Summary


We want to modify the Solow model to have human capital. This allows the skills of workers to increase separately from technological progress.

𝑌=𝐾1𝛼(𝐴𝐻)𝛼𝐻=𝑒𝜓𝑢𝑁

𝑢 is the amount of time spent acquiring human capital (i.e. years of schooling). If 𝜓=0.1, one year of schooling raises 𝐻 by 10%. In per-worker terms (not effective worker):

=𝑒𝜓𝑢𝑦=𝑘1𝛼(𝐴)𝛼𝑔𝑦=(1𝛼)𝑔𝑘+𝛼𝑔𝐴+𝛼𝑔

Balanced Growth

Assume in steady state 𝑔=0, then

𝑔𝑘=𝑔𝐴𝑔𝑦=𝑔𝐴

Adding human capital doesn’t change the determinants of steady-state growth.

From the capital accumulation equation (𝑘=𝐾𝑁):

𝑔𝐴=𝑠𝑦𝑘(𝛿+𝑛)=𝑠(𝐴)𝛼𝑘𝛼(𝛿+𝑛)

Solving for 𝑘𝐴:

𝑘𝐴=(𝑠𝛿+𝑛+𝑔𝐴)1𝛼𝑦(𝑡)=𝐴(𝑡)(𝑘𝐴)1𝛼=𝐴(𝑡)𝑒𝜓𝑢(𝑠𝛿+𝑛+𝑔𝐴)1𝛼𝛼

Human capital doesn’t affect steady-state growth but influences the level of output per worker.

Relative Output per Worker

Consider output per worker relative to the US:

̂𝑦𝑖=𝑦𝑖𝑦𝑈𝑆

Assuming the same rate of technological progress across countries:

̂𝑦𝑖=𝐴𝑖𝐴𝑈𝑆𝑒𝜓(𝑢𝑖𝑢𝑈𝑆)(𝑠𝑖𝑠𝑈𝑆)1𝛼𝛼(𝛿+𝑛𝑈𝑆+𝑔𝐴𝛿+𝑛𝑖+𝑔𝐴)1𝛼𝛼

The Solow Residual

Solow assumed 𝐴 was identical across countries, i.e. they could share technology, but that is a bold claim. If we assumed 𝐴𝑖=𝐴𝑈𝑆, the world would be a lot more equal: We can estimate technology from the Solow residual:

𝑦𝑖=𝑘1𝛼𝑖(𝐴𝑖𝑖)𝛼𝐴𝑖=(𝑦𝑖𝑘1𝛼𝑖𝛼𝑖)1𝛼

You can now compute

̂𝐴𝑖𝐴𝑖𝐴𝑈𝑆

Differences in 𝐴𝑖 explain about 1/2 to 2/3 of the differences in output per worker across countries.

Conditional Convergence

Some countries grow slowly despite being poor. Conditional convergence explains that countries grow faster the farther they are from their own steady state. Poor countries have low steady states, so they are already close to it.