Q:  A monopolist with MC=5 faces demand given by P=15-Q.  Calculate Dead Weight Loss.

A:  The first step is to determine what the monopolist will choose to do.
        TR=PQ=(15-Q)Q =15Q-Q2.  Therefore MR=15-2Q.
        Setting MR=MC gives the optimal quantity.  So 5=15-2Q or 5=Q.  Hence qm=5.
        The monopoly price is found by plugging qm into the demand curve. Pm=15-qm=15-5=10.

    The next step is to determine the efficient quantity.  
        This is where P=MC or 15-Q=5 or Q=10. So qe=10.
        The height of the demand curve at this quantity is P=15-10=5.  
        (MC could also have been used to find this height.)

    The Dead Weight Loss is due to the monopoly reducing the quantity traded from 10 to 5. 
        Graphically, this is the area below demand and above MC between qm and qe.
        As long as demand and MC are linear this area is given by:
        DWL=area of this triangle=1/2*(Pm-MC(qm))*(qe-qm)
                where MC(qm) is the marginal cost at qm, which is 5 in this problem.
        Therefore DWL=.5(10-5)(10-5)=12.5