Consider a utility function v(co, c). The marginal rate of substitution (MRS) is defined to be...

 

  

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Consider a utility function v(co, c₁). The marginal rate of substitution (MRS) is defined to be the negative of the slope of an indifference curve and is equal to av(co, c₁) /aco av(co, c₁)/ac₁ The elasticity of intertemporal substitution (EIS) is defined as MRS (CO, C₁): dlog(c₁/co) " dlog MRS (co, c₁) where the MRS is varied holding utility constant. Show that, if 1 1-P then the EIS is 1/p. v(co, c₁): = CO + 8 1-p C1 1-p " Consider the portfolio choice problem with only a risk-free asset and with consumption at both the beginning and end of the period. Assume the investor has time-additive power utility, so she solves 1 1 1-p max +8. 1- 1-p -C1 subject to P As shown in Exercise 2.6, the investor's EIS is 1/p. 1 co += c₁ = Wo. Rf (a) Show that the optimal consumption-to-wealth ratio co/wo is a decreasing function of Rf if the EIS is greater than 1 and an increasing function of Rf if the EIS is less than 1. Note: The effect of changing Rf is commonly broken down into an income effect and a substitution effect. This shows that the substitution effect dominates when the EIS is high and the income effect dominates when the EIS is low. (b) For given co and ₁, show that the solution of the investor's optimization problem implies that Rf must be lower when the EIS is higher. Consider a utility function v(co, c₁). The marginal rate of substitution (MRS) is defined to be the negative of the slope of an indifference curve and is equal to av(co, c₁) /aco av(co, c₁)/ac₁ The elasticity of intertemporal substitution (EIS) is defined as MRS (CO, C₁): dlog(c₁/co) " dlog MRS (co, c₁) where the MRS is varied holding utility constant. Show that, if 1 1-P then the EIS is 1/p. v(co, c₁): = CO + 8 1-p C1 1-p " Consider the portfolio choice problem with only a risk-free asset and with consumption at both the beginning and end of the period. Assume the investor has time-additive power utility, so she solves 1 1 1-p max +8. 1- 1-p -C1 subject to P As shown in Exercise 2.6, the investor's EIS is 1/p. 1 co += c₁ = Wo. Rf (a) Show that the optimal consumption-to-wealth ratio co/wo is a decreasing function of Rf if the EIS is greater than 1 and an increasing function of Rf if the EIS is less than 1. Note: The effect of changing Rf is commonly broken down into an income effect and a substitution effect. This shows that the substitution effect dominates when the EIS is high and the income effect dominates when the EIS is low. (b) For given co and ₁, show that the solution of the investor's optimization problem implies that Rf must be lower when the EIS is higher.