Lesson preview · Solow Model

The Solow Growth Model

~14 min · Free to read

Give a farmer one tractor and the harvest leaps. Give the same farmer a second tractor and the harvest rises again — but by less. A third helps less still. This single fact, that extra machines help but help by less each time, is the heart of the most famous model of economic growth. Robert Solow built it in 1956. This lesson walks through it from scratch.

Start with one number: capital per worker, written $k$ . This is the stock of tools, machines, and buildings each worker has to work with. More capital per worker means more output per worker — a builder with a crane lays more brick than a builder with a trowel. We measure output per worker as $y$ .

But the help shrinks as capital piles up — that is diminishing returns (each extra machine adds less than the one before). To capture this, Solow uses a curved production function. Here it is the square root: $y = k$ . Output still rises with capital, but at a slowing rate. Four units of capital give two of output; nine units give three; sixteen units give four. Quadruple the capital, and output only doubles.

Where does new capital come from? Saving. Households save a fixed slice of their income — the saving rate, written $s$ (a number like 0.3, meaning 30% of output is saved and invested). Saved output is poured back in as new machines. So investment per worker is $s k$ — the production curve scaled down by the saving fraction.

Capital also wears out. Machines rust, buildings age, tools break. The fraction that wears out each year is the depreciation rate, written $δ$ (the Greek letter delta). Here $δ = 0.1$ , so 10% of the capital stock is lost every year. Capital lost is therefore $δ k$ — a straight line, because losing a fixed 10% means the more capital you have, the more you lose.

Now the two forces meet. Investment $s k$ adds capital; depreciation $δ k$ subtracts it. When investment beats depreciation, the capital stock grows. When depreciation wins, it shrinks. The economy settles at the one level where the two exactly balance — the steady state. There, new investment just replaces what wears out, and capital per worker stops changing. Call that level $k^{*}$ .

We can solve for the steady state directly. Set investment equal to depreciation and read off the answer. The equation below does this; the popover on its card spells out every symbol in plain English.

The production function: output per worker rises with capital per worker, but with diminishing returns.

The steady-state condition: investment equals depreciation, which pins down capital and output per worker.

Interactive — Find the steady state

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The production curve, the investment curve, and the depreciation line, with their crossing point marking the steady state.

Tip

So what

Saving more lifts the economy to a higher steady state with more capital and more output — but because of diminishing returns it cannot make the economy grow forever; capital simply climbs to a new resting point and stops.

Worked Example

An economy has saving rate $s = 0.2$ and depreciation rate $δ = 0.1$ , with production function $y = k$ . Find its steady-state capital and output per worker. Then the country doubles its saving rate to $s = 0.4$ . Find the new steady state, and compare how much capital and output each rose.

Key Term

The golden-rule saving rate

Saving more raises steady-state output, but it also means consuming less of that output today. So there is a sweet spot — the golden-rule saving rate — that maximises steady-state consumption per worker (the output left over after replacing worn-out capital).

For the production function used here, that sweet spot is $s = α = 0.5$ . It gives $k^{*} = 25$ , $y^{*} = 5$ , and consumption per worker $c^{*} = 2.5$ . Saving more than this would raise output but lower consumption — you would be working to feed the machines rather than yourselves. (This level is not drawn on the graph; the buttons there top out at 40%.)

Trap · Common Misconception

"Just save more and the economy keeps growing"

It is tempting to read the model as: raise the saving rate and growth speeds up forever. It does not.

A higher saving rate lifts the economy to a higher steady state — more capital, more output — but once there, capital per worker stops rising. Growth from extra saving is a one-time climb to a new resting point, not a permanent faster pace. The reason is diminishing returns: each new machine adds less output, so investment eventually only manages to replace what wears out. To grow without end, you need something that does not run into diminishing returns — better technology (a higher $A$ ), which is the subject of the next lesson.

Check yourself · no marks

An economy currently has less capital per worker than its steady-state level $k^{*}$ . Is its capital stock growing, shrinking, or steady? Why?

Commit to one first. Guessing and being wrong beats reading the answer cold.

Practice · 1 / 4

No penalty for a wrong answer

In the Solow model, an economy currently has investment per worker that exceeds depreciation per worker. What happens to its capital stock per worker?

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