The American Dream opens in a nondescript apartment in post-WWII America. The protagonists, Mommy and Daddy, are seated in their armchairs. The couple is waiting for some expected visitors, but their guests are late, and Mommy remarks that people can get away with anything these days. She starts to tell a story about purchasing a hat the previous day, but Daddy does not seem to be paying attention to his garrulous wife. Mommy feels disturbed, and by taunting her husband repeatedly, she makes sure that he actually lends an ear to her. Mommy then resumes her story of hat-buying, telling Daddy how happy she was with her beige hat until she met the Chairwoman of her women’s club, who insisted that the hat was wheat. Mommy felt cheated, and went back to the hat shop for a replacement. They insisted that the hat was white, but Mommy made a scene in the shop, and eventually got a new hat and thus derived "satisfaction."
Meanwhile, Mommy’s mother, Grandma, enters from the archway, “loaded down with boxes, large and small, neatly wrapped and tied." She does not seem to be very polite to her daughter, as evident from her straightforward reply to Mommy’s inquiry regarding the contents of the box: “That’s nobody’s damn business." Dumping her boxes at Daddy’s feet, Grandma starts complaining how the old are not comfortable talking to anyone, as some people can be disrespectful to the elderly. She says, “That’s why old people die, eventually. People talk to them that way."
As Grandma goes out to fetch the rest of her boxes, Mommy recalls how Grandma has always been adept at wrapping boxes beautifully. When she was young and poor, Grandma would wrap her lunchbox for school so nicely that Mommy would not have the heart to open it. Her remembrance invokes the old times when the mother and her daughter had a good relationship, and Grandma would fill Mommy’s lunch boxes with her own uneaten dinner from the previous night, and Mommy would bring back her lunch from school for Grandma to eat. Mommy has become rich after marrying Daddy, and she thinks she has a right to use Daddy’s money as she pleases in exchange for letting her husband have sex with her. Grandma calls Mommy a tramp, and recalls how she has been an opportunist since childhood, always wanting to marry a rich man.
The doorbell rings, and the couple thinks that the expected visitors—the "van people"—have finally arrived. The bell rings for a second time, as Daddy takes time to reconsider his decision, but Mommy is firm that they should make up their minds about whether or not to open the door. As Daddy opens the door, Mrs. Barker enters, and amidst the absurdist chit-chat, Grandma says that she is aware of the purpose of Mrs. Barker’s visit.
As Mommy exits the stage to fetch some water for their guest, Mrs. Barker asks Grandma to explain her visit. The old lady gives the guest a subtle hint: twenty years ago, a couple like Mommy and Daddy lived in an apartment quite like their apartment with an old woman very similar to Grandma. The only difference was that “they were all somewhat younger." They contacted an adoption agent, akin to Mrs. Barker, at an adoption agency very similar to Mrs. Barker’s Bye-Bye Adoption Service, and were overjoyed to receive their "bumble of joy."
Things did not turn out as planned for the couple similar to Mommy and Daddy. The baby, or "bumble," did not look like either of its parents. That was manageable, but one night it cried its heart out, and its mommy said it only had eyes for its daddy. Mrs. Barker responds that in such a case, any self-respecting woman would have gouged those eyes right out of its head. Grandma tells them that the mommy actually did so, but then the baby “kept its nose up in the air” and even worse, "began to develop an interest in its you-know-what." Mrs. Barker replies that in such a case the parents should have cut off the baby’s hands; Grandma tells her that they first cut off its you-know-what. The adamant child, however, still put its hand under the covers, looking for its you-know-what, so the parents had to cut off its hands at the wrists as well. Similarly, its tongue had to be cut off as it had called its mommy dirty names one day. Eventually, the resentful bumble died; its parents, looking for satisfaction, called the adoption agent to their apartment to demand a refund. Even after this story, Mrs. Barker fails to realize its relevance, and exits to fetch a glass of water.
The doorbell rings again, and this time a Young Man presents himself. Grandma announces, in an affectionate and approving voice, that he is the American Dream. He looks familiar to Grandma. The Young Man explains that he is looking for a job, would do almost anything for money, and the apartment looked like a "likely building" to him. Grandma asks why he is ready to do any work for money, and he replies that he must compensate for his incompleteness. Grandma asks what that means, and the Young Man tells her that his mother died at childbirth, and he never knew his father. In birth, he came to know that he was not alone in the placenta, and had an identical twin, from whom he was separated while still very young. The Young Man recalls the close bond he shared with his twin brother: their hearts beat at the same cadence, and their stomachs ached at the same time as they cried for feeding. After the separation, he has suffered countless unexplainable losses. He feels like his heart has been wrenched from his body, and he has become unable to love; a "specific agony" has made him incapable of physical lovemaking, and he has been unable to feel and touch as if his own hands have been removed.
Grandma murmurs in pity, and says that she was wrong earlier and never knew the Young Man, but probably she knew someone very much similar to him “who might have turned out to be very much like [the Young Man has] turned out to be." Grandma feels that the Young Man is the perfect replacement for the faulty bumble from her tale. She says that he is about to get the job he is looking for.
Mrs. Barker enters, and Grandma introduces the Young Man as the "Van Man" who has come to take Grandma away along with her boxes. As he carries her boxes outside, Grandma whispers to Mrs. Barker how she has planned to solve the dilemma of Mommy and Daddy. Grandma then leaves, heading towards the elevator, and assuring the Young Man that everything will be explained to him. Mrs. Barker announces to Mommy and Daddy that their problem is settled, and that the Van Man has come and taken Grandma away with all her belongings. Near tears, Mommy says that it is impossible, because the Van Man is not a reality but their invention. While Daddy comforts Mommy, Grandma comes out near the footlights on the stage, visible only to the audience, looking forward to watch the ensuing events.
Mrs. Barker announces that the Young Man is the replacement for the faulty bumble, which pleases Mommy, who calls for a celebration. Everyone is apparently happy, as they have gotten what they have been looking for; Grandma addresses the audience, saying that she should go now, leaving the others as they are, and bids farewell to the audience.
