§12 Convergence and Cross-Country Variation

Adding Human Capital to the Solow Model

• Modify the Solow model to include human capital. This allows the skills of workers to increase, separately from technological progress.

  $$Y = K^{1-\alpha}(AH)^{\alpha}$$

  $$H = e^{\psi u} N$$

  where $u$ is the amount of time spent acquiring human capital (think of it as years of schooling).

• If $\psi = 0.1$, one extra year of schooling raises $H$ by 10% (consistent with micro-evidence)

Balanced Growth (Steady State)

• In per-worker terms (warning: not effective worker):

  $$h = e^{\psi u}$$

  $$y = k^{1-\alpha}(Ah)^{\alpha}$$

  $$g_y = (1 - \alpha) g_k + \alpha g_A + \alpha g_h$$

• Assume in steady state $g_h = 0$, then:

  $$g_k = g_A \Rightarrow g_y = g_A$$

• Adding human capital doesn’t change the conclusion wrt to the determinants of steady state growth.

• From the capital accumulation equation (recall that in this lecture $k$ is $K/N$ not $K/AN$), we have:

  $$\begin{aligned} g_A &= s \frac{y}{k} - (\delta + n) \\ & =s \frac{(Ah)^\alpha}{k^\alpha} - (\delta + n) \end{aligned}$$

  to imply:

  $$\begin{aligned} \frac{k}{Ah} &= \left( \frac{s}{\delta + n + g_A} \right)^{1/\alpha} \\ y(t) &= A(t)h \left( \frac{k}{Ah} \right)^{1-\alpha} \\ &= A(t)e^{\psi u} \left( \frac{s}{\delta + n + g_A} \right)^{(1-\alpha)/\alpha} \end{aligned}$$

• So human capital doesn’t affect steady state growth but it influences the level of output per worker.

Relative Output per Worker

• Consider output-per-worker relative to the US:

  $$\hat{y}_i \equiv \frac{y_i}{y_{US}}$$

• If we assume the same rate of technological progress across countries (big if), we have:

  $$\hat{y}_i = \frac{A_i}{A_{US}} e^{\psi (u_i - u_{US})} \left( \frac{s_i}{s_{US}} \right)^{(1-\alpha)/\alpha} \left( \frac{\delta + n_{US} + g_A}{\delta + n_i + g_A} \right)^{(1-\alpha)/\alpha}$$

• Solow initially assumed that $A$ was the identical across countries, as they could share technology… but are the differences in $u$, $s$, and $n$ sufficient to explain the great dispersion we see in output-per-worker across countries?

• If you plug in estimates of $u$, $s$, and $n$ for different countries, and assume $A_i = A_{US}$… the world would be a lot more equal…

The Solow Residual

• Savings, education, and population growth do not explain all of the variation in output per worker. What are we missing?

• Technology/productivity differences: $A_i \neq A_{US}$.

• We can estimate technology from:

  $$y_i = k_i^{1-\alpha}(A_i h_i)^\alpha \Rightarrow A_i = \left( \frac{y_i}{k_i^{1-\alpha} h_i^\alpha} \right)^{1/\alpha}$$

• This is known as “The Solow Residual.”

• Compute:

  $$\hat{A}i \equiv \frac{A_i}{A_{US}}$$

• Differences in $A_i$ explain about 1/2 to 2/3 of the differences in output per worker across countries!

Conditional Convergence

• Some countries grow very slowly even though they are poor.

• Conditional convergence: countries grow faster, the farther they are from their own steady state. Poor countries have low steady states, so they are already close to their steady state, and grow slowly.

• Can we see this in the data? Compute the steady state for each country from:

  $$y_i^* = A_i e^{\psi u_i} \left( \frac{s_i}{\delta + n_i + g_A} \right)^{(1-\alpha)/\alpha}$$

  using data on $u$, $s$ and $n$ as before (still assuming the same $g_A$…). Use the value of $A_i$ for 1970.

• Compute how far each country is from steady state, $y_i / y_i^*$, and graph growth from 1960-2008 versus this relative value.

Lack of Convergence

• That is, we see the forces of convergence across similar countries and within each country, but there are enormous differences in the level of output per worker and rates of growth. Both are mostly due to differences in the level and growth rate of $A$.

— Apr 20, 2025

§12 Convergence and Cross-Country Variation by Lu Meng is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Permissions beyond the scope of this license may be available at About.