Understanding the Lagrangian Function for Maximization Problems

[arfie]

Hi everyone!

I really need help for this. I have to read a paper in economics where some parts I don't understand.

Suppose:

$S \equiv [\alpha,\bar{\alpha}]x[y,\bar{y}]$
$V^e(p_j,g_j,y+r(r\alpha)-T_j,\alpha)\equiv \underset{h}{\text{max}}U(y+r(r,\alpha)-T_j,p_jh,h,g_j;\alpha)$

And then from maximization problem I have a Lagrangian function as follows:

$L= \sum_{i = 1}^{J} \Big[\int_S \omega (y,\alpha) V^e(p_i,y+r(y,\alpha)-T_i,g_i,\alpha)$ $a_i(y,\alpha)f(y,\alpha)\,dy\,d\alpha + \omega_R(R/J+\int_0^{p_i/(1+t_i)}H^i_s(z)\,dz)\Big]$

$\lambda_1[R+\int_S r(y,\alpha)f(y,\alpha)\,dy\,d\alpha]+\lambda_2[\int_S h_d(p_i,y+r(y,\alpha)-T_i,g_i,\alpha)a_i(y,\alpha)f(y,\alpha)\,dy\,d \alpha-H_S^i]$

How can I solve for FOC for the lagrangian function with respect to $r(y,\alpha)$? What rules should I use? I guess it has something to do with Leibniz's rule and chain rule but I'm not sure. Thanks!

 

[joeblow]

Definitely not enough info here for anyone to understand what's going on. For instance, "solve for FOC for the lagrangian function with respect to r(y,a)" is meaningless to me.

 

[arfie]

Definitely not enough info here for anyone to understand what's going on. For instance, "solve for FOC for the lagrangian function with respect to r(y,a)" is meaningless to me.

Thanks for the reply.
I mean "how to get first derivative of L w.r.t r(y,a) which is dL/dr(y,a)?"

 

[micromass]

Thanks for the reply.
I mean "how to get first derivative of L w.r.t r(y,a) which is dL/dr(y,a)?"

Sorry, but you're going to have to give way more information. Start by defining all the notation you use. Describe the actual problem that's given. Maybe give a reference to where you have seen this.

 

[arfie]

Sorry, but you're going to have to give way more information. Start by defining all the notation you use. Describe the actual problem that's given. Maybe give a reference to where you have seen this.

Sorry if it's not that clear. The reference is here: http://www.nber.org/papers/w17251.pdf
My problem is in page 14, equation (20). I don't know how to get it.