Wage Growth Rate in the R&D Model

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.

• Can you show the R&D model for context? It will make the question more clear – Brennan Apr 23 '20 at 21:29

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