IV. Consider a series of two-person two-commodity exchange economy. In each case, the two agents have the same endowments: each has one unit of good 1, and one unit of good 2. We normalize the price of good 2 to \( p_2 = 1 \), and look for the competitive equilibrium, that is, both agents are price takers. Find out all, if any, of the equilibrium price of good 1, \( p_1^* \), according to the utility function specified in each case (where \( x_j^i \) denote Agent \( i \)'s consumption of good \( j \)).