How would I compare the role of women in The Great Gatsby and Macbeth?
Before comparing "roles" of the female characters, one must first find a commonality shared by all characters.
When first examining two main female characters of F.Scott Fitzgerald's The Great Gatsby, Daisy and Jordan, one can conclude that both females manipulate the men in their lives and conduct themselves amorally for power and love.
Both women use and dispose of people, as Gatsby and Nick experience firsthand. In Fitzgerald's long line of sensual, modern flapper characters, Jordan is one of the most well-known. There is an amoral aura about her, and her world revolves around herself and false material values. Jordan is distinguished from Daisy in her hard, unsentimental view of romance.
Lady Macbeth, main female character of William Shakespeare's classic play, MacBeth, conducts herself similarly--a manipulative wife partaking in amoral actions doing whatever necessary in the name of power. Just like Jordan, Lady MacBeth has a "hard, unsentimental view of romance" as she "goads her husband into the action" belittling him and pushing him to kill to bring him the crown, and likewise, to her.
Ultimately, the roles of Gatsby's and MacBeth's women are to provide internal and external conflicts with the male characters. Nick, although fully aware of Jordan's dishonesty and ambivalent attitude, tolerates her selfish and power hungry attitude. Daisy provides a conflict between Gatsby and Tom. While Gatsby struggles to obtain her love, Tom has his "prize", struggling to keep it far from Gatsby's reach.
Lady MacBeth, just "like [the] two princesses in an unreal world," similarly creates conflict for her leading man. MacBeth, struggles between his loyalty to his king and his hunger for power.
However, when his wife argues with him, attacking his manhood, Macbeth resolves to follow through with the murder.
Lady MacBeth's actions, in turn, create an internal guilt for MacBeth which in turn causes him to react with violence. Of course, we all know that this act in the base of the play and the beginning of his demise.
So perhaps, one can simply conclude that the roles of these female women complicate the lives of the male characters causing amoral actions and responses.
This is the kind of question that one should ideally spend a great deal of time thinking about. I've never thought of The Great Gatsby and Macbeth in relation to one another.
That said, since you probably don't want to wait for an answer while an editor spends a week thinking about your question, I'll mention some obvious comparisons or contrasts.
Lady Macbeth and Daisy are both in subordinate roles as women in male-centered, or patriarchal, societies. No matter how intelligent or talented or capable they might be, their roles are limited. Lady Macbeth is forced to rule through her husband. She is the planner, the intelligent one in the family. If Macbeth would have stuck to her plan, he'd have gotten away with killing King Duncan. Daisy is forced to do what most women in 1930s America had to do if they wanted to improve their social and economic positions: marry a wealthy man. They are both women stuck in a society that confines them.
Women in both works are second-class citizens. That may be your best point of comparison.
In terms of the women themselves, and looking at their behaviors instead of their uncontrollable roles in society, Lady Macbeth is more active or verbally aggressive when she manipulates. They both are forced, at least in a sense, to manipulate in order to fulfill their needs, desires, and dreams. But Lady Macbeth is more direct and aggressive. Daisy is more passive. While Lady Macbeth manipulates her husband into going through with what she knows he really wants to do, and what she wants them to do, Daisy passively allows Gatsby to take the blame for Myrtle's death. She accepts Gatsby's advances, as well. She doesn't start their affair, etc. She probably manipulates just as well as Lady Macbeth, but she is more passive in her approach.
On the surface, Lady Macbeth seems to be in a league by herself, as opposed to Daisy. And she probably is. At the same time, she eventually suffers from devastating guilt for her evil behavior. Daisy's killing Myrtle is an accident, of course, but as far as the reader knows, she doesn't develop any feelings of guilt for the deaths she causes.
I would definitely agree with the previous point in that there has to be a great deal of reflection present here. I would say that one of the similarities I find between the women in both works is that they are shown to be able to manipulate men. Lady Macbeth's hold and sway over her husband is fairly well documented, especially in the early stages of the drama. She is able to wield influence so that he is able to bend to her will. The women in Fitzgerald's world are similar in that they are able to get men to do what they want. Daisy has a hold over Gatsby, that while might not be as driven as Lady Macbeth's, is still evident. Jordan makes her state of being as able to drive men to do what she wants and Myrtle's control over George is fairly dominant. It might be interesting to probe the level of how women in such works are depicted, in that the more powerful or in control a woman is, the more venom with which she is treated and the more blame she receives. It seems to be a no win situation for the depiction of women in many works of Classical Literature. Either they are victimized by their settings or they are shown to be villainous.