Question

The elasticity of substitution with constant-relative-risk-aversion utility. Consider an individual who lives for two periods and whose utility is given by equation (2.43). 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-maximizing choices of C1 and C2, given P1, P2, and W?

(b) The elasticity of substitution between consumption in the two periods is −[( P1/P2)/(C1/C2)][∂(C1/C2)/∂( P1/P2)], or −∂ ln (C1/C2)/∂ ln ( P1/P2). Show that with the utility function (2.43), the elasticity of substitution between C1 and C2 is 1/θ.

Answer 1

(a) What are the individual’s utility-maximizing choices of C1 and C2, given P1, P2, and W?

Maximizing lifetime utility given by the following:

U = (C₁^{1- θ} / 1- θ) +(1/1+ ρ) (C₂^{1- θ} / 1- θ)

When subject to the lifetime budget constraint by:

P1C1 + P2C2 = W

where W represents lifetime income. Solving for C2:

C2 = W/P2 – C1P1 / P2

With substitution,

U = (C₁^{1- θ} / 1- θ) + [1/(1θ +ρ)] [W/P₂ – C₁P₁/P₂] ^{1- θ} / 1- θ

Solving the unconstrained problem of maximizing utility, as given above, with respect to first period consumption C1. The first-order condition is:

∂U/∂C_{1 =} C₁^-θ + (1/1 + ρ) C₂ ^-θ (-P₁ /P₂) = 0

Solving C1, C₁^{1- θ} = (1/1 + ρ) (P₁ /P₂) C₂ ^-θ, and simplified,

C_{1 =} (1 + ρ)^{1/ θ} (P₂/P₁) ^{1/ θ} C_2, now we solve for C2 with substitution;

C₂ = W/P₂ – (1 + ρ) ^1/θ (P₂/P₁) ^1/θ C₂(P₁/P₂) or

C₂ = (W/ P₂) / [1 + (1 + ρ) ^1/θ (P₂/P₁) ^{(1-θ)/ θ}]

Optimal C1 requires further substitution:

C₁ = (1 + ρ) ^1/θ(P₂/P₁) ^1/θ (W/ P₂₎ / [1 + (1 + ρ) ^1/θ (P₂/P₁) ^{(1-θ)/ θ}]

(b) The elasticity of substitution between consumption in the two periods is −[(P1/P2)/(C1/C2)] [∂(C1/C2)/∂(P1/P2)], or −∂ln(C1/C2)/∂ln(P1/P2).

Show that with the utility function (2.43), the elasticity of substitution between C1 and C2 is 1/θ.

The optimal ratio of first-period to second-period consumption is

C₁/C₂ = (1 + ρ) ^1/θ (P₂/P₁) ^{1/ θ}

Taking the natural log of both sides of the equation gives us

1n(C₁/C₂) = (1/ θ) 1n (1 + ρ) + (1/ θ)1n(P₂/P₁)

The elasticity of substitution between C1 and C2 as positive is:

∂(C₁/C₂) (P₂/P₁) / ∂(P₂/P₁) / ∂ (C₁/C₂) =

∂[1n(C₁/C₂)] / ∂[1n(P₂/P₁)] = 1/ θ

Here we used the previous equation to find the derivative. Higher values of θ imply that the individual is less willing to substitute consumption between periods.