\f{c} Show that in a symmetric equilibrium. aggregate output. 1'. is a function of aggregate labor. L = j: LI[3':I di. Using this result. write the optimality condition for an intermediate goods producer in terms of aggregate variables. {d} The preferences of the representative household over consumption. C}. and labor hours. L*._ are given by no . 1 En(EH3'[ln{C*jX1_HIL:H])_ lilo-3:1. 1.31. x30- Households receive labor income and prots from rms. They spend their income on consumption and riskfree bonds. Let Es denote the quantity of bonds bought at time t. measured in terms of output. 1 unit of bonds delivers {l rs] units of output at time t l. Households also face the usual initial. non-negativity and NoPonziGame conditions. Find the rst order conditions for utility maximization. 3 lie} Assume that there is no capital in this economy. so that bonds represent trades between consumers. Imposing equilibrium. nd the laborleisure condition and the resource constraint. Suppose further that the parameters is and .--1 both vary other time. \"'rite consumption as a function of In. and .rlt. {f} Interpret your answer to part {e}. Is the coefficient on In" positive or negatitre";I How does it depend on a and \"3"." Briey interpret. \f1. 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. 10. Mortensen-Pissarides with ex ante surplus sharing Time: Discrete, infinite horizon Demography: A mass of 1 of ex ante identical workers with infinite lives and a large mass of firms who create individual vacancies. Preferences: Workers and firms are risk neutral (i.e. u(x) = a). The common discount rate is r. The value of leisure for workers is b utils per period. The cost of holding a vacancy for firms is a utils per period. Productive Technology: Matched firm/ worker pairs produce p units of the con- sumption good per period. With probability A each period, jobs (filled or vacant) experience a catastrophic productivity shock and the job is destroyed. Assume p > b. Matching Technology: Unemployed workers encounter vacancies at the rate m() where e = v/u, v is the mass of vacancies and u is the mass of unemployed workers. The function m(.) is increasing and concave, and m() 0m'(0). The rate at which vacancies encounter unemployed workers is then m(8)/8. Institutions: The terms of trade are determined by symmetric division of the nego- tiable surplus which means that the wage w = (p + b)/2. (a) Write down the set of flow value equations or Bellman equations for workers and firms. (b) Show that the outside option for the worker cannot bind (i.e. that w > row where Uw is the value to unemployment for a worker). (c) Define a steady state free-entry equilibrium and solve for a single equation in 8. (d) Obtain an expression for the unemployment rate, & in terms of e. ( e) How do changes in a and b affect the level of unemployment? Provide intuition for your results
