Consider an individual with a time-additive expected utility characterized by a utility function u (c) and a time preference rate . The individual lives in

Consider an individual with a time-additive expected utility characterized by a utility function u

(c) and a time preference rate δ. The individual lives in a continuous-time economy. Let us write the dynamics of the optimal consumption process of the individual as dct = ct



μct dt + σ

ct dzt

.

(a) What is the dynamics of the state-price deflator induced by this individual? Determine an expression for the continuously compounded short-term risk-free rate and a market price of risk for this economy in terms of the preferences and consumption process of this individual.

Bruce lives in a continuous-time complete market economy. Bruce has time-additive logarithmic utility, uB

(c) = ln

c, with a time preference rate of δB = 0.02, and his optimal consumption process cB = (cBt) has dynamics dcBt = cBt [0.03 dt + 0.1 dzt], where z = (zt) is a standard Brownian motion.

(b) Identify the continuously compounded short-term risk-free interest rate and the market price of risk associated with z. Can there be any other ‘priced risks’ in the economy? What is the instantaneous Sharpe ratio of any risky asset?

Patti lives in the same economy as Bruce. She has time-additive expected utility with a HARA utility function uP

(c) = 1 1−γ (c − c¯)1−γ and a time preference rate identical to Bruce’s, that is δP = 0.02.

(c) Explain why Patti’s optimal consumption strategy cP = (cPt) must satisfy cPt = c¯ +

 cBt cB0

1/γ

(cP0 − c¯).

Find the dynamics of Patti’s optimal consumption process.