Questioning Samuelson’s dogma: Schelling on growth and cycles

Can the economy grow continuously or is it condemned to cycle like Sisyphus to make roll a rock which returns perpetually to its starting point?

In the context of the 1937 recession, Paul Samuelson (1915-2009) provided a dynamic framework (1939) for studying the effects of a temporary fiscal stimulus on the path of the economy, which, he thought, could take three forms (permanent stimulus, cyclical policy or temporary shock). In the case of a stable system, he concluded that a one-time increase in government spending has only a temporary effect on income. Moreover, he pointed out that the return to the stationary state would be accompanied by a recession. For him, this case, confirmed by 1937 recession, invalidated the arguments of the pump priming supporters, who thought that a fiscal stimulus could be sufficient to permanently change the path of the economy. Of course, Samuelson was aware that if the economy had been in an unstable position, his model would have led to the opposite conclusion. But he did not think that such a case was relevant. Later, he would say that he was, at the time, like Lloyd Metzler (1913-1980), under the influence of the dogma forged by Ragnar Frisch (1895-1973) according to which only damped cyclical solutions were empirically relevant [Samuelson, 1988: 17].

After having defended his PhD thesis, Samuelson left Harvard in 1940, and obtained a position of professor at MIT, but the geographical and intellectual proximity between the two institutions did not keep him away from the Seminar on Fiscal Policy. In 1941 and 1942, he presented his work at sessions entitled “The Multiplier, a Hansenian Interest” [Salant, 1976: 20], always in close collaboration with Alvin Hansen (1887-1975), to whom he paid regular tribute thereafter [Samuelson, History Of Political Economy, 2002: 221]. Meanwhile a young economist, Evsey Domar (1914-1997), burst in on the scene and reoriented thinking on growth issues.

Domar joined Harvard in 1941 and discovered its seminars there, alongside his PhD supervisor, who was none other than Alvin Hansen. After attending it for three years, he presented his first paper in 1944. Like Roy Forbes Harrod (1900-1978), who was on the other side of the Atlantic, at Oxford, Domar wondered about the conditions under which an economy could grow steadily. Given the duality of investment (both a component of aggregate demand: the higher the investment, the higher the demand, but also a component of aggregate supply: the higher the investment, the greater the productive capacity and the higher the supply), he shows that the growth rate of income can be described by a simple ordinary differential equation. According to the Keynesian multiplier, income Y is equal to the product of investment and the inverse of the propensity to save $1-\alpha$ [Domar, 1946]:

$\begin{equation*} Y= \frac{1}{1 - \alpha}I \end{equation*}$

Assuming then that:

$\begin{equation*} I= \frac{1}{\sigma}\frac{dY}{dt} \end{equation*}$

He finds (see the similar equation of Erik Lundberg written $\dot Y= \frac{\Lambda}{\mu}Y$ ):

$\begin{equation*} \frac{dY}{dt} = (1-\alpha) \sigma \end{equation*}$

From which he deduces the “required rate of increase in investment”, i.e. the rate of growth of investment ensuring that all the additional production resulting from the increase in productive capacity is sold to consumers and businesses:

$\begin{equation*} I = I_o \text{e}^{(1-\alpha) \sigma t} \end{equation*}$

Because of its simplicity, such an equation can only generate exponential growth trajectories. A few years earlier, Jan Tinbergen (1903-1994) deplored the fact that Harrod’s analysis (1936), when reduced to an ordinary differential equation, could not generate cycles. Familiar with Samuelson’s model, what appeared to Tinbergen to be a limitation, was seen by economists such as Thomas Schelling (1921-2016), Sydney S. Alexander (1916-2005) and others at Harvard as an opportunity, a way to extend dynamic analysis to growth analysis.

For Schelling, this meant clarifying the links between Domar’s (but also Harrod’s) and Samuelson’s analyses. Schelling was aware that Harrod, unlike Domar, believed that growth paths are unstable because market imbalances (between supply and demand) would tend to increase as the economy moves away from its growth path. In contrast, Domar and Samuelson made the assumption that the market for goods is always balanced, i.e. at equilibrium. In his attempt to reconcile Domar and Samuelson, Schelling kept this hypothesis. In his American Economic Review article “Capital Growth and Equilibrium” (1947), he studied these growth paths by including them in Samuelson’s model. He indicated, however, that it is essential to amend Samuelson’s assumptions in order to make this connection. Unlike Samuelson, Schelling accepted that consumption depends on current income and not on income in the previous period:

$\begin{equation*} C_t = \alpha Y_t \end{equation*}$

Instead of making investment depend on changes in consumption, as Samuelson does ( $I_t = \beta(C_t - C_{t-1})$ ), Schelling makes investment depend on changes in income. He assigns Domar the idea that investment depends on the level of full employment income (FY) and the ratio \sigma between added productive capacity and added capital:

$\begin{equation*} I = \frac{1}{\sigma} \Delta (FY) \end{equation*}$

By positing that $\sigma = 1 / \beta$ and going from the change in the full employment income level $\Delta (FY)$ to the difference between present and past income. It leads to the following equation:

$\begin{equation*} I_t = \beta(Y_t - Y_{t-1}) \end{equation*}$

From the equilibrium condition in the goods market, he finally derives an ordinary recurrence equation:

$\begin{equation*} Y_t = C_t + I_t = \alpha Y_t + \beta (Y_t - Y_{t-1}) \end{equation*}$

That gives:

$\begin{equation*} Y_t = \frac{\alpha - \beta}{1- \beta} Y_{t-1} \end{equation*}$

And shows that the growth rate of the economy is equal to $\alpha \sigma / 1-\sigma$ .

Schelling thus does not find the same growth rate as defined by Domar, $\alpha \sigma$ . He notes, however, that for small values of $\sigma$ , these two rates are similar:

$\begin{equation*} \lim\limits_{\sigma \rightarrow 0} \alpha \sigma = \lim\limits_{\sigma \rightarrow 0} \frac{\alpha\sigma}{1 - \sigma} \end{equation*}$

Schelling points out two limits to this recurrence equation. The first concerns the treatment of investment. In his view, it is unreasonable to assume that there is no lag between the investment decision and production. It is therefore irrelevant to assume that It depends on $C_t$ or $Y_t$ .

The second limitation, linked to the first, is the simplicity of this first-order dynamic equation, which assumes that a shock can have a permanent effect on the economy’s paths, something he found questionable [Schelling, 1947: 876].

To overcome these two limits, he suggests the following modifications. First, he introduces a new time frame and posits that investment depends on the difference between income in $t-1$ and income in $t-2$ :

$\begin{equation*} I_t = \beta(Y_{t-1} - Y_{t-2}) \end{equation*}$

Without changing the consumption function in his first model, he arrives at the following equation:

$\begin{equation*} Y_t = C_t + I_t = \alpha Y_t + \beta(Y_{t-1} - Y_{t-2}) = \frac{\beta}{1-\alpha} (Y_{t-1} - Y_{t-2}) \end{equation*}$

This equation differs from Samuelson’s in two respects: the absence of a lag on the consumption side and the disappearance of the alpha term in the expression for investment as a function of income.

Solving the model reveals that depending on the value of the discriminant of the characteristic equation, $r^2 - (\beta/1-\alpha)r + (\beta/1-\alpha) = 0$ , such an equation has two real roots, a double root, or two imaginary conjugate roots. The condition for having two real roots and thus defining a monotonic growth path is:

$\begin{equation*} \beta > 4(1- \alpha) \end{equation*}$

“But the rate of growth in that case is rather frightening-more than 100 per cent per year!” [Schelling, 1947 : 875]. Moreover, if this condition holds, a temporary shock has permanent effects on the growth rate (See application). At the end of this examination, Schelling sides with Samuelson and dismisses the growth solutions, thus reinforcing the dogma. It will in fact be up to Alexander to challenge it (See forthcoming post).