Consider a representative infinitely lived agent maximization problem 8 max {Ct, Ht,Kt+1} t=0 subject to the...

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Consider a representative infinitely lived agent maximization problem 8 max {Ct, Ht,Kt+1} t=0 subject to the budget constraint and physical capital law of motion, respectively: Blog(Ct) + plog(1 — H₂)] (1 + Tc) Ct = WtH₂ + RK + Tt - It Kt+1 = = (1 - 8) Kt + It where K is the physical capital stock; C, consumption; I, investment; H, labor; (1 H): leisure; relative weight consumption-leisure; > 0 is the rate of depreciation of physical capital, R, 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 (TC) Ct = Tt, where TC is the consumption tax rate and T₁ represents a lump sum transfers in period t. A representative firm rents physical capital and hires workers on competitive markets at prices R₁, W₁, to maximize profits where output (Y) is as follows Y₁ = AKH, where 1 > > 0, and A, 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 (Ĥ), 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. Consider a representative infinitely lived agent maximization problem 8 max {Ct, Ht,Kt+1} t=0 subject to the budget constraint and physical capital law of motion, respectively: Blog(Ct) + plog(1 — H₂)] (1 + Tc) Ct = WtH₂ + RK + Tt - It Kt+1 = = (1 - 8) Kt + It where K is the physical capital stock; C, consumption; I, investment; H, labor; (1 H): leisure; relative weight consumption-leisure; > 0 is the rate of depreciation of physical capital, R, 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 (TC) Ct = Tt, where TC is the consumption tax rate and T₁ represents a lump sum transfers in period t. A representative firm rents physical capital and hires workers on competitive markets at prices R₁, W₁, to maximize profits where output (Y) is as follows Y₁ = AKH, where 1 > > 0, and A, 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 (Ĥ), 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.