In the simple consumption-based asset pricing model, the growth rate of aggregate consumption is assumed to have a constant expectation and standard deviation

(volatility). For example, in the continuous-time version aggregate consumption is assumed to follow a geometric Brownian motion. Consider the following alternative process for aggregate consumption:

dct = ct[μdt + σc

α−1 t dzt], where μ, σ, and α are positive constants, and z = (zt) is a standard Brownian motion. As in the simple model, assume that a representative individual exists and that this individual has time-additive expected utility exhibiting constant relative risk aversion given by the parameter γ > 0 and a constant time preference rate δ > 0.

(a) State an equation linking the expected excess return on an arbitrary risky asset to the level of aggregate consumption and the parameters of the aggregate consumption process. How does the expected excess return vary with the consumption level?

(b) State an equation linking the short-term continuously compounded risk-free interest rate r f

t to the level of aggregate consumption and the parameters of the aggregate consumption process. How does the interest rate vary with the consumption level?

(c) Use Itô’s Lemma to find the dynamics of the interest rate, drf t . Can you write the drift and the volatility of the interest rate as functions of the interest rate level only?