We now extend our discussion of first-order analysis to a model where time and uncertainty appear explicitly. As long as we can retain an assumption of complete markets, the apparatus of Chapters 9 and 10 carries over directly and leads to a criterion of discounted present value for evaluating projects. In this case, interest focuses on measuring shadow values of intertemporal prices (discount factors). However, once we take uncertainty seriously, we must drop the assumption of complete markets, and some corresponding modifications are required in our measures.

Extension of shadow-pricing principles to an intertemporal context enables us to define and quantify a social rate of discount, and we use this apparatus to discuss and evaluate some of the controversies as to its appropriate size. We find that the decomposition procedures of Chapter 9 are relatively more helpful in dealing with uncertainty and use them to identify and measure the social risk premium. Further, the fact of missing “uncertainty markets” leads to distributional second-best welfare measures in much the same way that failure to redistribute income leads to a second-best distributional term in the static model; the associated measure generalizes the concept of “pecuniary externality” that has been appearing in the literature on incomplete markets.

Temporal rate of discount

We begin with a discussion of issues involving time but not uncertainty. Later we will see how to incorporate the current discussion as a building block in general models involving time and uncertainty.