Consider the model y = x 0 + f*(z) + , where [d over =] N(0, 02). We calculate the smallest asymptotic variance that n1/2 consistent regular (n1/2CR) estimators of 0 can have when the only information we possess about f* is that it has a certain shape. We focus on three particular cases: (i) when f* is homogeneous of degree r, (ii) when f* is concave, (iii) when f* is decreasing. Our results show that in the class of all n1/2CR estimators of 0, homogeneity of f* may lead to substantial asymptotic efficiency gains in estimating 0. In contrast, at least asymptotically, concavity and monotonicity of f* do not help in estimating 0 more efficiently, at least for n1/2CR estimators of 0.