Ramsey–Cass–Koopmans model
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.
[note 1] Originally, Ramsey defined the model as a social planner's problem of maximizing consumption levels over successive generations.
[4] Only later was a model adopted by Cass and Koopmans as a description of a decentralized dynamic economy with a representative agent.
The Ramsey–Cass–Koopmans model aims only at explaining long-run economic growth rather than business cycle fluctuations and does not include sources of disturbances like market imperfections, heterogeneity among households, or exogenous shocks.
Subsequent researchers extended the model, allowing for government purchases, employment variations, and other shocks, notably in real business cycle theory.
The labour force, which makes up the entire population, is assumed to grow at a constant rate
The variables that the Ramsey–Cass–Koopmans model ultimately aims to describe are the per capita (or more accurately, per labour) consumption:
is required to be homogeneous of degree 1, which economically implies constant returns to scale.
Assume that the economy is populated by identical immortal individuals with unchanging utility functions
The utility function is assumed to be strictly increasing (i.e., there is no bliss point) and concave in
Thus, we have the social planner's problem: where an initial non-zero capital stock
The quantity reflects the curvature of the utility function and indicates how much the representative agent wishes to smooth consumption over time.
If the agent has high relative risk aversion, it has low EIS and thus would be more willing to smooth consumption over time.
A higher value implies that the agent prioritizes saving over consuming today, thereby deferring consumption later.
As it turns out, the optimal trajectory is the unique one that converges to the interior equilibrium point.
[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 6] 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).
The priority issue became a discussion point because, 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 he finds.
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 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.
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 that do not converge to the optimal steady-state would hit either a zero consumption or zero capital boundary in finite time.
At the end of the presentation, Koopmans asks Malinvaud whether it is not the case that Condition I guarantees that solutions to the Euler equations that do not converge to the optimal steady-state hit a boundary in finite time.
However, 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.
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 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 are the one that converges to the optimal steady-state equilibrium of the model.
[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.
