Solow’s model explains why a poor country grows fast as it catches up. It cannot explain why the richest economies keep growing, decade after decade, with no sign of slowing. Making long-run growth something the economy produces, not something handed down from outside, is the work of endogenous growth theory — the topic second-years bring most often to an economics tutor in the Bay Area.
1 · Why ideas are not like other goods
Growth runs on knowledge, a strange good. An idea is non-rival: a formula or a line of code can be used by everyone at once, and your using it does not use it up. It is also only partially excludable — a patent or secrecy fences it off for a while, but once known it leaks.
An idea costs a great deal to create and almost nothing to copy, so under perfect competition price falls to the near-zero cost of a copy — and the inventor never recovers the fixed cost of inventing. Perfect competition cannot pay for ideas. Some market power, such as a patent’s temporary monopoly, is what makes research worth doing. This appropriability problem is why innovation and growth are one subject.
2 · Where the Solow benchmark runs out
In Solow, capital accumulates but runs into diminishing returns, so the economy settles at a steady state. The limitation that matters here: the only source of permanent growth in output per worker is exogenous technological progress — productivity rising for unexplained reasons.
Raise the saving rate and you reach a richer steady state, a higher income level, but growth soon returns to that exogenous rate. Saving is a level effect, never a growth effect. Endogenous growth asks: what if the driver of productivity is produced inside the economy, and does not diminish?
3 · The AK model: the simplest engine of sustained growth
Take the sharpest version. Let output be proportional to capital, Y = A K, with “capital” read broadly — machines plus the human capital and knowledge that accumulate alongside them. Read that way, the bundle escapes diminishing returns.
Track capital through one year: saving adds sY = sAK, depreciation removes δK, so ΔK = (sA − δ)K. Dividing by K, capital — and therefore output, since Y = AK — grows at the constant rate
g = sA − δ
Notice what is missing: no capital stock K. In Solow, more capital lowers the marginal product and growth peters out; here it is fixed at A, so accumulation drives growth forever. And because g depends on s, a higher saving rate lifts growth permanently — saving is now a growth effect. Constant returns to a reproducible factor is the whole difference.
4 · Ideas, researchers, and the honest caveats
The AK model is a reduced form; it does not say where A comes from. Romer’s answer is: from people. Put a share of the workforce into research; each new idea raises everyone’s productivity, because ideas are non-rival — one blueprint lifts every factory that uses it. More researchers, faster growth.
Two caveats keep you honest. The scale effect: simple versions imply a bigger economy should grow faster — yet large countries do not outgrow small ones, and rising research effort has not lifted growth (later “semi-endogenous” models fix this by letting ideas get harder to find). And policy: because ideas are non-rival and only partly excludable, private research falls short of the social optimum — the case for R&D subsidies and for patents, which grant temporary monopoly power so invention pays, at the cost of the deadweight loss.
Worked example — two economies, the same saving rate
Two economies start at output per worker 100. Both save 20% (s = 0.20) and depreciate capital at 5% a year (δ = 0.05). They differ in one respect: economy K has constant returns to broad capital (output-capital ratio A = 0.5); economy S is Solow, with diminishing returns.
Step 1 — AK growth. In economy K, g = sA − δ = 0.20 × 0.5 − 0.05 = 0.05. Output grows 5% a year: 100 → 105 → 110.25 → 115.76, without slowing.
Step 2 — Why it never slows. The rate sA − δ holds no capital stock — double K‘s capital and growth is still 5%. Nothing diminishes to brake it.
Step 3 — Solow growth. Economy S saves the same 20%, but each new unit of capital is less productive than the last. Growth starts brisk — about 4.8% in year one — then fades to zero as output nears its steady state of about 200.
Step 4 — The perturbation. Both economies now permanently double their saving rate to 40% (s = 0.40), everything else fixed.
Step 5 — The two responses. Economy K: g = 0.40 × 0.5 − 0.05 = 0.15 — growth jumps from 5% to 15% and stays there. Economy S: its steady state climbs to about 283, then growth returns to zero.
Step 6 — Interpretation. Same policy, two different effects. In Solow it bought a one-time enrichment — a higher level — then growth died: a level effect. In the AK economy it bought permanently faster growth. That is the payoff of endogenous growth theory: to move the long-run growth rate, not just the level, some reproducible factor must escape diminishing returns.
Can you say why doubling the saving rate buys permanently faster growth in one economy and only a one-off level gain in the other? That contrast — g = sA − δ against the Solow steady state — is the distinction growth questions reward most. A one-on-one economics tutor derives both responses with you from the law of motion until the level-versus-growth line is second nature. Book a trial session.
Practice
Q1. An AK economy has A = 0.4, s = 0.25 and δ = 0.03. (a) Find the long-run growth rate of output. (b) An R&D subsidy raises effective A to 0.5, saving unchanged — find the new rate.
Q2. Two economies start at output per worker 100 and both raise their saving rate. Economy S (Solow) sees steady-state output rise from 100 to 150, with zero long-run growth; economy K (AK) grows at 3% a year, and the same rise lifts it to 5%. (a) Economy S‘s long-run growth rate afterwards? (b) Economy K‘s output after 20 years at each rate? (c) Which effect compounds?
Answers. Q1: (a) 0.25 × 0.4 − 0.03 = 7%; (b) 0.25 × 0.5 − 0.03 = 9.5% — lifting A raises growth exactly as lifting s would, because only the product sA enters g. Q2: (a) 0%, a pure level effect; (b) 100 × 1.03²⁰ = 180.6, 100 × 1.05²⁰ = 265.3; (c) the growth effect (economy K) — economy S‘s level gain is one-off.
Key takeaways
- Ideas are non-rival and only partly excludable, so competitive markets underpay for them — the appropriability problem that ties innovation to growth.
- Solow’s limit: without exogenous technology, growth stops; saving changes the income level, not the growth rate.
- The AK model removes the brake: constant returns to broad capital give g = sA − δ, with no capital stock in it — so growth is self-sustaining, and policy that raises s or A raises the growth rate itself, not just the level.
- Scale effects are the weak point; non-rivalry is the case for R&D subsidies and patents.
Why Bay Area students choose our economics tutoring
- Models rebuilt from the ground up: you derive g = sA − δ from the capital-accumulation identity, so you can reconstruct it under exam pressure, not quote a half-remembered formula.
- The distinctions examiners reward, drilled: level versus growth effect, rivalry versus excludability, Solow versus AK, private versus social return to research.
- One-on-one, matched to your syllabus: tutors work from your own notation and past papers, whether your course follows Jones, Weil or Acemoglu.
- In person or fully online: exam prep that fits around your course and timetable.
FAQ
Q: What is the difference between the Solow model and endogenous growth?
A: In Solow, long-run growth comes only from unexplained technological progress, and saving lifts just the income level. Endogenous growth makes productivity growth come from inside the economy — from research and accumulated knowledge — so saving and policy can move the growth rate itself.
Q: Why can’t perfect competition reward innovation?
A: An idea costs a great deal to create but almost nothing to copy, so competition drives price to near zero and the inventor never recovers the fixed cost of inventing. Some market power, such as a patent, is needed for research to pay.
Q: What does the “AK” in the AK model mean?
A: It is the production function Y = AK — output proportional to capital, with A the constant output-capital ratio. Because returns to capital do not diminish, accumulation raises output without limit, giving the constant growth rate g = sA − δ.
Q: What is a scale effect, and why is it a problem?
A: Simple endogenous growth models predict a larger economy, with more researchers, should grow faster. The data do not show that, so it is a known limitation later models correct by making ideas harder to find.
Q: How do patents fit in?
A: A patent grants temporary monopoly power so an inventor can charge above marginal cost and recover research costs — at the price of the deadweight loss while the patent lasts. Patent length balances rewarding invention against restricting use.
Book an economics tutor in the Bay Area or online
Endogenous growth rewards the student who can build the engine — the appropriability problem, the AK law of motion, g = sA − δ, and the level-versus-growth distinction — not just recite a result. One-on-one sessions build that fluency on your own course and past papers. Tell us your module and exam date, and we will match you with the right tutor this week.
