Set up a lifetime lagrangian formulation for the

Question

Question -

Consider the two-period economy (with zero government spending and zero taxation), in which the representative consumer has no control over his real income (y₁ in period 1 and y₂ in period 2). The lifetime utility function of the representative consumer is: u(c₁, c₂) = (c₁^σ + c₂^σ)^1/σ.

y₁ and y₂ are high enough and parameters are such that both consumption bundles take values that are always positive.

The lifetime budget constraint is:

c₁ + c²/1+r = y₁ + y₂/1+r + (1+r)a₀,

with zero net assets (a₀ = 0).

a) Set up a lifetime Lagrangian formulation for the representative consumer's lifetime utility maximization problem for the given utility function. Define any new notation you introduce.

b) Take first-order conditions with respect to c₁ and c₂.

c) Combine the first-order conditions derived in part (b) into a single equation which is a function of c₁, c₂, r, and σ.

d) If σ = 2 and r = 0.05. What is the value of c₁/c₂? Assuming consumption is always greater than zero, is c₁ > c₂ or is c₁ ≤ c₂?

e) If σ = 2 and r = 0.1. What is the value of c₁/c₂? If σ = 2, holding fixed y₁ and y₂, is consumption in period 1 relative to period 2 higher when r = .1 or when r = .05?