Optimize Using Lagrangian $U_A=\min ({X_A},Y_A)$ and $U_B=\min(\frac{X_B}{4}, Y_B)$ Utility functions of the two individuals is defined as $U_A=\min (X_A,Y_A)$ and $U_B=\min(\frac{X_B}{4}, Y_B)$ where $X$ and $Y$ denote individual $A$ and $B$'s consumption of good $X$ and $Y$ respectively.
How do I maximise for the utility of the two individuals subject to the budget constraint
$$pX_A+Y_A=2p+1$$
and
$$pX_B+Y_B=2p+1$$
where $p$ is a constant
 A: Well, in this particular case, the utility functions are not differentiable, so I can't see how you are trying to tackle the problem with Lagrange multipliers.
Here I'll show how I would solve the problem for consumer $A$ (as there are no explicit restrictions on the supply of goods $X$ and $Y$, I am assuming that these are two separate problems).
The utility $U_A=\min(X_A,Y_A)$ changes behavior whenever crossing the line $X_A=Y_A$. Given the budget constraint
$$pX_A+Y_A=2p+1,$$
the crossing poing between the constraint and the behavior change of $U_A$ is
$$X_A(p+1)=2p+1\qquad\Rightarrow\qquad X_A=\frac{2p+1}{p+1},$$
and we can rewrite the utility as
$$U_A=\begin{cases}X_A&X_A\leq\frac{2p+1}{p+1}\\2p+1-pX_A&X_A>\frac{2p+1}{p+1}\end{cases} $$
The first branch increases with $X_A$, with a maximum value of $\frac{2p+1}{p+1}$. The second decreases with $X_A$, with a maximum value of $\frac{2p+1}{p+1}$.
Therefore, the maximum utility of consumer $A$ subject to the budget contraint is
$$\max\,\, U_A=\frac{2p+1}{p+1}.$$
A similar approach can be done with the second consumer.
