Section 6 - Arrow-Debreu Pricing and General Equilibrium

Arrow-Debreu Security

• Pays 1 unit of consumption in one single state of nature, and nothing in any other state.

Completeness

Financial market for a contingent claim that pays off one unit of consumption in that data-state and zero otherwise

• provides an alternative deviation of the SDF

Complex Securty

• Pays off in more than one state of nature

CAPM motivation

• Puts variance on one axis and mean on the other

• Useful for deriving portfolio-management and asset-pricing results, but the theory proved difficult to generalise as it requires either quadratic utility or normally distributed returns

Arrow-Debreu Approach

• Puts consumption in the good state on one axis and consumption in the bad state on the other axis

• Does not require the restrictive underlying assumptions of CAPM

• Took much longer to be recognised for its implications on pricing

Advantages of AD

1. Investors do not have quadratic utility - or even expected utility functions

2. Returns don’t have to be normally distributed

3. Theory draws very explicit links between asset prices and the rest of the economy

Disadvantage of AD

While conceptually useful it is too abstract and not easy to use

• Difficult to identify individual states of nature, and when you can, their realisation cannot always be verified. Why we dont see pure AD secruties being traded

First and Second Welfare theorems hold in the AD economy

• Competitive equilibrium allocations are Pareto-optimal and any pareto-optimal allocation can be decentralised as a competitive equilibrium - this gives a practical shortcut

Risk Sharing

• Without aggregate risk, competitive markets fully diversify away all idiosyncratic risk regardless of agent’s risk-aversion.