Linearize a CES production function

Dynare help (legacy posts)
1

Hello,

I am trying to linearize a CES production function of the form:

Y_t = a * L_t^p + (1-a) * K_t^p] ^(1/p)

Y_t ^p= a * L_t^p + (1-a) * K_t^p

p*ln(Y_t) = ln(a L_t^p + (1-a) K_t^p)

p(ln(Y) + (Y_t – Y)/Y) ~ ln(a L^p + (1-a) K^p) + ((paL^(p-1)/ (a L^p + (1-a) K^p))(L_t –L) + ((p*(1-a)*K^(p-1)/ (a L^p + (1-a) K^p))(K_t –K)

(Y_t – Y)/Y) ~ ((a*L^(p-1)/ (a L^p + (1-a) K^p))(L_t –L) + (((1-a)*K^(p-1)/ (a L^p + (1-a) K^p))(K_t –K)

y_t ~ ((a*L^p/ (a L^p + (1-a) K^p)) l_t + (((1-a)*K^p/ (a L^p + (1-a) K^p))k_t

Is this correct?

I am concerned about the L^p and K^p , I know that they are constants but I am not sure if I can use this equation in my dynare code in this form.

I am use to seeing the C-D form

y_t = a*l_t + (1-a)*k_t

Any insight would be greatly appreciated.

Thank you in advance.

Regards,
Richard

2

Dear Richard, please have a look at Cantore/Levine (2012): Getting normalization right: Dealing with ‘dimensional constants’ in macroeconomics, dx.doi.org/10.1016/j.jedc.2012.05.009 and their Appendix A.

3

Is there any sample dynare code online for estimating a DSGE model assuming we have a CES production function as per Cantore et. al (2010)?