- #1
arfie
- 4
- 0
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=<br /> \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<br /> + \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!
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=<br /> \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<br /> + \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!