Section 10

Steady State

Plugging $k^*$ from Eq. (eq:olg-kstar) back into the production and factor-price equations gives a complete characterisation:

Quantity	Formula	Note
Capital	$k^* = \phi^{1/(1-\alpha)}$	Eq. (eq:olg-kstar).
Output	$y^* = A (k^*)^{\alpha}$	CD production.
Wage	$w^* = (1-\alpha) y^*$	Labour share is $1-\alpha$ .
Gross return	$1 + r^* = \alpha A (k^*)^{\alpha - 1}$	MPK at $k^*$ (with $\delta = 1$ ).
Savings	$s^* = \dfrac{\beta}{1+\beta} w^*$	Log utility ⇒ constant savings rate.
Young cons.	$c_1^* = \dfrac{1}{1+\beta} w^*$	Residual from young budget.
Old cons.	$c_2^* = (1 + r^) s^$	From the old budget.

Steady-state values under log utility, Cobb–Douglas, full depreciation. Define

\phi \equiv \beta(1-\alpha)A / [(1+n)(1+\beta)]

so that

k^* = \phi^{1/(1-\alpha)}

It is often useful to express the steady-state aggregate savings rate $S^*/Y^*$ in terms of primitives:

Step 1
$\frac{S^*}{Y^*} = \frac{L_t s^*}{L_t y^*} = \frac{s^*}{y^*}$
Aggregate over the young cohort.
Step 2
$= \frac{\beta}{1+\beta} \cdot \frac{w^*}{y^*}$
Substitute the savings function.
Step 3
$= \frac{\beta(1-\alpha)}{1+\beta}$
Use $w^*/y^* = 1 - \alpha$ (labour share).

\boxed{\; \frac{S^*}{Y^*} = \frac{\beta(1-\alpha)}{1+\beta} \;}

eq:olg-savings-rate

The steady-state savings rate is a pure function of preferences (

\beta

) and technology (

\alpha

). It is *independent* of

n

A

, and

k_0

The values below recompute live as you adjust parameters. Compare them with the comparative-statics table in Section 12.

No scalar found for key: steady_state_k

No scalar found for key: steady_state_y

No scalar found for key: steady_state_w

No scalar found for key: steady_state_r

No scalar found for key: steady_state_c1

No scalar found for key: steady_state_c2