Economic growth: Further thoughts

EC 235 | Fall 2023

Materials



Required readings:

  • Blanchard, ch. 11.

Prologue

Prologue


Last time, we derived two relations involving aggregate output per worker (Y/N) and an economy’s capital stock per worker (K/N):



  1. \(\dfrac{Y}{N} = F\bigg(\dfrac{K}{N}, 1\bigg) = f\bigg(\dfrac{K}{N}\bigg)\)


  1. \(\dfrac{K_{t+1}}{N} - \dfrac{K_t}{N} = s\dfrac{Y_t}{N} - \delta\dfrac{K_t}{N}\)

Prologue


\[ \dfrac{Y}{N} = F\bigg(\dfrac{K}{N}, 1\bigg) = f\bigg(\dfrac{K}{N}\bigg) \]


The first relation defines the long-run aggregate production function only depending on the economy’s capital intensity (or capital per worker).

Prologue


\[ \dfrac{K_{t+1}}{N} - \dfrac{K_t}{N} = s\dfrac{Y_t}{N} - \delta\dfrac{K_t}{N} \]


The second relation gives the law of motion of capital: capital per worker depends on two factors:


  1. Investment per worker: since I = sYt and given the saving rate s, output per worker determines the amount of saving per worker and thus the investment per worker;
  1. Depreciation per worker: the capital stock per worker determines the amount of depreciation per worker.

Capital and output dynamics

Capital and output dynamics


Given the two relations, we can further understand the dynamics of capital and output over time.



The easiest way to do so is through a graph.

Capital and output dynamics


Capital and output dynamics


The state in which output per worker and capital per worker are no longer changing is called the steady state of the economy.


In mathematical terms, the steady state is reached when there is no change in the capital stock per worker:

\[ \dfrac{K_{t+1}}{N} - \dfrac{K_t}{N} = s\dfrac{Y_t}{N} - \delta\dfrac{K_t}{N} \implies 0 = s\dfrac{Y_t}{N} - \delta\dfrac{K_t}{N} \]


\[ s\dfrac{Y^*_t}{N} = \delta\dfrac{K^*_t}{N} \]

Capital and output dynamics


As \(\dfrac{Y}{N} = f\bigg(\dfrac{K}{N}\bigg)\), we can write the steady state as:


\[ sf\bigg(\dfrac{K^*}{N}\bigg) = \delta\dfrac{K^*}{N} \]

Sources of growth

Sources of growth


Now that we are aware that there exists a possible steady state of economic growth, what are the sources of economic growth over the long-run?


A few candidates:

  1. The capital intensity;

  2. The savings propensity s;

  3. Consumption (“golden rule”);

  4. Technological progress.


Let us think about each of them.

EC 235 - Prof. Santetti