In
macroeconomics
Macroeconomics is a branch of economics that deals with the performance, structure, behavior, and decision-making of an economy as a whole. This includes regional, national, and global economies. Macroeconomists study topics such as output (econ ...
, 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 (economics), consumption and saving over the course of their lives. The earliest work on the subject was by Irving Fisher and Roy Harrod ...
choice. Usually it is expressed as a
differential equation relating the rate of change of
consumption 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, ...
s,
time preference
In behavioral economics, time preference (or time discounting,. delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good at an earlier date compared with receiving it at a late ...
, and (intertemporal)
elasticity of substitution. If derived from a basic
Ramsey–Cass–Koopmans model, the Keynes–Ramsey rule may look like
:
where
is consumption and
its change over time (in
Newton notation),
is the
discount rate,
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 appro ...
, and
is the (intertemporal)
elasticity of substitution.
The Keynes–Ramsey rule is named after
Frank P. Ramsey
Frank Plumpton Ramsey (; 22 February 1903 – 19 January 1930) was a British people, 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 ...
, 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 and philosopher whose ideas fundamentally changed the theory and practice of macroeconomics and the economic policies of governments. Originall ...
, 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 control theory 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 operations ...
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
*
Ramsey–Cass–Koopmans model
References
Further reading
*
Economic growth
Intertemporal economics
Mathematical optimization
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