In my previous post, I argued that an accounting identity, which tells us that two expressions are defined to be the same, must hold in every state of the world, and therefore could not be disproved by any conceivable observation. So if I define savings and investment (or income and expenditure) to be the same thing, I am simply restricting my semantic description of the world, I am not restricting in any way the set of observable states of the world that conform to my semantic convention. An accounting identity therefore has no empirical content, which means that the accounting identity between savings and investment cannot explain the process by which a macroeconomic model adjusts to a parametric change in the model, traversing from a pre-existing equilibrium with savings and investment being equal to a new equilibrium with savings and investment equal.
In his paper, “The Foundations of the Theory of National Income,” which I am attempting to summarize and explain in this series of posts, R. G. Lipsey provides a numerical example of such an adjustment path. And it will be instructive to follow that path in some detail. The key point about this model is the assumption that households decide how much to save and consume in the current period based on the disposable income received in the previous period. The assumption that all receipts of the business firms are paid out to owners and providers of factor services at the end of each period is a behavioral assumption (not an accounting identity) that rules out any change in the retained earnings held by firms. If firms were accumulating financial assets, then their payments to households would not match their receipts. The following simple model reflects a one-period lag (known as a Robertsonian lag) between household earnings and household consumption.
C(t) = aY(t-1) (behavioral assumption)
I(t) = I* (behavioral assumption)
E(t) ≡ C(t) + I(t) (accounting identity)
Y(t) ≡ C(t) + S(t) (accounting identity)
Y(t) = E(t) (behavioral assumption)
Y(t-1) = Y(t) (equilibrium condition)
Assume that the economy starts off with a = .9 and I(t) = 100. The system is easily solved for E = Y = 1000, with C = 900 and I = 100. Savings, which is the difference between Y and C, is 100, just equal to I. The definition of saving will have to be fleshed out further below. Now assume that there is a parametric change in a (the marginal propensity to consume) to .8 from .9. This change causes equilibrium income to fall from 1000 to 500. By assumption, investment is constant, so that in the new equilibrium saving remains equal to 100. The change in income is reflected in a drop in consumption from 900 to 400. But given the one-period lag between earnings and expenditure, we can follow how the system changes over time, moving closer and closer to the new equilibrium in each successive period, as shown in the following table.
Consider the following questions.
First, in the course of this period-by-period adjustment, will there be any unplanned investment?
Second, in this example, the parametric change — an increase in the propensity of households to save — may be described as an increase in planned savings by households. Planned investment is unchanged. With planned savings greater than planned investment, will the household plans to increase savings be frustrated (implying positive or negative unplanned savings) as alleged in proposition 3 in the list of erroneous propositions provided earlier in the first installment in this series (see appendix below).
The answer to the first question is: not necessarily. There is nothing to prevent us from assuming that all firms correctly anticipate the reduction in consumer demand, so that production falls along with consumption with no change in inventories. It is not necessary to assume that firms can foresee the future; it could be that all consumption is in the form of services, or that production is undertaken only in response to consumer orders. With inventories unchanged, there is no unplanned investment.
The answer to the second question is that it depends on what is meant by unplanned savings. Unplanned savings could mean that households wind up saving an amount other than the amount that they had intended to save at the beginning of the period; households intended to save 200 at the beginning of period 0, but because their income turned out to be only 900, instead of 1000, in period 0, household savings, under the accounting identity, is only 100 instead of 200. However, households intended to consume 800 in period 0, and that is the amount that they actually consumed. The only sense in which households did not execute their intended plans is that household income in period 0 was less than households had expected. Lipsey calls this a distinction between plans in the point sense, and plans in the schedule sense. In this scenario, while plans in the schedule sense are carried out, plans in the point sense are not, because households do not end up at the point on their consumption functions that they had expected to be on.
So the equilibrium condition above that income does not change from one period to the next can be restated as follows: the system is in equilibrium when planned savings equals realized savings. Planned savings is the unconsumed portion of households’ expected income, which is the income households earned in the previous period. The definition embodies a specific behavioral hypothesis about how households formulate their expectations of income in the future.
S_p_(t) ≡ Y(t-1) – C(t).
Realized savings is the unconsumed portion of households’ actual income in the current period. It can be written as
S_r_(t) ≡ Y(t) – C(t).
Or restated differently yet again, the equilibrium condition is that actual disposable income in period t equals expected disposable income in period t.
Let’s flesh out the behavioral assumptions behind this model in a bit more detail. Business firms disburse income to households (owners and providers of factor services) at the end of each period. Households decide how much to save and consume in the upcoming period after receiving their incomes from firms at the close of the previous period. Savings are in the form of bond purchases made at the start of the period. Based on the consumption and savings plans formulated by households at the start of the new period, firms decide how much output to produce and how much labor to hire to produce that output, firms immediately notifying households how many hours they will work in the upcoming period. However, households are committed to the consumption plans already made at the beginning of the period, so they must execute those plans even if the incomes earned during the period are less than anticipated.
In our example, by choosing to increase their savings to 200 through bond purchases at the beginning of the upcoming period, while reducing consumption from 900 to 800, households cause business firms to reduce output from 1000 to 900 (investment being unchanged), and to reduce employment (measured in terms of total hours worked) by 11.1%. After buying bonds equal to 200, households have 800 left in cash, with which they finance their purchases for the rest of the month. So it is not obvious that households were unable to execute any of their plans during the period. However, at the end of the period, households receive only 900 in income from business firms, so although households did buy bonds equal to 200 at the start of the period, they carry over only 900 in cash into the next period, not 1000 as expected. Thus, realized savings are only 100 instead of 200, because household cash holding at the endo f the period turned out to be 100 less than expected. Nevertheless, it is difficult to identify any plan to save that was frustrated, inasmuch as households did purchase bonds equal to 200 at the beginning of the period, and did reduce consumption as planned. As Lipsey puts it:
[W]hether or not the actual real plans laid by households are frustrated depends on what plans households lay, i.e., it depends on our behaviour assumption, not on our definitions. If we assume that households make point plans about their bonds, and schedules plans about their transactions and precautionary balances, then no frustration of plans occurs.
If the statement quoted in (3) [see appendix below] is meant to have empirical content, it depends on a very specific hypothesis about households’ savings plans. These plans must be made in the point and not in the schedule sense, and the plans must include not only additions to the stock of income-earning assets, but also point-plans concerning transactions balances even though the household does not now know what level of transactions the balances will be required to facilitate. . . .
[W]e are now in a position to see what is wrong with statement (2), that actual savings must always equal actual investment, and statement (5), which draws the analogy with demand and supply analysis. Consider statement (2) first.
In the General Theory, Keynes stressed the fact that savings and investment decisions are made by different groups and that there is thus, no reason why planned investment should equal planned savings. [It has been argued] that, although plans can differ, actual realised saving must always be equal to actual realised investment, and, therefore, when planned savings does not equal planned investment, either the plans of savers, or of investors, must be frustrated. Of course, it is quite possible to define savings and investment so that they are the same thing, but it is a basic error to equate the magnitude so defined with the magnitude about which savers actually lay plans. Since ex post S and I as defined bear no relation to the magnitudes about which savers actually make plans, we can deduce nothing about what happens when ex ante S is not equal to ex ante I from the fact that we chose to use the terms ex post S and ex post I to refer to a single, and different, magnitude. The basic error arises from the assumption that households and firms make plans about the same magnitude when they are planning their savings and investment. The traditional theory defines investment as goods produced and not sold to households (= capital goods plus changes in inventories). According to our theory of the behaviour of firms, this is what firms do lay plans about: they plan to add so many capital goods and so many inventories to their existing holdings. The theory then says I ≡ S, and , thus, builds in the implicit assumption that households lay plans about the same magnitude. But according to the standard theory of household behaviour, they do not do so! Households, not subject to money illusion, are assumed to wish to lay aside a certain quantity of real purchasing power which is either used to increase the holdings of cash or used to purchase bonds. There is nothing in the standard theory of household behaviour that leads us to hypothesise that households care whether or not there exists – produced but unconsumed – a physical stock of goods which is the counterpart of the money they have laid aside. Indeed why should they? All they are assumed to care about is the potential real purchasing power of their savings, and this depends only on the amount of money saved, the present price level, and the expected future price level.
This is one of the keys to the whole present confusion: households lay plans about a magnitude that is different from the one that firms lay plans about. Firms plan to have produced and unconsumed a certain quantity of goods, while households plan to leave unspent a certain quantity of purchasing power. This means that it is quite possible for planned investment to differ from planned savings and to have both sets of plans fulfilled so that actual, realized investment differs from actual realized savings. [footnote: Now, of course, we mean by realised S and I the realised magnitudes about which firms and households are actually laying plans. This, of course, does not interfere with the statistician saying that realised savings is identical with realised investment since he refers to a different magnitude when he speaks of realised savings.]
Now consider another variation of the numerical example in Table 1. Instead of a change in the propensity to consume in period 0, assume instead that planned investment drops from 100 to 0. Starting with period -1, Table 2 displays the same initial equilibrium as in Table 1. Because we make a behavioral hypothesis that inventories do not change, planned and realized investment must be zero in period 0 and in all subsequent periods.
According to the national-income identities, savings must equal zero because investment is zero. But what is the actual behavior that corresponds to zero saving? In period 0, households carried over 1000 in cash from period -1. From that 1000, they used 100 to buy bonds and spent the remainder of their disposable incomes on consumption goods. So households planned to save 100 and consume 900, and it appears that they succeeded in executing their plans. But according to the national-income identities, they failed to execute their plan to save 100, and saved only 0, presumably because there were unintended savings of -100 that cancelled out the planned (and executed) savings of 100. So it appears that we have come up against something of a paradox. Here is Lipsey’s solution of the paradox.
[A]ny definitions are possible if consistently used, but this use of the word “unintended” has nothing to do with intended and unintended behaviour. To preserve the identity we must say that the plans of households were frustrated because a real counterpart of the saving they successfully made was not produced. We may say this if we wish, but the danger is that we will think we have said something about the world, and about the actual experiences of households. Indeed, a perusal of established textbooks shows that this confusion has occurred over and over again.
Thus, we conclude that, when we define investment as production not consumed, and savings as income [not consumed] . . . there is no reason why actual savings should not differ from actual investment.
Finally, what about the analogy between savings and investment in macro analysis and demand and supply in micro analysis as in erroneous statement (4) (see appendix)? If we write demand for some good as a function of the price of the good
D = D(p),
and write the supply of some good as a function of the price of the good
S = S(p),
then our equilibrium condition is simply D = S, where D represents desired purchases of the good, and S represents desired sales of the good. Because the act of selling logically entails the activity of purchasing, a purchase and a sale are merely different names for the same thing. So the plans of demanders to buy and the plans of suppliers to sell are plans about the same thing. The plans of demanders to buy and the plans of suppliers to sell cannot be fulfilled simultaneously unless there is an equilibrium in which demand equals supply. The difference between the microeconomic equilibrium in which demand equals supply and the macroeconomic equilibrium in which savings equals investment is that suppliers and demanders in a market are making plans about the same magnitude: sales (aka purchases) of a good. However,
in the national income case the two sets of real plans (savers’ and investors’) are laid about two different magnitudes. Thus the analogy often draw between the two theories in respect of plans and realized quantities is an incorrect one.
Appendix: List of Erroneous Propositions
1 The equilibrium of the basic Keynesian model is given by the intersection of the aggregate demand (i.e., expenditure) function and the 45-degree line representing the accounting identity E ≡ Y.
2 Although people may try to save different amounts from what people try to invest, savings can’t be different from investment; realized (ex post) savings necessarily always equals realized (ex post) investment.
3 Out of equilibrium, planned savings do not equal planned investment, so it follows from (2) that someone’s plans are being disappointed, and there must be either unplanned savings or dissavings, or unplanned investment or disinvestment
4 The simultaneous fulfilment of the plans of savers and investors occurs only when income is at its equilibrium level just as the plans of buyers and sellers can be simultaneously fulfilled only at the equilibrium price.
5 Whenever savers (households) plan to save an amount different from what investors (business firms) plan to invest, a mechanism operates to ensure that realized savings remain equal to realized investment, despite the attempts of savers and investors to make it otherwise. Indeed, this mechanism is what causes dynamic change in the circular flow of income and expenditure.
6 Since the real world, unlike the simple textbook model, contains a very complex set of interactions, it is not easy to see how savings stay equal to investment even in the worst disequilibrium and the most rapid change.
7 The dynamic behavior of the Keynesian circular flow model in which disequilibrium implies unintended investment or disinvestment can be shown by moving upwards or downwards along the gap between the expenditure function and the 45-degree line in the basic Keynesian model.