How to maximise the constraint: $U(C_0,C_1)= \sum \frac{1}{3}\log(C_{1}^1)+\frac{1}{3}\log(C_{2}^1)+\frac{1}{3}\log(C_{3}^1)$ Similarly, for the other agent. by using the constraints as: $p_1h_1+p_2h_2+p_3h_3+p_4h_4=0$ (Budget constraint) For agent-1, the consumption bundles can be written in the form of $C_1 \leq w_1 + hX$: (here, subscript denotes portfolio and superscript denotes agent) $C_1 = \begin{bmatrix} 3+2h_1^1+h_2^1+h_3^1+h_4^1\\ 3h_1^1+h_2^1+2h_3^1+3h_4^1\\ h_1^1+2h_2^1+h_3^1+2h_4^1 \end{bmatrix}$ Similarly, for agent-2, it can be written as: $C_2 = \begin{bmatrix} 2h_1^2+h_2^2+h_3^2+h_4^2\\ 3h_1^2+h_2^2+2h_3^2+3h_4^2\\ 3+h_1^2+2h_2^2+h_3^2+2h_4^2 \end{bmatrix}$ Other, constraints are given by: $h_1^1+h_1^2=0, \\ h_2^1+h_2^2=0, \\ h_3^1+h_3^2=0,\\ h_4^1+h_4^2=0$
Now, consumption of both the agents are to be maximised with respect to their portfolios and put in the budget constraints, but I'm unable to proceed further. How to solve so many equations with these variables? Can anyone help me in finding the values of $h_1^1,h_2^1,h_3^1,h_4^1,h_1^2,h_2^2,h_3^2,h_4^2$?
Can anyone help? Can any online website be used to solve this?
Thank you
Basic concepts can be referred to Principles of Financial Economics by Stephen L Roy