econ.studio
Diamond Overlapping Generations Model
Section 11 of 16
Section 11

Dynamic Inefficiency

The golden rule recap

Sustainable consumption per young worker is c(k)=f(k)−(n+δ)kc(k) = f(k) - (n+\delta)k. The level of kk that maximises cc — the *golden rule* — solves f′(kGR)=n+δf'(k_{GR}) = n + \delta. With δ=1\delta = 1 this simplifies to f′(kGR)=1+nf'(k_{GR}) = 1 + n. Under Cobb–Douglas:

kGR=(αAn+δ)1/(1−α),kGR=(αA1+n)1/(1−α) (if δ=1).k_{GR} = \left(\frac{\alpha A}{n + \delta}\right)^{1/(1-\alpha)}, \qquad k_{GR} = \left(\frac{\alpha A}{1+n}\right)^{1/(1-\alpha)} \text{ (if }\delta = 1\text{)}.
eq:olg-kgr

When is the OLG steady state inefficient?

Compare k∗k^* from Eq. (eq:olg-kstar) with kGRk_{GR} (taking δ=1\delta = 1 for the closed form):

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

    Steady-state capital (from Section 9).

  2. Step 2
    kGR=[αA1+n]1/(1−α)k_{GR} = \left[\frac{\alpha A}{1+n}\right]^{1/(1-\alpha)}

    Golden-rule capital.

  3. Step 3
    k∗kGR=[β(1−α)α(1+β)]1/(1−α)\frac{k^*}{k_{GR}} = \left[\frac{\beta(1-\alpha)}{\alpha(1+\beta)}\right]^{1/(1-\alpha)}

    Take the ratio; AA and (1+n)(1+n) cancel.

Therefore the economy is **dynamically inefficient** (k∗>kGRk^* > k_{GR}) if and only if the *bracket* exceeds one:

  k∗>kGR    ⟺    β(1−α)>α(1+β)    ⟺    β>α1−2α(α<1/2)  \boxed{\; k^* > k_{GR} \;\iff\; \beta(1-\alpha) > \alpha(1+\beta) \;\iff\; \beta > \frac{\alpha}{1 - 2\alpha}\quad (\alpha < 1/2) \;}
eq:olg-inefficiency-condition
The condition for dynamic inefficiency depends *only* on the preference parameter β\beta and the capital share α\alpha — not on TFP, population growth, or initial conditions.
α\alphaThreshold β†\beta^\daggerRegion of dynamic inefficiency
0.200.333β>0.333\beta > 0.333 ⇒ inefficient.
0.250.500β>0.500\beta > 0.500 ⇒ inefficient.
0.300.750β>0.750\beta > 0.750 ⇒ inefficient.
0.330.971β>0.971\beta > 0.971 ⇒ inefficient (just barely possible).
0.40∞\inftyNever inefficient — 1−2α<01 - 2\alpha < 0.
0.50—Threshold breaks down; k∗k^* always below kGRk_{GR} for β<∞\beta < \infty.
Numerical sanity check: which (α,β)(\alpha, \beta) combinations produce dynamic inefficiency? The threshold for β\beta is α/(1−2α)\alpha / (1 - 2\alpha).

Why does inefficiency arise?

Recall the Ramsey modified-golden-rule condition: f′(kRCK∗)=ρ+δ>n+δf'(k^*_\text{RCK}) = \rho + \delta > n + \delta, so kRCK∗<kGRk^*_\text{RCK} < k_{GR} — RCK never over-accumulates. The infinitely-lived household *internalises* the trade-off between saving and future generations' consumption because *its own* consumption is what is at stake.

Diamond agents see no such trade-off. Each cohort decides only how much to consume today versus when old — never how much to consume versus how much to leave for unborn cohorts. They save *only* for their own retirement, and that motive can be strong enough to drive aggregate k∗k^* above the golden rule.

The dynamic-efficiency gap (live)

No scalar found for key: golden_rule_k
No scalar found for key: dynamic_efficiency_gap

Visualising the inefficient region

Sustainable consumption $c(k) = f(k) - (n+\delta)k$