# 12 Economic Growth

## 12.1 Economic Growth

Economic growth is usually measured by its

__final output__or__final output per capita__?

To understand its change, we need to focus on its determinants.

What are they?

Which one do you think is the most important driving force?

## 12.2 The Accumulation of Capital

The accumulation of capital \(\Delta K\) is called net investment, which is financed by a country’s saving. There is also \(\delta K\) amount of replacement investment.

Given a country’s real output \(Y\), consumption \(C\), and government purchase \(G\), how much national saving is used to finance net investment?

Assume no government, i.e. \(G=0\). There are two types of simplifying assumptions regarding national saving: (1) it is \(s\) proportion of real output, (2) it is \(s\) proprotion of real output net of replacement investment. For either assumptions, write down the relationship between net investment and national saving.

We take the second assumption of saving.

Do you think capital accumulation can grow indefinitely (i.e. \(K\) goes to infinity)?

Do you think capital accumulation rate can grow indefinitely (i.e. \(\Delta K/K\) goes to infinity)?

## 12.3 Growth Accounting

We usually assume that \[Y=AF(L,K).\] With proper assumption on production function, real output growth rate \(\Delta Y/Y\) can be decomposed into three parts: technology growth rate \(\Delta A/A\), capital accumulation rate \(\Delta K/K\), and labor growth rate \(\Delta L/L\), such that

\[\Delta Y/Y = \Delta A/A + \alpha \Delta K/K + \beta \Delta L/L\]

Keep \(K\) constant. If L doubles, what would happen to \(Y\)? Do you think it will double? What does it means for \(\beta\)?

If both K and L double, what would happen to \(Y\)? Do you think it will double? What does it means for \(\alpha+\beta\)?

In general we agree on constant returns to scale such that \(\alpha+\beta=1\). Therefore,

\[\Delta Y/Y = \Delta A/A + \alpha \Delta K/K + (1-\alpha) \Delta L/L\]

- Mathematically, if \(z=w/y\), then \(\Delta z/z=\Delta w/w-\Delta y/y\). Show that the economic growth rate on \(y=Y/L\) (i.e. \(\Delta y/y\)) is determined by the growth rate on capital per capita \(k=K/L\).

## 12.4 Solow Growth Model

Given proper assumption on production function, we lean that \[\Delta y/y=\Delta A/A+\alpha \Delta k/k\] Assume no technological progress, i.e. \(\Delta A/A=0\).

Argue that per capita economic growth rate is solely determined by per capita capital accumulation rate.

How is per capita capital accumulation rate (\(\Delta k/k\)) related with national saving?

How is per capita capital accumulation rate (\(\Delta k/k\)) related with average productivity of capital, i.e. \(Y/K\)?

\(Y/K=(Y/L)/(K/L)=y/k\). Argue that the marginal productivity of k to y is deminishing given that \(\Delta y/y=\alpha \Delta k/k\) with \(0<\alpha<1\).

Convergence. Other thins equal, argue that countries with lower k will grow faster.

For countries with the same income level, what makes one grow faster?