Indifference Pricing for Power Utilities

We study utility indifference pricing of claim streams with intertem- poral consumption and power (CRRA) utilities. We derive explicit for- mulas for the derivatives of the utility indifference price with respect to claims and wealth. The simple structure of these formulas is a reflec- tion of surprising operator identities for the derivatives of the optimal consumption stream. For example, the partial derivative of the opti- mal consumption stream with respect to the endowment is always a pro jection. Furthermore, it is an orthogonal pro jection with respect to a natural ”economic inner product”. We find that the indifference price is a highly non-linear function of the wealth and obtain its asymptotic behavior when the claims to wealth ratio is either very small or very large. In the large claims to wealth ratio limit, the asymptotic expansion of the indifference price is given in terms of fractional powers of the wealth. We also obtain sharp global bounds for the indifference price.