Question

Inter-temporal elasticity of substitution Consider an individual who lives for two periods and whose utility is given by U=C¹⁻⁰₁ /1−θ + 1/1+rho C¹⁻⁰₂ /1−θ ​,θ>0,rho>−1. Let P₁ and P₂

denote the prices of consumption in the two periods, and let W denote the value of the individual's lifetime income; thus the budget constraint is P₁C₁+P₂C₂=W (a) What are the individual s utility-maximising choices of C₁ and C₂, given P₁, P₂, and W? (b) The elasticity of substitution between consumption in the two periods is Show that with the above utility function, the elasticity of substitution between C₁ and C₂ is 1/θ
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Answer 1

The individual's utility-maximizing choices of C₁ and C₂ can be found by setting up and solving the Lagrangian equation. The elasticity of substitution between C₁ and C₂ is 1/θ.

Step-by-step explanation:

The utility-maximizing choices of C₁ and C₂ can be found by setting up the Lagrangian equation and solving for the optimal values. The Lagrangian equation for this problem is:

L = C₁⁻θ/1−θ + 1/1+ρ C₂⁻θ/1−θ + λ(W - P₁C₁ - P₂C₂)

By taking the partial derivatives of the Lagrangian equation with respect to C₁, C₂, and λ, and setting them equal to zero, we can find the optimal values of C₁ and C₂. In this case, the inter-temporal elasticity of substitution (IES) is equal to 1/θ.