§11 Technological Progress and Growth

Technological Progress and the Rate of Growth

• Technological progress takes many forms:

  • larger quantities of output for given quantities of capital and labor;
  • better products;
  • new products;
  • a large variety within products.

• In this course we will (primarily) capture the state of technology $A$ in terms of labor-equivalence (as if we had more labor):

  $$Y = F(K, AN)$$

• So $AN$ is the amount of effective labor.

• Recall that constant returns to scale implies

  $$xY = F(xK, xAN)$$

• Set: $x = 1/AN$, to obtain:

  $$\frac{Y}{AN} = f\left(\frac{K}{AN}\right)$$

Capital per Effective Worker (Dynamics)

$$K_{t+1} = I_t + (1 - \delta)K_t$$

$$\begin{aligned}\frac{K_{t+1}}{A_{t+1}N_{t+1}} &= \left(s \frac{Y_t}{A_tN_t} + (1 - \delta) \frac{K_t}{A_tN_t}\right) \frac{A_tN_t}{A_{t+1}N_{t+1}} \\&\approx \left(s \frac{Y_t}{A_tN_t} + (1 - \delta) \frac{K_t}{A_tN_t}\right)(1 - g_{AN}) \\&\approx s \frac{Y_t}{A_tN_t} + (1 - \delta - g_{AN}) \frac{K_t}{A_tN_t}\end{aligned}$$

• Note that in the last step we set $s \cdot g_{AN}$ and $\delta \cdot g_{AN}$ to zero since these are small numbers. We can now write:

  $$\frac{K_{t+1}}{A_{t+1}N_{t+1}} - \frac{K_t}{A_tN_t} \approx s \frac{Y_t}{A_tN_t} - (\delta + g_{AN}) \frac{K_t}{A_tN_t}$$

Capital per Effective Worker (Balanced growth)

$$\frac{K_{t+1}}{A_{t+1}N_{t+1}} - \frac{K_t}{A_tN_t} \approx s \frac{Y_t}{A_tN_t} - (\delta + g_{AN}) \frac{K_t}{A_tN_t}$$

• The balanced growth (or steady state) of this economy can be found by setting the LHS to zero.

• Conceptually, this requires that investment is exactly what is needed to cover the depreciation of the existing capital stock and to catch up with the growth in effective labor (the denominator).

• Mathematically:

  $$s \frac{Y_t}{A_tN_t} = (\delta + g_{AN}) \frac{K_t}{A_tN_t}$$

Balanced Growth

For example, $Y = K^{1-\alpha}(AN)^{\alpha}$

$$g_Y = (1 - \alpha)g_K + \alpha(g_A + g_N)$$

balanced growth (steady state)

$$(g_A + g_N) = (1 - \alpha)(g_A + g_N) + \alpha(g_A + g_N)$$

Measuring Technological Progress

• How do we measure the rate of technological progress? (Solow 1957)

• Suppose each factor of production is paid its marginal product (14.01).

  • Under this assumption, it is easy to compute the contribution of an increase in any factor of production to the increase in output.
  • For example, if a worker is paid $30k a year, her contribution to output is $30k.
  • If this worker increase the amount of hours she works by 10%, the increase in output is $3k.

  $$\Delta Y^N = \frac{W}{P} \Delta N$$

  $$\frac{\Delta Y^N}{Y} = \frac{WN}{PY} \frac{\Delta N}{N}$$

  $$g_Y^N = \alpha g_N$$

• We can do the same for capital

  $$\begin{aligned}\frac{\Delta Y^K}{Y} &= \left( \frac{PY - WN}{PY} \right) \frac{\Delta K}{K} \\&= \left( 1 - \frac{WN}{PY} \right) \frac{\Delta K}{K} \\g_Y^K &= (1 - \alpha) g_K\end{aligned}$$

• [The Solow residual] The residual between actual GDP growth and the growth due to labor and capital, must be due to technological progress

  $$\text{residual} = g_Y - [\alpha g_N + (1 - \alpha)g_K]$$

— Apr 19, 2025

§11 Technological Progress and Growth by Lu Meng is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Permissions beyond the scope of this license may be available at About.