Question on vertical integration:
Consider the market for a good with an inverse demand curve given by
p(q) = 100 - 1/3q
The market is served by a quantity-setting monopolist retailer. The retailer faces the cost function
cr(q) = q(10 + w)
where w is the wholesale price charged to the retailer by the manufacturer of the good. Note that the 10 in the cost function above does not go to the manufacturer, only the w (you can think of this as a per-unit fee charged by an exogenous shipping company who transports the goods between the manufacturing plant and the retailer). The manufacturer is also a monopoly and sells the good exclusively to the retailer. The manufacturer sets the wholesale price, and faces the cost function
cm(q) = 18q.
(a) Derive the wholesale demand curve faced by the manufacturer, as a function of w.
(b) Calculate the equilibrium w, q, and p.
(c) Calculate manufacturer profit, retailer profit, and deadweight loss. Hint - this is tricky to graph, so you
need to be very careful here.
(d) Calculate the equilibrium q and p if the retailer and manufacturer merge into a single, vertically integrated
firm.
(e) Calculate total profit and deadweight loss in the merged case. How does this compare to (c)?