The Harrod–Domar model is a Keynesian model of economic growth. It is used in development economics to explain an economy's growth rate in terms of the level of saving and of capital. It suggests that there is no natural reason for an economy to have balanced growth. The model was developed independently by Roy F. Harrod in 1939, and Evsey Domar in 1946, although a similar model had been proposed by Gustav Cassel in 1924. The Harrod–Domar model was the precursor to the exogenous growth model. Neoclassical economists claimed shortcomings in the Harrod–Domar model—in particular the instability of its solution—and, by the late 1950s, started an academic dialogue that led to the development of the Solow–Swan model. According to the Harrod–Domar model there are three kinds of growth: warranted growth, actual growth and natural rate of growth. Warranted growth rate is the rate of growth at which the economy does not expand indefinitely or go into recession. Actual growth is the real rate increase in a country's GDP per year. (See also: Gross domestic product and Natural gross domestic product). Natural growth is the growth an economy requires to maintain full employment. For example, If the labor force grows at 3 percent per year, then to maintain full employment, the economy’s annual growth rate must be 3 percent. Mathematical formalism Definitions Let Y represent output, which equals income, and let K equal the capital stock. S is total saving, s is the savings rate, and I is investment. δ stands for the rate of depreciation of the capital stock. The Harrod–Domar model makes the following a priori assumptions: Derivations Derivation of output growth rate: A derivation with calculus is as follows, using dot notation (for example, ) for the derivative of a variable with respect to time. First, assumptions (1)–(3) imply that output and capital are linearly related (for readers with an economics background, this proportionality implies a capital-elasticity of output equal to unity). These assumptions thus generate equal growth rates between the two variables. That is, Since the marginal product of capital, c, is a constant, we have Next, with assumptions (4) and (5), we can find capital's growth rate as, In summation, the savings rate times the marginal product of capital minus the depreciation rate equals the output growth rate. Increasing the savings rate, increasing the marginal product of capital, or decreasing the depreciation rate will increase the