Pop Art Design, curated by the Vitra Design Museum and currently at the Barbican, opens with Richard Hamilton’s 1956 ‘Just what makes today’s homes so different, so appealing?’. Made as a poster for the Whitechapel show This is Tomorrow, it’s a witty collage of consumer fantasies scissored out of magazines, reminding us that interest in popular culture among British artists operated as a humorous, semi-anthropological collegiate research project. In part, British Pop was a riposte to the lushness of American consumerism from a small island that had won the war but had lost the peace.

Pop Art in the United States got under way later and many American Pop artists had had first-hand experience of commercial art. Andy Warhol had been an illustrator, James Rosenquist a billboard artist, Edward Ruscha a typographer, while the sculptor Richard Artschwager had worked as a trade furniture-maker. Their work dominates the early part of Pop Art Design, together with some of their source material — a nicely battered Coca-Cola vending machine, for instance. The British work — sculpture by Eduardo Paolozzi, painting by Richard Hamilton and a magazine cover by John McHale — seems quirky and handmade beside the big glossy guns of American Pop.

If we are familiar with the synergy between Pop Art and the iconography of everyday consumerism — advertising, cheap comics, B-movie film posters, Coca-Cola bottles and Campbell soup tins — what is less well understood are the ways in which designers were in turn inspired by Pop Art’s scrutiny of consumerism. Pop Art, at least in the United States, reacted against the sombre grandeur of Abstract Expressionism. ‘Pop Design’, on the other hand, was questioning the tasteful austerity of design modernism, exemplified by the sleek form-follows-function aesthetic of the German electronics manufacturer Braun. Here it all gets complicated and I am not sure that Pop Art Design offers much clarity.

Though billed as the first exhibition to put Pop Art alongside related design, Pop Art Design is a surprisingly solemn exhibition that gives a lot of space to iconic pop art and sculpture and a selection of desirable design classics — quite a few of which are still in production under licence to Vitra’s manufacturing arm. Maybe this explains the presence of Ray and Charles Eames’s 1951 DKR chair and George Nelson’s 1956 Marshmallow sofa, whose originals now fetch astounding prices in the salerooms. Can we call these objects proto-Pop as the show’s curator Mathias Schwartz-Clauss argues? I am not so sure.

Because the show is top-heavy with gilt-edge objects from the Vitra Design Museum we miss a sense of the free-floating, throwaway playfulness of the visual culture of the 1960s and 1970s. Bernard Holdaway’s Tom Tom cardboard furniture has been added in for the Barbican showing, but from a British perspective it seems incredible to find no Biba clothing, no Mathmos lava lamps, no Mary Quant, no humble bean bags, no anarchic geodesic domes, no mention of Habitat and no nod in the direction of the daringly experimental unbuilt 1960s architecture of Cedric Price, Peter Cook and Ron Herron.

Affordable everyday goods are best covered in Pop Art Design in a section on graphics which takes in Peter Blake’s famous collaged cover for the Beatles’ Sergeant Pepper LP, and Milton Glaser’s famous art nouveau-inspired Bob Dylan poster. We learn less from images of Gunther Sachs’s apartment in the Palace Hotel, St Moritz, its bathroom lined with made-to-order enamelled panels by Roy Lichtenstein, a kitchen with a wall of Andy Warhol screen prints and a living room complete with a flock of sheep by French designer François-Xavier Lalanne.

Some of the most beguiling design objects in the show come out of 1960s and 1970s Italy, made in that country’s flexible small workshops. Gruppo Strum’s Pratone, monstrously scaled-up blades of grass intended as informal seating, Studio 65’s armchair in the form of an Ionic capital, Ettore Sottsass’s archaic-looking Malatesta chair that turns out to be made of polyester, foam rubber and vinyl and Achille and Pier Giacomo Castiglioni’s ad hoc tractor-seat stool arguably owe little to transatlantic Pop Art. They destabilise the premise of this exhibition, which sees design as mostly following in the wake of art. Caught between folk art and surrealism, the Italian work still looks political, a series of subtle critiques of American dominance. It is worth a visit to the Barbican just to study some of these remarkable objects.