Economic Growth

by Professor Throckmorton
for Intermediate Macro
W&M ECON 304
Slides

World Bank Data¶

• Growth economics is primarily concerned with how the average standard of living is changing over time.

• GDP per person is a widely available measure of standard of living.

• But volatile exchange rates and differences in price levels make it difficult to compare countries.

• Thus, we use purchasing power parity adjusted GDP, which corrects for those differences.

• The World Bank reports PPP GDP per capita for many countries.

• USA and Japan have more than double the income per person than China, Vitenam, or India

• Regardless, the gap between these high income and emerging economies is shrinking, i.e., they are converging

• Like China/Vietnam/India, income per person in Ethiopia has increased as rapidly over the last 20 years.

• Growth rates in the USA are higher on average than in Japan.

• Growth rates in the USA are less volatile than in Japan.

• USA and Japan’s growth rates are mostly positively correlated, especially when the USA is in recession.

• Growth rates in China were much higher than India or Vietnam up until about 2015.

• All three countries maintained high positive growth rates through the global financial crisis.

• Recent growth rates are more similar than in the past, particulary because China has slowed down.

• China has grown more quickly than the USA.

• Despite Ethiopia being low income, it grew as fast as China in the 2000s.

• Growth rates in China and Ethiopia have fallen together, but they are still higher than the USA.

Growth Facts¶

• The following table is from Blanchard, Macroeconomics (9th Edition, 2025), Chapter 10, Page 203
  

1. “There has been a large increase in output per person.” (compounding)

2. “There has been a convergence of output per person across countries.” (convergence)

3. “The growth rate of output per person has declined.” (falling productivity growth)

Production Function¶

• The core economic growth model is attributed to economist Robert Solow and is often called the “Solow Model” (1950s).

• At the heart of the Solow Model is the production function, which relates aggregate output to the factors of production, primarily, capital, technology, and labor, e.g., $Y = F(K,AN)$.

• The Cobb-Douglas production function is

  $$Y = K^\alpha (A N)^\beta$$

  (1)

  where $Y$ is aggregate output (which we measure with GDP), $K$ is capital, $A$ is technology, $N$ is labor, $\alpha$ is the output elasticity of capital, and $\beta$ is the output elasticity of labor.

• This function is used to model production. For example, in the context of a factory, $Y$ could be the number of trucks produced, $K$ the factory size, and $N$ the number of employees. $A$ could reflect bench or assembly line production.

Diminishing Returns to Capital (or Labor)

• If we double the amount of capital, but not labor, then it is reasonable to assume output will less than double. Why?

• Let’s assume $A = 1$. Suppose we increase $K$ by a factor $x$ (e.g., $x=2$ would double capital)

  $$Y(xK,N) = (xK)^\alpha N^\beta = x^\alpha K^\alpha N^\beta = x^\alpha Y(K,N)$$

  (2)

• If $\alpha < 1$, then $Y$ increases by less than $x$, which is the case of diminising returns to capital.

• Likewise, there are diminishing returns to labor if $\beta <1$.

Returns to Scale

• Suppose we scale up both $K$ and $N$ by a factor $x$

  $$Y(xK,xN) = (xK)^\alpha (xN)^\beta = x^{\alpha+\beta} K^\alpha N^\beta = x^{\alpha+\beta} Y(K,N)$$

  (3)

• There are three possible returns to scale for the Cobb-Douglas production function.

  1. If $\alpha + \beta = 1$, there are constant returns to scale (CRS)

  2. If $\alpha + \beta > 1$, there are increasing returns to scale.

  3. If $\alpha + \beta < 1$, there are decreasing returns to scale.

• E.g., if a firms’s production function is CRS, then doubling the amount of both capital and labor will double its output.

Output per Worker¶

• Assuming $A = 1$ (we will revisit this later) and CRS ($\alpha + \beta = 1$), the Cobb-Douglas production function becomes

  $$Y = K^\alpha N^{1-\alpha}$$

  (4)

• Divide both sides by $N$ to get

  $$\frac{Y}{N} = \left(\frac{K}{N}\right)^\alpha$$

  (5)

• Thus, output per worker is just a function of the capital stock available to each worker, where the output elasticty to capital is $\alpha$.

• The San Francisco Fed estimates $\alpha$ to be about 0.40 over the last decade (see Total Factor Productivity).

• The production function is concave.

• If capital is zero, then output is zero.

• Increasing capital increases output (per worker).

• Increasing capital increases output at a diminishing rate.

• i.e., capital accumulation alone should not sustain growth

• If a technology doubles output, then the curve shifts up, twice as high as the original curve.

• If technology were to continue growing each year, then technological progress should sustain economic growth.