The Solow–Swan Growth Model is the workhorse model for understanding long-run living standards: why some countries are richer than others, what determines the long-run level of output per person, and why simply “saving more” doesn’t create permanent growth (without technology). By the end of this tutorial, you’ll be able to (1) write down the core equations, (2) draw and interpret the key diagram, and (3) explain steady state, convergence, and the roles of savings, population growth, depreciation, and technological progress.

Goal: what you should be able to do

After working through this, you should be able to:

Define capital per worker (or per effective worker) and the production function.
Derive the capital accumulation equation.
Explain the steady state and how the economy moves toward it.
Predict what happens to income per worker when s, n, δ, or A changes.
Distinguish level effects (one-time increase in income) from growth effects (permanent increase in growth rate).

Prerequisites (quick check)

You’ll be most comfortable if you already know:

Basic production functions and diminishing marginal returns
The difference between levels and growth rates
Simple differential/difference equations intuition (how a variable changes over time)

1) The economic setup (what the model is trying to explain)

The Solow–Swan model is a neoclassical growth model with:

A country produces a single good that can be consumed or invested.
Capital accumulates through investment, but it also depreciates.
Labor grows with population.
Technology can be treated as constant (basic Solow) or growing (augmented Solow with technological progress).
The key mechanism is diminishing returns to capital: adding more machines to a fixed workforce raises output, but by smaller and smaller amounts.

This is why the model is so useful: it delivers a clear story for convergence and steady-state living standards.

2) Core variables and the production function

Let:

(Y(t)) = output (GDP)
(K(t)) = physical capital
(L(t)) = labor (population/workers)
(A(t)) = technology (labor-augmenting efficiency), sometimes held constant in the simplest version

A common production function is Cobb–Douglas:

[ Y(t) = K(t)^{\alpha},[A(t)L(t)]^{1-\alpha}, \quad 0<\alpha<1 ]

Convert everything into “per worker” (or “per effective worker”)

The model becomes much easier when you express variables per person.

Without technological progress (A constant):

Output per worker: (y = Y/L)
Capital per worker: (k = K/L)

With Cobb–Douglas (and constant (A)), you get:

[ y = f(k) = k^{\alpha} ]

Key idea: (f(k)) is increasing but concave (diminishing marginal product of capital).

3) How capital accumulates: the engine of the model

The economy saves a constant fraction (s) of output:

Investment: (I = sY)
Consumption: (C = (1-s)Y)

Capital changes according to:

[ \dot K = I - \delta K = sY - \delta K ]

where ( \delta ) is the depreciation rate.

Bring in population growth

Suppose labor grows at rate (n):

[ \dot L / L = n ]

We care about (k = K/L). When (L) grows, existing capital is “spread thinner” across workers. The law of motion for capital per worker is:

[ \dot k = s f(k) - (\delta + n)k ]

This is the single most important equation in the basic Solow model.

(s f(k)) = investment per worker
((\delta+n)k) = break-even investment per worker (what you need just to keep (k) constant)

4) The Solow diagram (your main visual tool)

The classic diagram plots quantities per worker against (k):

Curve 1: (y=f(k)) (concave)
Curve 2: (i = s f(k)) (same shape, scaled down/up by (s))
Line: break-even investment ((\delta+n)k) (straight line through the origin)

How to read it

If (s f(k) > (\delta+n)k), then (\dot k > 0): capital per worker rises
If (s f(k) < (\delta+n)k), then (\dot k < 0): capital per worker falls
Where they intersect, (\dot k = 0): that is the steady state (k^*)

In words: the economy accumulates capital until diminishing returns make investment just enough to cover depreciation and the needs of a growing population.

5) Steady state: what it is and why it matters

The steady state (k^*) solves:

[ s f(k^) = (\delta + n)k^ ]

At (k^*), capital per worker is constant. So:

Output per worker (y^* = f(k^*)) is constant
Consumption per worker (c^* = (1-s)f(k^*)) is constant

Important implication (basic model, no tech growth)

In steady state, output per worker does not grow. So long-run growth in living standards cannot come from capital accumulation alone.

This is a central Solow message: saving more raises the level of income per worker, not the long-run growth rate (unless technology improves).

6) Comparative statics (what happens when parameters change?)

This is where Solow becomes extremely “exam friendly”: you can predict shifts in the diagram.

A) Increase in the savings rate (s)

(s f(k)) shifts up
New steady state (k^{*}_{new}) is higher
(y^*) is higher, and so is investment per worker
During the transition, growth in (y) is temporarily higher as the economy moves to the new steady state

But: once the new steady state is reached, growth returns to zero (in the no-tech model). This is the difference between transition dynamics and steady-state growth.

B) Increase in population growth (n)

Break-even line ((\delta+n)k) becomes steeper
New steady state (k^*) is lower
Lower (y^) and lower (c^) (typically)

Intuition: faster labor growth dilutes capital, making it harder to maintain high capital per worker.

C) Increase in depreciation ( \delta )

Same effect as higher (n): break-even line shifts up/steepens
Lower (k^) and (y^)

D) Improvement in technology (A) (preview)

If technology improves, workers become more productive, and the entire production function effectively shifts upward. This is how Solow explains sustained growth in output per worker.

7) Adding technological progress (the version that generates long-run growth)

To get persistent growth in living standards, we usually assume labor-augmenting technological progress:

[ \dot A / A = g ]

Now define effective labor: (A L). Rewrite variables per effective worker:

Capital per effective worker: (\tilde{k} = K/(AL))
Output per effective worker: (\tilde{y} = Y/(AL))

With Cobb–Douglas:

[ \tilde{y} = f(\tilde{k}) = \tilde{k}^{\alpha} ]

The law of motion becomes:

[ \dot{\tilde{k}} = s f(\tilde{k}) - (\delta + n + g)\tilde{k} ]

This looks exactly like the earlier model, except the break-even term is now ((\delta+n+g)\tilde{k}). The logic is the same, but the interpretation changes:

There is a steady state in per effective worker terms: (\tilde{k}^*)
In that steady state, output per effective worker is constant
But output per worker grows at rate (g) (the technology growth rate)

The key takeaway

Long-run growth in (y = Y/L) comes from technology growth (g).
Savings affects the level of (y), not the long-run growth rate (in the standard Solow framework).

8) Convergence: why poorer countries might catch up

Because of diminishing returns, countries with low initial capital per worker tend to have high marginal returns to capital, so they grow faster—if they share similar parameters and access to technology.

This gives conditional convergence:

Countries converge to their own steady states determined by (s), (n), (δ), education/human capital (in extended models), institutions, etc.
If two countries have different fundamentals, they converge to different (k^) and (y^).

A common mistake is to claim Solow predicts unconditional convergence for all countries. In practice, differences in savings, demographics, and technology/institutions mean convergence is conditional.

9) The Golden Rule (best steady state for consumption)

Higher saving raises (k^) and (y^), but it also reduces consumption today. The “best” saving rate depends on what steady-state consumption you want.

Steady-state consumption per worker:

[ c^* = f(k^) - (\delta+n)k^ ]

The Golden Rule level of capital (k_{GR}) maximizes (c^*). The condition is:

[ MPK = f'(k_{GR}) = \delta + n ]

Interpretation: at the optimal steady state, the extra output from one more unit of capital equals the extra break-even cost of maintaining that capital per worker.

If an economy saves too much (capital is very high), diminishing returns make extra capital not worth its maintenance cost, and consumption could actually be lower.

Common pitfalls (quick fixes)

Mixing up levels and growth rates: Higher (s) increases the level of (y), not the long-run growth rate (with exogenous tech growth).
Forgetting population growth in break-even investment: It’s ((\delta+n)k), not just (\delta k).
Thinking “steady state” means “nothing changes”: In the tech-progress model, variables per effective worker are constant, but output per worker grows at rate (g).
Not explaining dynamics: Always say what happens when the economy is left of or right of (k^*).

Summary and next steps

You now have the Solow–Swan model’s full toolkit: the production function in per worker terms, the capital accumulation equation, the Solow diagram, steady state, comparative statics, convergence logic, and the Golden Rule.

Next steps to deepen your understanding:

Practice drawing the Solow diagram from memory and narrating the adjustment process.
Work through a numerical example (pick (s, \alpha, n, \delta)) to solve for (k^*) and compare two countries.
Extend the model by adding human capital (the “augmented Solow model”) to connect more directly to cross-country income differences.

The Solow–Swan Growth Model (300-Level): A Visual, Step-by-Step Guide