The Solow Growth Model Explained: Steady State, Saving and Technology

The Solow growth model explains how saving, depreciation, population growth and technology determine output per worker, using the equation Δk = s·f(k) − (n + d)·k, where k is capital per worker. The economy settles at a steady state where saving exactly covers break-even investment, s·f(k) = (n + d)·k, and from there only technological progress can keep output per worker growing.

This guide builds the model from zero, solves a full numeric steady state with f(k) = k^0.5, and then shows what happens when the saving rate, population growth or technology changes. If you want the one-paragraph version first, the Solow growth model glossary entry has it.

The building blocks

The model tracks one economy that produces a single good, which can be eaten (consumption) or saved and turned into machines (investment). Everything is measured per worker, which is what lets the model talk about living standards rather than the size of the economy.

The production function and diminishing returns

Output per worker depends on physical capital per worker through y = f(k). The key assumption is diminishing returns: each extra unit of capital adds less output than the one before it. With f(k) = k^0.5, raising k from 4 to 9 (five more units) lifts output from 2 to 3, but it takes seven more units, from 9 to 16, to lift output by the same amount again, from 3 to 4. This is the same law of diminishing marginal returns you meet in the theory of the firm, applied to a whole economy.

Investment adds capital, and two forces wear it down

Each period, workers save the fraction s of their output, so investment per worker is s·f(k). That investment raises capital per worker. Two things pull capital per worker down:

• Depreciation: machines wear out at rate d, so d·k of capital per worker disappears.
• Workforce growth: new workers arrive at rate n, and the existing capital is spread across more people, so n·k of investment is needed just to equip them at the current level.

Together, (n + d)·k is break-even investment: the amount needed to keep k constant. The change in capital per worker is the gap between what is invested and what is needed to stand still:

Δk = s·f(k) − (n + d)·k

When investment exceeds break-even investment, k rises. When it falls short, k falls. When the two are equal, k stops changing, and that point is the steady state.

Worked example: solving the steady state

Use f(k) = k^0.5, s = 0.2, n = 0.02 and d = 0.03.

Step 1. Write the steady state condition. Set Δk = 0:

0.2 × k^0.5 = (0.02 + 0.03) × k = 0.05 × k

Step 2. Divide both sides by k^0.5. Since k / k^0.5 = k^0.5:

0.2 = 0.05 × k^0.5

Step 3. Solve for k^0.5. k^0.5 = 0.2 ÷ 0.05 = 4.

Step 4. Square both sides. k* = 4² = 16.

Step 5. Output per worker. y* = 16^0.5 = 4.

Step 6. Investment and consumption per worker. Investment is s × y* = 0.2 × 4 = 0.8. Consumption is c* = (1 − 0.2) × 4 = 0.8 × 4 = 3.2.

Check one: does investment cover break-even investment? Break-even investment is 0.05 × 16 = 0.8, and investment is 0.8. They match, so k does not change.

Check two: does output add up? Consumption plus investment is 3.2 + 0.8 = 4.0, which equals y* = 4. The answer holds.

A useful shortcut for f(k) = k^0.5: k* = (s ÷ (n + d))² and y* = s ÷ (n + d). Here s ÷ (n + d) = 0.2 ÷ 0.05 = 4, so y* = 4 and k* = 16, the same result.

How the economy gets to the steady state

The steady state is stable. Start the economy anywhere and it moves toward k* = 16. The table uses the same parameters and shows one period of adjustment from several starting points.

Read down the last column. An economy with very little capital (k = 4) grows its capital per worker at 5 percent, one with more capital (k = 9) grows at about 1.7 percent, and one at the steady state does not grow at all. An economy that starts above the steady state (k = 25) is investing less than it needs to maintain its capital, so k shrinks back toward 16.

Convergence

That pattern is the model's convergence prediction. Poorer economies, with less capital per worker, earn a high return on each new machine and grow faster than richer ones. But the prediction is conditional: each economy converges to its own steady state, set by its own s, n, d and technology. Two countries with the same parameters should end up with the same output per worker. A country that saves less or has faster population growth converges to a lower level and stays there. The convergence hypothesis entry covers the difference between absolute and conditional convergence and what the evidence says.

What changes in s, n and technology do

Each change below starts from the baseline (s = 0.2, n + d = 0.05, f(k) = k^0.5) and solves the new steady state the same way.

A higher saving rate raises the level, not the long-run growth rate

With s = 0.3: 0.3 = 0.05 × k^0.5, so k^0.5 = 6, k* = 36 and y* = 6. Consumption is 0.7 × 6 = 4.2. Check: investment is 0.3 × 6 = 1.8 and break-even is 0.05 × 36 = 1.8.

Output per worker rises by 50 percent, from 4 to 6. During the move from k = 16 to k = 36 the economy grows faster than before. Once it arrives, growth in output per worker returns to zero. Saving changes where the economy ends up, not how fast it grows forever. Graphically, the s·f(k) curve shifts up and crosses the (n + d)·k line further to the right.

Faster population growth lowers the level

With n = 0.03, break-even investment becomes (0.03 + 0.03)·k = 0.06k. Then 0.2 = 0.06 × k^0.5, so k^0.5 = 3.333, k* = 11.11 and y* = 3.333. Check: 0.2 × 3.333 = 0.667 and 0.06 × 11.11 = 0.667. More of each period's investment goes to equipping new workers, so less goes to deepening capital per worker. The break-even line gets steeper and the steady state moves left.

Better technology raises output at every level of capital

Write production as y = A·k^0.5 and raise A from 1 to 1.5. The condition becomes 0.2 × 1.5 × k^0.5 = 0.05 × k, so 0.3 = 0.05 × k^0.5, k^0.5 = 6 and k* = 36. Output is y* = 1.5 × 6 = 9, and consumption is 0.8 × 9 = 7.2. Check: investment is 0.2 × 9 = 1.8 and break-even is 0.05 × 36 = 1.8.

Compare this row with the higher-saving row. Both end with k* = 36, but better technology delivers output of 9 instead of 6, because it raises output both directly (through A) and indirectly (more output means more saving, which supports a larger capital stock).

Why only technology sustains long-run growth per worker

Saving cannot produce permanent growth because of diminishing returns. Each round of extra capital adds less output, while break-even investment rises in a straight line with k. Eventually the added output only just pays for maintaining the larger capital stock, and accumulation stops. You could keep raising the saving rate, but it cannot go above 100 percent, and in practice it stops far short of that.

Technology escapes this limit because it keeps shifting the production function up. In the standard textbook version, technology raises the effectiveness of each worker at a steady rate g. The steady state then describes capital per effective worker, break-even investment becomes (n + d + g)·k with k measured per effective worker, and output per worker grows at exactly g forever. Total output grows at n + g. If g is 2 percent, the rule of 70 says output per worker doubles roughly every 35 years (70 ÷ 2 = 35).

This is why growth accounting splits measured growth into contributions from capital, labor and a residual for technology. The model also explains why capital deepening produces fast catch-up growth that then fades. What the model does not explain is where g comes from. That question drives endogenous growth theory, which links the rate of technological progress to research, education and human capital.

The golden rule saving rate

If saving raises output, is more saving always better? No. Consumption is what households actually enjoy, and at the steady state c* = (1 − s) × y* in every case. With our parameters, y* = s ÷ 0.05, so c* = (1 − s) × s ÷ 0.05.

Consumption peaks at s = 0.5, where k* = 100 and y* = 10. At that point the marginal product of capital, 0.5 × k^(−0.5) = 0.5 ÷ 10 = 0.05, equals n + d = 0.05, which is the golden rule condition. Saving beyond 0.5 gives more output but less consumption, because the extra output goes into maintaining an oversized capital stock.

How this fits AP Macroeconomics

AP Macroeconomics treats long-run growth through the production possibilities curve shifting outward, long-run aggregate supply shifting right, and the roles of physical capital, human capital and technology. It does not ask you to solve a Solow steady state. The Solow model is the formal version of the same story, usually taught in college intermediate macroeconomics, and it explains why the AP course emphasizes productivity and technology as the sources of sustained growth. The economic growth module covers the AP-level material, and for the short-run side of college macro, see the IS-LM model explained.

Common mistakes

• Saying higher saving raises the long-run growth rate. It raises the steady-state level of output per worker. Growth is faster only during the transition.
• Forgetting population growth in break-even investment. Capital per worker falls when the workforce grows, even if no machine wears out. Break-even investment is (n + d)·k, not d·k.
• Mixing up output and consumption. At the steady state, y* = 4 but c* = 3.2. Questions about living standards often want c*.
• Reading convergence as unconditional. The model predicts each country approaches its own steady state, so a poor country with low saving can stay poor.
• Treating the steady state as zero total growth. With no technological progress, output per worker is constant, but total output still grows at rate n because the workforce grows.