Consider an economy consists of three types of economic agents who live for two periods. Their utililty function is given by ᵤ₍C₁, C₂₎ = ₗₒC₁ +ᵦ ₗₒᴄ₂ where ᵦ ∈ (0,1). Workers differ by their income stream, which is exogenous. {type-X ᵧ₁=2 and ᵧ₂=2
{type-Y ᵧ₁=3 and ᵧ₂=1
{type-Z ᵧ₁=1 and ᵧ₂=3
There is a capital market with interest rate r = ₁
—
₁ ₊ ᵦ
(a) Solve for the optimal consumption of each type. What is the ratio between c₁ and c₂? Does this ratio differ by type? Explain. (b) Compare the optimal consumption level of each type. Who consumes the most? Provide economic interpretation of your results.