Keynes–Ramsey rule

In macroeconomics, the Keynes–Ramsey rule is a necessary condition for the optimality of intertemporal consumption choice.^[1] Usually it is expressed as a differential equation relating the rate of change of consumption with interest rates, time preference, and (intertemporal) elasticity of substitution. If derived from a basic Ramsey–Cass–Koopmans model, the Keynes–Ramsey rule may look like

{\dot {c}}(t)=\sigma \cdot (r-\rho )\cdot c(t)

where $c(t)$ is consumption and ${\dot {c}}(t)$ its change over time (in Newton notation), $\rho \in (0,1)$ is the discount rate, $r\in (0,1)$ is the real interest rate, and $\sigma >0$ is the (intertemporal) elasticity of substitution.^[2]

The Keynes–Ramsey rule is named after Frank P. Ramsey, who derived it in 1928,^[3] and his mentor John Maynard Keynes, who provided an economic interpretation.^[4]

Mathematically, the Keynes–Ramsey rule is a necessary first-order condition for an optimal control problem, also known as an Euler–Lagrange equation.^[5]

References

^ Blanchard, Olivier Jean; Fischer, Stanley (1989). Lectures on Macroeconomics. Cambridge: MIT Press. pp. 41–43. ISBN 0-262-02283-4.
^ Barro, Robert J.; Sala-i-Martin, Xavier (2004). "Growth Models with Consumer Optimization". Economic Growth (Second ed.). New York: McGraw-Hill. p. 91. ISBN 978-0-262-02553-9.
^ Ramsey, F. P. (1928). "A Mathematical Theory of Saving". Economic Journal. 38 (152): 543–559. doi:10.2307/2224098. JSTOR 2224098.
^ See Ramsey (1928, p. 545): “Enough must therefore be saved to reach or approach bliss some time, but this does not mean that our whole income should be saved. The more we save the sooner we shall reach bliss, but the less enjoyment we shall have now, and we have to set the one against the other. Mr. Keynes has shown me that the rule governing the amount to be saved can be determined at once from these considerations.”
^ Intriligator, Michael D. (1971). Mathematical Optimization and Economic Theory. Englewood Cliffs: Prentice-Hall. pp. 308–311. ISBN 0-13-561753-7.