Homework 3
October 13, 2021
1. Growth model with investment-specific productivity shocks. Consider the social
planner formulation of the following stochastic growth model:
V (ks, ke, q) = max
c,h,is,ie
θ log c+ (1− θ) log(1− h) + βEq[V (k′e, k
′
s, q
′)] s.t.
c+ is + ie = k
αs
s k
αe
e h
1−αs−αe
k′s = (1− δs)ks + is
k′e = (1− δe)ke + qie
c ≥ 0, is ≥ 0, ie ≥ 0
log q = ρq log q−1 + ε, ε ∼ N(0, σ2q ) (1)
where c is consumption, h is leisure, is is investment in structures and ie is investment in
equipment. Here, the primes ′ denote next-period value and the subscripts −1 denote past
values. The non-negativity constraint on each type of investment means that machines
cannot be removed once they are installed.
System (1) is a recursive formulation. One can also write the problem as an infinite
horizon as follows:
max
ct,ht,ist,iet
∞∑
t=0
βtEt [θ log ct + (1− θ) log(1− ht)] s.t.
ct + ist + iet = k
αs
st k
αe
et h
1−αs−αe
t
ks,t+1 = (1− δs)kst + ist
ke,t+1 = (1− δe)ke,t + qtiet
ct ≥ 0, ist ≥ 0, iet ≥ 0 for all t
log qt = ρq log qt−1 + εt, εt ∼ N(0, σ2q )
1
This economy features a shock to the efficiency of new capital produced (aka investment-
specific technological change). A rise in q is an improvement in the quality of new
capital goods because the same raw units of investment goods have a larger impact on
the capital stock used in production. In a competitive equilibrium, q is the ratio of
the price of consumption and structures pc relative to the price of equipment pe, i.e.
q = pc/pe. You will not need to solve the business cycle version of the model. I will
discuss how to do so in a program later on.
(a) What are the state and control variables?
(b) Characterize the equilibrium of the economy. The equilibrium conditions involve a
resource constraint, labor leisure tradeoff, and two (intertemporal) Euler equations.
(Hint: take first order conditions with respect to ct, ht, ks,t+1 and ke,t+1 and rear-
range. You can follow the logic of the real business model closely but now note that
there are two types of investment).
(c) Carefully interpret the equilibrium conditions, in particular the intertemporal con-
dition with respect to investment in capital equipment.
(d) Suppose there is a positive realization to εt. Describe the transmission of the shock
into the economy. Compare the transmission to that of a total factor productivity
shock in the standard real business cycle model.
(e) Write the steady-state conditions of the model. (Take the equilibrium conditions
and remove the time indices. You can simplify by using the rate of time preference
r = (1−β)/β. The steady state can be written as a function of c, ke, ks and h alone,
where output satisfies y = kαss k
αe
e h
1−αs−αe .)
(f) The parameter space is Θ = {β, θ, αs, αe, δs, δe}. Solve for the steady state recur-
sively. That is, solve for the steady state in a sequence of steps for variables xt such
that x0 = f0(Θ), x1 = f1(Θ, x1),… Note that the steady state can be solved without
the use of any numerical solver. (Hint: start by solving for the ratio c/ks.)
(g) Suppose the law of motion for equipment is
k′e = (1− δe(u))ke + qie
where
δe(u) = δ0 +
b
ω
uω, ω > 1
Provide intuition on the effects of an endogenous utilization rate on the transmission
mechanism. How does a q shock change the utilization rate?
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