U(c, c' ) = u(c) + u(c 0 ) where (0, 1) and u is a concave increasing differentiable function. Assume there is a government who uses a sales tax to...

U(c, c' ) = u(c) + βu(c 0 ) where β ∈ (0, 1) and u is a concave increasing differentiable function. Assume there is a government who uses a sales tax to finance G and G' (there is no lump-sum tax). That is, for each unit of c the household purchases, they must pay τ c to the government and for each c' must pay τ ' c '.

Answer the following: 1. Derive the two-period budget constraint of the household. 2. Derive the lifetime budget constraint of the household. 3. Derive the two-period budget constraint of the government. 4. Derive the lifetime budget constraint of the government. 5. Write down the household problem. 6. What is the household's FOC? This is also called the Euler equation. 7. If β(1 + r) = 1 and τ = τ '> 0, what must be the relationship between c and c ' ? How does this relationship compare with the zero tax environment, τ = τ '= 0? 8. What is credit/bond market clearing here? 9. What is the definition of competitive equilibrium here? 10. Does Ricardian equivalence hold in this environment? That is, suppose τ and τ' change, G and G' remain the same, and the government budget constraint still holds. Is there any effect on c, c' , s, or r? What is the economic reason for your answer? HINT: here MRS(c,c')= (1 + r) ((1 + τ ')/ (1 + τ)) If τ ↑, then τ '↓. Does this change the household's consumption decision?