rrr economics

1. DiarnondMortensenPissarides with onthejob search Time: Discrete, innite horizon Demography: A mass of 1 of workers with innite lives. There is a large mass of rms who create individual and identical vacancies. The number of vacancies, v, is controlled by fresentry. Preferences: 1Workers and rms are risk neutral {i.e. abs} = 3:}. The common discount rate is r. The value of leisure for workers is b. The cost of holding a vacancy for rms is o utils per period. Productive Technology: A rm matched to a worker produces 13 units of the consump- tion good per period. 'With probability 9: each period, jobs {lled or vacant} experience a catastrophic productivity shock and the job is destroyed. Matching Technology: In this arrangement, workers are always in the market. 1Whether they have a job or not does not stop them getting another job. As they can only have one job at a time if an employed worker meets a rm with a vacancy, the worker quits the current job and switches employment to the new rm. Firms cannot commit to paying a higher wage than the current rm. {'Wages are determined by Nash bargaining and symmetry will mean they all pay the same wage.) 'W'ith probability m{v} each period workers encolmter vacancies where again 'L' is the mass of vacancies. The function mi.) is increasing concave and ml[i:} s: 1 for all 1:. Also limb-nu 37:11:) = 1, limanm WITH} = l}, and sh!) 2':- L'mTL'}. The rate at which vacancies encounter workers is then m{c},-'r1: which is decreasing in 1:. [Assume that job destruction and matching are mutually exclusive so mlEL'} + I". if. 1.} Institutions: The terms of trade are determined by generalized Nash bargaining where rms have bargaining power a. This will mean that in every match the wage is determined from 1-} 1-; = [1-3: 1-; + 1-;1-:,] where 1-} is the value to the rm of having a worker, if\" is the value to holding a vacancy, i; is the worker value of employment, and H, is the worker value to unemployment. Institutions: The terms of trade are determined by generalized Nash bargaining where rms have bargaining power or. This will mean that in every match the wage is determined from 1-} 1-1, = [1-3: I; + v. 1-1,] where 1-} is the value to the rm of having a worker, I"; is the value to holding a vacancy, l; is the worker value of employment, and l-'u is the worker value to unemployment. {a} Let to represent the wage and obtain the ow value or Bellman type equations for workers and rms. {b} Dene a freeentry search equilibrium. {c} Solve for an expression that characterizes equilibrium in terms of the mass of vacancies, s. {d} Under what parameter restriction does a unique interior [i-e. c 3:- } equilibrium exist? Explain. {e} llbtain an expression for steadystate unemployment. [f] How does unemployment change with the separation rate, A? Briey explain. 2. Consider a real business cycle model in which the representative agent chooses capital and and labor to maximize the utility of consumption {c} and leisure {(1 31}. where the time endowment is unity and labor is 5. u[c_. l if} subject to stochastic productivity shocks {A}- Output {3;} is given by y= Aklil 5}1_\"= where the rm rents capital from the household at rental rate r". (a) {b} \"'rite the rm\"s prot maximization problem and solve for the values of the wage (to) and the rental rate {r} . \"'rite the expression for the agent\"s budget constraint using recursive notation [primes for oneperiodahead values} Let the rate of depreciation on capital be :3. \"Thy can't the representative agent in a closed economy use bonds to smooth consumption? 'What are the state variables in the consumer's optimization problem? \"Trite the value function for the consumer, using recursive notation and take rst order conditions. W'rite the expression for the envelope condition and write an expression for the Euler equation and one for the labor supply decision- Explain the permanent income theory of consumption. Use this theory to compare the e'ect of a transitory increase in A on consumption with a permanent increase- Now, consider three different specications of utilityT each of which is used in macro models. 1 (1 if} d" . u{c.1}=lnc+u1 viral??? 1 {BL} _ F). u[c._ l if} = lnc tit {IDL} 1 at 1\"\" where (BL) represents the baseline specification, (IDL) is the specification with indi- visible labor, and the (GHH) is due to Greenwood, Hercowitz and Huffman. Write the equations for the equilibrium relationship between consumption and leisure for each specification. f) Define a balanced growth equilibrium. Which, if any, of the specifications have a labor- leisure choice which is consistent with balanced growth? Explain. g) Compare the response of labor supply to a transitory increase in A which raises the wage using the baseline model and the GHH model