Macroeconomic Volatilities and Long-Run Risks of Asset Prices

This is an on-line appendix with more details and analysis for the readers of the paper. B.1 Derivation for the A i 's, risk-free rate and market price of risk First, we rewrite the normalized aggregator f defined in Equation (5) as f (C, J) = β 1 − 1 ψ (1 − γ)J[G − 1], where G ≡ (C ((1 − γ)J) 1 1−γ) 1− 1 ψ. (B1) Then, taking partial derivatives of f (C, J) with respect to J and C , we have f J = (θ − 1)βG − βθ (B2) and f C = β G C (1 − γ)J. (B3) where we use the notation θ = 1−γ 1− 1 ψ. Theoretically, the aggregator f (C, J) should be an increasing function of the value function the monotonicity axiom of preferences will be violated. This places joint restrictions on γ and ψ such that θ ≥ 1 or θ < 0. If θ > 1, the first inequality is possible to have solutions. However, if 0 < θ < 1, the second inequality is impossible as G > 0 always. Hence, the necessary restriction on γ and ψ is either θ > 1 or θ < 0. If θ = 1, as shown by Duffie and Epstein (1992), we obtain the standard additive expected utility of constant relative risk aversion (CRRA). So θ > 1 can be extended to θ ≥ 1.