Question 1 Consider a representative infinitely lived agent maximization problem max {Ct, Ho,Kit)to Bt [log (Ct) + blog(1 - H.)] subject to the budget constraint and physical capital law of motion, respectively: (1 + TC) Ct = W.H, + RK. + Ti -It Ktti = (1 - 8) Ko + It where K, is the physical capital stock; Ct consumption; I, investment; H, labor; (1 - H.): leisure; o relative weight consumption-leisure; o > 0 is the rate of depreciation of physical capital, Re is the capital rental rate, and W, is the real wage. A government imposes a consumption tax and rebate the tax revenue by giving identical lump sum transfers to the representative agent. Hence, the government budget constraint is (To) Ct = Tt, where To is the consumption tax rate and T represents a lump sum transfers in period t. A representative firm rents physical capital and hires workers on competitive markets at prices Ry, We, to maximize profits where output (Yt) is as follows Yt = At K. H! -, where 1 > 0 > 0, and At is the economy's TFP. 1. Solve the household's optimization problem and present the first order conditions. Present and briefly explain the household's intertemporal and intratemporal equilibrium conditions. Solve the firm's profit maximization problem. List the set of equations that describe the competitive equilibrium of this economy. 2. Present the conditions for the stationary state and solve for the stationary state labor supply (H), consumption (C) and capital (K). 3. Explain the following claim: "The tax on consumption changes the equilibrium so that the decen- tralized equilibrium is no longer the same as that of the Robinson Crusoe (centralized) economy. "Be as precise (i.e., refer to your answers above) as possible. 1Question 2 Consider an agent characterized by a log-utility log ct, discounting the future (so that B 0; . Capital law of motion: ky = (1 -5)k, + it; where o is depreciation rate and i, is investment. . Economy's resource constraint: y = q + i + gt, where g, is government (wasteful) expenditures given by gi = Oye; 0 e (0, 1). . Production function is yt = Akell-, where 1 > o > 0, and A is the economy's TFP. Let B E (0, 1) be the agent's discount factor. Assume o = 1. 1. Define the state and control variables. 2. Write out the recursive optimization problem (Bellman equation). 3. Find the first-order and envelope conditions. 4. Present and briefly explain the agent's intertemporal and intratemporal equilibrium conditions. 5. Find the value function (guess and verify) and the related policy functions. 6. Briefly explain how 0 affects the agent's optimal consumption and labor choices. *************** **************:*#* * ***:4. Suppose the government wants to offer the representative agent compensation by means of a subsidy either on his/her labor income or capital income (not both) such that, in the stationary state and together with a tax on consumption, the decentralized equilibrium is equivalent to the centralized equilibrium. In other words, for a given To, find a condition for one of the additional tax/subsidy instruments - either related to labor income or capital income (not both), such that the representative agent can reach the Pareto efficient (first-best) allocation in the decentralized economy. Briefly explain the intuition of your tax policy. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *