# Ramsey–Cass–Koopmans model

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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, ${\displaystyle F(K,L)}$, with factors capital ${\displaystyle K}$ and labour ${\displaystyle L}$. Since this production function is assumed to be homogeneous of degree 1, one can express it in per capita terms, ${\displaystyle F(K,L)=L\cdot F\left({\frac {K}{L}},1\right)=L\cdot f(k)}$. The amount of labour is equal to the population in the economy, and grows at a constant rate ${\displaystyle n}$, i.e. ${\displaystyle L=L_{0}e^{nt}}$ where ${\displaystyle L_{0}>0}$ was the population in the initial period.

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

${\displaystyle {\dot {k}}=f(k)-(n+\delta )k-c}$

a non-linear differential equation akin to the Solow–Swan model, where ${\displaystyle k}$ is capital intensity (i.e., capital per worker), ${\displaystyle {\dot {k}}}$ is change in capital intensity over time ${\displaystyle \left({\tfrac {dk}{dt}}\right)}$, ${\displaystyle c}$ is consumption per worker, ${\displaystyle f(k)}$ is output per worker for a given ${\displaystyle k}$, and ${\displaystyle \delta \,}$ 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, ${\displaystyle U_{0}=\int _{0}^{\infty }e^{-\rho t}U(C)\,\mathrm {d} t}$, that consists of the stream of exponentially discounted instantaneous utility from consumption, where ${\displaystyle \rho \in (0,\infty )}$ 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: ${\displaystyle U(C)=Lu(c)=L_{0}e^{nt}u(c)}$. The utility function is assumed to be strictly increasing (i.e., there is no bliss point) and concave in ${\displaystyle c}$, with ${\displaystyle \lim _{c\to 0}u_{c}=\infty }$, where ${\displaystyle u_{c}}$ is short-hand notation for the marginal utility of consumption ${\displaystyle {\tfrac {\partial u}{\partial c}}}$. Normalizing the initial population ${\displaystyle L_{0}}$ to one, the problem can be stated as:

${\displaystyle \max _{c}U_{0}=\int _{0}^{\infty }e^{-(\rho -n)t}u(c)\,\mathrm {d} t}$
${\displaystyle {\text{subject to}}\quad c=f(k)-(n+\delta )k-{\dot {k}}}$

where an initial non-zero capital stock ${\displaystyle k(0)=k_{0}>0}$ 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,

${\displaystyle {\dot {c}}=-{\frac {u_{c}(c)}{c\cdot u_{cc}(c)}}\left[f_{k}(k)-\delta -\rho \right]\cdot c}$

which is known as the Keynes–Ramsey rule. [5] The term ${\displaystyle f_{k}(k)-\delta }$, where ${\displaystyle f_{k}}$ is short-hand notation for the marginal product of capital ${\displaystyle {\tfrac {\partial f}{\partial k}}}$, reflects the marginal return on net investment. The expression ${\displaystyle -\left.u_{c}(c)\right/c\cdot u_{cc}(c)}$ 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. ${\displaystyle \sigma =-\left.c\cdot u_{cc}(c)\right/u_{c}(c)>0}$.

Together both differential equations describe the Ramsey–Cass–Koopmans dynamical system. The steady state, which is found by setting ${\displaystyle {\dot {k}}}$ and ${\displaystyle {\dot {c}}}$ equal to zero, is given by the pair ${\displaystyle (k^{\ast },c^{\ast })}$ implicitly defined by

${\displaystyle f_{k}\left(k^{\ast }\right)=\delta +\rho \quad {\text{and}}\quad c^{\ast }=f\left(k^{\ast }\right)-(n+\delta )k^{\ast }}$

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 ${\displaystyle (k^{\ast },c^{\ast })}$ by a first-order Taylor polynomial

${\displaystyle {\begin{bmatrix}{\dot {(k-k^{\ast })}}\\{\dot {(c-c^{\ast })}}\end{bmatrix}}\approx \mathbf {J} (k^{\ast },c^{\ast }){\begin{bmatrix}(k-k^{\ast })\\(c-c^{\ast })\end{bmatrix}}}$

where ${\displaystyle \mathbf {J} (k^{\ast },c^{\ast })}$ is the Jacobian matrix evaluated at steady state, [note 4] given by

${\displaystyle \mathbf {J} \left(k^{\ast },c^{\ast }\right)={\begin{bmatrix}\rho -n&-1\\{\frac {1}{\sigma }}f_{kk}(k)\cdot c^{\ast }&0\end{bmatrix}}}$

which has determinant ${\displaystyle \left|\mathbf {J} \left(k^{\ast },c^{\ast }\right)\right|={\frac {1}{\sigma }}f_{kk}(k)\cdot c^{\ast }<0}$ since ${\displaystyle c}$ is always positive, ${\displaystyle \sigma }$ is positive by assumption, and only ${\displaystyle f_{kk}}$ is negative since ${\displaystyle f}$ 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]

${\displaystyle \lim _{t\to \infty }k\cdot e^{-\int _{0}^{t}\left(f_{k}-n-\delta \right)\mathrm {d} s}\geq 0}$

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
${\displaystyle H=e^{-\rho t}u(c)+\mu \left[f(k)-(n+\delta )k-c\right]}$
where ${\displaystyle \mu }$ is the costate variable usually economically interpreted as the shadow price. Because the terminal value of ${\displaystyle k}$ is free but may not be negative, a transversality condition ${\displaystyle \lim _{t\to \infty }\mu \cdot k=0}$ 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
${\displaystyle \mathbf {J} \left(k,c\right)={\begin{bmatrix}{\frac {\partial {\dot {k}}}{\partial k}}&{\frac {\partial {\dot {k}}}{\partial c}}\\{\frac {\partial {\dot {c}}}{\partial k}}&{\frac {\partial {\dot {c}}}{\partial c}}\end{bmatrix}}={\begin{bmatrix}f_{k}(k)-(n+\delta )&-1\\{\frac {1}{\sigma }}f_{kk}(k)\cdot c&{\frac {1}{\sigma }}\left[f_{k}(k)-\delta -\rho \right]\end{bmatrix}}}$
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.

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## References

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3. Koopmans, T. C. (1965). "On the Concept of Optimal Economic Growth". The Economic Approach to Development Planning. Chicago: Rand McNally. pp. 225–287.
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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.
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