Essays on ambiguity and optimal growth with renewable resources.

Summary In the first two chapters, we study the optimal contract problem in the presence of risk and ambiguity in the context of an optimal control problem. Ambiguity is modeled according to Klibanoff et al. (2005). Our approach generalizes the analyses performed so far by considering the insurance contract as a pair of a premium and an indemnity function to be solved simultaneously. We prove the existence of an optimal contract in the most general case where all agents can be simultaneously averse to ambiguity and risk, which includes all the cases previously considered. We characterize not only the risk sharing but also the ambiguity sharing rule between the contracting parties. In the case of unilateral ambiguity aversion, we show that a direct franchise policy cannot be an optimal insurance contract. Instead, under the assumption that conditional densities can be ranked according to the monotonic likelihood ratio, a contract with vanishing deductibles is optimal, a result that is consistent with Gollier (2014). In particular, the implemented methodology complements Raviv's (1979) analysis for the pure risk case with a risk-neutral insurer, showing that an upper limit coverage cannot be an optimum. This result is robust to ambiguity neutrality.In the third chapter, I examined the impact of risk and ambiguity on optimal investment in human and physical capital using the two-period Ben-Porath (1967) model. Uncertainty (both in the sense of risk and ambiguity) is introduced to human capital accumulation in two ways. When uncertainty is about the rate of depreciation of human capital (uncertain obsolescence of skills), I found that the optimal investment in human capital always increases regardless of whether physical capital is present. This response to uncertainty in a household represents typical self-insurance behavior. In contrast, when the uncertainty is about the efficiency of human capital accumulation, the optimal investment in human capital decreases among households with constant relative risk aversion less than one. This response to uncertainty is typical of a household that views investment as an asset with a risky return instead of insurance.The final chapter (relatively independent of the previous chapters) examines an important issue in growth theory: the role of renewable resources and externalities in the economy. The introduction of a regenerative function (of a natural resource) that is non-concave with respect to one of the arguments makes the problem non-convex. As a consequence, we can no longer use traditional dynamic programming techniques. By attacking this problem, we propose a new method to study a two-sector economy in the presence of externalities. In this case, we introduce the concept of "net stock gain", which is a notion similar to the "net investment gain" introduced by Kamihigashi et al. (2007). In the absence of the usual convex or supermodular properties, we prove that the economy evolves to increase the net stock gain and establish the conditions ensuring the convergence of the economy in the long run. This approach can be applied to similar problems posed above, or be extended to the analysis of multi-sector economies in general.