We revisit the correspondence of competitive partial equilibrium with a social optimum in markets where risk-averse agents solve multistage stochastic optimization problems formulated in scenario trees. The agents trade a commodity that is produced from an uncertain supply of resources which can be stored. The agents can also trade risk using Arrow-Debreu securities. In this setting we define a risk-trading competitive market equilibrium and prove a welfare theorem: competitive equilibrium will yield a social optimum (with a suitably defined social risk measure) when agents have nested coherent risk measures with intersecting polyhedral risk sets, and there are enough Arrow-Debreu securities to hedge the uncertainty in resource supply. We also give a proof of the converse result: a social optimum with an appropriately chosen risk measure will yield a risk-trading competitive market equilibrium when all agents have nested strictly monotone coherent risk measures with intersecting polyhedral risk sets, and there are enough Arrow-Debreu securities to hedge the uncertainty in resource supply.

## Citation

Technical Report, Electric Power Optimization Centre, University of Auckland.