"...the notion that a social system moved by independent actions in pursuit of different values is consistent with a final coherent state of balance, and one in which the outcomes may be quite different from those intended by the agents, is surely the most important intellectual contribution that economic thought has made to the general understanding of social processes. [Adam] Smith also perceived the most important implication of general equilibrium theory, the ability of a competitive system to achieve an allocation of resources that is efficient in some sense. Nothing resembling a rigorous argument for, or even a careful statement of the efficiency propositions can be found in Smith, however."
- Arrow and Hahn, 1971Arrow of course proceeds to provide some efficiency propositions that most economists would find relevant to think about, and he rigorously demonstrates the point that individuals pursuing their own interests can generate beneficial results for society in a system of market exchange. Adam Smith took an enormous step forward and provided arguments that had been fairly convincing. Walras, Arrow, Hahn, Debreu, etc. picked up where Smith left off. I'm sure they found Smith's arguments plenty convincing, but they were nevertheless able to provide new arguments - and they did.
Is Arrow's efficiency criteria Smith's? No, of course not. They didn't even talk about "efficiency criteria" back then and Pareto hadn't even been born yet (much less Kaldor or Hicks). To point out that this is not precisely how Adam Smith addressed the question misses the point. We don't want to keep providing the same answer, we want to make this Smithian channel of thought deeper, broader, and stronger.
Are the parameters of Arrow's inquiry exactly the same as Smith's? No, of course not. The analytical toolbox gave primacy to some assumptions and made others less relevant. In the end Arrow did not speak to all that Smith spoke to, and vice versa. But again, we don't want to keep providing the same answer in science. We want to take a good answer and make it deeper, broader, and stronger. And that's what Arrow did.
Mark Blaug has called this a "travesty", but Mark Blaug seems to think Arrow's citation of Smith indicates that Arrow thought he was reproducing Smith's argument as opposed to doing constructive work in the same channel of thought. Mark Blaug is a historian of thought, but in calling Arrow's citation of Smith a "travesty" he demonstrates that tendency that we all succumb to sometimes when we hold a hammer: the tendency to see nails everywhere.
I, for one, am always in awe whenever I even dip my toes into the work of someone like Arrow. The suggestion that he is not moving forward Smithian economics because his framing of the issue is not identical to Smith's seems to miss the whole point of "moving forward Smithian economics" to me. The most important element here is that private choices in market exchange are socially beneficial. Arrow has that. The idea that he brings in certain efficiency criteria and assumptions to talk rigorously about this phenomenon doesn't change the fact that he's doing Smithian economics. Smithian economics is about the heart of the theory, not the minutiae of approaching the question that change over time. On the flip side, the idea that Arrow is not moving forward Smithian economics because he only provided his answer to a relatively restricted case is laughable. You try to do better if you are tempted to throw this at him. I know I can't!