Growth · Interactive
The Solow model and the steady state
An economy with too little capital saves more than it needs to replace, so capital per worker rises on its own and settles exactly where saving stops outrunning break-even investment. Move the saving rate, population growth or depreciation and watch that resting point move.
k* 5.20 · y* 1.73
Output, f(k)Saving, s·f(k)Break-even, (n+δ)kCurrent point
The maths, with your numbers
Output per workerf(k) = k^(1/3)y = k^0.33
Steady-state capitalk* = (s / (n+δ))^1.55.20
Steady-state outputy* = (k*)^(1/3)1.73
Net accumulation at k₀Δk = s·f(k₀) − (n+δ)k₀+0.12
1
Start at rest
At k*, saving per worker exactly covers break-even investment, what's needed to replace depreciated capital and equip new workers. Capital per worker neither rises nor falls.
2
Below the steady state, capital grows
Start with less capital than that and saving outruns break-even investment. Capital per worker rises every period, pulling the economy back up toward k*.
3
Raise the saving rate
Raise s and the steady state itself moves out. The capital stock that used to be at rest is now below the new k*, so convergence starts again, this time toward a permanently higher level of capital and output.
4
More capital isn't automatically more consumption
Let the economy settle at the new steady state and compare consumption, not just output, with where it started. Raising s always means saving a bigger slice of a bigger pie, not banking the whole gain. Check c*; don't assume it.
Reading the diagram
What moves, and why
Output, f(k), never moves
How much a worker produces from a given amount of capital is a property of technology, not policy. Changing the saving rate, population growth or depreciation shifts the other two curves; this one stays put.
The gap between the curves is what moves capital
Wherever saving, s·f(k), sits above the break-even line, (n+δ)k, capital per worker is rising. Wherever it sits below, capital is falling. The steady state, k*, is the one point where they exactly meet.
A policy change moves the target, not just the position
Raising the saving rate lifts the whole saving curve, so k* itself moves out. The economy doesn't jump there. It starts converging again, from wherever it currently is, toward the new resting point.
Watch out for
Where this trips people up on the paper
- A higher saving rate raises output, not automatically consumption. More is saved out of a bigger pie, but not all of the gain: check c* = (1−s)y*, not just y*, before calling a higher saving rate an improvement.
- The steady state is a resting point, not a ceiling. An economy above k* is not “doing well”. It is running capital down toward k*, same as one below it is building capital up. Distance from k* says nothing about which side is better off.
- “Growth” in this model is a one-time move to a new level. Raising s makes the economy grow faster only while converging to the new, higher k*. Once there, growth per worker stops again, without ongoing technological progress.
Go deeper
The full lessons behind this diagram
This page is the quick look. The Solow–Swan model, lesson 5 derives the capital accumulation equation, works past-paper steady states by hand and explains why the economy is stable. It sits in the economic growth module, which also covers Harrod–Domar and growth accounting.
Try it on real questions
From CUET-PG and GATE
Five questions on exactly this: what holds at the steady state, the Golden Rule, and two numeric steady-state problems.
These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.
Read next
All resources →MacroeconomicsGrowth 5: the Solow–Swan model, capital accumulation and the steady stateWhy diminishing returns bring a steady state, the capital accumulation equation k̇ = sf(k) − (n + δ)k, stability on the Solow diagram, steady-state growth rates, and the difference from Harrod–Domar.MacroeconomicsIS–LM: fiscal and monetary shocksRaise government spending or the money supply and watch IS and LM shift, then see exactly how much gets crowded out.MacroeconomicsThe Phillips curve, short run and long runPush unemployment below its natural rate, then watch expectations catch up and pull it back, step by step.