Many jokes were told about New York City’s dangerous economic times in the 1970s. Comedian Marty Allen told this joke in 1975:
 
“A visitor to New York was asked at the airport by a stranger, ‘Would you like to share a cab?’ He said he would. ‘O.K.,’ said the first party, ‘you take the hub caps and I’ll take the motor.’”
 
British comedian Tommy Cooper (1921-1984) told a similar joke, but it’s uncertain when he first said it.
 
   
Wikipedia: Marty Allen
Marty Allen (born Morton David Alpern; March 23, 1922) is an American stand-up comedian, actor, and veteran of World War II. He has worked as a comedy headliner in night clubs and as a dramatic actor in TV roles.
 
Google News Archive
29 July 1975, Pittsburgh (PA) Post-Gazette, “It Happened Last Night” by Earl Wilson, pg. 11, col. 1:
NEW YORK—There’s laughter around the political and economic scenes today if you find comedian Marty Allen out at Westbury Music Fair and impersonator David Frye at Dangerfield’s. They seem to have a good time in our bad times.
(...)
A VISITOR to New York was asked at the airport by a stranger, “Would you like to share a cab?” He said he would. “O.K.,” said the first party, “you take the hub caps and I’ll take the motor.”
 
6 November 1979, San Diego (CA) Union, Frank Rhoades column, pg. 18, col. 1:
Facing a $45 charge to the Hilton in suburban Northfield, San Diego’s Jack Canaan turned to a traveler he’d never seen before, asking, “Want to share a cab?” The guy said, “Yeah, you take the hubcaps and I’ll take the fenders.”
 
Google Books
Strictly for Laughs
By Joey Adams
New York, NY: A & W Publishers
1981
Pg. 267:
One said to the other: “You wanna share a cab?” The other guy said, “Sure— you take the hubcaps— I’ll take the battery.”
 
MumblingNerd’s Mumbling Blog
Tommy Cooper One-liners
18 OCTOBER 2011
Tommy Cooper (1921 – 1984) was a very popular British comedian and magician, who tragically died in the middle of his act on live television. A very funny but flawed man who made a lot of people laugh. I think it’s time to revisit some of his old, traditional but funny, one-liners.
(...)
I met this man at the airport and I asked him if he’d like to share a taxi with me. He said he would, I said: ‘You take the engine, I’ll take the wheels!’
   
Classic Telly
Tommy Cooper Classic TV Show
21 FRED from Illonois: Cooperisms…...
(...)
I met this man at the airport and I asked him if he`d like to share a taxi with me.
He said he would.
I said `You take the engine, I`ll take the wheels!`