Uniqueness of a First-Order Condition Solution from Concave Functions
In a constrained optimization problem, the concavity of the objective and constraint functions is a sufficient condition to ensure that the first-order condition has, at most, one solution. This uniqueness arises because the concavity of the utility component v(t) implies its derivative, v'(t), is a decreasing function of t. Similarly, the concavity of the feasible frontier function g implies that the marginal rate of transformation is an increasing function of t. Graphically, this means a downward-sloping curve intersects an upward-sloping curve, which can only occur at a single point, thus guaranteeing a unique solution if one exists.
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