Ramsey–Cass–Koopmans model

Last updated

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, [1] with significant extensions by David Cass and Tjalling Koopmans. [2] [3] The Ramsey–Cass–Koopmans model differs from the Solow–Swan model in that the choice of consumption is explicitly microfounded at a point in time and so endogenizes the savings rate. As a result, unlike in the Solow–Swan model, the saving rate may not be constant along the transition to the long run steady state. Another implication of the model is that the outcome is Pareto optimal or Pareto efficient. [note 1]

Contents

Originally Ramsey set out the model as a central planner's problem of maximizing levels of consumption over successive generations. [4] Only later was a model adopted by Cass and Koopmans as a description of a decentralized dynamic economy. The Ramsey–Cass–Koopmans model aims only at explaining long-run economic growth rather than business cycle fluctuations, and does not include any sources of disturbances like market imperfections, heterogeneity among households, or exogenous shocks. Subsequent researchers therefore extended the model, allowing for government-purchases shocks, variations in employment, and other sources of disturbances, which is known as real business cycle theory.

Key equations

The Ramsey–Cass–Koopmans model starts with an aggregate production function that satisfies the Inada conditions, often specified to be of Cobb–Douglas type, , with factors capital and labour . Since this production function is assumed to be homogeneous of degree 1, one can express it in per capita terms, . The amount of labour is equal to the population in the economy, and grows at a constant rate , i.e. where was the population in the initial period.

The first key equation of the Ramsey–Cass–Koopmans model is the state equation for capital accumulation:

a non-linear differential equation akin to the Solow–Swan model, where is capital intensity (i.e., capital per worker), is change in capital intensity over time , is consumption per worker, is output per worker for a given , and is the depreciation rate of capital. Under the simplifying assumption that there is no population growth, this equation states that investment, or increase in capital per worker is that part of output which is not consumed, minus the rate of depreciation of capital. Investment is, therefore, the same as savings.

The second equation of the model is the solution to the social planner's problem of maximizing a social welfare function, , that consists of the stream of exponentially discounted instantaneous utility from consumption, where is a discount rate reflecting time preference. It is assumed that the economy is populated by identical individuals, such that the optimal control problem can be stated in terms of an infinitely-lived representative agent with time-invariant utility: . The utility function is assumed to be strictly increasing (i.e., there is no bliss point) and concave in , with , where is short-hand notation for the marginal utility of consumption . Normalizing the initial population to one, the problem can be stated as:

where an initial non-zero capital stock is given. Solving this problem, for instance by converting it into a Hamiltonian function, [note 2] [note 3] yields a non-linear differential equation that describes the optimal evolution of consumption,

which is known as the Keynes–Ramsey rule. [5] The term , where is short-hand notation for the marginal product of capital , reflects the marginal return on net investment. The expression reflects the curvature of the utility function; its reciprocal is known as the (intertemporal) elasticity of substitution and indicates how much the representative agent wishes to smooth consumption over time. It is often assumed that this elasticity is a positive constant, i.e. .

Phase space graph (or phase diagram) of the Ramsey model. The blue line represents the dynamic adjustment (or saddle) path of the economy in which all the constraints present in the model are satisfied. It is a stable path of the dynamic system. The red lines represent dynamic paths which are ruled out by the transversality condition. Ramseypic.svg
Phase space graph (or phase diagram) of the Ramsey model. The blue line represents the dynamic adjustment (or saddle) path of the economy in which all the constraints present in the model are satisfied. It is a stable path of the dynamic system. The red lines represent dynamic paths which are ruled out by the transversality condition.

Together both differential equations describe the Ramsey–Cass–Koopmans dynamical system. The steady state, which is found by setting and equal to zero, is given by the pair implicitly defined by

The equilibrium has saddle point property as can be seen from the phase diagram. By the Hartman–Grobman theorem the non-linear system is topologically equivalent to a linearization of the system about by a first-order Taylor polynomial

where is the Jacobian matrix evaluated at steady state, [note 4] given by

which has determinant since is always positive, is positive by assumption, and only is negative since is concave. Since the determinant equals the product of the eigenvalues, the eigenvalues must be real and opposite in sign. [6] Hence by the stable manifold theorem, the equilibrium is a saddle point and there exists a unique stable arm, or “saddle path”, that converges on the equilibrium, indicated by the blue curve in the phase diagram. The system is called “saddle path stable” since all unstable trajectories are ruled out by the “no Ponzi scheme” condition: [7]

implying that the present value of the capital stock cannot be negative. [note 5]

History

Spear and Young re-examine the history of optimal growth during the 1950s and 1960s, [8] focusing in part on the veracity of the claimed simultaneous and independent development of Cass' "Optimum growth in an aggregative model of capital accumulation" (published in 1965 in the Review of Economic Studies ), and Tjalling Koopman's "On the concept of optimal economic growth" (published in Study Week on the Econometric Approach to Development Planning, 1965, Rome: Pontifical Academy of Science).

Over their lifetimes, neither Cass nor Koopmans ever suggested that their results characterizing optimal growth in the one-sector, continuous-time growth model were anything other than "simultaneous and independent". That the issue of priority ever became a discussion point was due only to the fact that in the published version of Koopmans' work, he cited the chapter from Cass' thesis that later became the RES paper. In his paper, Koopmans states in a footnote that Cass independently obtained conditions similar to what Koopmans finds, and that Cass also considers the limiting case where the discount rate goes to zero in his paper. For his part, Cass notes that "after the original version of this paper was completed, a very similar analysis by Koopmans came to our attention. We draw on his results in discussing the limiting case, where the effective social discount rate goes to zero". In the interview that Cass gave to Macroeconomic Dynamics, he credits Koopmans with pointing him to Frank Ramsey's previous work, claiming to have been embarrassed not to have known of it, but says nothing to dispel the basic claim that his work and Koopmans' were in fact independent.

Spear and Young dispute this history, based upon a previously overlooked working paper version of Koopmans' paper, [9] which was the basis for Koopmans' oft-cited presentation at a conference held by the Pontifical Academy of Sciences in October 1963. [10] In this Cowles Discussion paper, there is an error. Koopmans claims in his main result that the Euler equations are both necessary and sufficient to characterize optimal trajectories in the model because any solutions to the Euler equations which do not converge to the optimal steady-state would hit either a zero consumption or zero capital boundary in finite time. This error was apparently presented at the Vatican conference, although at the time of Koopmans' presenting it, no participant commented on the problem. This can be inferred because the discussion after each paper presentation at the Vatican conference is preserved verbatim in the conference volume.

In the Vatican volume discussion following the presentation of a paper by Edmond Malinvaud, the issue does arise because of Malinvaud's explicit inclusion of a so-called "transversality condition" (which Malinvaud calls Condition I) in his paper. At the end of the presentation, Koopmans asks Malinvaud whether it is not the case that Condition I simply guarantees that solutions to the Euler equations that do not converge to the optimal steady-state hit a boundary in finite time. Malinvaud replies that this is not the case, and suggests that Koopmans look at the example with log utility functions and Cobb-Douglas production functions.

At this point, Koopmans obviously recognizes he has a problem, but, based on a confusing appendix to a later version of the paper produced after the Vatican conference, he seems unable to decide how to deal with the issue raised by Malinvaud's Condition I.

From the Macroeconomic Dynamics interview with Cass, it is clear that Koopmans met with Cass' thesis advisor, Hirofumi Uzawa, at the winter meetings of the Econometric Society in January 1964, where Uzawa advised him that his student [Cass] had solved this problem already. Uzawa must have then provided Koopmans with the copy of Cass' thesis chapter, which he apparently sent along in the guise of the IMSSS Technical Report that Koopmans cited in the published version of his paper. The word "guise" is appropriate here, because the TR number listed in Koopmans' citation would have put the issue date of the report in the early 1950s, which it clearly was not.

In the published version of Koopmans' paper, he imposes a new Condition Alpha in addition to the Euler equations, stating that the only admissible trajectories among those satisfying the Euler equations is the one that converges to the optimal steady-state equilibrium of the model. This result is derived in Cass' paper via the imposition of a transversality condition that Cass deduced from relevant sections of a book by Lev Pontryagin. [11] Spear and Young conjecture that Koopmans took this route because he did not want to appear to be "borrowing" either Malinvaud's or Cass' transversality technology.

Based on this and other examination of Malinvaud's contributions in 1950s—specifically his intuition of the importance of the transversality condition—Spear and Young suggest that the neo-classical growth model might better be called the Ramsey–Malinvaud–Cass model than the established Ramsey–Cass–Koopmans honorific.

Notes

  1. This result is due not just to the endogeneity of the saving rate but also because of the infinite nature of the planning horizon of the agents in the model; it does not hold in other models with endogenous saving rates but more complex intergenerational dynamics, for example, in Samuelson's or Diamond's overlapping generations models.
  2. The Hamiltonian for the Ramsey–Cass–Koopmans problem is
    where is the costate variable usually economically interpreted as the shadow price. Because the terminal value of is free but may not be negative, a transversality condition similar to the Karush–Kuhn–Tucker “complementary slackness” condition is required. From the first-order conditions for maximization of the Hamiltonian one can derive the equation of motion for consumption, see Ferguson, Brian S.; Lim, G. C. (1998). Introduction to Dynamic Economic Models. Manchester University Press. pp. 174–175. ISBN   978-0-7190-4997-2 , or Gandolfo, Giancarlo (1996). Economic Dynamics (3rd ed.). Berlin: Springer. pp. 381–384. ISBN   978-3-540-60988-9.
  3. The problem can also be solved with classical calculus of variations methods, see Hadley, G.; Kemp, M. C. (1971). Variational Methods in Economics. New York: Elsevier. pp. 50–71. ISBN   978-0-444-10097-9.
  4. The Jacobian matrix of the Ramsey–Cass–Koopmans system is
    See Afonso, Oscar; Vasconcelos, Paulo B. (2016). Computational Economics : A Concise Introduction. New York: Routledge. p. 163. ISBN   978-1-138-85965-4.
  5. It can be shown that the “no Ponzi scheme” condition follows from the transversality condition on the Hamiltonian, see Barro, Robert J.; Sala-i-Martin, Xavier (2004). Economic Growth (Second ed.). New York: McGraw-Hill. pp. 91–92. ISBN   978-0-262-02553-9.

Related Research Articles

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given; each further term of the sequence or array is defined as a function of the preceding terms.

Heat equation partial differential equation for distribution of heat in a given region over time

In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation.

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

In the calculus of variations, a field of mathematical analysis, the functional derivative relates a change in a functional to a change in a function on which the functional depends.

The primitive equations are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models. They consist of three main sets of balance equations:

  1. A continuity equation: Representing the conservation of mass.
  2. Conservation of momentum: Consisting of a form of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere under the assumption that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere
  3. A thermal energy equation: Relating the overall temperature of the system to heat sources and sinks
Euler equations (fluid dynamics) set of quasilinear hyperbolic equations governing adiabatic and inviscid flow

In fluid dynamics, the Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. The equations represent Cauchy equations of conservation of mass (continuity), and balance of momentum and energy, and can be seen as particular Navier–Stokes equations with zero viscosity and zero thermal conductivity. In fact, Euler equations can be obtained by linearization of some more precise continuity equations like Navier–Stokes equations in a local equilibrium state given by a Maxwellian. The Euler equations can be applied to incompressible and to compressible flow – assuming the flow velocity is a solenoidal field, or using another appropriate energy equation respectively. Historically, only the incompressible equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy equation – of the more general compressible equations together as "the Euler equations".

Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that it is necessary for any optimal control along with the optimal state trajectory to solve the so-called Hamiltonian system, which is a two-point boundary value problem, plus a maximum condition of the Hamiltonian. These necessary conditions become sufficient under certain convexity conditions on the objective and constraint functions.

Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work.

Electromagnetic tensor Mathematical object that describes the electromagnetic field in spacetime

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely.

The Solow–Swan model is an economic model of long-run economic growth set within the framework of neoclassical economics. It attempts to explain long-run economic growth by looking at capital accumulation, labor or population growth, and increases in productivity, commonly referred to as technological progress. At its core is a neoclassical (aggregate) production function, often specified to be of Cobb–Douglas type, which enables the model "to make contact with microeconomics". The model was developed independently by Robert Solow and Trevor Swan in 1956, and superseded the Keynesian Harrod–Domar model.

Aeroacoustics is a branch of acoustics that studies noise generation via either turbulent fluid motion or aerodynamic forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. A notable example of this phenomenon is the Aeolian tones produced by wind blowing over fixed objects.

In quantum mechanics, the probability current is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area. Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. This is analogous to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The concept of a probability current is a useful formalism in quantum mechanics.

Hamiltons principle formulation of the principle of stationary action

In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories.

In numerical linear algebra, a Jacobi rotation is a rotation, Qk, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an n×n real symmetric matrix, A, when applied as a similarity transformation:

The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time horizon. Inspired by, but distinct from, the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian.

Nonlinear acoustics (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. Large amplitudes require using full systems of governing equations of fluid dynamics and elasticity. These equations are generally nonlinear, and their traditional linearization is no longer possible. The solutions of these equations show that, due to the effects of nonlinearity, sound waves are being distorted as they travel.

In fluid mechanics and mathematics, a capillary surface is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid interfaces.

In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument.

Mild-slope equation The combined effects of diffraction and refraction for water waves propagating over variable depth and with lateral boundaries

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

The Zwanzig projection operator is a mathematical device used in statistical mechanics. It operates in the linear space of phase space functions and projects onto the linear subspace of "slow" phase space functions. It was introduced by Robert Zwanzig to derive a generic master equation. It is mostly used in this or similar context in a formal way to derive equations of motion for some "slow" collective variables.

References

  1. Ramsey, Frank P. (1928). "A Mathematical Theory of Saving". Economic Journal . 38 (152): 543–559. doi:10.2307/2224098. JSTOR   2224098.
  2. Cass, David (1965). "Optimum Growth in an Aggregative Model of Capital Accumulation". Review of Economic Studies . 32 (3): 233–240. doi:10.2307/2295827. JSTOR   2295827.
  3. Koopmans, T. C. (1965). "On the Concept of Optimal Economic Growth". The Economic Approach to Development Planning. Chicago: Rand McNally. pp. 225–287.
  4. Collard, David A. (2011). "Ramsey, saving and the generations". Generations of Economists. London: Routledge. pp. 256–273. ISBN   978-0-415-56541-7.
  5. Blanchard, Olivier Jean; Fischer, Stanley (1989). Lectures on Macroeconomics. Cambridge: MIT Press. pp. 41–43. ISBN   978-0-262-02283-5.
  6. Beavis, Brian; Dobbs, Ian (1990). Optimization and Stability Theory for Economic Analysis. New York: Cambridge University Press. p. 157. ISBN   978-0-521-33605-5.
  7. Roe, Terry L.; Smith, Rodney B. W.; Saracoglu, D. Sirin (2009). Multisector Growth Models: Theory and Application. New York: Springer. p. 48. ISBN   978-0-387-77358-2.
  8. Spear, S. E.; Young, W. (2014). "Optimum Savings and Optimal Growth: The Cass–Malinvaud–Koopmans Nexus". Macroeconomic Dynamics. 18 (1): 215–243. doi:10.1017/S1365100513000291.
  9. Koopmans, Tjalling (December 1963). "On the Concept of Optimal Economic Growth" (PDF). Cowles Foundation Discussion paper 163.
  10. McKenzie, Lionel (2002). "Some Early Conferences on Growth Theory". In Bitros, George; Katsoulacos, Yannis (eds.). Essays in Economic Theory, Growth and Labor Markets. Cheltenham: Edward Elgar. pp. 3–18. ISBN   978-1-84064-739-6.
  11. Pontryagin, Lev; Boltyansky, Vladimir; Gamkrelidze, Revaz; Mishchenko, Evgenii (1962). The Mathematical Theory of Optimal Processes. New York: John Wiley.

Further reading