Economics 312/702 -Macroeconomics: Problem Set 4

Q1. This problem considers a simplified New Keynesian model, given by the IS equation:

x_t = -φ(R_t - E_tπ_t+1) + g_t

and the New Keynesian Phillips curve:

π_t = κx_t + βE_tπ_t+1 + u_t

Suppose the cost shock is given by:

u_t = ρ_uu_t-1 + ∈_t^u

while the demand shock g_t is i.i.d. Suppose that monetary policy is conducted by the policy rule:

R_t = au_t + (1/φ)g_t

where a > 0 is a constant and φ is the same constant as in the IS equation.

(a) Solve for the equilibrium levels of inflation π_t and the output gap x_t as a function of the shocks u_t and g_t. To do so, use the policy rule in the IS equation and solve for x_t in terms of the other variables. Substitute your answer into the Phillips curve, and guess that π_t = ku_t for some constant k. Then solve for k and determine π_t and x_t.

(b) If policymakers want to minimize expected losses, with a loss function:

L_t = ½(ωx²_t + π²_t)

what is the optimal value of the policy rule coefficient a?

Q2. Suppose a policymaker operates under commitment and wants to minimize the loss function:

E₀∑β^t(π²_t + λx²_t).

Inflation is determined by a New Keynesian Phillips curve:

π_t = βE_tπ_t+1 + κx_t + u_t,

where the cost push shock u_t is serially correlated:

u_t = ρu_t-1 + ε_t,

with 0 < ρ < 1 and where ε_t is an i.i.d. Normal random variable with E_tε_t+1 = 0.

(a) Suppose that the policymaker can commit implementing a rule x_t = bu_t for some b. Find the resulting equilibrium level of inflation.

(b) What is the optimal value of b, the one the minimizes loss? What are the resulting values of inflation and the output gap?

(c) Suppose the economy is also governed by an Euler equation with a demand shock:

x_t = E_tx_t+1 - φ(R_t - E_tπ_t+1) + g_t.

Find a policy rule for setting R_t which is implements the equilibrium for the optimal rule for b you found above. Interpret your answer.

Q3. Consider a search model of the labor market in which workers have linear utility U(c) = c, all jobs are the same and pay a wage of w units of consumption, unemployed workers get b < w in unemployment benefits, the separation rate is 50% (s = 1/2), the job finding rate is 50% (p = 1/2), and agents discount the future with factor β = 1/2.

(a) The equilibrium unemployment rate occurs when flows into and out of unemployment are equal. Find the equilibrium unemployment rate.

(b) Find the values (i.e. the total expected discounted utility) of an employed worker and an unemployed worker.

(c) Instead of the scenario outlined above, suppose that the job finding rate was p = 1, but that 1/2 of jobs paid a wage w₁ < b while 1/2 of jobs paid a wage w₂ > b. How would this change your results?

Q4. Consider the equilibrium search model from class, but instead of simply assuming that the wage is determined by equalizing the gains from the match, suppose that the wage is determined with a bargaining power 0 ≤ φ ≤ 1:

V_E - V_U = (φ/1 - φ)[V_F - V_V].

We previously assumed φ = 0.5.

(a) How are steady state unemployment and wages affected by an increase in φ? Interpret your answer.

(b) Suppose φ = 0. What is the wage? How is the equilibrium condition rV_v(E) = 0 affected? Interpret your answer.

(c) Now suppose φ = 1. What is the wage? How is the equilibrium condition rV_v(E) = 0 affected? Is there any value of E for which it is satisfied? What is the steady state of the model in this case?