Economic Growth

Economic growth is the general improvement in living standards, which is representable through an indicator like real GDP per capita.

Supposing $y=\frac{Y_{\text{real}}}{\text{Population}}$, the economic growth rate from $t-1$ to $t$ is

$$g=\frac{y_t-y_{t-1}}{y_{t-1}}.$$

Making $y_t$ the subject, and then extrapolating across time periods $[0,t]$, we have:

$$\begin{array}{rl}{y}_t& =(1+g)\cdot y_{t-1},\\ & =(1+g{)}^2\cdot y_{t-2}\\ & =(1+g{)}^t\cdot y_0.\end{array}$$

A small difference in $g$ across a long span of time will result in enormous differences because of the compounding effect.

Purchase Power Parity

The Purchase Power Parity exchange rate uses a standard basket of goods and services to assess economic productivity and living standards across different countries.

Aggregate Production Function

An aggregate production function is a function of variables such as aggregate physical capital stock, aggregate labour quantity used, general level of factor productivity, etc.

The Cobb-Douglas Production Function is an example of an aggregate production function that produces real GDP as a function of labour, physical capital and technology and has the the form:

$$Y_{\text{real}}=A{K}^{\alpha }{L}^{1-\alpha },$$

where $A$ is ‘factor productivity’, $K$ is physical capital, $L$ is labour input, and $0<\alpha <1$.

A notable property of this function is that if you were to double capital $K$ and labour $L$ (representing the quantity and quality of human capital), the production output doubles. This property is called constant returns to scale.

• Marginal product of labour: $MPL=\frac{\delta Y}{\delta L}=(1-\alpha )\frac{Y}{L}$
• Marginal product of capital: $MPK=\frac{\delta Y}{\delta L}=\alpha \frac{Y}{K}$

Capital Per Worker The average worker productivity is given by $\frac{Y}{L}=A(\frac{K}{L}{)}^{\alpha }$. Since $0<\alpha <1$, increases in $\frac{K}{L}$ result in diminishing increases in worker productivity. Letting $y=\frac{Y}{L}$ and $k=\frac{K}{L}$ (the capital per worker), we have $y=A{k}^{\alpha }$.

Growth of Productivity The change in aggregate production is given by

$$\Delta Y=\Delta A+\alpha \Delta K+(1-\alpha )\Delta L,$$

which can be rearranged to get the growth in productivity, $\Delta A$,

$$\Delta A=\Delta Y-\alpha \Delta K-(1-\alpha )\Delta L.$$