## Tuesday, May 27, 2014

### The Solow growth model and information transfer

Since Piketty has been in the news about economic growth and its relationship with the return on capital, I thought an information-theoretic take on the Solow growth model would be in order.

The Solow growth model basically posits that output is given by a Cobb-Douglas form equation

$$Y = K^{\alpha} L^{\beta}$$

I previously applied the information transfer model to Cobb-Douglas form models in matching theory. The same math at that link gives us the information transfer model version of the Solow growth model:

$$NGDP = K^{\kappa - 1} L^{1 - 1/\kappa} - \lambda_{ref}$$

The matching theory gives us an interpretation: labor ($L$) is matched with capital ($K$) and creates NGDP. In the information transfer model that becomes: capital transfers information to labor that is detected by NGDP. Let's see how this model does empirically.

I used the real capital stock data from FRED and adjusted it by the CPI (less food, energy) to give the nominal capital stock. Labor is simply the total non-farm employees. The fit parameters for the duration of the data (1957 to 2011, set by the CPI and capital stock data limits, respectively) are $\kappa = 1.51$ and $\lambda_{ref} = 1081$ billion dollars. This means that the exponents don't exactly fit the "constant returns to scale" assumption $\alpha + \beta = 1$. We have $\alpha = 0.51$ and $\beta = 0.34$. The model doesn't do too badly for such a simple model:

It does better on shorter time scales -- here it is fit to 1980-2010:

In the real Solow growth model, $L$ is actually $A \cdot L$ where $A$ represents phlogiston aether technology and knowledge. If we assume that $A$ is reponsible for the deviation from the constant returns to scale between $K$ and $L$, the imputed value of $A$ is given in this graph (normalized to 1 in 1957):

Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead.