Section 2

Historical Context

The Ramsey-Cass-Koopmans model has the unusual distinction of being named for one author who wrote a brilliant paper in 1928 and two authors who rediscovered and extended it three and a half decades later.

Timeline

Year	Author	Contribution
1928	Frank P. Ramsey	A planner's problem of choosing the saving rate to maximise an undiscounted utility integral, with a finite 'bliss' level of consumption. Published in the Economic Journal at age 25.
1956	Robert Solow & Trevor Swan	Independently derive the exogenous-savings growth model. Becomes the dominant framework but is silent on why households save what they save.
1958	Paul Samuelson	Overlapping generations model (Section 2.1.3). A different way to microfound saving, with finite-lived agents.
1965	David Cass	Reformulates Ramsey's problem with discounting and a neoclassical production function. The modern phase-diagram treatment originates here.
1965	Tjalling Koopmans	Independently delivers an equivalent formulation, with axiomatic foundations for discounted utility. Published in the same volume as Cass.
1970s-80s	Real Business Cycle pioneers	Kydland & Prescott (1982) embed RCK preferences and technology into stochastic dynamic general equilibrium. RCK becomes the backbone of quantitative macroeconomics.
1986-88	Romer, Lucas	Endogenous growth - relax the assumption that long-run growth is exogenous (Section 14).

Ramsey's 1928 paper was so ahead of its time that the field had to catch up to it. Keynes called it 'one of the most remarkable contributions to mathematical economics ever made'.

Why Ramsey's original problem was hard

Ramsey did not discount future utility. He considered it 'ethically indefensible' for a planner to weight her own welfare more heavily than that of her descendants. Without discounting, $\int_0^\infty u(c_t) dt$ diverges, so he assumed a *bliss point* $B$ - a finite maximum utility - and minimised $\int_0^\infty (B - u(c_t))\, dt$ . The famous **Keynes-Ramsey rule** falls out of this minimisation.

Ramsey (1928) result: The marginal utility of consumption falls at rate $\dot u'(c)/u'(c) = -(f'(k) - \delta)$ along the optimal path. Without discounting, the economy saves enough to drive capital all the way to the *golden rule* level $f'(k_{GR}) = n + \delta$ .
Cass-Koopmans (1965) modification: Add discounting at rate $\rho > 0$ . The Euler equation becomes $\dot c/c = (1/\theta)(f'(k) - \delta - \rho)$ and the steady state shifts to $f'(k^*) = \rho + \delta > n + \delta$ . The economy stops short of the golden rule - hence the **modified** golden rule.