Econ 33220
Final Exam
Problem 1 (Total: [ 30 pt ]) Consider the New Keynesian model.
1. ([ 10 pt ]) State the model, assuming that there are no demand or supply shocks and writing the disturbance to the Taylor rule as zt instead of ∈t(M) .
2. ([ 10 pt ]) Suppose that Φx = 0, Φπ = 2, κ = 0.5, σ = 1, ρ = 0, β = 0
and rt(f) = 0 for all t. Suppose that zt is autocorrelated,
zt = αzt-1 + ∈t (1)
where ∈t is now the unpredicted shock to the Taylor rule, Et [∈t+1] = 0 and where 0 ≤ α ≤ 1 is the persistence of zt . Suppose zt-1 = 0 and ∈t = 1 . By how much do xt , πt and it move, as a function of α? (Hint: let yt = [xt , πt , it]. Conjecture that yt = Qzt for some vector Q = [Qx , Qπ , Qi].)
3. ([ 10 pt ]) For what value of the persistence α does the NK model become “Fisherian”, i. e. when does the reaction of πt and it have the same sign in response to ∈t ?
Problem 2 (Total: [ 30 pt ]) Consider the continuous-time problem
(2)
subject to
(3)
and kt ≥ 0, for a given initial value for k0 > 0, where ρ > 0, 0 < α < ρ and k0 > 0 are parameters.
1. ( [ 5 pt ]) State the Hamiltonian (be explicit) .
2. ( [ 5 pt ]) Find the first-order conditions.
3. ( [ 10 pt ]) Calculate the path for consumption, starting from some c0 .
4. ( [ 5 pt ]) Given c0 , calculate the resulting path for capital. (Hint: let
xt = e-Qtkt. Calculate x.t and xt = x0 + ∫0(t) x.sds.)
5. ( [ 5 pt ]) Given c0 , calculate the limit k∞ = limt→∞ kt . Calculate c0 so that k∞ = 0 .
6. ( [ For extra 10 pt ]) Draw the state space diagram.
Problem 3 (Total: [ 60 pt ]) Consider the following equilibrium. There are two exogenous process At and Qt , given by
for constants 0 ≤ ρQ < 1, Q > 0, where
(4)
Consider a small open economy. Households supply a constant amount of labor nt 三 1 and receive the wage wt . Households can invest in capital as well as international one-period discount bonds bt ∈ RI, i. e. we allow borrowing as well as lending. Capital earns the rental rate rt and then depreciates at rate δ . The discount bonds sell for the exogenous discount price Qt in period t and pay of one unit in period t + 1, all in terms of the consumption goods. Thus, the per-period budget constraint of the household is
ct + kt + Qt bt = wt + (rt + 1 — δ)kt-1 + bt-1 (5)
Given initial capital, bond holdings and the processes for wages, rental rates and bond discount prices, the representative household maximizes
(6)
subject to the budget constraints (5) above. Define
Rt = rt + 1 - δ (7)
Production takes place by a competitive sector of firms. Given wages wt and capital rental rates rt , the representative firm maximizes profits
(8)
Aggregate output is therefore
(9)
The parameters are ρQ, Q, σA(2), σQ(2), δ, β, θ . Your tasks:
1. ( [ 10 pt ]) State the Lagrangian for the representative household and find the resulting first-order conditions.
2. ( [ 5 pt ]) State a list of equations characterizing the equilibrium. When you do, rewrite the left hand side of (5) using yt and without wt , Rt and rt .
3. ( [ 5 pt ]) State the steady state version of these equations.
4. ( [ 5 pt ]) Is there a restriction on the parameters that must be satisfied
for a steady state to exist? What do you think happens (absent shocks, say), if that condition is not satisfied? Provide a brief description.
From here, suppose the conditions for the existence of a steady state are satisfied. Assume that b = 0 .
5. ( [ 5 pt ]) Calculate the steady state in terms of the parameters or in terms of steady state values calculated in a prior step.
6. ( [ 10 pt ]) Log-linearize the equations around that steady state, but use dt = c/dt (think: “percent of steady state consumption”), since bt can take positive as well as negative values. For the log-linearized version of the (rewritten) budget constraint (5), divide by .
7. ( [ 5 pt ]) Solve for k(^)t in terms of Q(^)t . (Hint: exploit that Et [A(^)t+1] = 0.) Solve for ^(y)t in terms of A(^)t and Q(^)t-1 .
8. ( [ 5 pt ]) Use the last results to show that the log-linearized budget constraint can be written in the form.
(10)
for some ψ 1 , ψ2 and φj , j = 1, 2, 3. Explicitly state ψ 1 and ψ2 .
9. ( [ 10 pt ]) Assume that ^(c)t has a recursive law of motion, which we shall write as
(11)
for coe cients P and √j , j = 1, 2, 3. Calculate P which avoids explosive debt dynamics, i. e. b(^)t should have at most a unit root. Interpret your solution. (Hint: even if you cannot derive (10), proceed as far as you can, assuming (10) to be true.)