Problem Set 2 (Business Cycle Models)
ECON 6002
Due date: Monday, 8 May, 6pm
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NOTE: To receive full marks, it is crucial to show all of your workings and not just
provide a final algebraic or numerical answer. Solve to a higher number of decimals,
but report final numerical answers to two decimal and place a box around the answer.
It is also important that you provide answers in your own words. Any quotations
from the textbook or other sources must be in quotation marks and attributed to the
original source.
1. Abstracting from long-run growth by setting n = g = 0 and from persistent shocks by setting
ρA = ρG = 0, with A˜t ≡ lnAt − lnA¯ and G˜t ≡ lnGt − lnG¯, and normalizing the population
to N = 1, the following nine equations describe the “baseline” RBC model in Chapter 5:
Yt = Ct + It +Gt (1)
Yt = K
α
t (AtLt)
1−α (2)
Kt+1 = Kt + It − δKt (3)
A˜t = A,t (4)
G˜t = G,t (5)
rt = α(AtLt/Kt)
1−α − δ (6)
wt = (1− α)(Kt/AtLt)αAt (7)
1
Ct
= e−ρEt
[
1
Ct+1
(1 + rt+1)
]
(8)
Ct
1− Lt =
wt
b
(9)
(a) Find the steady state for this economy under the following calibration: α = 13 , δ = 0.05,
r¯ = 0.03, A¯ = 1, and L¯ = 0.5 and choose G¯ such that G¯/Y¯ = 0.2.In particular, you
should find the remaining parameters values b and ρ that are consistent with steady
state and determine steady-state values for the endogenous variables, Y¯ , C¯, I¯, G¯, K¯,
and w¯. (Hint: first solve for ρ using (8), then solve for K¯ using (6), then I¯ using (3),
then w¯ using (7), then Y¯ using (2), then C¯ using (1), then b using (9).)
(b) Now consider the special case of the model where δ = 1 instead of δ = 0.05 and Gt = 0
for all t (note: ρ will remain the same and b will be different, but you do not need
to solve for it). Solve for Yt, Ct, It, Kt+1, rt, and wt as analytical expressions of
exogenous and predetermined variables At and Kt and constants. (Hint: with 100%
depreciation, there is a constant saving rate s = αe−ρ and constant labour supply
Lt = L¯. Given this solution to the household optimization problem, first solve for Yt, rt,
and wt from equations (2), (6), and (7) and then the solutions for Ct, It, and Kt+1 are
straightforward.)
(c) Again, for the special case of the model, what is the percentage change in output and
percentage point change in the interest rate if the economy is at steady state at time
t − 1, but there is a shock A,t = 0.0.25 (i.e., 25%) at time t? Explain the economic
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intuition behind the responses of output and the interest rate in terms of the marginal
products of labour and capital. (Hint: note that the A,t = 0.0.25 shock is to lnAt, but
the model solution is for the level of At. First solve for the steady-state level of output
and then solve for output and the real interest rate given the shock.)
2. Consider Calvo price-setting firms with partial indexation. That is, if a firm is not visited
by the Calvo tooth fairy in period t, its price in t is the previous period’s price plus γpit−1,
0 ≤ γ ≤ 1. The average price in period t is pt = αxt + (1− α)(pt−1 + γpit−1), where α is the
fraction of firms visited by the Calvo tooth fairy in any given period and xt is the price they
set. The resulting Phillips curve is a hybrid one: pit =
γ
1+βγpit−1 +
β
1+βγEtpit+1 +
1
1+βγκy˜t,
where β > 0 is the discount factor, κ = α1−α [1 − β(1 − α)]φ > 0 determines the slope of
the Phillips curve, y˜t is the output gap, and Etpit+1 is the expectation (taken at time t) of
inflation at t+ 1.
(a) Show that xt − pt = 1−αα (pit − γpit−1).
(b) Use the result in (a) and the representative firm’s optimal price under Calvo pricing
with partial indexation being xt = pt + (1− β(1− α))φy˜t + β(1− α)(Et(xt+1 − pt+1) +
Etpit+1 − γpit) to derive the hybrid Phillips curve.
(c) What value of γ would lead to the highest degree of inflation persistence? Why?
(d) Now assume that β = 0.9, γ = 0.5, and κ = 0.1. Assume that the central bank has some
control of the evolution of y˜t. Suppose that the central bank announces a permanent
and fully credible reduction in its target or steady-state inflation rate from 7% to 2%
at t = 1 (prior to this, the economy was at the steady state with 7% inflation and zero
output gap). Determine the cost of this disinflation episode. How much is the output
gap reduced?
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