A Growth Theoretical Approach
Chapter 9: Welfare Measurement under Uncertainty
It has been established in Chapter 2 that a static equivalent of welfare is embedded in a deterministic autonomous Ramsey problem. It has also been shown that technological progress and imperfect market conditions will complicate welfare measurement, as well as add terms that contain forward looking components. In this chapter we show that the results derived under an assumption of perfect certainty are special cases of more general results which form part of the toolkit of stochastic dynamic optimization. More precisely, such results follow as special cases of the ﬁrst order conditions of a stochastic Ramsey problem. Here, as in Chapter 7, we introduce population growth explicitly into the analysis. We also provide intuition for some of the technicalities created by introducing growth as a continuous-time stochastic process known as Brownian motion (often called a Wiener process). The chapter1 is organized as follows. After brieﬂy reviewing some of the mathematical tools of stochastic control theory in Section 9.1, we use these tools in Section 9.2 to analyze a stochastic Ramsey problem originally introduced by Merton (1975). In Section 9.3 we derive stochastic versions of previous welfare measures. Section 9.4 contains a paralleled analysis for a stochastic version of our workhorse model. The derivation of a cost–beneﬁt rule is dealt with in Section 9.5, and a general principle for obtaining a closed form solution is examined in Section 9.6. We illustrate these principles and the cost–beneﬁt rule by a numerical example. Section 9.7 sums up the...
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