AEC 705:
Time: discrete innite horizon Demography: A mass N: E Nu + n): of newborns enter in every period. Everyone lives for 2 periods excem for the rst generation of old people. Preferences: for the generations horn in and aI'tEr period I]; U:{511t5 CE,:+1:l = 11631;} + {32,t+l:l where my is consumption in period t and stage i of life. n[.} is in creasing strictly concave and twice differentiable limc_,l u.'(c} = co. limcnm 11:11:} 2 [1. For the initial old generation Ulzaul 2 Milan}- Productive technology: A constant returns to scale technology implies that an it} is available to each household. In; is per worker capital stock. The usual \"loads\" cmditions apply. Endowments: Everyone has one unit of labor services when young. [Old cannot worl: so they have to rely on earnings from renting capital.) The rst Eneration of old have [1 + elk\" units of capital each. Each old person has H units {dollars} of money which provides no benet per as but it is xed in supply and recognizable as money. Institutions: There are competitive markets, for labor and physical cap ital. consumption goods and money. Using the consumption good as the numeraire, let the per unit wage in period t be wt and the gross return on capital rented in period t be H:- The price of money in terms of the consumption good is 1] 13:. So that acquiring one unit of the the consumption good costs pt dollars. (a) Write out the problems faced by generation t workers and firms in this economy (ignore inside money). What condition on the prices p, and interest rate R, must be met if money is to circulate (i.e. be used as means of saving)? (b) Write down the market clearing conditions and define a competitive equilibrium in which money circulates. (c) In a steady-state of the economy where real money demand, , is fixed Pt for all f what condition must be met by the return on holding money, Pt Pt+1 (d) Derive the condition that a Social Planner, who weights all generations equally, puts on the steady-state per capita stock of capital. What does this imply for the optimality of competitive equilibria? (e) What can we conclude from (c) and (d) about the efficiency properties of any steady-state monetary equilibrium? What does this imply for the existence of steady-state equilibria with over investment. Explain your answer.Consider the following economy: Time: Discrete; infinite horizon Demography: Continuum of mass 1 of (representative) consumer/worker households, and a large number of profit maximizing firms, owned jointly by the households. Preferences: the instantaneous household utility function is u(c) where c is household consumption and u(.) is strictly increasing and strictly concave. The discount factor is Be (0, 1). Technology: There is a constant returns to scale technology for which labor is the only input so that a firm 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 expenditures, {g} . Government spending is thrown into the ocean. The government 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 t, the tax revenue from a household which provides labor services he is then Truth, where it is the period t tax rate and w is the wage rate. Every period there are markets for labor, government bonds and consumption goods. (a) Write down and solve the problems faced by the representative household and the representative firm. (b) Write down the government's (period by period) budget constraint.(c) Define and a characterize a competitive equilibrium. (d) Does Ricardian equivalence hold? Explain (e) How would your answer to part (d) change if the utility function was replaced by u(ct, 1 - he) and u(., .) is strictly increasing in both argu- ments? Explain your answer. (f) Solve for the equilibrium values of c and he in terms of the exogenous variables and model parameters if u(ct, 1 - h;) = Alogo + log(1 - he) where A is a preference parameter. 2. Simple Dynamic system Let reperesent a dynamic system in a, where ? is a postive constant. (a) Given ? > e (the exponential constant) draw a graph of It+1 against It with a 450 line. (b) How many steady states are there? (c) Chracterize the dynamic properties of each steady state. (d) Provide a mathematical argument to support your assertion as to the dynamic properties of each steady state
