Solve the following.

1. An increase in government spending will raise interest rates. 2. Tax cuts are better for stimulating the economy than spending increases because the}: can be implemented more quickly. 3. As long as the study is not focussed on the transactions role of money: putting moneuv in the utility function is simply a convenient shortcut. 4-. Ine''icient levels of unemployment cannot occur in a competitive general equilibrium. 5. Hyperiltation is impossible in an economy populated by rational agents. (1'. During the past fen: _vears_. household savings rates have increased. even as the economy has grotvn V21} slowly. This is evidence against the permanent income hypothesis. i". Optimal Growth with depreciation spillovers. Time: Discrete. innite horizon Demography: :"L continuum. mass normalized to l. of {representative} innite lived consumer_.u"v.'orlcer households. There is a large number. mass I. of rms ownedjointly and equally by the households. Preferences: The instantaneous household utility function over. consumption. at; is nlici}. The function n{.} is twice differentiable. strictly increasing and strictly concave. The discount factor is 3 E [l]. 1}. Technology: There is a constant returns to scale technology over capital and labor such that the output y; associated with the input of capital. leg. and labor. by. is given by y; = FHQ. Ill-g}. The production function. F{.. .:I is twice differentiable. strictly increasing in both arguments. concave and exhibits constant returns to scale. This means that aggregate output. 1; is given by F{K*.H} where R} is the aggregate capital stock and Hi is the aggregate labor supply. lCapital depreciates with usage. The extent of depreciation. however. depends on the aggregate amount of labor with which it is employed. Thus a rm that employs in; units of capital will have {1 Edie units remaining at the end of the period where 15 = H and l] s: 5 s: 1. Endowments: Each households" initial capital stock is kg. Each household has 1 unit of time which it can devote in any proportions to labor or leisure. {This means that for any household. all-g E [ll 1] and. because the number of households is normalized to 1__ H. E [11. 1].} {a} 1|I."|."rite down the Planner's problem for this economy and solve for the system of difference equations {in C}. If; and Hi} that characterize its solution. {b} Assuming the Planner is not at a corner solution. what {local} condition controls dH the relationship between H; and If; {i.e. what determines the sign of dIgj'? i Now consider the decentralized economy. {c} 1|I."|."rite down and solve the representative household's optimization problem {given 1"; and mg. the interest rate and hourly wage. respectively}. {d} 1|I."|."rite down and solve the representative firms"s optimization problem. {e} 1|I."|."rite down the market clearing conditions. dene a competitive equilibrium and solve for the system of equations in g. If; and H; that specify equilibrium dy namics. {C} is aggregate consumption.) {fl 1|I."|."hat is the relationship between H; and K; in the competitive equilibrium? \fwith Ht following an exogenous stochastic process. In equilibrium: lumpsum transfers equal seigniorage and consumption tax revenues: M M s; = A no... {CBC} {a} Solve the producer's problem. {b} Let W; = PL."P_1 1 denote the rate of ination. and let no = \"LIP; denote the real wage. Derive the consumer's 'otv budget constraint in both nominal and real terms. Combining budget constraints: derive the aggregate {physical} resource constraint for the overall economy. {c} Set up the household's problem and nd the optimality conditions. 4 {d} Find the equilibrium labor optimality condition. Is money neutral in this model"! 1it'l'hy or tvhy not".r Conversely: does the consumption tax aect inflation'iI \"ihy or why not'." {e} Let letters with carats H\" denote deviations of logged variables around their steady st ate talua. Show that the loglinearized expressions for labor hours and output are: a eel5.}. a 3-D. ; = .k{c1j: {lose-9. {f} Netv York City Mayor Michael Bloomberg has proposed to prohibit the sale of large soda drinks in an eort to reduce the consumption of unhealthy. sugary beverages. Because smaller sodas have a higher per unit {per ounce} cost: this effectively a tax on a consumption good. According to the analysis above. would Mayor Bloomberg's proposal work".r