Economic Growth: Growth Accounting and the Solow Model

The puzzle

An average worker in the United States produces several times what an average worker in India produces, and dozens of times what a worker in Niger produces. In 1960, South Korea and Ghana had similar incomes per person. Two generations later, Korea was many times richer.

Differences this large dominate everything else in economics. As Robert Lucas put it, once you start thinking about them, it is hard to think about anything else.

Robert Solow’s 1956 model is the first step toward an answer. It asks a narrow question: how does a country’s stock of capital (machines, buildings, roads) grow over time, and where does it stop? The answer turns out to depend on three numbers: how much people save, how fast capital wears out, and how fast the population grows.

Why growth matters

Any factor that changes the long-run growth rate, even slightly, has a huge effect on living standards once it compounds:

Half a percentage point more growth a year, sustained for a century, is the difference between a sevenfold and a twelvefold rise in living standards.

Measuring growth

Statistical offices report quarterly GDP growth three ways:

• Quarterly growth at an annual rate: the quarter-on-quarter change, compounded to a year. A 0.3% quarterly change is roughly 1.2% at an annual rate.
• Year-over-year (four-quarter) growth: this quarter against the same quarter last year. It removes seasonal effects.
• Full-year growth: this year’s total against last year’s.

Over long spans, compound growth links two levels: the lecture writes $GDP_{2022} = GDP_{1998}(1 + g)^{96}$ for Türkiye’s 96 quarters, where $g$ is the average quarterly growth rate.

Growth accounting

Before building a model, measure: how much of growth came from more capital, more labour, or better productivity? Starting from $Y = AF(K, N)$ the lecture (following ABC) derives the growth accounting equation:

With Cobb-Douglas, $a_K = \alpha$ and $a_N = 1 - \alpha$: the elasticities are the income shares from Lecture 3. The four steps from the slides:

The Solow model: two questions

Robert Solow’s 1956 paper (Nobel Prize 1987) asks two questions:

1. How does a nation’s long-run standard of living depend on its saving rate, population growth and technical progress?
2. How does growth evolve over time: does it stabilise, accelerate or stop?

Output per worker

Start from the aggregate production function from Lecture 3, with constant returns to scale:

Constant returns means that if we multiply both inputs by the same number $z$, output is multiplied by $z$ too: $zY = F(zK, zL)$. Pick $z = 1/L$:

Now write lowercase letters for per-worker values: $y = Y/L$ and $k = K/L$. The production function becomes a function of one variable:

This is the trick that makes the whole model work. The size of the economy no longer matters. Only capital per worker matters for output per worker.

The shape of $f(k)$ is what drives everything. Output rises with $k$, but each extra unit of capital adds less than the one before. The slope of $f(k)$ is the marginal product of capital, $MPK = f'(k)$, and it falls as $k$ rises.

In exam problems you will almost always use Cobb-Douglas. With $Y = K^{\alpha}L^{1-\alpha}$, divide by $L$:

Recall from Lecture 3 that $\alpha$ is capital’s share of income. In real economies it is close to $1/3$.

How capital accumulates

Output per worker is split between consumption and investment. People save a constant fraction $s$ of income, and in a closed economy saving equals investment:

Capital also wears out. A constant fraction $\delta$ of the capital stock depreciates every year, so depreciation per worker is $\delta k$.

The change in the capital stock is what gets added minus what wears out:

The steady state

Now plot both terms against $k$. Investment $s\,f(k)$ has the same concave shape as $f(k)$, scaled down by $s$. Depreciation $\delta k$ is a straight line through the origin.

At low $k$, capital is scarce and highly productive, so investment is above depreciation and $k$ grows. At high $k$, depreciation (a straight line) overtakes investment (a flattening curve), and $k$ shrinks. There is exactly one point where they cross. That point is the steady state, $k^*$:

Drag the black point $k_0$ along the axis to choose a starting capital stock, then press Let time run. The shaded bar shows the gap between investment and depreciation, which is $\Delta k$.

Try these three experiments before moving on:

1. Start far to the left of $k^*$ and far to the right. Does the economy always end up at the same place?
2. Watch the speed. Is $k$ moving faster when it is far from $k^*$ or near it?
3. Raise $s$. Which curve moves, and where is the new $k^*$?

Solving for the steady state

With Cobb-Douglas, set $s\,k^{\alpha} = \delta k$ and solve for $k$:

Saving more: levels, not growth

This is the result your exam will test most often, and the one students most often get wrong.

When the saving rate rises, the investment curve shifts up. At the old $k^*$ investment now exceeds depreciation, so capital starts to grow. It keeps growing until it reaches a new, higher steady state. Then it stops.

Below, the economy starts in a steady state with $s = 0.20$. In year 10 something changes. Pick the shock on the right and watch output and consumption per worker over the next 70 years.

Turn on Growth rate with the saving shock selected. Growth jumps above zero after the shock and then slowly dies out. The economy ends up richer, but it does not end up growing faster.

Now try the War shock. Capital is destroyed, output falls, and then the economy grows very fast, because capital is now scarce and its marginal product is high. It returns to exactly the same steady state. This is how the Solow model explains the rapid growth of West Germany and Japan after 1945.

The golden rule

If saving more makes a country richer, should it save as much as possible? No. People care about consumption, not output. Saving everything ($s = 1$) would build a huge capital stock and leave nothing to eat.

In the steady state, consumption is output minus what must be invested to keep $k$ constant:

In the steady state, gross investment has two jobs: replacing worn-out capital ($dk$) and equipping new workers ($nk$). So steady-state investment is $(n + d)k$ and consumption is the rest of output.

Pick $k^*$ to make this as large as possible. Taking the derivative and setting it to zero gives the golden rule condition:

At the golden rule, the slope of the production function equals the slope of the depreciation line. Below it, one more unit of capital produces more than the $\delta$ needed to maintain it, so consumption can rise. Above it, the extra output does not even cover the extra depreciation.

In the diagram below the purple bar is steady-state consumption: the vertical distance between $f(k)$ and $s\,f(k)$ at $k^*$. Move $s$ and watch the bar. It is tallest when $k^*$ sits on the golden rule line.

Getting to the golden rule is not free

Go back to the time-path chart and set the new saving rate to 0.33, which is close to the golden rule for that economy ($\alpha = 1/3$). Consumption first drops, then climbs above its old level. A generation pays for the higher consumption of later generations.

The reverse case is different. An economy with too much capital can lower its saving rate and see consumption jump immediately and stay higher forever. There is no trade-off, which is why an economy above the golden rule is called dynamically inefficient.

Population growth

So far the labour force was constant. Now let it grow at rate $n$. Growth in $L$ spreads the existing capital over more workers, so it pushes $k$ down just like depreciation does. The key equation becomes:

The term $(\delta + n)k$ is called break-even investment: the investment needed just to keep $k$ constant, replacing worn-out capital ($\delta k$) and equipping new workers ($nk$).

Everything you learned still holds with $\delta$ replaced by $\delta + n$:

Raise $n$ above. The break-even line gets steeper and crosses investment at a lower $k^*$. This is the Solow model’s explanation for why countries with fast population growth tend to be poorer.

Endogenous growth

In the Solow model, productivity growth is the only source of long-run growth in output per person, and the model simply assumes it: productivity is exogenous. Endogenous growth theory tries to explain productivity growth inside the model.

The simplest endogenous growth model uses:

Why might the marginal product of capital not diminish for the economy as a whole?

• Human capital. As economies get richer they invest in people: nutrition, schooling, health, training. If physical capital grows while human capital stays fixed, returns diminish. If both grow together, they need not.
• Research and development. Firms in a growing economy invest in R&D, which raises the stock of commercially useful knowledge. The resulting productivity gains offset the tendency of the MPK to fall.

Paul Romer (1990, Nobel Prize 2018) listed five properties a model of long-run growth should have:

1. The accumulation of ideas is the source of long-run growth.
2. Ideas are non-rival: one person’s use does not reduce anyone else’s.
3. A larger stock of ideas makes it easier to find new ones.
4. Ideas are created by costly, purposeful activity.
5. Ideas can be owned, and the owner can sell the right to use them.

What the model leaves out

The Solow model explains a lot: why saving and population growth matter for living standards, why war-damaged economies grow fast, and why poor countries with the same fundamentals as rich ones should grow faster and catch up (conditional convergence).

It also leaves a gap. In its steady state, output per worker does not grow at all, yet in rich countries it has grown about 2% a year for over a century. Capital accumulation alone cannot produce that, because of diminishing returns. Sustained growth needs technological progress, which is where the next part of the growth material picks up.

Exam practice

These mirror the standard question types. Work them on paper first.

Summary and review

• Per-worker production: $y = f(k)$, with diminishing returns to capital.
• Key equation: $\Delta k = s\,f(k) - (\delta + n)k$.
• Steady state: $s\,f(k^*) = (\delta + n)k^*$. Every starting point converges to it.
• Cobb-Douglas: $k^* = \left(\frac{s}{\delta+n}\right)^{1/(1-\alpha)}$.
• Higher $s$ gives a higher level of $y^*$, not a higher long-run growth rate.
• Higher $n$ gives a lower $k^*$ and $y^*$.
• Golden rule: $MPK = \delta + n$. With Cobb-Douglas, $s_{gold} = \alpha$.
• Below the golden rule, reaching it costs consumption today. Above it, everyone gains.
• Rule of 70: doubling time $\approx 70/g$.
• Growth accounting: $\Delta Y/Y = \Delta A/A + a_K\,\Delta K/K + a_N\,\Delta N/N$; productivity is the residual.
• Endogenous growth: $Y = AK$, no diminishing returns, saving affects long-run growth. Romer: ideas drive growth.

Review these cards today. They will come back after 1, 3 and 7 days, which is what moves the material into long-term memory before the exam.