Introduction to Modern Economic Growth further assumptions are imposed on the utility and production functions In particular, let us suppose that the utility functions take the familiar CRRA form: Ã ! c1 (t)1−θ − c2 (t + 1)1−θ − +β , (9.10) U (t) = 1−θ 1−θ where θ > and β ∈ (0, 1) Furthermore, assume that technology is Cobb-Douglas, so that f (k) = kα The rest of the environment is as described above The CRRA utility simplifies the first-order condition for consumer optimization and implies c2 (t + 1) = (βR (t + 1))1/θ c1 (t) Once again, this expression is the discrete-time consumption Euler equation from Chapter 6, now for the CRRA utility function This Euler equation can be alternatively expressed in terms of savings as s (t)−θ βR (t + 1)1−θ = (w (t) − s (t))−θ , (9.11) which gives the following equation for the saving rate: (9.12) s (t) = w (t) , ψ (t + 1) where ψ (t + 1) ≡ [1 + β −1/θ R (t + 1)−(1−θ)/θ ] > 1, which ensures that savings are always less than earnings The impact of factor prices on savings is summarized by the following and derivatives: ∂s (t) = ∈ (0, 1) , ∂w (t) ψ (t + 1) ả s (t) s (t) = (βR (t + 1))−1/θ ∂R (t + 1) θ ψ (t + 1) sw ≡ sr Since ψ (t + 1) > 1, we also have that < sw < Moreover, in this case sr > if θ < 1, sr < if θ > 1, and sr = if θ = The relationship between the rate of return on savings and the level of savings reflects the counteracting influences of income and substitution effects The case of θ = (log preferences) is of special importance and is often used in many applied models This special case is sufficiently 425