The effect of the minimum wage in the short run on existing firms can be small. They mostly just suck it up and pay the higher wage. However, over time, there will be a tendency for processes that use low-skilled workers to be less profitable and processes that instead use capital and high-skilled workers to be relatively more profitable. So the patterns of specialization and trade that break up will tend to be those that have been employing low-skilled workers, and the new ones that form will tend to employ fewer low-skilled workers than would have been the case otherwise.
As for the other evidence – the evidence that finds no negative employment effects of minimum-wage hikes – the vast bulk of this evidence comes from studies that take a minimum-wage regime as a given. That is, these studies are done of economies in which minimum-wage legislation already exists and, typically, has existed for a long time. Further, these economies are ones not only in which there is no expectation that minimum-wage legislation will be permanently repealed, but likely contain widespread expectations that existing levels of minimum wages will be raised to higher levels sometime in the foreseeable future.
In short, nearly all existing studies that find no negative employment effects of minimum wages are studies of economies in which employers and employees have already adjusted their production arrangements and employment-contract terms (both formal and informal) to the realities of binding minimum wages and to the likelihood that existing minimum wages will be raised, if not today or tomorrow, the day after tomorrow. These empirical studies, therefore, tell us at best how employers and employees adjust to unexpected increases in existing minimum wages. (See here and here.) And very few of these studies even attempt to distinguish unexpected hikes in the minimum wage from expected hikes. So even if such studies correctly find no negative employment effects of minimum-wage hikes, those findings do not tell us that imposing a minimum wage where none before existed (and where none had been reasonably anticipated) will not destroy jobs for some low-skilled workers.
No matter how scientific empirical investigations of minimum-wage changes appear to be – no matter how well-adorned are such studies with econometric bells, whistles, and impressive jargon – if they fail to account for the long-run effects on what Arnold calls “patterns of specialization and trade,” their conclusions supply no guidance for assessing the full, long-run impact of minimum wages. Alas, the economics profession has within it many people are skilled econometricians but unskilled economists – and no amount of the former begins to make up for the latter.