macroeconomics Macroeconomics (from the Greek prefix ''makro-'' meaning "large" + ''economics'') is a branch of economics dealing with performance, structure, behavior, and decision-making of an economy as a whole. For example, using interest rates, taxes, and ...

, the Keynes–Ramsey rule is a necessary condition for the optimality of

intertemporal consumption Economic theories of intertemporal consumption seek to explain people's preferences in relation to consumption and saving over the course of their lives. The earliest work on the subject was by Irving Fisher and Roy Harrod, who described 'hump sav ...

choice. Usually it is express as a

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

relating the rate of change of

consumption Consumption may refer to: *Resource consumption *Tuberculosis, an infectious disease, historically * Consumption (ecology), receipt of energy by consuming other organisms * Consumption (economics), the purchasing of newly produced goods for curren ...

with

interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...

time preference In economics, time preference (or time discounting, delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good or some cash at an earlier date compared with receiving it at a later ...

, and (intertemporal)

elasticity of substitution Elasticity of substitution is the ratio of percentage change in capital-labour ratio with the percentage change in Marginal Rate of Technical Substitution. In a competitive market, it measures the percentage change in the two inputs used in respons ...

. If derived from a basic

Ramsey–Cass–Koopmans model The Ramsey–Cass–Koopmans model, or Ramsey growth model, is a neoclassical model of economic growth based primarily on the work of Frank P. Ramsey, with significant extensions by David Cass and Tjalling Koopmans. The Ramsey–Cass–Koopmans mo ...

, the Keynes–Ramsey rule may look like :

\dot(t) = \frac \cdot (r - \rho) \cdot c(t)

where

c(t)

is consumption and

\dot(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 The real interest rate is the rate of interest an investor, saver or lender receives (or expects to receive) after allowing for inflation. It can be described more formally by the Fisher equation, which states that the real interest rate is approx ...

, and

\sigma > 0

is the (intertemporal)

. The Keynes–Ramsey rule is named after

Frank P. Ramsey Frank Plumpton Ramsey (; 22 February 1903 – 19 January 1930) was a British philosopher, mathematician, and economist who made major contributions to all three fields before his death at the age of 26. He was a close friend of Ludwig Wittgenste ...

, who derived it in 1928, and his mentor

John Maynard Keynes John Maynard Keynes, 1st Baron Keynes, ( ; 5 June 1883 – 21 April 1946), was an English economist whose ideas fundamentally changed the theory and practice of macroeconomics and the economic policies of governments. Originally trained in ...

, who provided an economic interpretation. Mathematically, the Keynes–Ramsey rule is a necessary first-order condition for an

optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...

problem, also known as an

Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...

See also

References

Further reading