Q: A monopolist has TC=q2/2+10q. Its demand is given by P=40-2Q. Find dead weight loss.
A: First we need to determine the monopolist quantity and the efficient quantity. To find the monopoly quantity set MR=MC. MC=q+10 and MR=dTR/dQ=d/dQ(40q-2q2)=40-4q. Thus the monopoly quantity is 40-4q=q+10 or 30=5q or q=6. The efficient quantity is where Value (price)=MC or 40-2q=q+10 or q=10. There is a deadweight loss because the monopolist keeps the quantity below the efficient level. So nobody will get the value for the lost units and of course society does not incur the cost for those units.
The last piece we need to know is the difference between value and MC at the monopoly quantity. The value is the price the monopolist will charge. To find this plug q=6 into demand. This gives P=40-2(6)=28. MC for the 6th unit is q+10=16. Thus, DWL=(1/2)*(10-6)*(28-16)=24.