Consider an individual facing the prospect of having high incomE, y}; > 0.. with probability 11' and low income. ye. with probability 1 7r, yr; > 9-1,. Prior to learning whether realized income is high or low, the individual is able to go into the market and purchase (or sell) two types of assets. Let the Asset 1 have a return structure such that it pays Ru; units of goods if y = y}; and pays Ru, units of goods if y 2 yr. Similarly, let Asset 2 have a return structure such that it pays Ra units of goods if y = ya and pays RQ'L units of goods if y = yL. The individual is endowed with w units of wealth to spend in the asset market but this wealth is not storable and hence cannot be save to purchase consumption goods. Denote by all the amount of Asset 1 purchased by the individual and a; the amount of Asset 2 purchased by the individual. Denition 1 Au asset's returri structure is state-contingent if the realized return of the asset varies acruss statesofnature. If at! asset pays yields the some return. across stotesofnature, then it has a return structure that is noncontingent. In our exercise: as the the structure of return for asset i is (Rug, 85,3} and depends on the individual's income (which is the realized stateofnature) then if Rh; # RU! then the asset's return structure is statecontingent (the return depends on the realized stateofnature}. The individual's problem is to maximize the expected utility from consumption sub ject to the constraints that consumption must be nanced out of income and the realized return from the asset portfolio as well as a constraint that spending on the asset portfolio must be nanced out of the nonstorable endowment wealth to. Consumption and portfolio spending satises the following constraints, CH = EH + R11Hav]. + Raj-{12 (1) Cr. = yr. + 31,1501 + Rule; (2) Lu 2 a1 + a2. (3) Note that a1 and a; can be positive (purchase) or negative [sell]. Denition 2 Au asset structure is complete from the perspectiue of an. economic actor if the actor is, in effect, unrestricted (up to their budget constraints) in the ability to transfer wealth across statesofnatur'e. One way to think about this is that if there are N statesofnature, then a. complete asset structure allows the individual the exibility to construct combinations of payoffs that can vary in N different directions If the asset structure is incomplete then the individual can only choose from payoffs that vary in M