The rationale for Schopenhauer’s renunciation as the basis for happiness lies in the mutability of the physical world:
The chief obstacle to our arriving at these salutary views is that hypocrisy of the world to which I have already alluded–an hypocrisy which should be early revealed to the young. Most of the glories of the world are mere outward show, like the scenes on a stage: there is nothing real about them. Ships festooned and hung with pennants, firing of cannon, illuminations, beating of drums and blowing of trumpets, shouting and applauding–these are all the outward sign, the pretence and suggestion,–as it were the hieroglyphic,–of joy: but just there, joy is, as a rule, not to be found; it is the only guest who has declined to be present at the festival.
While Schopenhauer derived his concept of renunciation due to mutability based upon a Buddhist (which is consonant with Hindu concepts) understanding of reality. However, in evaluating his reading, we should also compare this language with Western responses to mutability. First, concepts of mutability were being addressed in contemporary Western Romanticism as well as Western thought generally. Second, by considering a different consideration of the same proposition from a different direction, we have a greater perspective to evaluate the matter.
Schopenhauer’s metaphor to life being a play (“Most of the glories of the world are mere outward show, like the scenes on a stage: there is nothing real about them), appears much in Shakespeare. Perhaps the closest analogy is found in Macbeth’s speech upon hearing that his wife has died and now his downfall is in view:
(FTLN 2278)  The Queen, my lord, is dead.
(FTLN 2279)  She should have died hereafter.
(FTLN 2280)  There would have been a time for such a word.
(FTLN 2281)  Tomorrow and tomorrow and tomorrow
(FTLN 2282)  Creeps in this petty pace from day to day
(FTLN 2283)  To the last syllable of recorded time,
(FTLN 2284)  And all our yesterdays have lighted fools
(FTLN 2285)  The way to dusty death. Out, out, brief candle!
(FTLN 2286)  Life’s but a walking shadow, a poor player
(FTLN 2287)  That struts and frets his hour upon the stage
(FTLN 2288)  And then is heard no more. It is a tale
(FTLN 2289)  Told by an idiot, full of sound and fury,
(FTLN 2290)  Signifying nothing.
Act V, 5, 19-31. Schopenhauer has a similar view to Macbeth when it comes to the matter of the mutability of the world. Seeing that the world is impermanent, life is meaningless and is best not trusted. This is especially poignant in Macbeth, in that he has been destroying a kingdom and committing murder upon murder to obtain something which he hoped would be permanent: a kingdom. But even in that, he was told that the throne would not pass to his son.
There is thus the irony of seeking to obtain something seemingly powerful and permanent: a throne; and to seek it by proving that neither the throne nor life is permanent.
A similar but different view of the matter is found in The Tempest, Shakespeare’s farewell to the stage.
The Magician Prospero has put a play acted by spirits who appear from nowhere and then vanish just as quickly when Prospero needs to attend to other business. The young man who marry is daughter is distraught at the sudden disappearance of the spirits. Prospero then turns to him and uses the doctrine of mutability to explain the matter:
Prospero, ⌜to Ferdinand⌝
(FTLN 1832)  You do look, my son, in a moved sort,
(FTLN 1833)  As if you were dismayed. Be cheerful, sir.
(FTLN 1834)  Our revels now are ended. These our actors,
(FTLN 1835)  As I foretold you, were all spirits and
(FTLN 1836)  Are melted into air, into thin air;
(FTLN 1837)  And like the baseless fabric of this vision,
(FTLN 1838)  The cloud-capped towers, the gorgeous palaces,
(FTLN 1839)  The solemn temples, the great globe itself,
(FTLN 1840)  Yea, all which it inherit, shall dissolve,
(FTLN 1841)  And, like this insubstantial pageant faded,
(FTLN 1842)  Leave not a rack behind. We are such stuff
(FTLN 1843)  As dreams are made on, and our little life
(FTLN 1844)  Is rounded with a sleep. Sir, I am vexed.
(FTLN 1845)  Bear with my weakness. My old brain is troubled.
(FTLN 1846)  Be not disturbed with my infirmity.
(FTLN 1847)  If you be pleased, retire into my cell
(FTLN 1848)  And there repose. A turn or two I’ll walk
(FTLN 1849)  To still my beating mind.
Act IV, Scene 1, lines 163-180. This physical world is insubstantial, so why should a plain realization of this fact trouble you so. Shakespeare does not resolve this tension immediately, but does it work it out through the action of the play.
In Macbeth the usurpation of the king by murder results in the destruction of Macbeth and enormous sorrow for the kingdom. In the Tempest, the usurpation of the Duke (who becomes the magician on the island) is resolved by the restoration of his throne through the marriage of his daughter to the Prince of Naples.
When the play is read against the most common understanding which is Shakespeare giving his leave to the theater, there are various levels of irony. The play within the play is dissolved at the word of the Magician. The play itself is dissolved and the characters are released from their duty to the audience:
(FTLN 2344)  Now my charms are all o’erthrown,
(FTLN 2345)  And what strength I have ’s mine own,
(FTLN 2346)  Which is most faint. Now ’tis true
(FTLN 2347)  I must be here confined by you,
(FTLN 2348)  Or sent to Naples. Let me not,
(FTLN 2349)  Since I have my dukedom got
(FTLN 2350)  And pardoned the deceiver, dwell
(FTLN 2351)  In this bare island by your spell,
(FTLN 2352)  But release me from my bands
(FTLN 2353)  With the help of your good hands.
(FTLN 2354)  Gentle breath of yours my sails
(FTLN 2355)  Must fill, or else my project fails,
(FTLN 2356)  Which was to please. Now I want
(FTLN 2357)  Spirits to enforce, art to enchant,
(FTLN 2358)  And my ending is despair,
(FTLN 2359)  Unless I be relieved by prayer,
(FTLN 2360)  Which pierces so that it assaults
(FTLN 2361)  Mercy itself, and frees all faults.
(FTLN 2362)  As you from crimes would pardoned be,
(FTLN 2363)  Let your indulgence set me free.
It was magic which presented the momentary world of the play, “Now my charms are all o’erthrown”. But within the logic of the play, he would be forced to stay on a “baren” island rather than returned to his dukedom: By the audience letting lose of the illusion of the play, the Magician is permitted to return to his “real” or proper life. When we take a step back and put this into the context of Shakespeare’s life, Shakespeare then would leave the false and magic world for the “real” life at his home.
But notice here that the temporality of the world in this play is not a cause for despair, but rather of release and rest. The play ends with “mercy” and “forgiveness”.
In Sonnet 73, Shakespeare uses the temporality of life as the basis to drive the intensity of the love:
 That time of year thou mayst in me behold
 When yellow leaves, or none, or few, do hang
 Upon those boughs which shake against the cold,
 Bare ruined choirs where late the sweet birds sang.
 In me thou see’st the twilight of such day
 As after sunset fadeth in the west,
 Which by and by black night doth take away,
 Death’s second self, that seals up all in rest.
 In me thou see’st the glowing of such fire
 That on the ashes of his youth doth lie,
 As the death-bed whereon it must expire,
 Consumed with that which it was nourished by.
 This thou perceiv’st, which makes thy love more strong,
 To love that well which thou must leave ere long.
In short, the mutability of life is seen as a catastrophe and tragedy in Macbeth, whose life is marked by murder and usurping the crown. But in other circumstances, the brevity of life can be seen as a basis to better cherish and love the instant world.
In fact, he points to an even deeper reality, a truer life which stands behind the changeable play of this world.