econ.studio
Diamond Overlapping Generations Model
Section 9 of 16
Section 9

Capital Accumulation

Aggregating savings into capital

Aggregate savings of the young cohort at tt is LtstL_t s_t. With full depreciation (δ=1\delta = 1), the old eat everything they have and the entire next-period capital stock comes from the young's savings:

Kt+1=Ltst.K_{t+1} = L_t s_t.
eq:agg-capital

Divide both sides by Lt+1=(1+n)LtL_{t+1} = (1+n) L_t to convert to per-young-worker terms:

kt+1=Kt+1Lt+1=Ltst(1+n)Lt=st1+n.k_{t+1} = \frac{K_{t+1}}{L_{t+1}} = \frac{L_t s_t}{(1+n)L_t} = \frac{s_t}{1+n}.
eq:per-worker-capital
Tomorrow's capital per young worker = today's savings, scaled down by cohort growth.

The law of motion (log utility + Cobb–Douglas)

Plug the log-utility savings rule st=β/(1+β) wts_t = \beta/(1+\beta)\, w_t and the CD wage wt=(1−α)Aktαw_t = (1-\alpha)A k_t^{\alpha} into Eq. (eq:per-worker-capital):

  1. Step 1
    kt+1=st1+nk_{t+1} = \frac{s_t}{1+n}

    Start from per-young-worker capital.

  2. Step 2
    =11+n⋅β1+βwt= \frac{1}{1+n} \cdot \frac{\beta}{1+\beta} w_t

    Substitute the log-utility savings function.

  3. Step 3
    =11+n⋅β1+β(1−α)Aktα= \frac{1}{1+n} \cdot \frac{\beta}{1+\beta} (1-\alpha) A k_t^{\alpha}

    Substitute the CD wage.

  kt+1=β(1−α)A(1+n)(1+β)ktα  \boxed{\; k_{t+1} = \frac{\beta (1-\alpha) A}{(1+n)(1+\beta)} k_t^{\alpha} \;}
eq:olg-law-of-motion
The Diamond OLG law of motion under log utility, Cobb–Douglas, and full depreciation. One nonlinear equation in ktk_t.

Define the constant ϕ:=β(1−α)A/[(1+n)(1+β)]\phi := \beta(1-\alpha)A / [(1+n)(1+\beta)]. Then the law of motion is simply kt+1=ϕ ktαk_{t+1} = \phi\, k_t^{\alpha} — a power-function map of the unit interval into itself.

The transition function: shape and properties

PropertyValueImplication
g(0)g(0)00Origin is a fixed point.
g′(k)g'(k)αϕkα−1\alpha \phi k^{\alpha-1}Strictly positive — capital is monotone increasing in itself.
g′(0+)g'(0^+)+∞+\inftySteep at the origin — small kk grows fast (Inada).
g′(∞)g'(\infty)00Flat asymptotically — diminishing returns dominate.
g′′(k)g''(k)α(α−1)ϕkα−2<0\alpha(\alpha-1)\phi k^{\alpha-2} < 0Concave — single intersection with the 45° line.
Fixed pointsk=0k = 0 and k=k∗k = k^*Two fixed points; only k∗k^* is interior and stable.
Properties of g(k):=ϕkαg(k) := \phi k^{\alpha} with α∈(0,1)\alpha \in (0, 1).

Cobweb diagram: the law of motion vs. the 45° line

Existence and uniqueness of the interior steady state

A steady state k∗k^* satisfies k∗=ϕ(k∗)αk^* = \phi (k^*)^{\alpha}. The trivial solution k∗=0k^* = 0 exists. For the interior:

  1. Step 1
    k∗=ϕ(k∗)αk^* = \phi (k^*)^{\alpha}

    Fixed-point condition.

  2. Step 2
    (k∗)1−α=ϕ(k^*)^{1-\alpha} = \phi

    Divide both sides by (k∗)α(k^*)^{\alpha}.

  3. Step 3
    k∗=ϕ1/(1−α)=[β(1−α)A(1+n)(1+β)]1/(1−α)k^* = \phi^{1/(1-\alpha)} = \left[\frac{\beta(1-\alpha)A}{(1+n)(1+\beta)}\right]^{1/(1-\alpha)}

    Solve for k∗k^* — closed form.

  k∗=[β(1−α)A(1+n)(1+β)]1/(1−α)  \boxed{\; k^* = \left[\frac{\beta(1-\alpha)A}{(1+n)(1+\beta)}\right]^{1/(1-\alpha)} \;}
eq:olg-kstar

Stability of k∗k^*

Linearise the law of motion around the steady state by computing g′(k∗)g'(k^*):

  1. Step 1
    g′(k∗)=αϕ(k∗)α−1g'(k^*) = \alpha \phi (k^*)^{\alpha - 1}

    Derivative of the law of motion.

  2. Step 2
    =α⋅k∗k∗=α= \alpha \cdot \frac{k^*}{k^*} = \alpha

    Use ϕ(k∗)α−1=k∗/k∗=1\phi (k^*)^{\alpha-1} = k^*/k^* = 1 from the fixed-point condition.

Since α∈(0,1)\alpha \in (0, 1), we have ∣g′(k∗)∣<1|g'(k^*)| < 1, so k∗k^* is **locally asymptotically stable**. Because gg is monotone, concave and crosses the 45° line only once, k∗k^* is in fact **globally** stable for any k0>0k_0 > 0.

Transition trajectory

Capital per young worker over time