Hello! everyone.
There are several questions about detrending the Jermann(1998) model. JohannesPfeifer_Jermann(1998)_algebra
Why do we need to detrend the model? And how should we do that? Is it to eliminate the labor-augment technology?
For example, I can’t understand why the capital accumulation equation becomes \gamma K_{t+1} = (1-\delta)K_t - \cdots in the paper(see the linked pdf). Why is there a \gamma in the left-hand side? And why the household objective write in that manner? It seems simple algebra can’t lead to that and I can’t understand it.
I do find a paragraph written by professor Pfeifer. So what is the meaning in the Jermann(1998) example?
Yes, it’s about labor augmenting technology. The whole model is supposed to be in intensive form, i.e all variables are divided by ((1+g_y)*(1+g_n))^t.
gamma_x=(1+n)(1+g_y), i.e. it is the gross growth rate of the economy. It comes from
K_{t+1}=(1-\delta)K_t+I_t
Where K_{+1}=k_{t+1}((1+n)(1+g_y))^{t+1} and similarly for the other variables. Dividing by ((1+n)(1+g_y))^{t} on both sides gives the result. Economically, g_y=x, i.e. the empirically observed growth rate of output per capita g_y is equal to the growth rate of labor-augmenting technology x.
Thank you professor Pfeifer!
Your answer reminds me of the Solow growth model. It seems the g_n is the population growth rate and the g_y is the long-run(steady state) growth rate of labor-augmenting technology.
In this manner, I guess the gross growth rate of the economy is the growth rate of aggregate variables like aggregate output Y, or like aggregate capital K, whose growth rate is \gamma = (1+g_y)*(1+g_n) \approx 1+g_y+g_n .
Further, I guess I can detrend asset pricing or RBC models in Solow’s spirit as long as the production function and capital accumulation equation are set same as Solow model. Is it right?
And one more question about the algebra file. You say the model is supposed to be in intensive form. Then, do the upper case letters(after detrending) in the file still represent per effective worker level?
Yes, but they represent values per effective worker along the BGP, i.e. technology growth is separated into a linear deterministic component and stochastic fluctuations around the trend.