Let's imagine that the representative household lives for two periods only. They have to decide how much they are going to consume in period 1 and in period 2. Let's assume that their preferences are described by the utility function: logc₁ + Blog 0₂ The household takes as given its current and future income 3₁ and 32, lump-sum taxes t₁ and t2 and the interest rate and simply solves a consumer optimization problem: max logc₁ + ₂log ₂ {c₁.ca.ba) s.t. (1) (2) C₁ + b₂ = y₁-t₁ C₂ = 92-1₂+(1+r)b₂ (3) b₂ is a one-period real bond that gives real return r. There is no cash in this economy. 1. Set up Lagrangian problem and derive the consumption Euler equation. 2. Rearrange your Euler equation to write c₂ as a function c₁ and interest rate. Substitute this resulting equation into the household budget constraint to solve for consumption in period 1. 3. What is the marginal propensity to consume out of period one income. i.e. 2. How does it depend on $₂? Firms hire workers to produce y₁ in first period. Firm labor demand condition gives W₁/P₁ = 1, i.e. real wage in period 1 is one. We will assume that W₁ = 1, nominal wages are fixed at one. We also assume that output in period 2 is fixed at y2 = 1. Furthermore, assume that all output must be consumed either by the government or by agents. In particular, we will assume that government spending is only done in the first period. ₁=₁+9₁. As a result in the second period y/2 = C₂. 4. What is the price level in period 1, i.e. P₁? 5. Let's define the natural interest rate to be real interest rate such that output in period one is equal to one. Write down an expression for the natural interest rate. 6. Write down a condition for 2 that has to be satisfied for the natural interest rate to be negative?