Keynes
Keynes' General Theory (1936) is a book with a multitude of conclusions, specially disequilibrium and that saving cannot be forced. The reasoning of change in savings as a function of labor income growth can be found in Keynes' General Theory (1936) as a "simple principle":
“if he does adjust his expenditure to changes in his income, he will over short periods do so imperfectly. Thus a rising income will often be accompanied by increased saving, and a falling income by decreased saving, on a greater scale at first than subsequently [emphasis supplied]” (Keynes, p. 96-97).
Keynes extended its short period conclusion to involuntary unemployment and long term changes in equilibrium: “On the other hand, a decline in income due to a decline in the level of employment, if it goes far, may even cause consumption to exceed income not only by some individuals and institutions using up the financial reserves which they have accumulated in better times, but also by the government, which will be liable, willingly or unwillingly, to run into a budgetary deficit or will provide unemployment relief; for example, out of borrowed money. Thus, when employment falls to a low level, aggregate consumption will decline by a smaller amount than that by which real income has declined, by reason both of the habitual behaviour of individuals and also of the probable policy of governments; which is the explanation why a new position of equilibrium can usually be reached within a modest range of fluctuation. Otherwise a fall in employment and income, once started, might proceed to extreme lengths (Keynes, p. 97-98).”
Translating Keynes' extensive wording into a mathematical equation: Change in savings (S) is a function of change in labor income (Y) or
Δ S = f (ΔY)
What is amazing is that Keynes' principle cannot be mathematically derived from his consumption function, instead we will see later that it can be derived from the Euler equation approach.
How do we confirm that Keynes reached the mathematical conclusion above? In General Theory, saving and dissaving are effectively the result of errors from decision making. Keynes thought that saving is a ‘determinate’ and the result of the system’s determinants. Thus, there can’t be ‘forced’ saving. In his words:
"Saving and investment are the determinates of the system, not the determinants. They are the twin results of the system's determinants, namely, the propensity to consume, the schedule of the marginal efficiency of capital and the rate of interest. These determinants are, indeed, themselves complex and each is capable of being affected by prospective changes in the others. But they remain independent in the sense that their values cannot be inferred from one another. The traditional analysis has been aware that saving depends on income but it has overlooked the fact that income depends on investment, in such fashion that, when investment changes, income must necessarily change in just that degree which is necessary to make the change in saving equal to the change in investment [emphasis supplied]" (Keynes, p 118-119).
"Thus 'forced saving' has no meaning until we have specified some standard rate of saving. If we select (as might be reasonable) the rate of saving which corresponds to an established state of full employment, the above definition would become: 'Forced saving is the excess of actual saving over what would be saved if there were full employment in a position of long-period equilibrium'. This definition would make good sense, but a sense in which a forced excess of saving would be a very rare and a very unstable phenomenon, and a forced deficiency of saving the usual state of affairs [emphasis supplied]" (Keynes, p. 62).
Curiously, in Lucas' Critique, consumption is a function of unpredicable policy changes and therefore unforecastable. As we saw from above, Keynes reached this conclusion 40 years earlier, blaming on unemployment and disequilibrium. And, as a matter of fact, neither one of them applied micro economic equations and Euler optimization technique to achieve their conclusion.
Keynes, J. M. (1936), The General Theory of Employment, Interest, and Money, Hartcourt Brace Jovanovich.
Lucas, Robert (1976). "Econometric Policy Evaluation: A Critique" (PDF). In Brunner, K.; Meltzer, A. (eds.). The Phillips Curve and Labor Markets. Carnegie-Rochester Conference Series on Public Policy. Vol. 1. New York: American Elsevier. pp. 19–46. ISBN 0-444-11007-0. Archived (PDF) from the original on 2021-11-05.
Wu, Cheng (2017): "Does Clower’s Dual-Decision Hypothesis lead to the change in saving conclusion in Keynes’s General Theory?" https://mpra.ub.uni-muenchen.de/82840/
https://journals.econsciences.com/index.php/JEL/article/view/1497