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.

**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.