Ramsey-Cass-Koopmans Model

Section 4 of 16

Ramsey-Cass-Koopmans Model

Section 4 of 16

Section 4

Assumptions

RCK inherits Solow's technological assumptions and adds preference assumptions. Each assumption below has a specific job in the proof.

Technology (production side)

Label	Assumption	Why it matters
T1	$F(K, L)$ is twice continuously differentiable.	Marginal products are well-defined.
T2	$F$ exhibits constant returns to scale: $F(\lambda K, \lambda L) = \lambda F(K, L)$ .	Permits the per-worker reduction $y = f(k)$ .
T3	Positive and diminishing marginal products: $f'(k) > 0$ , $f''(k) < 0$ .	Ensures a unique steady state and concavity of the planner's value function.
T4	Inada conditions: $\lim_{k\to 0} f'(k) = \infty$ , $\lim_{k\to\infty} f'(k) = 0$ .	Guarantees an interior steady state with $k^* > 0$ .
T5	Capital depreciates at constant rate $\delta \in (0, 1]$ .	Closes the capital accumulation identity $\dot K = I - \delta K$ .
T6	For the canonical exposition, no technological growth ( $g = 0$ ). Easily relaxed.	Keeps the algebra clean. With $g > 0$ , work in effective-worker units and replace $\rho$ with $\rho - (1-\theta) g$ in the Euler equation.

Demography and markets

Label	Assumption	Why it matters
M1	Population $L(t) = L_0 e^{n t}$ , with $n \geq 0$ .	Per-worker reduction; one source of dilution.
M2	Perfectly competitive factor markets.	Factor prices equal marginal products: $r = f'(k) - \delta$ , $w = f(k) - k f'(k)$ .
M3	Complete asset markets with one safe asset (capital).	Household holds wealth $a = k$ ; arbitrage gives $r$ unique.
M4	No taxes, no government - in the baseline.	Added later for policy analysis.

Preferences (household side)

Label	Assumption	Why it matters
P1	A representative dynastic household of mass $L(t)$ at time $t$ .	Internalises all future generations - no overlapping-generations issues. Aggregation is trivial.
P2	Lifetime utility: $U_0 = \int_0^\infty e^{-\rho t}\, u(c(t)) L(t)\, dt$ .	Discounted time-separable preferences. The discount rate $\rho > 0$ encodes pure impatience.
P3	Per-period utility is CRRA: $u(c) = (c^{1-\theta} - 1)/(1-\theta)$ for $\theta > 0$ , $\theta \neq 1$ , and $u(c) = \log c$ for $\theta = 1$ .	Constant intertemporal elasticity of substitution $1/\theta$ . Required for balanced-growth compatibility.
P4	Transversality / finiteness: $\rho - n - (1 - \theta) g > 0$ .	Without this, the utility integral diverges and no optimum exists. For $g = 0$ this reduces to $\rho > n$ .
P5	Perfect foresight.	Household knows the entire path of prices $\{r(t), w(t)\}$ . In equilibrium these paths are the ones it conjectures.