A three-period model of intertemporal optimization Consider an agent who lives for three periods denoted t = 0, 1, 2. The agent can take long- and short-positions in bonds. That is, she can freely borrow and lend. The only exception is that the agent is not allowed to leave the final period with any debt. The maximization problem of the agent is given by '2 . 010 _1 co,c1..1ffilbhb2U = Z', 0 0 at. (:0 +b0 2 en Cl +51 = 81+(1-l-Telbe c2+b2 = 82+(1+r1)b1 b2 2 0 where et > 0 denotes the endowment in period t = 0,1.2, e, is period-t consumption, or is net bond holdings in period t, and n is the net interest rate on bonds purchased in period t. Note that. the agent will optimally set b2 2 0. (a) Write down the rst-order necessary conditions, denoting the Lagrange multipliers on the period-t budget constraint by A,. \"That is the interpretation of At? (b) Derive the Euler equations for period 0 and period 1. What is the interpretation of these equations? (c) Now suppose that in period 0, the agent can invest in a long-term bond that matures only after two periods, and per unit. pays off 1 + mg in period 2. Adjust the agent's optimization problem to allow for the long-term bond and derive the corresponding Euler equation. ((1) Provide a condition on the interest rates that rules out. arbitrage between bonds. That is, under what conditions is it not possible for the agent to make an innite return by going short in long-term bonds and buying short-term bonds, or vice versa