A patriarchal society is one that is controlled by men: as heads of government, religion, society and family. In other words, Elizabethan society (ironically—in that England had a female monarch) was a male-dominated society.
In Shakespeare's The Taming of the Shrew, Kate is a headstrong young woman. She has a vile temper and refuses to marry. Her sister is the exact opposite, much sought after because she is amiable and demur. However, Kate's father (Baptista) will not allow Bianca to marry until Kate is married.
The story is about Petruchio, a clever and strong-willed man, who vows to get Kate to marry him. When they first meet, Kate fights him violently. Soon, Petruchio has won Kate's consent to marry by "clever repartee." In the process of "taming her," he insults her by showing up to their wedding dressed like a slob. He takes her home and denies her food or clothing until she submits to his control over her. This is all a part of his plan to "tame" his bride.
I am no child, no babe.
Your betters have endured me say my mind,
And if you cannot, best you stop your ears.
My tongue will tell the anger of my heart
Or else my heart, concealing it, will break,
And, rather than it shall, I will be free
Even to the uttermost, as I please, in words. (IV.iii.78-84)
The two are arguing about the hat Kate shall have. Among other things, she speaks her mind and says that if she were to conceal her anger in her heart, it would break. (One senses as the play progresses that breaking Kate's spirit is not something that Petruchio desires.)
Petruchio does all he can to antagonize Kate. He tells her that he shall decide the hat she wears and he will gain pleasure in that she does not like it. This is all a part of wearing her down.
By degrees, Petruchio gains ground. Eventually Kate is tamed. In order to please her husband (and in public) she agrees that the sun is shining (as her husband says) when it is really nighttime. Ironically, the other wives at the wedding that Kate and Petruchio are attending, who are called to come out by their husbands to attend to them (as a bet to their superiority to Kate), do not respond. Ironically, it is Kate that scolds the other wives to be more submissive to their husbands, and be good wives.
Thy husband is thy lord, thy life, thy keeper,
Thy head, thy sovereign, one that cares for thee,
And for thy maintenance commits his body
To painful labor both by sea and land,…
…Whilst thou liest warm at home, secure and safe,
And craves no other tribute at thy hands
But love, fair looks and true obedience—
Too little payment for so great a debt. (V.ii.157-161, 163-65)
For the Elizabethan man, the ideal woman should know her place and see to all of her husband's needs, putting her husband above all else. Petruchio is applauded for his ability to tame his wife, especially the once-shrewish Kate. This behavior supports the Elizabethan expectation of submissive women, a patriarchal expectation of the time.
...deals to some extent with the play's treatment of gender roles: that is, what it has to say about socially accepted definitions of appropriate male and female behavior.
It is however, interesting to note that while other playwrights of the time wrote about shrews, Shakespeare adopts a rather unusual perception of Kate—he provides logical reasons why she might act as she does. Petruchio still secures Kate's submissiveness, but for more laudable reasons than once might first expect.
Shakespeare creates characters that for all their fighting are well suited. There is a sexual tension between Petruchio and Kate—we can infer that there is an attraction between these strong characters. The two are closely matched in intelligence. Over time, Petruchio wins Kate, sometimes by force—denying her food, clothing and freedom until she acquiesces. In the meantime, they develop a relationship that draws them together—perhaps it is because Petruchio admires the very strength and spirit that he tries to control. Kate does come around to her husband's way of thinking, but not without the reward of finding a companion that is equal to her spirited personality. And perhaps Petruchio's statement in Act Two, scene one, foreshadows the truth that will ultimately define their relationship. He tells Bianca's suitors:
If she and I be pleased, what's that to you? (309)
On the other hand, Bianca's suitors leave much to be desired.
[T]he prospective suitors are shallow and rude; father and suitors alike tend to treat marriage as a purely commercial transaction.
While Bianca may be happy with these pickings, Kate would never settle for such foppish characters. This is not to say that Kate's character needs a man to tame her—Shakespeare seems to accept the idea of having an accommodating wife, but the author also portrays Petruchio as a man who appreciates his wife's finer qualities. He is not interested in Kate's money, it would seem, as much as he is interested in Kate and the woman who is so unlike her peers—even her sister. At the beginning of their marriage he is harsh in trying to control her, but it develops into something vastly more desirable than what the other men have achieved in their marriages, and I don't believe it because their wives aren't as submissive as Kate. It is Kate's depth, spirit and honesty.
Petruchio explains that is an ideal life to his way of thinking:
Marry, peace it bodes, and love, and quiet life,
And awful rule, and right supremacy,
And, to be short, what not that’s sweet and happy? (V.ii.108-110)
Translations show that "awful rule, and right supremacy" refer to "reverence, and legitimate supremacy." Petruchio allows that "awful rule" (which would not refer to Kate who has no control) is translated into modern English as "reverence" that he will have for his wife, along with peace, love and a quiet way of life. "Supremacy" is not used in a negative context—right supremacy is that justified by society, but he makes no indication of the need for brutal force. These lines seem to define the relationship that Petruchio and Kate have cultivated from their previous grievances with one another. While society may be satisfied by Kate's submissiveness, Petruchio reflects on important aspects of a happily married life that are not based on bullying, beating or oppression. This would seem to be Shakespeare's forward-thinking in seeing his characters from outside contemporary society's lens.
The play addresses the Elizabethan patriarchal society's desire that women—like Kate, and all others—be tamed. However, while Shakespeare allows for a change in Kate's character to meet society's expectations, her spirit is not destroyed in the process.