Section 7

Comparative Statics, Evidence, and Extensions

An increase in $s$ shifts the investment curve upward, raising $k^*$ and $y^*$ . It raises the level of output per worker permanently but not the long-run growth rate. Elasticity: a 1% rise in $s$ raises $y^*$ by $\alpha/(1-\alpha)\%$ — about 0.5% with $\alpha = 1/3$ .

Higher $n$ steepens the break-even line, lowering $k^*$ and $y^*$ . Capital must be spread over more workers; each ends up with less. Countries with high population growth tend to have lower per-capita incomes.

Faster $g$ also lowers $k^*$ in effective-worker terms but raises the growth rate of $Y/L$ in the long run — living standards rise faster even though each unit of technology requires more capital to maintain.

Higher $\delta$ raises the effective cost of maintaining capital, lowering $k^*$ and $y^*$ . Countries with faster-depreciating capital stocks will be poorer in steady state.

Comparative statics summary

Parameter	Effect on k*	Effect on y*	Effect on growth rate
$s \uparrow$	+	+	0 (temporary + during transition)
$n \uparrow$	−	−	0
$g \uparrow$	−	−	+
$\delta \uparrow$	−	−	0
$\alpha \uparrow$	+	+	0

Quick comparative-statics recap

Higher savings rate s → savings curve shifts up → higher k* and y*. A 1% rise in s raises k* by 1/(1-α)%. Countries that save more accumulate more capital per worker in steady state. Higher depreciation δ → break-even line steepens → lower k* and y*. Capital wears out faster, requiring more gross investment just to stand still, leaving less net accumulation. Faster population growth n → break-even line steepens → lower k* and y*. Capital must be spread over more workers; each worker ends up with less equipment. Faster technology growth g → lower k* per effective worker, but higher growth rate of output per worker. This resolves the apparent paradox: living standards rise faster even though each unit of technology requires more capital to maintain.

Growth accounting and the Solow residual

Solow (1957) showed how to decompose observed output growth into contributions from capital, labour, and technology.

The decomposition

Starting from $Y = K^\alpha (AL)^{1-\alpha}$ , take logs and differentiate:

\hat{Y} = \alpha \hat{K} + (1-\alpha)\hat{L} + (1-\alpha)g

Isolating technology:

\text{TFP growth} = \hat{Y} - \alpha \hat{K} - (1-\alpha)\hat{L}

This is the Solow residual: the portion of output growth not explained by measured capital and labour inputs.

Solow's Finding (1957): For the US from 1909–1949, roughly 87% of output growth per worker was attributable to the residual (technological progress), and only 13% to capital deepening.
Limitations: The residual measures our ignorance as much as technology. It reflects mismeasurement of inputs (especially human capital), factor utilisation changes, returns to scale assumptions, and sectoral reallocation.
Human capital correction: Adding human capital (Mankiw-Romer-Weil) dramatically reduces the residual.

Implications and predictions

No long-run growth without technology

Capital accumulation alone cannot sustain long-run per-worker growth. As $k$ rises, $f'(k)$ falls. Eventually the return to additional investment falls below the break-even rate and growth stops.

Sustained growth in living standards requires $g > 0$ . But $g$ is exogenous — the model tells us that technology matters, not why it grows.

Conditional convergence

Countries with the same $s, n, g, \delta, \alpha$ but different initial $k_0$ converge to the same steady state. Poorer countries grow faster.

Convergence is conditional: a poor country with a low savings rate is converging to its own lower steady state, not the US.

The Lucas Paradox

The model predicts very large returns to capital in low- $k$ countries. If the US-India capital ratio is 20:1, the predicted MPK ratio is $20^{2/3} \approx 7.4$ . Capital should flood into poor countries.

In reality it does not. Possible explanations: institutional quality, human capital complementarities, sovereign risk, and information asymmetries.

Critiques and limitations

Exogenous Technology: The model takes $g$ as given. It tells us long-run growth requires technological progress but gives no theory of where progress comes from. This motivated endogenous growth theory (Romer 1986, 1990; Lucas 1988).
Exogenous Savings Rate: Real households optimise intertemporally. The Ramsey-Cass-Koopmans model replaces the fixed $s$ with a utility-maximising household.
No Institutions, Geography, or Culture: Cross-country income differences span a factor of 60 between the richest and poorest nations. The model attributes these entirely to differences in $s$ , $n$ , and $A_0$ . Institutions, geography, culture, and history are absent.
No Natural Resources or Environment: Capital and labour are the only inputs. Natural resources, land, and environmental constraints are absent.

Extensions

Human Capital: Mankiw-Romer-Weil (1992)

Augment the model with human capital $H$ :

Y = K^\alpha H^\beta (AL)^{1-\alpha-\beta}, \quad \alpha + \beta < 1

With $\alpha \approx \beta \approx 1/3$ , the model explains about 78% of cross-country income variation, versus 59% for the basic model. Implied convergence speed $\approx 2.7\%$ per year, much closer to the 2% empirical estimate.

Endogenous Growth: the AK Model

If the marginal product of capital does not fall:

Y = AK \quad \Rightarrow \quad \hat{k} = sA - n - \delta

Growth is constant and perpetual. The savings rate now affects the long-run growth rate, not just the level. This is the foundation Romer (1986) built on.

Open Economy

In an open economy with perfect capital mobility, the domestic capital stock is determined by the world interest rate $r_w$ :

f'(k^*) = r_w + \delta

Changes in the domestic savings rate affect the current account, not $k^*$ .

End-of-chapter questions

Why does the Solow-Swan model converge to a unique steady state?
What happens to k* when the savings rate s permanently increases?
Why can faster technology growth lower k* per effective worker but still raise living standards?