The study compares how public and private companies in a mixed duopoly (a market with both private and public firms) decide on prices or quantities while dealing with a union. The public firm has an advantage in choosing contract types. In this setup, having companies compete on price leads to better overall welfare than when they compete on quantities. Social welfare is higher in Bertrand competition than in Cournot competition in this mixed duopoly. There are multiple possible outcomes when companies negotiate contracts. Overall, these results hold true regardless of the type of goods being sold, except when dealing with complementary goods, in which case private firms' profits don't always switch positions as expected.