| This dissertation studies conditions under which optimal solutions to stochastic optimization problems--problems of the form {dollar}maxlimitssb{lcub}rm x,theta{rcub}V({lcub}bf x{rcub},{lcub}bftheta{rcub},{lcub}bf t{rcub},{lcub}bf tau{rcub})equiv intpi{dollar}(x,s;t)dF(s;{dollar}theta,tau{dollar})--are monotone nondecreasing in exogenous parameters. We characterize properties of stochastic objective functions based on properties of the payoff function ({dollar}pi{dollar}) and the probability distribution (F).; In Chapter 1, we focus on problems of the form {dollar}V({lcub}bf theta{rcub};{lcub}bf tau{rcub})equiv intpi{dollar}(s)dF(s;{dollar}{lcub}bf theta{rcub},{lcub}bf tau{rcub}{dollar}) (where all of the variables are vectors). We first develop a result which unifies and extends the existing stochastic dominance literature, proving a class of theorems about monotonicity of V({dollar}{lcub}bf theta{rcub};{lcub}bf tau{rcub}{dollar}). We then show that the method which is used to prove stochastic dominance theorems is exactly the right method for characterizing when V({dollar}{lcub}bf theta{rcub};{lcub}bf tau{rcub}{dollar}) satisfies any property in a class which we call "linear difference properties," a class which includes the properties supermodular and concave. The results about supermodularity of the stochastic objective function can be applied to derive monotone comparative statics predictions.; Chapter 2 studies a second class of optimization problems, which can be written {dollar}intsb{lcub}s{rcub}pi(x,s)dF(s,theta{dollar}), where all variables are real numbers. We characterize Milgrom and Shannon's single crossing property, providing necessary and sufficient conditions so that the optimal x is nondecreasing in {dollar}theta{dollar}. We find that single crossing of the payoff function {dollar}pi{dollar} and the Monotone Likelihood Ratio Order (MLR) on the distribution F together are sufficient for the conclusion--single crossing of the expectation--and further, neither one of the hypotheses can be weakened without strengthening the other. We also prove an analogous result about the Spence-Mirrlees single crossing condition for problems of the form {dollar}intsb{lcub}s{rcub}nu(x,y,s)dF(s,theta{dollar}). The results are applied to signaling games, auction games, and mechanism design.; Chapter 3 applies the results of Chapter 1 to a firm's long-run choices of organization and technology, choices which affect the firm's short-run innovative activity. We focus on flexibility, which affects the future costs of implementing innovations, and information gathering, which affects the future opportunities for innovation. We consider two dimensions of innovation: demand-enhancing (product) and cost-reducing (process). Our analysis reveals that short-run complementarities between these two types of innovation lead to complementarities between its long-run decisions about product and process flexibility and information gathering. |
