Wage Growth Rate in the R&D Model

Question

I am trying to understand the R&D model and how to calculate the growth rate of wage in this model.

There is no equation given, but it’s stated that : household income must equal total consumption. as income is total profit plus wage income, and the two are fixed in proportions, wages grow at the same rate as output $g_w=g_y$

I tried to write down the equation of this statement, however, I don‘t really think it’s right: $\pi+w= C_t$.

Can anyone help with the equation and it’s intuition?

Addition: the R&D model introduces final good producers that sell their good in a competitive market and intermediate goods producers that produce either a produced intermediate good or “ideas” that are sold to the final good producers and used to produce the final good. The intermediate good and “ideas” are substitutes.

Answer 1

income is total profit plus wage income, and the two are fixed in proportions

I.e. $y_t = w_t + \pi_t$ and $w_t = \alpha y_t$ and $\pi_t = (1-\alpha)y_t$, where $\alpha$ and $(1-\alpha)$ are the fixed proportions.

For $y_{t+1}$ we have $y_{t+1}=(1+g_y)y_t$ and $w_{t+1} = \alpha y_{t+1} = \alpha (1+g_y)y_t = (1+g_y)\alpha y_t = (1+g_y) w_t$ and $\pi_{t+1} = (1-\alpha) y_{t+1} = (1-\alpha)(1+g_y)y_t = (1+g_y)(1-\alpha) y_t = (1+g_y) \pi_t.$

I.e. $g_y = g_w = g_{\pi}$