Two Models of Growth

Both models emerged in a brief window between 1939 and 1956, each trying to explain why some economies sustain growth while others stagnate or collapse. They share a question but diverge immediately on the most fundamental assumption in production theory: whether capital and labour can substitute for each other.

Roy Harrod (1939) and Evsey Domar (1946) were both working in the Keynesian tradition, trying to make Keynes's static multiplier analysis dynamic — to ask not just what level of output equilibrium gives, but at what rate an economy can grow while keeping that equilibrium intact. Their answer rests on a Leontief production function: capital and labour combine in fixed proportions, with no possibility of substitution. If a factory requires exactly two workers per machine, adding a third worker or a second machine on its own produces nothing. The ratio of capital to output is rigid.

That rigid ratio is formalised as the Incremental Capital-Output Ratio, $v$: to raise output permanently by one unit, you need exactly $v$ units of new capital. If $s$ is the fraction of income saved (and invested), the economy can sustain a warranted growth rate $G_w = s/v$ — the rate at which the capital stock grows fast enough to keep all installed capacity in use. A separate natural growth rate $G_n = n$ is set by labour force growth. The model's notorious instability follows directly: if actual growth $G$ drifts even slightly above or below $G_w$, the deviation amplifies rather than self-corrects — the so-called knife-edge. Likewise, if $G_w \neq G_n$, the economy tends toward either chronic over-capacity or chronic labour shortage, with no mechanism to close the gap.

Robert Solow (1956) opened his paper with a direct critique of Harrod-Domar: the knife-edge instability is an artefact of the fixed-coefficient assumption, not an intrinsic feature of capitalist economies. His fix was to replace the Leontief function with a neoclassical production function — most commonly Cobb-Douglas — where capital and labour substitute smoothly. As the capital stock grows relative to the workforce, the capital-output ratio rises and the marginal product of capital falls. This is not a failure of the economy; it is the stabilising mechanism. Trevor Swan reached the same result independently in the same year.

Solow reformulates the problem in terms of capital per effective worker, $k = K/(AL)$, where $A$ captures the level of technology and $L$ is the workforce. The dynamics of $k$ are governed by a single equation: $k$ rises when actual investment $sf(k)$ exceeds the break-even investment $(n + g + \delta)k$ needed to keep $k$ constant in the face of population growth $n$, technological progress $g$, and depreciation $\delta$. Under standard Inada conditions, a unique stable steady state $k^*$ exists, and the economy converges to it from any starting point. In steady state, output per worker grows at rate $g$ regardless of the savings rate — a stark result with direct policy implications: saving more raises the level of income per capita, not its long-run growth rate.