The evidence that unions substantially increase productivity is contradicted by the evidence that the effect of unions on employment is small. It is shown in this paper that if the union is constrained by the firm's demand function, and it is the efficiency units of labor that are in the demand function, then essentially the only way to resolve this contradiction is for unions to raise the productivity of capital (and not of labor) under conditions where the substitutability between labor and capital is very limited. But estimates of the substitutability parameter are near unity, so that one of these effects must be wrong: either unions do not substantially increase productivity or they substantially reduce employment.