Labour-saving vs. Labour-augmenting technical change

This is how I'd approach the problem. Please point out any issues on this method as it is based on my own approach (I have no textbook to reference this to).

Based on the information you have, you would run a regression of log output on log-labor, log-capital and log-human capital. This would give you a model like this.

$$ln(Y)=beta_0+beta_1ln(L)+beta_2ln(K)+beta_3ln(H)+mu$$

in terms of a more "economic looking" equation, we take the expectation of this equation and take $e$ and raise it to the power of both sides giving us our production function.

$$mathbb{E}[ln(Y)]=mathbb{E[}beta_0+beta_1ln(L)+beta_2ln(K)+beta_3ln(H)+mu]$$

$$ln(Y)=beta_0+beta_1ln(L)+beta_2ln(K)+beta_3ln(H)$$

Recall that we view $beta_0$s the co-efficient on omitted variable $ln(A)$ as the rate of technological change^1,2 $$exp{ln(Y)}=exp{beta_0+beta_1ln(L)+beta_2ln(K)+beta_3ln(H)}$$

$$Y=A^{beta_0}L^{beta_1}K^{beta_2}H^{beta_3}$$

Using this form you can more comfortably calculate elasticity of subsitution between $L$ and $K$. If your elasticity of substitution is greater than or equal to 1 you have a labor saving process, however if elasticity of substitution is less than 1, we either have a process which is either a Human capital augmented process of TFP augmented process ³.

Hope this helps

---

_{1. https://en.wikipedia.org/wiki/Solow_residual#Regression_analysis_and_the_Solow_residual}

_{2. the actual "quantity" of $A$ can be calculated by $$A=left(frac{Y}{L^{beta_1}K^{beta_2}H^{beta_3}}right)^{frac{1}{beta_0}}$$}

_{3. this is of course assuming that either $beta_3>0$ and/or $beta_0>0$.}