Keynes–Ramsey rule explained

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

• c (t) = \sigma ⋅ (r - \rho) ⋅ c(t) where c(t) is consumption and • 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]

See also

• Ramsey–Cass–Koopmans model

Further reading

• Book: Bliss, C. . Christopher Bliss . 1984 . Notes on the Keynes–Ramsey Rule . Ingham . A. . Ulph . A. M. . Demand, Equilibrium and Trade . Palgrave Macmillan . London . 93–104 . 0-333-33184-2 .

Notes and References

1. Book: Olivier Jean . Blanchard . Olivier Blanchard . Stanley . Fischer . Stanley Fischer . Lectures on Macroeconomics . Cambridge . MIT Press . 1989 . 0-262-02283-4 . 41–43 .
2. Book: Robert J. . Barro . Robert J. Barro . Xavier . Xavier Sala-i-Martin . Sala-i-Martin . Growth Models with Consumer Optimization . Economic Growth . New York . McGraw-Hill . 2004 . Second . 978-0-262-02553-9 . https://books.google.com/books?id=jD3ASoSQJ-AC&pg=PA91 . 91 .
3. F. P. . Ramsey . 1928 . A Mathematical Theory of Saving . . 38 . 152 . 543–559 . 10.2307/2224098 . 2224098 .
4. See : “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.”
5. Book: Intriligator, Michael D. . Michael Intriligator . Mathematical Optimization and Economic Theory . Englewood Cliffs . Prentice-Hall . 1971 . 0-13-561753-7 . 308–311 .