PURPOSE: In this problem set, you are to draw asset portfolio returns to highlight the concept of complete vs incomplete asset return structures. This is meant to illustrate an individual’s ability to use asset portfolios to achieved desired returns across di↵erent states-of-nature. It is directly related to our discussion about the ability of an individual to completely insure their consumption risk or only partially insure their consumption risk.

Consider an individual facing the prospect of having high income, yH   > 0, with

probability ⇡ and low income, yL , with probability 1 − ⇡ , yH   > yL . 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 R1,H  units of goods if y = yH  and pays R1,L  units of goods if y = yL . Similarly, let Asset 2 have a return structure such that it pays R2,H  units of goods if y = yH  and pays R2,L  units of goods if y = yL . The individual is endowed with ! 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 a1  the amount of Asset 1 purchased by the individual and a2  the amount of Asset 2 purchased by the individual.

Definition 1  An asset’s return structure is state- contingent if the realized return of the asset varies across states-of-nature. If an asset pays yields the same return across states-of-nature, then it has a return structure that is non-contingent.

In our exercise, as the the structure of return for asset i is (Ri,L ,Ri,H ) and depends on the individual’s income (which is the realized state-of-nature) then if Ri,L     Ri,H then the asset’s return structure is state-contingent (the return depends on the realized state-of-nature).

The individual’s problem is to maximize the expected utility from consumption sub- ject to the constraints that consumption must be financed out of income and the realized return from the asset portfolio as well as a constraint that spending on the asset portfolio must be financed out of the non-storable endowment wealth !.

Consumption and portfolio spending satisfies the following constraints,

cH     = yH + R1,Ha1 + R2,Ha2

cL     = yL + R1,La1 + R2,La2

! = a1 + a2 .

Note that a1  and a2  can be positive (purchase) or negative (sell).

Definition 2  An asset structure is complete from the perspective of an economic actor if the actor is, in e↵ect, unrestricted (up to their budget constraints) in the ability to transfer wealth across states-of-nature.

One way to think about this is that if there are N states-of-nature, then a complete asset structure allows the individual the flexibility to construct combinations of payo↵s that can vary in N di↵erent directions. If the asset structure is incomplete then the individual can only choose from payo↵s that vary in M < N directions. Note that in referring to the asset structure, we are referring to the possibility of constructing portfolio returns to acheive any combination of portfolio returns across the N states-of-nature using all the available assets; we are not referring to the asset return structure of a single asset.

1. Suppose that R1,H  = ↵R2,H  and R1,L  = ↵R2,L . Then if ↵ < 1 the individual only invests in a1 , otherwise, only invests in a2  because one of the assets dominates the other in return across both states-of-nature.

Consider the case ↵ < 1. Using a 2-D diagram with RL  on the horizontal axis and RH  on the vertical axis, plot a vector (arrow) with its head at (R1,L,R1,H). Plot another vector with its head at (R2,L,R2,H). These represent the returns that can be achieved by purchasing a single unit of Asset 1 and a single unit of Asset 2.

The return from purchasing a fraction of either of these assets can be illustrated as a vector that has a fraction of the length of each of these vectors. Purchasing a portfolio can then be illustrated by adding two asset return vectors together. This is done by putting the tail of one vector at the head of the other. See Figure 1 for examples of adding two vectors together, multiplying a vector by a scalar and subtracting a vector from another.

Show that if the return of one asset is a scalar multiple of the other then the return vectors of the two assets can only reach points in the direction (or opposite direction) of one of the vectors. What does this mean and relate it to our model of consumption under risk with state-contingent claims?

2. Suppose that (R1,L,R1,H) = (1, 0) and (R2,L,R2,H) = (0, 1). Again, illustrate the asset returns for each of the assets as a return vector. Show that sums of asset combinations can reach any point in the (RL ,RH )-plane. What does this mean and relate it to our model of consumption under risk state-contingent claims?

3. Finally, consider the asset return structure (R1,L,R1,H) and (R2,L,R2,H) with R1,H  = xR2,H  and R1,L  = zR2,L , x  z. Specifically, assume that neither of the two return vectors lie on top of either of the axes. Show that sums of asset combinations can reach any point in the (RL ,RH )-plane. What does this mean?

4. Given these three cases, which case is the asset return structure incomplete and with of these three cases are examples of complete asset return structures?