Overlapping Generations and the Golden Rule

See life in generations: earn while young, spend while old. The young generation and the old never actually meet.

The young generation has plenty and the old generation has little. Their lifetimes never overlap, so they cannot trade directly. 💰 Money builds the bridge between them. The young trade goods for money, and spend that money when old.

For example, a grandfather sold 🍚 rice for money when he was young. Now old, he buys rice from a young farmer with that money. When that farmer grows old, he will buy from the next generation. This is how money links the generations.

s=δe1+δ,(cy,co)=(e1+δ,δe1+δ)s = \frac{\delta e}{1+\delta}, \quad (c_y, c_o) = \left(\frac{e}{1+\delta}, \frac{\delta e}{1+\delta}\right)

Examples & Exercises

In an OLG economy with young endowment e=10e = 10 and δ=0.6\delta = 0.6, find the monetary steady state and the golden rule.

s=δe1+δ=3.75,(cy,co)CE=(6.25, 3.75)s = \frac{\delta e}{1+\delta} = 3.75, \quad (c_y, c_o)^{CE} = (6.25,\ 3.75)
(cy,co)GR=(5, 5)(c_y, c_o)^{GR} = (5,\ 5)
e=10e = 10, δ=0.6\delta = 0.6 — CE (6.25,3.75)(6.25, 3.75) vs golden rule (5,5)(5, 5).
Show proof

  1. The young's problem max⁡log⁡(e−s)+δlog⁡s\max \log(e-s) + \delta \log s has FOC −1/(e−s)+δ/s=0-1/(e-s) + \delta/s = 0, giving s=δe/(1+δ)=3.75s = \delta e/(1+\delta) = 3.75. In steady state, (cy,co)=(10−3.75, 3.75)=(6.25,3.75)(c_y, c_o) = (10 - 3.75,\ 3.75) = (6.25, 3.75).

  2. The golden rule maximizes log⁡cy+log⁡co\log c_y + \log c_o subject to cy+co=10c_y + c_o = 10, giving (5,5)(5, 5). Since δ=0.6<1\delta = 0.6 < 1, the two differ.

In the example above, compare steady-state welfare under the golden-rule (equal-weight) criterion and explain what the gap suggests.

WGR=log⁡cy+log⁡coW^{GR} = \log c_y + \log c_o
WGR(5,5)=2log⁡5≈3.219>log⁡6.25+log⁡3.75≈3.154=WGR(CE)W^{GR}(5,5) = 2\log 5 \approx 3.219 > \log 6.25 + \log 3.75 \approx 3.154 = W^{GR}(CE)
Welfare gap ≈0.064\approx 0.064 — the infinite-horizon warning.
Show proof

  1. Golden-rule welfare is 2log⁡5≈3.2192 \log 5 \approx 3.219 at the golden rule vs. log⁡6.25+log⁡3.75≈3.154\log 6.25 + \log 3.75 \approx 3.154 at the monetary equilibrium — a gap of about 0.0640.064.

  2. In finite economies competitive equilibrium is Pareto efficient (1st theorem), but in infinite-horizon OLG the steady state can fall short of the golden rule — infinite consumers and goods break the 1st theorem's proof.

16. Equilibrium and Time