Question $3............................................................................................0$ points

A monopolist is serving a market that is divided between two segments; a high income and a low income segments. The

following is the demand for each consumer in the segments.

demand for high income: ph $=33$ qh

demand for low income: pl $=24$ ql

Furthermore, the monopolist$$s marginal cost is given by mc $=23.$ Suppose there are nh $=16$ consumers of type h and

nl $=201$ consumers of type l

Q $3-1$: At quantity q$,$ what is the maximum willingness to pay for each consumer in these segments? draw and shade the demand

curve and the area of the willingness to pay.

Q $3-2$: Suppose the monopolist cannot price discriminate, what is the price and quantity in each segment.

Q $3-3$: Suppose the monopolist knows which segment a consumer belongs to and can prevent consumers trading with each others.

What is the profit maximizing strategy by the monopolist? What is the quantity demanded by the consumers in both

markets and what are the profits of the monopolist from each segment?

Q $3-4$: Now suppose the monopolist only knows there are two segments in the market but cannot tell which segment a consumer

belongs to$.$ Let Vh be the total price paid by consumers h for qh units. Similarly, let Vl be the total price by consumers

l for ql units

i$.$ In terms of Vh$,$ Vl$,$ qh$,$ ql$,$ nl$,$ nh$,$ what are the profits of the monopolist

ii$.$ Since the monopolist does not know which segment a consumer belongs to$,$ it needs to make sure whatever $$package$$

it designs for consumers h is bought by consumers h$.$ It also needs to make sure whatever $$package$$ it designs for

consumers l is bought by consumers l$.$ Write down the constraints that the monopolist has when optimizing profits.

iii. Show that if the participation constraint of l type and the incentive compatibility constraint of the h type are satisfied,

the h type participation constraint will automatically be satisfied. Put it in another way: if the l type is willing to

buy and the h type prefer their package to the l type package, then the h type will also be willing to buy their own

package.

iv$.$ Suppose the monopolist offers consumer l a package with ql $=1$

$10$ and Vl $=479$

$200($note: this might not be the

optimal package but pretend it is$).$ What package will the monopolist offer to consumers h that are willing to buy

instead of the $($Vl$,$ ql$).$ Explain your reasoning