Section 3

First Principles - Why Endogenise Savings

Solow's exogenous savings rate $s$ is convenient but unsatisfying for three reasons. Each motivates a piece of the RCK construction.

Problem 1 - Solow is silent on welfare

In Solow, any $s \in (0, 1)$ delivers a steady state. We can compute consumption at each, and we know that consumption is maximised at the golden rule savings rate $s_{GR}$ where $f'(k^*) = n + g + \delta$ . But Solow gives no reason households would choose $s_{GR}$ rather than 0.1 or 0.9. Welfare statements are assertions, not consequences of the model.

\text{Solow:}\quad k^* \text{ depends on } s.\qquad \text{RCK:}\quad k^* \text{ depends on } \rho.

In RCK, the long-run capital stock is a function of household impatience, not of an arbitrary saving propensity.

Problem 2 - Solow cannot handle policy

Consider an investment subsidy or a capital-income tax. In Solow, the response of the savings rate to the policy is whatever the modeller assumes. In RCK, the response is derived from the Euler equation, so policy analysis can be done rigorously.

Question	Solow can say?	RCK can say?
Effect of a capital-income tax	Only if you stipulate how $s$ responds.	Tax enters Euler equation directly; comparative statics in closed form.
Optimal long-run policy	Cannot define optimality without preferences.	Optimal policy implements the unrestricted first-best.
Effect of an expected productivity boom	$s$ is fixed, no anticipation effects.	$c$ jumps today on news about tomorrow - forward-looking.
Welfare gain from converging to steady state	Not defined.	Direct from the utility integral.

Endogenising savings is what makes welfare and policy analysis possible.

Problem 3 - Solow has no transversality

The Solow model is a single first-order ODE in $k$ . Given $k_0$ , the path is determined. RCK is a system of two ODEs in $(k, c)$ with one boundary condition $k(0) = k_0$ . We need one more condition to pin down $c(0)$ . That condition is the transversality condition - a no-Ponzi-game restriction that the household cannot run unbounded debt forever. Without it, the household over-saves or over-consumes and the optimisation is ill-defined.

The conceptual change in three steps

Step 1
$\text{Solow:}\quad s\,\text{ given} \;\;\Rightarrow\;\; \dot k = s f(k) - (n + g + \delta) k$
A saving rate yields a path of capital.
Step 2
$\text{RCK step 1:}\quad U = \int_0^\infty e^{-\rho t} u(c)\, dt$
Treat the household as an optimiser. Discount future utility at rate $\rho$ .
Step 3
$\text{RCK step 2:}\quad \max U \;\;\text{subject to}\;\; \dot k = f(k) - c - (n + \delta) k$
Choose the consumption path that maximises lifetime utility given the capital accumulation constraint.
Step 4
$\text{RCK step 3:}\quad \dot c / c = (1/\theta)\bigl(f'(k) - \delta - \rho\bigr)$
The Euler equation - the central result. Consumption grows when the (net) return on capital exceeds the discount rate. Here $\theta > 0$ is the coefficient of relative risk aversion, equivalently the inverse of the intertemporal elasticity of substitution: it measures how strongly the household prefers a smooth consumption path. A high $\theta$ means consumption responds weakly to the return gap (strong smoothing motive); a low $\theta$ means it responds sharply.

Replaces $s$ with $(\rho, \theta)$ - preferences instead of behaviour.
Adds the Euler equation as a second dynamic equation.
Adds the transversality condition as a second boundary condition.
Gains welfare analysis, anticipation effects, and rigorous policy comparative statics.
Loses simplicity - saddle paths are harder than monotone convergence.