Comparative Statics, Evidence, and Extensions

The steady-state formulas $k^* = \bigl(s/(n+g+\delta)\bigr)^{1/(1-\alpha)}$ and $y^* = \bigl(s/(n+g+\delta)\bigr)^{\alpha/(1-\alpha)}$ let us read off how each parameter moves the long-run equilibrium.

**Savings rate $s$**: a higher $s$ shifts the investment curve $sf(k)$ upward. At the old $k^*$ actual investment now exceeds break-even investment, so $k$ rises until a new, higher $k^*$ is reached. Output per worker is permanently higher, but the long-run growth rate of $y$ returns to $g$. Saving more lifts the level, not the growth rate.

**Population growth $n$**: a higher $n$ steepens the break-even line $(n+g+\delta)k$. Each unit of capital must now be spread across more workers, so less remains per worker in steady state. High population growth $\Rightarrow$ lower $k^*$ and $y^*$.

**Technology growth $g$**: a higher $g$ also steepens the break-even line — each unit of technology needs more capital to stay current — so $k^*$ in effective-worker terms falls. But $g$ directly raises the growth rate of $Y/L$, so living standards rise faster. Higher $g$ lowers the level of $k^*$ but raises the long-run growth rate of per-worker output.

**Depreciation $\delta$**: faster depreciation raises the effective cost of maintaining capital, steepening the break-even line and lowering $k^*$ and $y^*$. Countries with faster-depreciating capital stocks are poorer in steady state.

The richest countries are roughly 30 to 60 times richer than the poorest. Can the Solow model explain this through differences in saving rates? The answer is a clear no — and understanding why is the single most important lesson the model teaches.

From $y^* = \bigl(s/(n+g+\delta)\bigr)^{\alpha/(1-\alpha)}$, the elasticity of $y^*$ with respect to $s$ is:

This means a **1% rise in $s$ raises $y^*$ by only about $\tfrac{1}{2}$%.** The square-root relationship is a direct consequence of diminishing returns to capital.

Now run the numbers. Suppose a rich country saves 40% and a poor country saves 10% — a 4-to-1 ratio in saving rates. The predicted income ratio is $4^{1/2} = 2$. That is a 2× income gap, not 30–60×.

To generate a 30× income gap through saving alone, you would need $s_{\text{rich}}/s_{\text{poor}} = 30^2 = 900$. Saving rates are bounded between 0 and 1. The math is impossible.

There is a second way to see the same problem. If income gaps were really due to capital differences, poor countries would have very little capital and therefore — by diminishing returns — very high marginal products of capital (MPK).

With $\alpha = 1/3$, MPK $= \alpha k^{\alpha - 1} = \alpha y / k$. If the US–India capital-per-worker ratio is 20:1, the predicted MPK ratio is $20^{2/3} \approx 7.4$. Capital should flood into poor countries, equalising returns.

In reality, capital does not flow to poor countries at anything like this rate. Robert Lucas (1990) called this the Lucas Paradox. Possible resolutions: institutional weakness, sovereign risk, missing human capital, and information asymmetries all reduce the effective return to capital in poor countries — consistent with the view that $A$ differs, not just $k$.

Solow (1957) turned the model into a measurement tool. Starting from $Y = K^\alpha (AL)^{1-\alpha}$, take logs and differentiate with respect to time:

Rearranging gives the Solow residual — the portion of output growth not explained by measured capital and labour:

For the US from 1909–1949, Solow found that roughly 87% of output growth per worker came from the residual (technology), and only 13% from capital deepening. This result both confirmed the model's prediction that technology is the engine of long-run growth and reinforced the central question: if capital explains only 13% of US growth, it cannot explain 30–60× cross-country income gaps.

Capital accumulation alone cannot sustain long-run per-worker growth. As $k$ rises, diminishing returns drive $f'(k)$ down. Eventually the return to additional investment falls below the break-even rate and $\dot k$ drops to zero.

Sustained growth in living standards requires $g > 0$. But $g$ is exogenous in this model — it arrives from outside, like manna. The model tells us technology matters enormously; it does not explain where technology comes from. This silence is the primary motivation for endogenous growth theory.

Countries with the same fundamentals ($s$, $n$, $g$, $\delta$, $\alpha$) but different starting points converge to the same steady state. Poorer ones grow faster because they are further below $k^*$, where diminishing returns have not yet fully bitten.

This is conditional convergence: a country grows faster the further it is below its own $k^*$. A poor country with a low savings rate is racing toward its own lower steady state, not toward the US. Cross-country regressions confirm conditional (but not unconditional) convergence.

The most influential extension adds human capital $H$ as a third factor:

With $\alpha \approx \beta \approx 1/3$, the augmented model explains about 78% of cross-country income variation (versus 59% for the basic model). The implied convergence speed is $\approx 2.7\%$ per year — much closer to the empirical estimate of $\approx 2\%$ per year than the basic model's $\approx 5.3\%$.

Crucially, MRW also shows that accounting for human capital reduces the implied capital share to reasonable levels and substantially shrinks the Solow residual, reconciling some — but not all — of the technology gap.

What if diminishing returns to capital are absent? Suppose the aggregate production function is linear in capital:

The long-run growth rate of $k$ is $sA - n - \delta$, which is now a function of $s$. The savings rate affects not just the level of output but the long-run growth rate — the opposite of the Solow result. This is the foundation Romer (1986) built on, by micro-founding why MPK need not fall.

With perfect capital mobility the domestic capital stock is pinned by the world interest rate $r_w$:

$k^*$ is now determined externally, not by the domestic savings rate. A rise in $s$ leads to a current-account surplus as saving flows abroad, not to higher domestic capital. The saving-income link that drives comparative statics in the closed economy disappears entirely.