Section 3

First Principles

The thought experiment

Consider an economy in steady state where the capital stock is *above* the golden rule level $k_{GR}$ . Reducing $k$ to $k_{GR}$ would raise sustainable consumption $c^* = f(k^*) - (n+\delta)k^*$ . If agents are infinitely lived (Ramsey), they will eventually internalise this and reduce savings. If agents live for only two periods, the chain is broken: nobody alive today *sees* the higher consumption that would prevail forever after the adjustment — that benefit accrues to cohorts not yet born. They cannot bid for it; they cannot pay current agents to save less. The market produces no mechanism to correct the over-accumulation.

Demographic structure

Time runs $t = 0, 1, 2, \ldots$ . At each date $t$ there are two generations alive: the *young* cohort born at $t$ , and the *old* cohort born at $t - 1$ . The young cohort at $t$ has size $L_t$ and grows at rate $n$ :

L_t = (1 + n) L_{t-1}, \qquad n \ge 0.

eq:popgrowth

Total population at date $t$ is $L_t + L_{t-1} = L_t(1 + 1/(1+n))$ . Each agent supplies one unit of labour when young and zero when old.

The life cycle

Period of life	Action	Constraint
Young (age 1)	Earn wage $w_t$ ; consume $c_{1,t}$ ; save $s_t$ .	$c_{1,t} + s_t = w_t$
Old (age 2)	Consume $c_{2,t+1}$ funded by savings and interest.	$c_{2,t+1} = (1 + r_{t+1}) s_t$

What each agent does in each period of life.

Critically, the old leave no bequests in the baseline model. They consume everything and die. This is the source of all the interesting departures from Ramsey: there is no infinite-horizon smoothing motive.

From individual to aggregate

Aggregate savings: Only the young save. The old dissave (consume their savings plus interest) but those funds were already in the capital stock. Net aggregate saving at $t$ is $L_t s_t$ — the savings of the young.
Next-period capital: With full depreciation per period (the standard textbook case), $K_{t+1} = L_t s_t$ . Per young person: $k_{t+1} = s_t / (1 + n)$ .
The fundamental dynamic equation: $k_{t+1}$ is determined by the savings of today's young, which depends on $w_t$ (their income) and $r_{t+1}$ (their return). Both are determined by $k_{t+1}$ in general equilibrium — leading to an implicit equation for $k_{t+1}$ as a function of $k_t$ .

Why this differs from Ramsey

Feature	Ramsey–Cass–Koopmans	Diamond OLG
Horizon	Infinite	Two periods per cohort
Agents alive at $t$	Single representative	Two overlapping cohorts
Savings driver	Intertemporal Euler over all $\infty$ dates	Two-period Euler within a life
Welfare theorem holds?	Yes (under TVC)	Not necessarily — market between current and unborn is missing
Steady state efficient?	Always (RCK saves less than GR)	Possibly not — can over-accumulate
Role for govt debt?	Ricardian: irrelevant	Non-Ricardian: alters capital path
Role for PAYG SS?	Reduces capital, lowers welfare	Can raise welfare if dynamically inefficient

Same building blocks, different welfare implications.