The book shows that the classical second-order sufficient condition
for the induced optimization problem (IOP), together with the strict bang-bang property, ensures second-order sufficient conditions
for the bang-bang control problem.
Intuitively, with a satisfying the first-order condition, an increase in wealth leads to an increase (decrease) in the willingness to bear risk when utility exhibits DARA (IARA) implying that the choice of [alpha] rises (falls), while the second-order sufficient condition implies that the direction of change in [alpha] has the same sign as [integral of] U" [Z.
Observe that the second-order sufficient condition implies that the sign of the short-run supply response to MLR improvements, [delta][y.
KL]) [greater than] 1 given the second-order sufficient conditions.
Hence, the second-order sufficient conditions imply that demand for labor decreases (increases) as the wage rate increases, [delta][L.
We use the method of Lagrange multipliers and apply the first- order necessary conditions as well as the second-order sufficient conditions for maximization.
Organization of remaining paper is as below: section 2 details the building of model, in section 3 we consider an explicit example and find optimal output, section 4 explains the interpretation of Lagrange multipliers, in section 5 second-order sufficient conditions are applied for maximization, section 6 enlightens the comparative static analysis, and concluding remarks are given in section 7.
Topics include the existence of optimal solutions, necessary optimality conditions and adjoint equations, second-order sufficient conditions
, and main principles of selected numerical techniques.