Petco held its IPO on January 14, 2020. The firm issued a total of 55.2 million shares at an offer price of $18 per share. The stock closed at the end of the first trading day at $29.40 per share. 

a) List the various direct costs the firm likely paid in association with this offering and, using historical averages for these costs, estimate the approximate dollar amount of proceeds to Petco after paying these costs. 

b) The stock price rose 63% on the first day of trading, suggesting that the firm may have leftover $600 million on the table. Briefly describe two reasons why the underwriter might choose to underprice the IPO.