Suppose the agent can be one of three types rather than just two as in the chapter. a. Return to the monopolist's problem of computing the optimal nonlinear price. Represent the first best in a schematic diagram by modifying Figure \(18.4 .\) Do the same for the second best by modifying Figure 18.6 b. Return to the monopolist's problem of designing optimal insurance policies. Represent the first best in a schematic diagram by modifying Figure \(18.7 .\) Do the same for the second best by modifying Figure 18.8

: We'll first consider the optimal nonlinear pricing problem for three agents and draw schematic diagrams for both the first best and second best solutions. Step 1: First Best Solution For the first best solution, we'll modify Figure 18.4 by adding another agent type. This will involve extending the axes, adding a new point for the third agent type, and connecting the dots to create the new first best allocation. Step 2: Second Best Solution To represent the second best solution, we'll modify Figure 18.6 by introducing the third agent type. In this case, we'll also have to extend the axes and add a new point for the third agent type. Then, we'll connect the points to show the second-best optimal nonlinear prices. Note that the lines representing the allocations may no longer be parallel due to the introduction of the third agent type.

: Now, let's analyze the optimal insurance policies when there are three agent types. We will draw schematic diagrams for both the first best and second best solutions. Step 1: First Best Solution for Insurance Policies For the first best solution, we'll modify Figure 18.7 by adding another agent type. We again need to extend the axes, add a new point for the third agent type, and draw a line connecting the agent types for the first best allocation. Step 2: Second Best Solution for Insurance Policies To represent the second best solution, we need to modify Figure 18.8 by introducing the third agent type. This involves extending the axes, adding a new point for the third agent type, and connecting the points to show the second-best optimal insurance policies. By completing these steps, we've successfully analyzed and drawn schematic diagrams for both optimal nonlinear prices and optimal insurance policies while considering three agent types.