Q $4-11$: Suppose we have a monopolist supplying two different markets. The demand in these markets

is given by two types of consumers, each buying exactly one unit of the product a monopolist is selling so long as their

consumer surplus is non$-$negative. If the consumer has a choice, she or he will buy the product that gives them the highest

consumer surplus. the monopolist has estimated the indirect utility $($CS$)$ of each type of consumer as

$V_1=7z_1-p_1$

$V_2=2z_2-p_2$

where $z=\{z_1,z_2\}$ is the quality chosen by the monopolist i$.$e$.$ vertical differentiation. The monopolist does not know

each consumer's type. There are $397$ type one consumers and $107$ type two consumers. Finally suppose the marginal cost

for all quantities is given by $c(z)=\frac{1}{2}{z}^2.$

Q $4-1$: What are the lowest qualities type $1$ and $2$ consumers will demand.

Q $4-3$: What are the incentive compatibility constraints for type $1$ and $2$

Q $4-4$: The monopolist has three options:

Sell only to high type consumers

Sell to both consumers the same quality product

Sell to both type of consumers but different quality products.

Solve for all these options and determine which one is best.