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

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

where ${\displaystyle c(t)}$ is consumption and ${\displaystyle {\dot {c}}(t)}$ its change over time (in Newton notation), ${\displaystyle \rho \in (0,1)}$ is the discount rate, ${\displaystyle r\in (0,1)}$ is the real interest rate, and ${\displaystyle \sigma >0}$ 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]

• Ramsey–Cass–Koopmans model