3. Proportional income tax in simple dynamic model with a government and specic utility functions Consider the following economy: Time: Discrete; innite horizon Demography: Continuum of mass 1 of {representative} oonsumery'rworker households, and a large number of prot maximizing rms, owned jointly by the households. Preferences: the instantaneous household utility function is Eel-\"'3 where c is household oonsumption. The discount ctor is .5 E {0,1}. Technology: There is a constant returns to scale technology for which labor is the only input so that a rm that hires h units of labor produces sh units of output. Endowments: Each household has 1 unit of time per period to allocate how ever they like between work and leisure. Institutions: There is a government that has to meet an exogenous stream of expen ditures: {go}. Government spending is thrown into the ocean. The govermnent can levy taxes and issue bonds in order to meet its expenditure requirement. Taxes are restricted to being proportional to labor income so that in period t1 the tax revenue from a household which provides lahor services In is then num'n where T1; is the period t tax rate and ts; is the wage rate. Every period there are markets for labor: government bonds and consumption goods. {a} 1Write down and solve the problems mood by the representative household and the representative rm. {b} 1Write down the govermnent's [period by period} budget constraint. {c} Dene and a characterize a competitive equilibrium. {:1} Does Ricandian equivalence hold? Explain {e} How would your answer to part {d} change if the utility function was replaced by Ecl'lzfl to)\": Explain your answer. 1. Monetary neutrality.r implies that monetary policy.r has no effect on output. 2. If recessions were an eicient response to changes in productivity. we should observe an increase in beach ball sale: [i.e. leisure activities} during downturns in the economy. 3. Permanent decreases in income taxes are better at stimulating consumption and labor supply.r than temporary.r decreases. 4. Procyclical average productivity of labor means that business cycles cannot be caused by demand shocks. 5. The labor supply of highwealth workers is more elastic than the labor supply of low wealth workers. 6. An increase in government spending reduces output. 'i". Sidrauslci model with seig'nior'age Time: Discrete, innite horizon Demography: PL continuum, mass normalized to l, of {representative} innite lived consumerlg'worker households. There is a large number, mass N, of rms owned jointly and equally by the households. Preferences: The instantaneous household utility flmction over consumption, ch and real money balances, rm, is \"{5}, arm]. The function u{., .I is twice dierentiable, strictly increasing in both arguments and strictly concave. The discount factor is ,3 E [I], 1}. Technology: Aggregate output, Y1; = FfflthsII where K} is the aggregate capital stock and L; = 1 is the aggregate labor supply. The function F{., .II is twice dierecn tiable, strictly increasing in both arguments, concave and exhibits constant returns to scale. It will be helpful to use Hist} as output per worker where in, is the capital stock per worker. Capital depreciates by the fraction 5 for each period it is in use. Endowments: Each household has one unit of labor and an initial endowment of capital kn. Each also has an initial nominal money holding H\". Information: Complete, perfect foresight. Institutions: Competitive markets in each period for capital, the consumption good, labor and money. There is a government that has to meet the exogenous sequence of per capita expenditures, 3*. To do so, it issues new money and buys output om the market. Its policy instrument is the nominal money growth rate, or, such that H; = {1 + J}H_1. {a} Using Mien-1 as period t nominal money demand, P; as the period t price of the consumption good, r; as the rental rate on capital and w; as the wage paid per effective unit of labor, write down and solve the household's problem. {b} Solve the problem faced by the rms and, write down the market clearing condi tions and the government budget constraint. {c} Dene a monetary equilibrium and solve for the equations that characterize the equilibrium. [dII Consider now 91., = g for all t. In steady state, for a given value of a", what are the maximum sustainable values of g and m' [steady state real money balances}. To what extent do these values depend on no"? Explain your answer. 4. Ono-sided search with learninghydoing Tim: Discrete1 innite horizon. Demographic A single worker who faces the possibility of death with probability .5 each period. 1|nil-"hen dead he gets ll utility forever. Preferences: The worker is risk neutral lie. BEE} = I] and discounts the future at the rate l\". Endowment s: The worker can be high or low' skilled. He is born unemployed and lowskilled. He becomes highskilled only by being employed as a Low-skilled worker. I \"men unelnployed [regardless of skill level} he receives the "benet", If, each period. I \"Then rst unemployed he has lD'H" skill level and with probability o he gets a lowskilled job that pays 10L each period. I Employed lowskilled workers become highskilled with probability "- each period. I High skilled employed workers receive war each period. I Once the worker becomes high-skilled he is high Skilled for life. a Employed workers [regardless ofskill level] loose their job with probability .'a. I Unemployed highskilled workers get highskilled jobs [paying mg] with probability 0.. N019 1: Events that ean happen. to an individual are mutually exclusive {i.e. not indepen dent}. Assume that 13+ 1- +115 of land I5+CL c: l. N019 I: The worker can be in one of 4 possible states: unemployed lowskilled, employed lowskilled__ unemployed highskilled and employed highskilled. a. 1|Write down the ow asset 1value [ow Bellman} equations for the worker. [Notatiul use I: for the 1value to being in skill level, I = L, H and employment status gr = 3,8.) -L E b. Obtain an expression for NEH Tl in terms of the model parameters. us. Use the result from part b. to obtain a restriction on the value of WL such that the lowskilled worker will not reject a job o'Eer. d. Now suppose there is mass one of such individuals and that anyone who dies is replaced by a newborn. {Newborns are unskilled and unemployed.) Draw a diagram to show the population ows between the 4 states. e. 1'Write down a system of equations that can be solved for the steady state populations [you don't need to solve them]