Keynes-Ramsey rule

The Keynes-Ramsey-Rule (English Keynes-Ramsey-Rule , KRR for short) describes the growth rate of consumption as a result of intertemporal utility maximization in dynamic macroeconomics . The rule is part of the neoclassical growth theory and describes the relationship between the growth rate of consumption, the interest rate , the time preference rate and the intertemporal elasticity of substitution .

The Keynes-Ramsey rule is a result of the Ramsey model and should be normative, i.e. H. from the point of view of a social planner , give an answer to the question of the optimal savings. Most of the time, advanced mathematical methods from the calculus of variations are needed to derive the rule.

Demarcation

The Euler equation or Euler-Lagrange equation is understood in the calculus of variations to mean the optimality conditions of a dynamic optimization problem. Euler's equation therefore has a general meaning, also outside of economics.

Nevertheless, some authors make no distinction here and use the terms Keynes-Ramsey rule and Euler equation synonymously. Others describe it as a special representative or economic interpretation of the Euler equation. In this context, the Euler equation of consumption is also used, which describes the optimal intertemporal consumption allocation of a utility-maximizing household.

General statement

The Keynes-Ramsey rule is as follows:

${\displaystyle g_{C}={\frac {\dot {c}}{c}}={\frac {1}{\sigma }}(r-\rho ).}$

The key messages of the KRR are:

• consumption increases ( ) if the interest rate ( ) is greater than the time preference rate ( ). ${\displaystyle g_{C}>0}$ ${\displaystyle r}$ ${\displaystyle \rho }$
• a lower willingness to substitute intertemporally (greater ) means a less strong reaction with regard to the difference between interest and time preference. ${\displaystyle \sigma }$

The growth rate is positive if the interest rate is greater than the time preference rate . In such a case, the household would save or forego consumption, because the interest earned on its non-consumption compensates it for the direct benefit it has lost. The time preference rate should describe the inclination of the household by how much more or less it estimates consumption in a later period (in the future) than consumption today. A positive time preference rate is usually assumed here. The elasticity parameter describes the curvature of the assumed utility function. The more the utility function is curved (concave), the more households prefer an even distribution of consumption over time. So if this value is very high, a lower growth rate would be optimal. ${\displaystyle g_{C}}$ ${\displaystyle r>\rho }$ ${\displaystyle \sigma }$

It is believed to be positive. However, empirical studies by Robert E. Hall showed some negative elasticity values. The parameter comes from a special utility function. The intertemporal elasticity of substitution is described by the reciprocal value . ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle {\frac {1}{\sigma }}}$

Formal representation

The KRR is the result of a dynamic optimization problem. Depending on the form of the objective function ( utility function , e.g. CIES utility function) and the restrictions (e.g. budget of the household) or the assumed production technology (e.g. Cobb-Douglas function ), it looks somewhat different.

Constant infinite time horizon

A well-meaning dictator ( social planner ) rules a country with infinitely long households. He wants to maximize the following utility function:

${\displaystyle \max \int _{0}^{\infty }{\frac {C^{1-\sigma }-1}{1-\sigma }}e^{-\rho t}dt.}$

Here stands for consumption, for constant elasticity and for a positive rate of time preference. He has to take into account a budget and a production technology: ${\displaystyle C}$ ${\displaystyle \sigma }$ ${\displaystyle \rho >0}$

${\displaystyle Y(t)=I(t)+C(t),\quad y^{*}=T^{\frac {1}{1-\alpha }}\left({\frac {s}{\delta +n}}\right){}^{\frac {\alpha }{1-\alpha }}}$

In addition, the capital accumulation equation applies:

${\displaystyle {\dot {K}}(t)=I(t)-\delta K(t)}$.

Here stands for a depreciation rate (which need not be assumed). The result of the optimization would then look like this (in per capita sizes): ${\displaystyle \delta }$

${\displaystyle {\frac {\dot {c}}{c}}={\frac {1}{\sigma }}\left[\alpha {\frac {y}{k}}-\rho -\delta \right].}$

Here is synonymous with the marginal product of capital of the assumed production function. For this growth rate to be positive, this marginal product must be greater than the time preference rate and the depreciation rate combined. ${\displaystyle \alpha {\frac {y}{k}}}$

Finite time horizon

Example: A household wants to create an optimal consumption plan in each period based on its income and the respective interest rate . The optimization problem arises as follows: ${\displaystyle t}$ ${\displaystyle y_{t}}$ ${\displaystyle r}$

${\displaystyle \max \sum _{t=0}^{T}\beta ^{t}u(c_{t})}$ under the secondary condition ${\displaystyle a_{t}+y_{t}=c_{t}+{\frac {a_{t+1}}{1+r}}\quad \forall t=1,...,T}$

The result shows:

${\displaystyle u'(c_{t})=\beta \cdot u'(c_{t+1})\cdot (1+r).}$

This describes the basic property of an optimal consumption path over time (it is a necessary condition). The current marginal utility corresponds to the discounted marginal utility of the following period in combination with the expected marginal return on savings . ${\displaystyle u'(c_{t})}$ ${\displaystyle u'(c_{t+1})}$ ${\displaystyle (1+r)}$

This form of KRR is sometimes referred to as the Euler equation in economics .

History and reception

Ramsey developed the rule in his 1928 essay A mathematical theory of saving . It is the answer to the question of how much a nation should save and it is:

"The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utlity falls short of the maximum possible rate of enjoyment."

- Frank Ramsey, 1928.

The Keynes-Ramsey rule is named after the British mathematician Frank Plumpton Ramsey and the British economist John Maynard Keynes . Ramsey created the basis of the rule in his 1928 article, but Keynes pointed out the current interpretation of his results, whereupon the rule was named after both.

"[Ramsey's 1928 article] is, I think, one of the most remarkable contributions to mathematical economics ever made, both in respect of the intrinsic importance and difficulty of its subject, the power and elegance of the technical methods employed, and the clear purity of illumination with which the writer's mind is felt by the reader to play about it subject. The article is terribly difficult reading for an economist, but it is not difficult to appreciate how scientific and aesthetic qualities are combined in it together. "

- John Maynard Keynes, "FP Ramsey", Economic Journal, 1930

Individual evidence

1. ^ Keynes-Ramsey-Rule - Gabler Wirtschaftslexikon.
2. ↑ Maik Heinemann: Dynamic Macroeconomics . Springer Gabler; Edition: 2015 (November 20, 2014). ISBN 978-3662441558 . P. 57.
3. ^ Hannula, Helena, Slavo Radošević, and GN Von Tunzelmann, eds. Estonia, the new EU economy: building a Baltic miracle ?. Ashgate Publishing, Ltd., 2006. p. 82.
4. ↑ Maik Heinemann: Dynamic Macroeconomics . Springer Gabler; Edition: 2015 (November 20, 2014). ISBN 978-3662441558 . P. 26.
5. ^ Frank Hettich: Economic Growth and Environmental Policy: A Theoretical Approach . Edward Elgar Publishing Ltd (August 2000). ISBN 978-1840643695 . P. 47.
6. ↑ Euler equation of consumption - definition in the Gabler Wirtschaftslexikon.
7. ↑ Xavier Sala-I Martin, Robert J. Barro: Economic Growth . MIT Press. 2003. ISBN 978-0262025539 . P. 91.
8. ^ Hall, Robert E. "Intertemporal substitution in consumption." (1988).
9. ↑ Maik Heinemann: Dynamic Macroeconomics . Springer Gabler; Edition: 2015 (November 20, 2014). ISBN 978-3662441558 . Pp. 25/26.
10. ↑ Generation models - definition in the Gabler Wirtschaftslexikon.
11. ↑ Ramsey, Frank Plumpton. "A mathematical theory of saving." The economic journal (1928). P. 543.
12. ^ Maria Frapolli, Maria Jose Frapolli: FP Ramsey: Critical Reassessments . Continnuum-3pl; Edition: First Edition (March 15, 2005). ISBN 978-0826476005 . Pp. 107/108.
13. ^ Collard, David A. Generations of Economists. Vol. 120. Routledge, 2011.

literature

Original literature

• Ramsey, Frank Plumpton. "A mathematical theory of saving." The economic journal (1928): 543-559.

Secondary literature

• Stanley Fischer, Olivier Blanchard: Lectures on Macroeconomics . MIT Press (January 1, 1989). ISBN 978-0262022835 . P. 41ff.