at least one factor is fixed. In other contexts it can refer to a situation in which all prices are flexible, whereas in the short run some prices may be ‘sticky’ ; or a situation in which expectations have fully adjusted, as opposed to the short run in which some expectational errors or ‘money illusion’ may persist.

    Productive capacity is the amount of output that it is possible to supply if all neoclassical factors are fully employed, given the current state of technology, size of the ‘capital stock’, size of the population, etc.

    Neil: Well, Magpie has asked that I occasionally put on a neoclassical hat while he works through a mainstream textbook as part of a Big Challenge. As they say, if the hat fits (which it doesn’t!), wear it. 🙂

  7. “Magpie has asked that I occasionally put on a neoclassical hat while he works through a mainstream textbook as part of a Big Challenge. ”

    Yes, it’s good that you’re doing that. It highlights the madness.

    My comment is just a side thought. What is the aggregation function they have in mind I wonder?

  8. I think that there is no doubt that sophisticated investors, which means the big money, pays close attention to the real interest rate relative to the rate of profit, both in the present and into the future through expectations, as well as geographically, which accounts for capital flows.

    The reality is that there are two sides to every trade and different people have opposite “rational expectations.” The assumption that “in the long run” this cancels out to equal max u at equilibrium seems like a logical jump to me. If there is “stickiness” in the short run, there are going to be different stickiness in the long run as the sum of short runs, only the stickiness will migrate. What is the reasoning that suggesting it will cancel and overcome lack of equilibrium through the action of market forces? This seems to me to be the characteristic of a model that is not representational given the prevalence of cycles where one would expect linearity to prevail based on their assumption. Do neoclassical economists not bother with empirical (historical) evidence?

  9. I wrote above, “What is the reasoning that suggesting it will cancel and overcome lack of equilibrium through the action of market forces?” Neoclassical economics is based on a conceptual model of a closed system with no friction (perfect market, perfect knowledge, etc.). The idea is the factor equalize through the rational pursuit of max u, which is reasonable conceptually. I doubt that any economist would claim that this is anymore than a conceptual model as an ideal base case for thinking about a closed economy.

    National economies are not closed, and there is lots of friction in the system. Moreover, it is not a matter of increasing complication as in natural system but of increasing complexity due to emergence, adaptation, etc, in complex systems, which social system are, especially large societies over time.

    The current state of economics since the fall of the Berlin and especially since the dissolution of the USSR is the introduction of over a billion people to the global economy and enormous resources that were not previously available. Moreover, technological innovation has hastened the speed of globalization. The entire world can now be considered from the POV of a functioning closed system, albeit with a great deal of imbalance to be worked through and a lot of friction involved.

    So, while the general notion of a tendency toward equalizing factors through market forces may be a useful assumption conceptually, the time frame is so unclear as to make “the long run” ambiguous to say the least. I would say that it is actually unhelpful in thinking about complex adaptive (non-ergodic) systems if it creates the impression that they function like even very complicated but simple in the sense of non-complex systems that are ergodic. This is Paul Davidson’s basic criticism that Keynes also suggested.

    Marx had already explained this more fully by including sociology in economics, which is necessary in analyzing complex social systems. For example, to ignore the power relationships involved in class structure and their dynamics is to completely miss the actual dynamic and falsely concluded that the chief driver is the market. That is what most modern economics does and therefore misses most of what is actually going on in the working of a system that is simultaneously social, political and economic due to the institution arrangements in terms of which it functions. Every anthropologist and sociologists knows this but somehow economists have missed it so far.

  10. Another thing I left about concerning the neoclassical model. It assumes only one type of atom, the “representative agent” whose chief property is known to be rationality. This is, of course, a huge oversimplification of agency theory. To base an equilibrium theory on a closed system with one type of atom acting with one type of force and no friction is a bit of an oversimplification. But to depart very much at all from the basic assumptions, let alone introduces realistic assumptions, makes the assumption of equilibrium “in the long run” a matter of faith rather than the model.

  11. Peter,

    Fair enough.

    I agree that the context in which long- and short-run are used is what defines their meaning. The dictionary, by the way, says the same: “It is a concept which can strictly be defined only in the particular context in which it is applied, because its meaning depends of which variables or processes the user of the term has in mind as flexible”.

    However (and, fair dinkum, I am not saying this just to rubbish mainstream theory) it all sounds kind of ambiguous and confusing, to me, and prone to create misunderstanding, too.

    I can imagine people arguing for hours on end over something being or not “short run”, just to settle with “it is so, because I say so”.

    And I just started with this business! Oh well.


    C.E. Ferguson and J.P. Gould (Teoría Microeconómica, one of my prehistoric books), speak of a “market period” which is a kind of a Very Short Run, where all inputs (and output) are fixed.

    I haven’t seen this market period mentioned by anybody else. Have you ever heard of it?

  12. I’m not familiar with Ferguson and Gould, specifically, but a “market period” could apply, for example, to sellers bringing perishables to a market or fair.

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