Is unemployment lower if unions bargain over employment?

We consider an economy in which all firms are unionized and bargain with their own union. (1) If unions bargain over employment as well as wages, employment will be the same as if they bargain over wages only, provided that the production function is Cobb-Douglas. (Employment will be higher if the elasticity of substitution between labor and capital is smaller than unity.) (2) If we start from a fully competitive labor market and then move to one of efficient bargaining (over wages and employment), employment falls. This is so even if the marginal utility of income is constant, so that bargaining is “strongly efficient.”