This paper derives necessary and sufficient conditions under which the firstorder characterization of Holmström (1979) robustly produces affine solutions: a risk-neutral principal, an agent with logarithmic utility, and output whose distribution depends on effort through a one-parameter exponential family with a linear sufficient statistic. For this primitive structure, I establish existence of an optimal contract and validity of the first-order approach in two cases: first, without a lower bound on compensation when the principal’s target action is below a threshold; and second, when the lower bound on compensation lies strictly above the agent’s subsistence consumption level, with the gap sufficiently small. In the latter case, the level of the principal’s target action determines whether fixed pay is optimally complemented with equity or call options.