10.1.1.3.9670

Lecture Notes on Dynamic Competitive Analysis1
Jeremy Greenwood
Department of Economics, University of Rochester, Rochester, NY 14627
Fall 2001 — Comments Welcome
1If you’d like to contribute to these on-line lecture notes then please let me know. Com-
ments on ambiguities, mistakes, omissions, etc. would all be appreciated.

Contents
1 Dynamic Programming 1
1.1 A Heuristic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Neoclassical Growth Model . . . . . . . . . . . . . . . . . . . . 1
1.1.2 The Envelope Theorem . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 A More Formal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Method of Successive Approximation . . . . . . . . . . . . . . . 7
1.2.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 The Contraction Mapping Theorem . . . . . . . . . . . . . . . . 14
1.2.6 Characterizing the Value Function . . . . . . . . . . . . . . . . . 18
1.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Business Cycle Analysis 30
ii

CONTENTS – MANUSCRIPT
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.1 Real Business Cycle Models – Kydland and Prescott (1982) and
Long and Plosser (1983) . . . . . . . . . . . . . . . . . . . . . . 30
2.1.2 Keynesian Investment Multiplier Model . . . . . . . . . . . . . . 32
2.1.3 Business Cycles with Investment-Specific Technological Progress 33
2.2 The Economic Environment . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 The Representative Agent’s Optimization Problem . . . . . . . 36
2.2.2 Impact Effect of Investment Shocks . . . . . . . . . . . . . . . . 37
2.2.3 Dynamic Effects of Investment Shocks . . . . . . . . . . . . . . 39
2.3 Applied General Equilibrium Analysis . . . . . . . . . . . . . . . . . . . 40
2.3.1 Sample Economy . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.2 Discrete-State-Space Dynamic Programming Problem . . . . . . 41
2.3.3 Construction of the Markov Chain . . . . . . . . . . . . . . . . 43
2.3.4 Calibration Procedure . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.6 The Importance of Capacity Utilization . . . . . . . . . . . . . . 50
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Economic Growth 54
3.1 Solow (1957) Growth Accounting . . . . . . . . . . . . . . . . . . . . . 54
iii

3.1.1 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.2 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Solow (1960) — Growth Accounting with Investment-Speciﬁc Technolog-
ical Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Some Motivating Observations . . . . . . . . . . . . . . . . . . . 59
3.2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.3 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . 61
3.2.4 Balanced Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.6 Growth Accounting . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.7 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.8 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2.9 Solow (1957) versus Solow (1960) . . . . . . . . . . . . . . . . . 72
3.3 Malthus to Solow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.2 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
iv

3.3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 Asset Pricing 91
4.1 Lucas (1978) Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.1 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.2 Dynamic Programming Problem . . . . . . . . . . . . . . . . . . 92
4.1.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 Complete Markets à la Arrow, Debreu and McKenzie . . . . . . . . . . 94
4.2.1 Dynamic Programming Problem . . . . . . . . . . . . . . . . . . 94
4.2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.3 Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 The Equity Premium: A Puzzle . . . . . . . . . . . . . . . . . . . . . . 96
4.3.1 The Problem à la Mehra and Prescott (1985) . . . . . . . . . . 96
4.3.2 The Environment . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.3 Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.4 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 Output Eﬀects of Government Purchases 107
v

5.1 The effects of Temporary versus Permanent changes in Government Spend-
ing à al Barro (1987) and Hall (1980) . . . . . . . . . . . . . . . . . . . 107
5.2 The Effects of Government Spending à al Aiyarari, Christiano and Eichen-
baum (1992) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3 The Economic Environment . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4 The Planner’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4.1 The Static Problem . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4.2 The Dynamic Problem . . . . . . . . . . . . . . . . . . . . . . . 114
5.5 The Output Effects of Government Spending — Theory . . . . . . . . . 115
5.5.1 Transitory Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.5.2 Permanent Shocks . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.6 Some Counter Examples — Quantitative Analysis . . . . . . . . . . . . 116
5.6.1 Steady-State Output and Employment Multipliers . . . . . . . . 117
5.6.2 Impact Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.6.3 Interest Rate Effects . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Policy Function Iteration Methods 120
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2 A Distorted Version of the Neoclassical Growth Model . . . . . . . . . 121
6.3 The Representative Agent’s Dynamic Programming Problem . . . . . . 123
6.4 Existence Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
vi

6.5 The Coleman Algorithm (1991) . . . . . . . . . . . . . . . . . . . . . . 128
6.5.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5.2 Linear Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5.3 Cubic Spline Interpolation . . . . . . . . . . . . . . . . . . . . . 129
6.5.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.6 Parameterized Expectations — den Haan and Marcet (1990) . . . . . . . 131
6.6.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.7 Parameterized Policy Functions . . . . . . . . . . . . . . . . . . . . . . 134
6.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7 Incomplete Markets 135
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.1.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.1.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.2 The Aiyagari (1994) Model . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.2.1 Heterogeneity and Aggregation . . . . . . . . . . . . . . . . . . 139
7.3 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.4 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
vii

8 Equilibrium Unemployment 147
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.1.1 Stylized facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.3 Choice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.4 Market Clearing Conditions . . . . . . . . . . . . . . . . . . . . . . . . 152
8.5 Results — steady state with no aggregate uncertainty . . . . . . . . . . 153
8.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.7 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.7.1 Micro-level Findings . . . . . . . . . . . . . . . . . . . . . . . . 156
8.7.2 Macro-Level Findings . . . . . . . . . . . . . . . . . . . . . . . . 158
viii

Chapter 1
Dynamic Programming
1.1 A Heuristic Approach
1.1.1 Neoclassical Growth Model
Consider the following optimization problem
max
{ct,kt+1}∞
t=1
∞P
t=1
βt−1
U(ct)
| {z }
U:R+→R
subject to
ct + kt+1 = F(kt)
| {z }
F:R+→R+
.
The above problem can be reformulated as
max
{kt+1}∞
t=1
∞P
t=1
βt−1
U(F(kt) − kt+1).
1

CHAPTER 1 – MANUSCRIPT
How can this problem be solved?
T-Period Problem
max
{kt+1}T
t=1
TP
t=1
βt−1
U(F(kt) − kt+1).
Period-T Problem
V 1
(kT ) ≡ max
kT +1
{U(F(kT ) − kT+1)} P(1)
= U(F(kT ) − k∗
T+1
|{z}
=0
),
where k∗
T+1 solves problem P(0).
V 1
(kT ) = value of entering last period with kT units of capital and behaving
optimally henceforth. The superscript refers to the number of periods remaining in the
planning problem. The function V 1
(kT ) is called the value function, while kT is known
as the state variable.
Period-(T-1) Problem
V 2
(kT−1) ≡ max
kT
{U(F(kT−1) − kT ) + βV 1
(kT )} P(2)
= U(F(kT−1) − k∗
T
|{z}
=G2(kT−1)
) + βV 1
(k∗
T ),
2

(a,b)
•
• 0
b • • a
Φ
φ
Rn
E⊂Rn+m
U⊂RnW⊂Rm
Figure 1.1: Implicit Function Theorem
where k∗
T = G2
(kT−1) solves problem P(2). The function G2
is called the decision rule
or policy function. Here k∗
T solves the first-order condition
−U1(F(kT−1) − k∗
T ) + βV 1
1 (k∗
T ) = 0. (1.1)
Question: What can be said about the functions G2
and V 2
? Note that
first-order condition (1.1) defines an implicit function determining kT as a function of
kT−1. More generally in economics one often comes across equation systems of the form
Φ(x, y) = 0, where x ∈ Rn
, y ∈ Rm
, and Φ : Rn+m
→ Rn
. Can a function φ be found
that solves for x in terms of y so that x = φ(y)?
Theorem 1 Implicit Function. Let Φ be a Cq
mapping from an open set E ⊂ Rn+m
into Rn
such that Φ(a, b) = 0 for some point (a, b) ∈ E. Suppose that the Jacobian
determinant |J| = |∂Φ(a,b)
∂x
| 6= 0. Then there exits a neighborhood U ⊂ Rn
around a and
a neighborhood W ⊂ Rm
around b and a unique function φ : W → U such that,
3

1. a = φ(b),
2. φ is class Cq
on W,
3. for all y ∈ W, (φ(y), y) ∈ E, and Φ(φ(y), y) = 0.
Now, applying the implicit function theorem to the ﬁrst-order condition (1.1)
it is apparent that under the standard conditions k∗
T = G2
(kT−1) will be C1
function
which implies that V 2
(kT−1) will be one too.
Period-t Problem
V T+1−t
(kt) ≡ max
kt+1
{U(F(kt) − kt+1) + βV T−t−1
(kt+1)} P(T+1-t)
= U(F(kt) − k∗
t+1) + βV T−t−1
(k∗
T+1),
where k∗
t+1 = GT+1−t
(kt) solves problem P(T+1-t).
Observe that dynamic programming has eﬀectively collapsed a single large
problem involving T + 1 − t choice variables into T + 1 − t smaller problems, each
involving one choice variable. To see this, solve out for V T−t
(kt+1) in P(T+1-t) to get
V T+1−t
(kt) ≡ max
kt+1
{U(F(kt) − kt+1) + βmax
kt+2
{U(F(kt+1) − kt+2) + βV T−t
(kt+2)}}
= max
kt+1,kt+2
{U(F(kt) − kt+1) + βU(F(kt+1) − kt+2) + β2
V T−t−1
(kt+2)}.
Solving out recursively for V T−t
(kt+2), V T−t−1
(kt+3), ..., yields
4

max
{kt+j+1}T−t
j=0
T−tP
j=0
βj
U(F(kt+j) − kt+j+1).
Infinite Horizon Problem
As T → ∞ one might expect that
V T+1−t
(kt) → V (kt)
and
GT+1−t
(kt) → G(kt).
This is true but it takes some effort to show it. Thus, the problem for the infinite
horizon will take the form:
V (kt) ≡ max
kt+1
{U(F(kt) − kt+1) + βV (kt+1)} P(∞)
= U(F(kt) − k∗
t+1) + βV (k∗
t+1)},
where k∗
t+1 = G(kt).
1.1.2 The Envelope Theorem
Assumption: V is continuously differentiable.
How is the solution to problem P(∞) characterized? The f.o.c. is
−U1(F(kt) − kt+1) + βV1(kt+1) = 0 (1.2)
5

or
U1(F(kt) − kt+1) = βV1(kt+1). (1.3)
Problem: This equation involves the unknown function V . What should be
done?
Answer: Diﬀerentiate both sides of P(∞) with respect to kt to get
V1(kt) = U1(F(kt) − kt+1)F1(kt) − U1(F(kt − kt+1)
∂kt+1
∂kt
+ V1(kt+1)
∂kt+1
∂kt
= U1(F(kt) − kt+1)F1(kt) + [−U1(F(kt − kt+1) + V1(kt+1)]
∂kt+1
∂kt
= U1(F(kt) − kt+1)F1(kt),
since the term in brackets on the second line is zero by the ﬁrst-order condition (1.2).
Updating this expression from period t to period t + 1 gives
V1(kt+1) = U1(F(kt+1) − kt+2)F1(kt+1).
This allows equation (1.3) to be rewritten as
U1(F(kt) − kt+1) = βU1(F(kt+1) − kt+2)F1(kt+1).
1.2 A More Formal Analysis
Dynamic Programming Representation
6

V (k) ≡ max
k0
{U(F(k) − k0
) + βV (k0
)} P(1)
The problem at hand is to get answers to the following questions:
1. Will V exist?
2. Is V unique?
3. Is V continuous?
4. Is V continuously diﬀerentiable?
5. Is V increasing in k?
6. Is V concave in k?
1.2.2 Method of Successive Approximation
Goal: To approximate the value function V by a sequence of successively better guesses,
denoted by V j
at stage j
Procedure:
• Stage 0. Make an initial guess for V . Call it V 0
.
• Stage 1. Construct a revised guess for V , denoted by V 1
.
V 1
(k) ≡ max
k0
{U(F(k) − k0
) + βV 0
(k0
)}
7

• Stage n + 1. Compute V n+1
given V n
, as follows
V n+1
(k) ≡ max
k0
{U(F(k) − k0
) + βV n
(k0
)}. P(2)
This procedure can be represented much more compactly using operator no-
tation.
V n+1
= TV n
.
The operator T is shorthand notation for the list of operations, described by P(2) that
are performed on the function V n
to transform it into the new one V n+1
. Often the
operator T maps some set of functions, say F, into itself. That is, T : F → F. The
hope is that as n gets large it transpires that V n
→ V , where V = TV .
1.2.3 Metric Spaces
Deﬁnition 2 A metric space is a set S, together with a metric ρ : S × S → R+, such
that for all x, y, z ∈ S (see Figure 2):
1. ρ(x, y) ≥ 0, with ρ(x, y) = 0 if and only if x = y,
2. ρ(x, y) = ρ(y, x),
3. ρ(x, z) ≤ ρ(x, y) + ρ(y, z).
Example 1
8

Vancouver, x
L.A., y
Rochester, z
ρ(y,z)
ρ(x,z)
ρ(x,y)
Figure 1.2: Distances Between Cities
-0.1 0.1 0.3 0.5 0.7 0.9 1.1
t
1.00
1.05
1.10
1.15
1.20
1.25
x(t)andy(t)
x(t)=1
y(t)=1+t-t2
ρ(x,y)=0.25
Figure 1.3: Uniform Metric
Space of continuous functions C : [a, b] → R+. See Figure 3.
ρ(x, y) = max
t∈[a,b]
|x(t) − y(t)|.
Deﬁnition 3 A sequence {xn}∞
n=0 in S converges to x ∈ S, if for each ε > 0 there
9

exists a Nε such that
ρ(xn, x) < ε, for all n ≥ Nε.
Definition 4 A sequence {xn}∞
n=0 in S is a Cauchy sequence if for each ε > 0 there
exists a Nε such that
ρ(xm, xn) < ε, for all m, n ≥ Nε.
Remark 5 A Cauchy sequence in S may not converge to a point in S.
Example 2
Let S = (0, 1], ρ(x, y) = |x − y|, and {xn}∞
n=0 = {1/n}∞
n=0. Clearly,
xn → 0 /∈ (0, 1]. this sequence satisfies the Cauchy criteria, though,
for
ρ(xn, xm) = |
1
m
−
1
n
| ≤
1
m
+
1
n
< ε, if m, n >
2
ε
.
Definition 6 A metric space (S, ρ) is complete if every Cauchy sequence in S converges
to a point in S.
Theorem 7 Let X ⊆ Rl
and C(X) be the set of bounded continuous functions V : X →
R with the uniform metric ρ(V, W) = max
x∈X
|V − W|. Then C(X) is a complete metric
space.
10

Proof. : Let {V n
}n be any Cauchy sequence in C(X). Now, for each x ∈ X the sequence
{V n
(x)}n is Cauchy since
|V n
(x) − V m
(x)| ≤ sup
y∈X
|V n
(y) − V m
(y)| = ρ(V n
, V m
).
By the completeness of the reals V n
(x) → V (x), as n → ∞. Deﬁne the function V by
V (x) for each x ∈ X.
It will now be shown that ρ(V n
, V ) → 0 as n → ∞. Choose an ε > 0. Now,
|V n
(x) − V (x)| ≤ |V n
(x) − V m
(x)| + |V m
(x) − V (x)|
≤ ρ(V n
, V m
)
| {z }
≤ε/2
+ |V m
(x) − V (x)|
| {z }
≤ε/2
.
The ﬁrst term on the left can be made smaller than ε/2 by the Cauchy criteria; that is,
there exists a Nε such that for all n, m ≥ Nε it transpires that ρ(V n
, V m
) ≤ ε/2. The
second term can be made smaller than ε/2 by the pointwise convergence of V m
to V ;
that is, there exists a Mε(x) such that for all m ≥ Mε(x) it follows that |V m
(x)−V (x)| ≤
ε/2. Observe that while Mε(x) depends on x, Nε does not. Also, note that for any
value of x such a Mε(x) will always exist. Therefore, |V n
(x) − V (x)| ≤ ε for all n ≥ Nε
independent of the value of x. It follows that ρ(V n
, V ) ≤ ε, the desired result.
The last step is to show that V is a continuous function. To do this, pick an
ε > 0. Does there exist a δ ≥ 0 such that |V (x) − V (x0)| ≤ ε whenever ρ(x, x0) ≤ δ?
11

Note that
|V (x) − V (x0)| ≤ |V (x) − V n
(x)|
| {z }
ε/3
+ |V n
(x) − V n
(x0)|
| {z }
ε/3
+ |V (x0) − V n
(x0)|
| {z }
.
ε/3
The ﬁrst and third terms can be made arbitrarily small by the uniform convergence of
V n
to V . The second term can be made to vanish by the fact that V n
is a continuous
function; that is, by picking a δ small enough such that this term is less than ε/3.
Remark 8 Pointwise convergence of a sequence of continuous functions does not imply
that the limiting function is continuous.
Example 3
Let {V n
}∞
n=1 in C[0, 1] be deﬁned by V n
(t) = tn
. As n → ∞ it
transpires that: (i) V n
(t) → 0 for t ∈ [0, 1) and (ii), V n
(t) → 1 for
t = 1. Thus,
V (t) =



0, for t ∈ [0, 1),
1, for t = 1.
Hence V (t) is a discontinuous function. See Figure 4. Clearly, by the
above theorem {V n
}∞
n=1 cannot describe a Cauchy sequence. This
can be shown directly too, however. In particular, for given any
Nε it is always possible to pick a m, n ≥ Nε and t ∈ [0, 1) so
|tn
− tm
| ≥ 1/2. To see this pick n = Nε and a t ∈ (0, 1) so that
12

-0.1 0.1 0.3 0.5 0.7 0.9 1.1
t
0.0
0.2
0.4
0.6
0.8
1.0
tn
t
t10t2
t3
Figure 1.4: Pointwise Convergence to a Discontinuous Function
tn
≥ 3/4; i.e, choose t ≥ (3/4)1/Nε
. Next, pick m large enough such
that tm
< 1/4 or m ≥ (ln 1/4)/(ln t). The desired results obtains.
Example 4
Consider the space of continuous functions C[−1, 1] with metric
ρ(x, y) =
1R
−1
|x(t) − y(t)|dt.
Let {V n
}∞
n=1 in C[−1, 1] be deﬁned by
V n
(t) =



0, if − 1 ≤ t ≤ 0,
nt, if 0 < t < 1/n,
1, if 1/n ≤ t ≤ 1.
Show that {V n
}∞
n=1 is a Cauchy sequence. Deduce that the space of
continuous functions is not complete with this metric.
13

1.2.4 The Contraction Mapping Theorem
Definition 9 Let (S, ρ) be a metric space and T : S → S be function mapping S into
itself. T is a contraction mapping (with modulus β) if for β ∈ (0, 1),
ρ(Tx, Ty) ≤ βρ(x, y), for all x, y ∈ S.
Theorem 10 (Contraction Mapping Theorem or Banach Fixed Point Theorem). If
(S, ρ) is a complete metric space and T : S → S is a contraction mapping with modulus
β, then
1. T has exactly one fixed point V ∈ S such that V = TV,
2. for any V 0
∈ S, ρ(Tn
V 0
, V ) < βn
ρ(V 0
, V ), n = 0, 1, 2... .
Proof. Define the sequence {V n
}∞
n=0 by
V n
= TV n−1
= TT|{z}
T2
V n−2
= Tn
V 0
.
It will be shown that {V n
}∞
n=0 is a Cauchy sequence. To this end, the contraction
property of T implies
ρ(V 2
, V 1
) = ρ(TV 1
, TV 0
) ≤ βρ(V 1
, V 0
).
Hence,
ρ(V n+1
, V n
) = ρ(TV n
, TV n−1
) ≤ βρ(V n
, V n−1
) ≤ βn
ρ(V 1
, V 0
).
14

Therefore for any m > n
ρ(V m
, V n
) ≤ ρ(V m
, V m−1
) + ρ(V m−1
, V m−2
) + ... + ρ(V n+1
, V n
)
| {z }
Triangle of Inequality
≤ (βm−1
+ βm−2
+ ... + βn
)ρ(V 1
, V 0
)
≤
βn
1 − β
ρ(V 1
, V 0
).
Therefore {V n
}∞
n=0 is a Cauchy sequence, since βn
1−β
→ 0 as n → ∞. Since S is complete
V n
→ V .
To show that V = TV note that for all ε > 0 and V 0
∈ S
ρ(V, TV ) ≤ ρ(V, Tn
V 0
) + ρ(Tn
V 0
, TV )
≤
ε
2
+
ε
2
,
for large enough n since {V n
}∞
n=0 is a Cauchy sequence. Therefore, V = TV .
Finally suppose that another function W ∈ S satisﬁes W = TW. Then,
ρ(V, W) = ρ(TV, TW) ≤ βρ(V, W),
a contradiction unless V = W.
ρ(Tn
V 0
, V ) = ρ(Tn
V 0
, TV ) ≤ βρ(Tn−1
V 0
, V ) ≤ βn
ρ(V 0
, V ).
15

Corollary 11 Let (S, ρ) be a complete metric space and let T : S → S be a contraction
mapping with ﬁxed point V ∈ S. If S0
is a closed subset of S and T(S0
) ⊆ S0
then
V ∈ S0
. If in addition T(S0
) ⊆ S00
⊆ S0
, then V ∈ S00
.
Proof. Choose V 0
∈ S0
and note that {Tn
V 0
} is a sequence in S0
converging to V . Since
S0
is closed, it follows that V ∈ S0
. If T(S0
) ⊆ S00
, it then follows that V = TV ∈ S00
.
Theorem 12 (Blackwell’s Suﬃciency Condition) Let X ⊆ Rl
and B(X) be the space
of bounded functions V : X → R with the uniform metric. Let T : B(X) → B(X) be
an operator satisfying
1. (Monotonicity) V, W ∈ B(X). If V ≤ W [i.e., V (x) ≤ W(x) for all x] then
TV ≤ TW.
2. (Discounting) There exists some constant β ∈ (0, 1) such that T(V +a) ≤ TV +βa,
for all V ∈ B(X) and a ≥ 0. Then T is a contraction with modulus β.
Proof. For every V, W ∈ B(X), V ≤ W + ρ(V, W). Thus, (1) and (2) imply
TV ≤ T(W + ρ(V, W))
| {z }
Monotonicity
≤ TW + βρ(V, W)
| {z }
Discounting
.
Thus,
TV − TW ≤ βρ(V, W).
16

By permuting the functions it is easy to show that
TW − TV ≤ βρ(V, W).
Consequently,
|TV − TW| ≤ βρ(V, W),
so that
ρ(TV, TW) ≤ βρ(V, W).
Therefore T is a contraction.
Consider the mapping
(TV )(k) = max
k0∈K
{U(F(k) − k0
) + βV (k0
)}, P(3)
where k, k0
∈ K = {k1, k2, ..., kn}. Is T a contraction?
1. Monotonicity. Suppose V (k) ≤ W(k) for all k. Need to show that (TV )(k) ≤
(TW)(k).
(TV )(k) = {U(F(k) − k0∗
) + βV (k0∗
)},
17

where k0∗
maximizes P(3). Now, clearly
(TV )(k) ≤ {U(F(k) − k0∗
) + βW(k0∗
)}
≤ max
k0∈K
{U(F(k) − k0
) + βW(k0
)}
= (TW)(k).
2. Discounting.
T(V + a)(k) = max
k0∈K
{U(F(k) − k0
) + β[V (k0
) + a]}
= max
k0∈K
{U(F(k) − k0
) + βV (k0
)} + βa
= (TV )(k) + βa.
1.2.6 Characterizing the Value Function
What can be said about the function V ?
1. Is V continuous in k?
2. Is V strictly increasing in k?
3. Is V strictly concave in k?
4. Is V diﬀerentiable in k?.
18

Deﬁnition 13 A function V : X → R is strictly increasing if x > y implies V (x) >
V (y). A function V : X → R is nondecreasing (or increasing) if x > y implies
V (x) ≥ V (y).
Deﬁnition 14 A function V : X → R is strictly concave if
V (θx + (1 − θ)y) > θV (x) + (1 − θ)V (y),
for all x, y ∈ X such that x 6= y and θ ∈ (0, 1). A function V : X → R is concave if
V (θx+(1 −θ)y) ≥ θV (x)+(1− θ)V (y),for all x, y ∈ X such that x 6= y and θ ∈ (0, 1).
Assumption: Let U and F be strictly increasing functions.
Assumption: Let U and F be strictly concave functions.
Theorem 15 The function V is both strictly increasing and strictly concave.
Proof. Consider again the mapping given by
(TV )(k) = max
k0
{U(F(k) − k0
) + βV (k0
)}. P(3)
It will be shown that the operator T maps concave functions into strictly concave ones.
It also maps increasing functions into strictly increasing ones. Let V be a concave
function. Take two points k0 6= k1 and let kθ = θk0 + (1 − θ)k1. Observe that F(kθ) >
θF(k0) + (1 − θ)F(k1), since F is strictly concave. It needs to be shown that
(TV )(kθ) > θ(TV )(k0) + (1 − θ)(TV )(k1).
19

To this end, define k0∗
0 as the maximizer for (TV )(k0), k0∗
1 as the maximizer for (TV )(k1),
and k0
θ = θk0∗
0 +(1−θ)k0∗
1 . Note that k0
θ is a feasible choice when k = kθ since k0∗
0 ≤ F(k0)
and k0∗
1 ≤ F(k1) while θF(k0) + (1 − θ)F(k1) < F(kθ). Now,
(TV )(kθ) ≥ U(F(kθ) − k0
θ) + βV (k0
θ), (k0
θ is nonoptimal)
> θ[U(F(k0) − k0∗
0 ) + βV (k0∗
0 )]
+(1 − θ)[U(F(k1) − k0∗
1 ) + βV (k0∗
1 )], (by strict concavity )
> θ(TV )(k0) + (1 − θ)(TV )(k1) (by definition).
Remark 16 The space of strictly concave is not complete — see figure 4. Hence, to
finish the argument an appeal to the corollary of the contraction mapping theorem can
be made.
Theorem 17 The function V is continuous in k.
Proof. It will be shown that the operator described by P(3) maps strictly increasing,
strictly concave C2
functions into strictly increasing, strictly concave C2
functions.
Suppose that V n
is a continuous, strictly increasing, strictly concave C2
function. The
the decision rule for k0
is determined from the first-order condition
U1(F(k) − k0
) = βV n
1 (k0
).
20

0 2 4 6 8
t
0.0
0.5
1.0
1.5
2.0
log(t)/n
y=log(t)/2
y=log(t)/10
y=log(t)
Figure 1.5: The space of strictly concave functions is not complete
This determines k0
as a continuously differentiable function of k by the implicit function
theorem. Note that 0 < dk0
/dk < F1(k). Therefore, V n+1
(k) is a strictly increasing,
strictly concave C2
function too since V n+1
1 (k) = U1(F(k)−k0
)F1(k). The limit of such
a sequence must be a continuous function. (It is does not have to be a C2
function)
Differentiability
Lemma 18 Let X ⊆ Rl
be a convex set, V : X → R be a concave function. Pick
an x0 ∈intX and let D be a neighborhood of x0. If there is a concave, differentiable
function W : D → R with W(x0) = V (x0) and W(x) ≤ V (x) for all x ∈ D then V is
differentiable at x0 and
Vi(x0) = Wi(x0), for i = 1, 2, ..., l.
Proof. See Figure 5
21

V
W
Figure 1.6: Differentiability of V
.
Theorem 19 (Benveniste and Scheinkman) Suppose that K is a convex set and that
U and F are strictly concave C1
functions. Let V : K → R in line with P(3) and
denote the decision rule associated with this problem by k0
= G(k). Pick k0 ∈intK and
assume that 0 < G(k0) < F(k0). Then V (k) is continuously differentiable at k0 with its
derivative given by
V1 = U1(F(k0) − G(k0))F1(k0).
Proof. Clearly, there exists some neighborhood D of k0 such that 0 < G(k0) < F(k)
for all k ∈ D. Define W on D by
W(k) = U(F(k) − G(k0)) + βV (G(k0)).
22

Now, W is concave and diﬀerentiable since U and F are. Furthermore, it follows that
W(k) ≤ max
k0
{U(F(k) − k0
) + βV (k0
)} = V (k),
with this inequality holding strictly at k = k0. The results then follows immediately
from the above lemma.
1.3 Problems
1. Consider the problem described by
V (k) = max
x1,x2,...,xn
F(x1, x2, ...xn; k).
Let F be a C2
function. Presume that a maximum exists. Show that Vn+1(k) =
Fn+1(x1, x2, ...xn; k).
2. Consider the following dynamic programming problem
V (ki, εr) = max
c,k0
j∈K
{U(c) + β
nP
j=1
πrsV (k0
j, εs)},
subject to
c + i = F(ki),
and
k0
j = (1 − δ)ki + iεr. (1.4)
23

Let U be a bounded, strictly increasing, strictly concave, continuous function.
Suppose that 0 < β < 1. The bounded positive random variable ε follows a
m-point Markov process. In particular, ε is drawn from the discrete set E ≡
{ε1, ε2,..., εm} according to the probability distribution speciﬁed by πrs = Pr{ε0
=
εr|ε = εs}, where 0 ≤ πrs ≤ 1, and
Pm
j=1 πrs = 1. Furthermore, suppose that
k ∈ K = {k1, k2, ..., kn}.
(a) Show that there exists a V that solves the above Bellman equation.
3. Show that the space of increasing functions with the uniform metric is complete.
4. Show that the space of concave function with the uniform metric is complete.
5. (Aiyagari, 1994) Consider the following dynamic programming problem
V (a, zi) = max
c,a0
{U(c) + β
nP
j=1
πijV (a0
, zj)},
subject to
c + a0
= zi + (1 + r)a,
and
a0
≥ 0. (1.5)
Let U be a bounded, strictly increasing, strictly concave, continuously diﬀeren-
tiable function. Suppose that 0 < r < β < 1. The bounded positive random
24

variable z follows a n-point Markov process. In particular, z is drawn from the
discrete set Z ≡ {z1, z2,..., zn} according to the probability distribution specified
by πij = Pr{z0
= zj|z = zi}, where 0 ≤ πij ≤ 1, and
Pn
j=1 πij = 1.
(a) Is the function V (a, z) continuously differentiable in a, for all a > 0, when-
ever a0
> 0?
(b) Is the function V (a, z) continuously differentiable in a, for all a > 0, when
a0
= 0. What is the issue here?
6. Capacity Choice (Harris, 1987): Here is a problem facing a monopolist. He faces
a demand curve each period given by
q = (1 − p),
That is, if the price is p he can sell the quantity q. Production is costless but at
each period in time the monopolist faces a capacity constraint,
q ≤ c,
where c is the upper bound on his production. Capacity can be increased with a
one period time delay according to the cost function
(c0
− c)2
,
where c0
≥ c is the level of capacity that the monopolist chooses for next period.
The monopolist faces the time-invariant gross interest rate r.
25

(a) Formulate the monopolist dynamic programming problem.
(b) Prove that the solution is given by
V (c) =



α, for c ≥ 1/2,
δ + γc(1 − c), for c ≤ 1/2,
where α and δ are some constants and
γ =
2(1/r) − 1 +
p
1 + 4(1/r)2
2(1/r)
,
and that the optimal policy is
c0
= c +



0, for c ≥ 1/2,
(1/r)γ(1/2 − c)/(1 + γ/r), for c ≤ 1/2.
Also solve for α and δ.
7. The Replacement Problem: Imagine a lot with an age-j building on it. Denote
this amount of capital in this building by k(j). The per period profit from the
lot with an age-j building on it is k(j)α
. Time flows continuously and the capital
stock depreciates with age according to the law of motion dk(j)/dj = δk(j). At
any point in time the owner is free to tear down his existing building and replace
it with a new one. The size of a new building is fixed at k. The interest rate is
always r.
(a) Write out the dynamic programming problem facing the owner.
26

(b) Does a continuous value function solving this problem exist?
(c) Is it concave?
(d) Is it diﬀerentiable?
8. Stochastic Goldmining (Bellman, 1957): Consider the problem of an entrepreneur
who owns two goldmines, Anaconda and Bonanza. Ananconda has x units of gold
in its bowls and Bonanza has y units, both measured in dollars. The entrepreneur
has a goldmining machine. The machine has the following properties: If it is used
in Ananconda in a period it will reap the fraction ra of the gold in the mine with
probability pa. With probability 1 − pa the machine breaks down, mines no gold,
and can never be used again. Getting another machine is impractical. Likewise,
in any given period the machine can be used in Bonanza. There it may mine the
fraction rb of the gold with probability pb and break down with probability 1−pb.
The entrepreneur’s discount factor is β.
(a) Let V (x, y) be the value of the mine. Write out the entrepreneur’s dynamic
programming problem
(b) Prove that V is increasing in x and y.
(c) Derive the locus of x and y combinations that yield the same payoﬀ. What
does this say about how the value function can be written?
27

9. Let X : R+ → R+ and Y : R+ → R+ be bounded, continuous, strictly concave
functions, H : R+ → [0, 1] be an increasing, C1
function, and 0 < β < 1.
Additionally, presume that X(0) > Y (0), X(∞) < Y (∞), and Y1(ε) > X1(ε).
Now, consider the bivariate functional equation shown described below:
S(ε) = max
T
{X(ε) + β
Z T
S(ε)H1(ε)dε + β
Z
T
W(ε)H1(ε)dε}
and
W(ε) = max
T
{Y (ε) + β
Z T
S(ε)H1(ε)dε + β
Z
T
W(ε)H1(ε)dε}.
Note that the function S depends on W while the function W depends on S.
Given this simultaneity problem, can Blackwell’s sufficient condition be used to
prove that this mapping defines a contraction? What, if anything, can be said
about S and W?
10. An economist, Noah Nomatt, is analyzing the following problem
max
t≥0
{ln(ψ − pe−rt
) + κ
e−βt
β
}.
Here t is that date that a representative consumer may adopt a new consumer
durable, p is the price of the good, β is the agent’s rate of time preference, r
is the market interest rate, ψ is the consumer’s permanent income, and κ is
the additional utility the agent will realize from the new good. The first-order
28

condition associated with this problem is
rpe−(r−β)t
= κ(ψ − pe−rt
) (for t > 0).
After displacing this ﬁrst-order condition, he ﬁnds that
dt
dψ
= −
κ
r(r − β)pe−(r−β)t + κrpe−rt
.
When r < β the denominator may be negative, so he concludes that it may tran-
spire that dt/dψ > 0; that is, an increase income may lead to the new technology
being adopted later.
29

Chapter 2
Business Cycle Analysis
2.1 Introduction
2.1.1 Real Business Cycle Models – Kydland and Prescott
(1982) and Long and Plosser (1983)
In the early 1980s one of the most important papers in modern macroeconomics was
published. This was Kydland and Prescott’s “Time to Build and Aggregate Fluctua-
tions.” Along with Long and Plosser’s “Real Business Cycle Theory”, this work demon-
stated how business cycles could be generated from the neoclassical model. While this
was an important contribution in and of itself, the Kydland and Prescott paper did two
additional things. It demonstrated how dynamic stochastic economies could be solved
30

on a computer using modern techniques from operations research. And, it suggested
a procedure for matching models with the data. These two things re-engineered the
macroeconomic landscape.
The economics underlying real business cycle theory is as follows:
• Let y = zF(k,l). Cycles are generated via exogenous contemporaneous produc-
tivity shocks, z.
• Dynamic optimizing behavior on the part of agents in the economy implies that
both consumption and investment react positively to supply shocks.
— Output, zF(k,l), increases.
— Consumption smoothing implies both current and future consumption should
rise. Additionally, the marginal product of capital, zF1(k,l), rises. This
should stimulate investment too.
• Labor productivity, zF2(k,l), is directly aﬀected. Results in employment and
measures of labor productivity being procyclical.
• Capital accumulation provides a channel of persistence, even if the technology
shocks are white noise. Note investment is reacting to a supply shock.
• Conclusion: Productivity shocks from a neoclassical perspective can generate
the observable co-movements in macroeconomic variables and the persistence of
31

economic fluctuations.
• Criticism: Do negative productivity shocks occur in reality? Oil shocks and
harvest failures are two obvious examples, but what is another one?
2.1.2 Keynesian Investment Multiplier Model
• “Animal spirits” cause investment fluctuations which generate the business cycle.
— Marginal efficiency of investment shifts exogenously affecting investment
demand and hence output, y = c + i, through the investment multiplier-
accelerator mechanism.
• Quintessential Case: Change in the expected future marginal productivity of
capital which does not affect the current production function. A positive shock
in the neoclassical growth model will cause:
— Investment to increase.1
— The real interest to rise to clear the goals market.
— Current consumption to fall and labor effort to rise (and hence leisure to
fall.)
— The marginal product of labor to fall.
1
If investment falls due to a strong income effect, then it is easy to show that consumption will rise
but labor effort will fall.
32

2.1.3 Business Cycles with Investment-Speciﬁc Technological
Progress
• Adopts the Keynesian view that direct shocks to investment are important for
business ﬂuctuations
• Incorporates them into a neoclassical framework where the rate of capacity uti-
lization is endogenous. Involves Keynes’ (1936) notion of ‘user cost’ in production.
33

2.2 The Economic Environment
Production Function
y = F(kh, l)
Here h represents the rate of capacity utilization, or the rate at which capital is utilized.
Law of Motion for Capital
k0
= k[1 − δ((h)] + i(1 + ε),
which implies
y0
= F ({k[1 − δ(h)] + i(1 + ε)}h0
, l0
) .
Observe that ε is a shock to the marginal eﬃciency of investment spending. A extra
unit of investment spending today can purchase more units of new capital for tomorrow.
Now, 1/(1+ε) can be thought of as the relative price of new capital in terms of forgone
consumption. That is, it costs 1/(1 + ε) units of consumption to purchase an extra
unit of capital. There is a cost of utilizing your capital today in terms of increased
depreciation. Assume that δ1, δ2 > 0.
Technology Shock
34

ε0
∼ Φ(ε0
|ε).
Tastes
E[
∞X
t=0
βt
U(ct, lt)],
with
U(c, l) = U(c − G(l)).
Implication:
∂c
∂l
¯
¯
¯
¯
U
=
U2(c, l)
U1(c, l)
= G1(l).
– Intertemporal substitution eﬀect on labor supply is eliminated. Early crit-
ics of real business cycle theory complained that the models required implausible high
elasticities of intertemporal substitution. This utility function implies that the supply
of labor in the current period can be written solely as a function of the current wage.
It is very convenient to use.
Resource Constraint
y = c + i.
35

2.2.1 The Representative Agent’s Optimization Problem
V (k, ε) = max
c,k0,h,l
[U(c, l) + β
Z
Q
V (k0
, ε0
)dΦ(ε0
|ε) P(1)
s.t.
c = F(kh, l) −
k0
(1 + ε)
+
k[1 − δ(h)]
(1 + ε)
| {z }
i/(1+ε)
.
The ﬁrst-order conditions are:
U1(c − G(l))/(1 + ε) = β
R
Q
V1(k0
,ε0
)dΦ(ε0
|ε)
= β
R
Q
U1(c0
− G(l0
))[F1(k0
h0
,l0
)h0
+ (1−δ(h0))
1+ε0 ]dΦ(ε0
|ε),
F1(kh, l) =
δ1(h)
(1 + ε)
, (2.1)
F2(kh, l) = G1(l). (2.2)
Equation (2.1) is the eﬃciency condition regulating capacity utilization. The righthand
side portrays Keynes’ notion of the user cost of capital. Suppose capacity utilization
is increased by a unit. This results in old capital depreciating by δ1(h). But a unit of
capital can be replaced at a cost of 1/(1 + ε) in terms of current consumption. Keynes
said (1936, pg. 69-70)
36

“User cost constitutes the link between the present and the future. For in
deciding the scale of his production an entrepreneur has to exercise a choice
between using up his equipment now or preserving it to be used later on ...
.”
2.2.2 Impact Eﬀect of Investment Shocks
Capacity Utilization and Labor Eﬀort
Consider the impact of a transitory shift in the current technology factor ε. Observe
that (2.1) and (2.2) represent a system of two equations in two unknowns. Using
Crammer’s rule yields
dh
dε
> 0,
dl
dε
> 0.
Interpretation
– An increase in ε reduces the cost of capacity utilization and induces a
higher h.
– Since F12 > 0 labor’s marginal product increases, resulting in a higher level
of employment.
Now,
dw
dε
=
dF2(kh, l)
dε
=
dF2(kh/l, 1)
dε
> 0,
37

if and only if
d(kh/l)
dε
> 0.
Hence,
dAPL
dε
=
d[F(kh,l)/l]
dε
=
d[F(kh/l,1)]
dε
= F1(kh/l,1)
dkh/l
dε
> 0.
2.1 Show that dw/dε > 0.
Capital accumulation
dk0
dε
=
−U1(·)
[U11(·) + (1 + ε)2β
R
Q
V11(·0)dΦ]
| {z }
Substitution Effect
+ i
U11(·)
[U11(·) + (1 + ε)2β
R
Q
V11(·0)dΦ]
| {z }
Income Effect
> 0.
Interpretation
• Substitution Effect – New capital is more productive, so invest more.
• Income Effect – More resources available for capital accumulation and consump-
tion, so invest more.
Alternative Specification of the Capital Evolution Equation (Capital-Augmenting
Technological Change).
Let
38

k0
= k[1 − δ(h)](1 + ε) + i(1 + ε)
⇒ c = F(kh,l) −
k0
1 + ε
+ [1 − δ(h)]
The efficiency condition governing the use of capital services now becomes
F1(kh, l) = δ1(h).
Observe that the techology shock no longer will effect h or l. Old capital can no longer
be replaced with less expensive new capital.
2.2.3 Dynamic Effects of Investment Shocks
The effect of a technology shock propagates into the next period via its effect on k0
.
dh0
dk0
< 0,
dl0
dk0
> 0,
d(k0
h0
)
dk0
> 0,
39

dk00
dk0
> 0.
2.3 Applied General Equilibrium Analysis
2.3.1 Sample Economy
Tastes and technology:
U(c, l) =
1
1 − γ
[(c −
l1+θ
1 + θ
)1−γ
] ,
F(kh, l) = A(kh)α
l1−α
,
δ(h) =
1
ω
hω
.
Technology shock
ε ∈ E = {eξ1 − 1, eξ2 − 1},
with
Pr[ε0
= eξs − 1 | ε = eξr
− 1] ≡ πrs for r, s = 1, 2.
40

π11π11
π12
π22
π22
1
2
1
2
π21
Figure 2.1: Two-Point Markov Chain
Figure 1 illustrates the situation.
The long-run (or unconditional) distribution function for technology shock
Pr[ε = eξs
− 1] ≡ φ∗
s =
πrs
π12 + π21
for r, s = 1, 2 and r 6= s.
2.3.2 Discrete-State-Space Dynamic Programming Problem
Finally, the capital stock in each period is constrained to be an element of the ﬁnite
time-invariant set, K, where
K = {k1, ...., kn}.
Representative Agent’s Dynamic Programming Problem
41

V (ki, ξr) = max
c,k0∈K
{
1
1 − γ
(c −
∧
l
1+θ
1 + θ
)1−γ
+ β
2X
s=1
πrsV (k0
,ξs)}, P(1)
subject to
c = A(ki,
∧
h)α
∧
l
1−α
− k0
e−ξr + ki(1 −
∧
h
ω
ω
)e−ξr , (2.3)
where
∧
h,
∧
l = arg max[A(ki,
∧
h)α
∧
l
1−α
− ki(1 −
∧
h
ω
ω
)e−ξr −
∧
l
1+θ
1 + θ
]. (2.4)
Observe that V : K ×E → R is merely a list of 2n values, one for each (ki, ξr) ∈ K ×E.
So, how can a solution V be obtained? The material in Chapter 1 suggests
the following algorithm.
1. Enter iteration j+1 with a guess for the V on the righthand side of P(1). Call this
guess V j
, it’s merely a list of 2n values. Note that contraction mapping theory
states that the initial guess for V is irrelevant. You could set V 0
= 0.
2. Next, compute the solution to the righthand side of P(1). Denote this solution
by V j+1
. To do this, note from (2.4) that
∧
h and
∧
l can be expressed as functions
of ki and ξr while (2.3) implies that c is a function of ki, ξr, and k0
. Hence, let
Mj
(ki, ξr, k0
) = {
1
1 − γ
(c −
∧
l
1+θ
1 + θ
)1−γ
+ β
2X
s=1
πrsV j
(k0
,ξs)}.
42

It’s easy to see that V j+1
is given by
V j+1
(k1, ξ1) = max
n elements in set
z }| {
{Mj
(k1, ξ1, k1), Mj
(k1, ξ1, k2), · · · , Mj
(k1, ξ1, kn)},
V j+1
(k2, ξ1) = max{Mj
(k2, ξ1, k1), Mj
(k2, ξ1, k2), · · · , Mj
(k2, ξ1, kn)},
...
...
V j+1
(kn, ξ2) = max{Mj
(kn, ξ2, k1), Mj
(ki, ξr, k2), · · · , Mj
(ki, ξr, kn)}.
This constitutes a revised guess for V .
3. Check whether |V j+1
− V j
| is suﬃciently small. If so, stop. If not, go back to
Step 1. Essentially, equation P(1) deﬁnes an operator T such that V j+1
= TV j
.
The contraction mapping theorem implies that limj→∞ V j
= V .
Decision Rule for Capital
k0
= K0
(ki, ξr) ∈ K.
2.3.3 Construction of the Markov Chain
Pr[k0
= kj | k = ki, ξ = ξr] =



1, for some j,
0, for the rest.
Trivially, then
nP
j=1
Pr[k0
= kj | k = ki, ξ = ξr] = 1 for all (k, ξ) ∈ K × E.
43

Transition Probabilities
Deﬁne the transition probability between (k, ξ) pairs by
pir,js = Pr[k0
= kj, ξ0
= ξs|k = ki, ξ = ξr] = Pr[k0
= kj | k = ki, ξ = ξr]πrs.
Now, load these transition probabilities into a matrix:
P = [pir,js ]
| {z }
2n×2n
.Given some initial probability distribution ρ0
|{z}
1×2n
over the state space K×E, next period’s
probability distribution is given will be given by
ρ1
= ρ0
P,
or
(ρ1
11, ..., ρ1
n2) = (ρ0
11, ..., ρ0
n2)








p11,11 ... p11,n2
...
...
pn2,11 ... pn2,n2








= (
2P
r=1
nP
i=1
ρ0
irpir,11, · · · ,
2P
r=1
nP
i=1
ρ0
irpir,n2).
It easy to see that the m-period-ahead probability distribution states will be given by
ρm
= ρm−1
P = ρm−2
P2
= · · · = ρ0
Pm
.
Stationary Distribution
44

Let P2n
represent the space of 2n-dimensional probability vectors. Think
about the transition matrix as defining an operator P : P2n
→ P2n
. The long-run or
stationary distribution, ρ∗
, will be given by the fixed point to this operator or
ρ∗
= ρ∗
P. (2.5)
One might guess that the existence of a unique, invariant long-run distribution might
be related to whether or not the operator P is a contraction mapping.
Lemma 20 limm→∞ ρ0
Pm
= ρ∗
for all ρ0
∈ P2n
if and only if for some m it occur
that Pm
defines a contraction on P2n
.
Proof. See Stokey, Lucas with Prescott (1987), chapter 11.
A sufficient condition for this to occur is given next — again see Stokey, Lucas with
Prescott (1987), chapter 11.
Condition 21 For some j there exists a m such that mini pm
ij > 0.
Note that this condition states that at interation m it is possible to get into state j
from any other state. This condition rules out the examples like the ones below.
Example 1
Let P =




1 0
0 1



. Check that with this transition matrix, if you
start out in state 1 you’ll stay there, while the same is true for
45

state 2. Pick any conjectured ρ∗
. The above fact implies that
limm→∞ ρ0
Pm
6= ρ∗
for all ρ0
∈ P2
. Here there are diﬀerent ρ∗
associated with diﬀerent starting values, ρ0
— try ρ0
= [1, 0] and
ρ0
= [0, 1].
Example 2
Let P =




0 1
1 0



. Check that with this transition matrix, if you
start out in state 1 you’ll switch to state 2 and vice versa. Here, it is
easy to deduce that limm→∞ ρ0
Pm
doesn’t exit for certain ρ0
— try
ρ0
= [1, 0].
Computation of Moments
Once the long run distribution, ρ∗
, has been obtained it is easy to compute
any moment of interest.
E[y] =
2P
r=1
nP
i=1
ρ∗
irY (ki, ξr),
E[cy] =
2P
r=1
nP
i=1
ρ∗
irC(ki, ξr)Y (ki,ξr),
E[y0
y] =
2P
s=1
nP
j=1
2P
r=1
nP
i=1
pir,jsρ∗
irY (k0
i, ξ0
s)Y (ki, ξr).
46

2.3.4 Calibration Procedure
Experimental Design
1. Pin down the parameter values for tastes, technology, and the stochastic structure
of the model by using
(a) a priori information from the literature or,
(b) so that various ﬁrst and second moments from the model match their coun-
terparts in the data.
2. The model is ‘tested’ so to speak, by comparing the standard deviations, corre-
lations with output, and serial correlations of the other variables (consumption,
investment, hours, and productivity) with the corresponding statistics in the US
data.
Taste and Technology Parameters
1. β = 0.96 — Kydland and Prescott(1982).
2. α = 0.29 — Capital’s share of income, as meassure by the National Income and
Product Accounts (NIPA).
3. 1/θ = 1.7 — Intertemporal elasticity of labor supply. MaCurdy (1981) estimated
this to be about 0.3 for adult males. Heckman and MaCurdy (1980,1982) found
the corresponding value for females to be about 2.2.
47

4. γ = 1.0 and 2.0 — Coefficient of relative risk aversion. A controversial parameter.
The first estimate is close to what Hansen and Singleton (1983) found, the second
is in accord with what Friend and Blume (1975) discovered.
5. ω = 1.42 — Elasticity of the depreciation function. This number was picked
because it implies a steady-state depreciation rate of 10%.
Stochastic Process Parameters
Let
π11 = π22 ≡ π and ξ1 = −ξ2 = Ξ. (2.6)
Then
σ = Ξ (standard deviation),
λ = 2π − 1 (autocorrelation coefficient).
(2.7)
The parameters σ and λ are picked so as to make the model generate the same standard
deviation and first-order serial correlation for output as is observed in the data.
2.3.5 Simulation Results
Cases: (Shocks Calibrated to Match US Output Fluctuations)
48

Case 1: γ = 1 σ = 0.047 λ = 0.43
Case 2: γ = 2 σ = 0.051 λ = 0.44
– Size of ﬂuctuations in shocks relative to output seems about the same as
Kydland and Prescott (1982) and Hansen (1985) 5.15
3.50
= 1.5.
– Required amount of persistence in shock much less than in Kydland and
Prescott (1982) and Hansen (1985).
Stylized Facts
1. (Volatility) Investment much more volatile than output, consumption less. The
model qualitatively mimics this behavior but quantitatively exaggerates it.
2. (Correlations) Hours has the highest correlation with output, but the other vari-
ables particularly consumption come fairly close. The procyclical behavior of
consumption, however, is critically dependent on the value of γ. When γ = 1 the
correlation of consumption with output is only 0.50. For γ = 2, this correlation
increases to 0.79 close to the 0.74 value with actual data. Also, increasing γ from
1 to 2 which corresponds to reducing the amount of intertemporal substitution ]
lowers the standard deviation of investment from 14.7% to 11.6%, closer to the
actual data value of 10.5%. Higher values of γ could do even better. Overall, the
best ﬁt for the model corresponds to γ = 2.
49

3. (Persistence) In the data, consumption and productivity have the highest auto-
correlations, and investment the lowest. In the sample economy consumption also
has the highest autocorrelations productivity the second, and investment the low-
est. the model displays a tendency though to overemphasize, though, the degree
of persistence in investment spending.
U.S. Data Model
Var. S.D. Corr Auto S.D. Corr. Auto
Output 3.5 1.00 .66 3.5 1.00 .66
Cons. 2.5 .74 .72 2.2 .79 .94
Inv. 10.5 .68 .25 11.6 .90 .50
Hours 2.1 .81 .39 2.2 1.00 .66
Prod. 2.2 .82 .77 1.3 1.00 .66
Util. 5.6 0.61 .52
2.3.6 The Importance of Capacity Utilization
So what role does capacity utilization play in the model’s transmission mechanism?
Figure 2 tells the story. Imagine that a positive investment-speciﬁc technology shock
hits the economy. The productivity of new capital goods jumps up. Investment should
rise (say from i to i0
). But, at the old wage rate, w, this will cause the consumer/worker’s
budget constraint to drop down from wl + rk − i to wl + rk − i0
. Provided that
consumption and leisure are normal goods this will cause an increase in work eﬀort but
50

Figure 2.2: The Eﬀect of Capacity Utilization on Consumption
a fall in consumption. Now, with capacity utilization the wage rate, w, increases to
w0
. This happens because: (i) the rate of capacity utilization increases and (ii), capital
services and labor are Edgeworth-Pareto complements in the production function. The
budget line rotates up from wl + rk − i0
to w0
l + rk − i0
, which permits consumption to
rise.
2.4 Conclusions
• Addressed the macroeconomic eﬀects of direct shocks to investment.
• A variable capacity utilization rate may be important for the understanding
of business cycles. The modelling apparatus employed provides a mechanism
51

through which investment shocks generate a higher utilization rate of the existing
capital stock, and hence higher labor demand. This mechanism stands in contrast
to the intertemporal substitution effect which works on labor supply.
2.5 Problems
2.1 Let F be a constant-returns-to-scale production function. Show that F12 ≥ 0 and
F11F22 −F2
12 = 0. Suppose that F is strictly concave in each of its two arguments.
Demonstrate that F12 > 0.
2.2 Suppose that Φ satisfies the Feller property: that is, for any continuous function
H(·, ε0
) it transpires that the function
R
H(·, ε0
)dΦ(ε0
|ε) is continuous too. Does
the operator defined by P(1) map C2
functions into C2
functions?
2.1 Show that O(k, l, ε) = max
h
{F(kh, l) + k[1 − δ(h)]} is concave in k and l. What
does this imply about the value function?
2.2 Show that dh/dε > 0 and dl/dε > 0.
2.3 It was never established that V is continuously twice differentiable. Let k0
=
K0
(k, ε) represent the decision rule for investment. Is it continuous? Can you
argue by induction that it must be nondecreasing in ε?
2.4 Consider the system of equation given by (2.5). Is (I − P) invertible? What is
52

the import of this fact? Suppose that the last equation of the system (2.5) is
replaced with ρ∗
n2 = 1 −
P
i6=n,r6=2 ρ∗
ir. Is there a direct way to compute ρ∗
?
2.5 Show that (i) φ∗
s = πrs/[ π12+π21] and (ii), that given (2.6) the standard deviation
and autocorrelation coeﬃcient are given by (2.7).
2.6 How long can a recession (the low technology state) be expected to last for?
53

Chapter 3
Economic Growth
3.1 Solow (1957) Growth Accounting
3.1.1 Goal
• To present an accounting framework where the contributions to economic growth
of capital accumulation and technological progress can be calculated.
3.1.2 Findings
1. Technological progress is neutral.
2. Total factor productivity grew at about 1.5 percent a year (over the 1909-1949).
3. 87% percent of growth was due to technological progress, the rest to capital
54

accumulation.
4. Output shows diminishing returns in the capital-to-labor ratio.
5. Cobb-Douglas production function works well.
3.1.3 Methodology
Consider a aggregate production function of the form
Y = F(K, L, t). (3.1)
In the case of neutral technological progress this would simplify to
Y = zf(K, L). (3.2)
Now, if the production exhibited constant returns to scale then
Y/L
|{z}
y
= zf(K/L
|{z}
k
, 1).
Taking the derivative with respect to time yields
1
y
dy
dt
=
1
z
dz
dt
+
z
y
∂f
∂k
dk
dt
.
Under perfect competition capital will get paid its marginal product. Thus,
the above expression can be rewritten as
1
y
dy
dt
=
1
z
dz
dt
+ wk
1
k
dk
dt
, (3.3)
55

where wk = z(∂f/∂k)k
y
represents capital’s share of income. Given the constant-returns-
to-scale assumption there are zero proﬁts, so that by Euler’s theorem wk = 1 − wl,
where wl is labors’ share of income. Hence,
1
z
dz
dt
=
1
y
dy
dt
− (1 − wl)
1
k
dk
dt
. (3.4)
Therefore, the rate of neutral technological progress,1
z
dz
dt
, can be computed by using
data on the rate of output growth, 1
y
dy
dt
, the capital stock’s growth rate, 1
k
dk
dt
, and
labors’ share of income, wl.
For the general production function (3.1), equation (3.3) reads
1
y
dy
dt
=
1
F
∂F
∂t
+ wk
1
k
dk
dt
.
If 1
F
∂F
∂t
is independent of K and L then F must have the form displayed by (3.2).
3.1.4 Results
1. z shows steady growth over the 1909-1949 sample period – Figure 3.1.
2. ∆F
F
and k show little relationship which Solow interprets as evidence that tech-
nological progress is neutral — Figure 3.2.
3. Real GNP per man hour rose from $.623 to $1.275 over the period. z rose from 1
to 1.809 over the same period. Thus, y/z rose from $.623 to $.704. Therefore of the
56

1910 1920 1930 1940 1950
Year
1.0
1.2
1.4
1.6
1.8
z
Figure 3.1: Measure of Neutral Technological Progress, z
.65 change in GNP approximately .08 (or 12%) was due to capital accumulation
the remainding (88%) being due to technological progress.
4. A plot of y/z against k shows evidence of decreasing returns – see Figure 3.3. A
Cobb-Douglas production function ﬁts the data well:
ln y = −.729 + .353 ln k, r = .9996,
where observe that the coeﬃcient on capital is probably very close to mean value
of capital’s share of income for the period.
57

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
k
-0.10
-0.05
0.00
0.05
∆z/z
Figure 3.2: The Relationship Between (1/z)dz/dt and k — technological progress is
neutral.
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
k
0.61
0.65
0.69
0.73
y/z=f(k)
Figure 3.3: The Production Function f(k) — diminishing returns in k.
58

1950 1960 1970 1980 1990
Year
0.5
1.0
1.5
2.0
2.5
RelativePriceofEquipment
0.04
0.05
0.06
0.07
Investment-to-GNPRatio
Price Quantity
Figure 3.4: Investment in New Equipment
3.2 Solow (1960) — Growth Accounting with Investment-
Specific Technological Progress
3.2.1 Some Motivating Observations
• The relative price of equipment has declined secularly – Figure 4.1.
• The equipment-to-GNP ratio has increased secularly – Figure 4.1.
• The detrended price of equipment and equipment investment are negatively cor-
related.
Intepretation A significant amount of technological progress is embodied in the pro-
duction of new capital goods; i.e., has been specific to the investment-goods sector.
59

Question
• What is the contribution of investment-speciﬁc technological progress to postwar
U.S. growth?
Answer: 60%.
3.2.2 The Model
Tastes
∞X
t=0
βt
U(ct, lt), (3.5)
with
U(c, l) = θ ln c + (1 − θ) ln(1 − l), 0 < θ < 1. (3.6)
Technology
y = zF(ke, ks, l) = zke
αe
kαs
s l1−αe−αs
, 0 < αe, αs < 1. (3.7)
Structures
k0
s = (1 − δs)ks + is, where 0 < δs < 1. (3.8)
Equipment
k0
e = (1 − δe)ke + ieq. (3.9)
60

Shocks
z0
= Z(z),
and
q0
= Q(q).
Government
τ = τk(reke + rsks) + τlwl. (3.10)
Resource Constraint
y = c + ie + is. (3.11)
3.2.3 Competitive Equilibrium
Household’s Problem
V (ke, ks; Ke, Ks, z, q) = max
c,k0
e,k0
s,l
{U(c, l) + β V (k0
e, k0
s; s0
, z0
, q0
)} P(1)
subject to
c + k0
e/q + k0
s = (1 − τk)[Re(λ)ke + Rs(λ)ks] + (1 − τ`)W(λ)l
+(1 − δe)ke/q + (1 − δs)ks + T(λ)
61

and (Ke, Ks) = K(Ke, Ks, z, q). Here λ = (Ke, Ks, z, q) represents the aggregate state
of the world. In equilibrium all factor prices and transfer payments will be functions of
the aggregate state of the world.
Firms’s Problem
max
eke,eks,el
πy = zF(eke, eks,el) − Re(λ)eke − Rs(λ)eks − W(λ)el. P(2)
Plus Market Clearing Conditions such as (3.11) and
eke = ke, eks = ks and el = l.
3.2.4 Balanced Growth
Assumption: qt = γt
q, zt = γt
z.
– l constant
– y = c + ie + is =⇒ y, c, ie, is all grow at common rate g.
– k0
s/ks = (1 − δs) + is/ks =⇒ ks also grows at g.
62

– k0
e/ke = (1 − δe) + ieq/ke =⇒ ke grows at rate gγq.
y = zke
αe
kαs
s l1−αe−αs
=⇒ g = γz(gγq)αe
gαs
Along a balanced growth path, the stock of equipment grows faster than GNP.
g = output growth = γ
1
1−αe−αs
z
| {z }
neutral
γ
αe
1−αe−αs
q
| {z }
investment-speciﬁc
, (3.12)
and
growth in stock of equipment = γ
1
1−αe−αs
z γ
1−αs
1−αe−αs
q . (3.13)
• Equipment-to-GNP ratio rises.
Equipment Euler Equation
zF1(ke, ks, l) = αe
y
hke
– declines at rate γq.
Thus, the marginal product of capital continuously declines.
zF1(ke, ks, l)q – remains constant.
While this is true, the price of capital 1/q continuously declining. Therefore, one unit
of forgone consumption can buy everincreasing amounts of capital. A unit of forgone
consumption returns zF1(ke, ks, l)q units of extra output next period.
63

3.2.5 Calibration
Values for the following parameters need to be determined:
Preferences: β and θ.
Technology: αe, αs, δs, δe.
Tax Rates: τk, τl.
Some parameter values are set on the basis of a priori information. Speciﬁcally,
δe = 0.124, δs = 0.056, and τl = 0.40.
Long-Run Restrictions
g = 1.0124,
l = 0.24,
αe + αs = 0.30,
ˆıe/ˆy = 0.073,
ˆıs/ˆy = 0.041,
(β/g) = 1/1.07.
This gives 6 restrictions that allow for 6 parameters values to be obtained: αe, αs, β,
θ, g, and τk.
64

Balanced Growth Equations
Let ˆxt = xt/gt
x. This transformation implies that ˆxt is stationary.
γq = (β/g)[(1 − τk)αe ˆy/ˆke + (1 − δe)] (Euler, equipment), (3.14)
1 = (β/g)[(1 − τk)αs ˆy/ˆks + (1 − δs)] (Euler, structures), (3.15)
ˆıe/ˆy = (ˆke/ˆy)[gγq − (1 − δe)] (equipment accumulation), (3.16)
ˆıs/ˆy = (ˆks/ˆy)[g − (1 − δs)] (structure accumulation), (3.17)
(1 − τ`)(1 − αe − αs)
θ(1 − l)
(1 − θ)(ˆc/ˆy)
= l (consumption/leisure choice), (3.18)
and
ˆc/ˆy + ˆıe/ˆy + ˆıs/ˆy = 1 (resource constraint). (3.19)
This represents 6 equations in 6 unknowns: ˆc/ˆy, ˆıe/ˆy, ˆıs/ˆy, ˆke/ˆy, ˆke/ˆy, and l.
65

Parameter Values
– The above constitutes a system of 12 equations in 12 unknowns. The solution
obtained for the parameter values is:
θ = .40,
αe = 0.17,
αs = 0.13,
τk = 0.44,
β = 0.95.
3.2.6 Growth Accounting
Growth accounting is based on equation (3.12). To use this equation two things are
needed: measures of γz and γq.
Measure of Investment-Specific Technological Progress
In the model the price of new equipment is given by the relationship
p = 1/q.
Hence, investment-specific technological progress can be identified from the relative
price of equipment.
66

1969 1973 1977 1981 1985 1989 1993 1997
0
10
20
30
40
50
60
70
80
90
100
Percent of newcars
Features of New Cars
Figure 3.5: Quality Improvement in Cars: 1, power steering; 2, antilock brakes; 3, power
door locks; 4, power seats; 5, power windows; 6, sun roof; 7, air bags; 8, windshield
wiper delay; 9, tinted glass; 10, air conditioning; 11 adjustable steering column; 12,
cruise control; 13, remote control sideview mirror.
The Importance of Adjusting Prices For Quality Improvement When com-
puting a price series for the new producer durables it is important to adjust for the
quality improvement in these goods that has taken place over time. Consider a car.
This is by no means a homogenous product. Figure 4.2 shows the improvement in
quality that has taken place since 1969. Getting more car for a given amount of money
is really the same as thing as a reduction in price for cars. Figure 4.3 compares the
quality adjusted price series with the standard national income and product one.
• γq is computed directly from Gordon’s price data using the relation p = 1/q –
67

RelativePriceof Equipment
Figure 3.6: The Importance of Quality Improvement
Figure 3.7. Here γq = 1.032
Measure of Residual or Neutral Technological Progress
• γz is computed using an estimate of z obtained from Solow residual accounting
– Figure 3.7. Here γz = 1.004
– An equipment series is constructed by iterating on the law of motion for
equipment
k0
e = (1 − δe)ke + ieq.
– A measure of z can then be obtained from the production function rela-
tionship (3.7)
z =
y
ke
αe kαs
s l1−αe−αs
.
68

1950 1960 1970 1980 1990
Year
0.4
0.6
0.8
1.0
1.2
Investment-Specific
1.0
1.1
1.2
1.3
1.4
Neutral
q
z
Figure 3.7: Technological Progress
3.2.7 Findings
γy = 1.0124, where γ
αe
1−αe−αs
q = 1.008, where γ
1
1−αe−αs
z = 1.006.
• q contributes 60% of output growth.
• z contributes 40% of output growth.
Comparison with Traditional Growth Accounting
There: c + qie = F(ke, l)
Here: c + ie = F(ke, l)
Both: k0
e = (1 − δe)ke + qie.
Substituting the last equation into the ﬁrst yields
69

c + k0
e = F(ke, l) + (1 − δe)ke.
• Traditional formulation fails to capture the notion of investment-speciﬁc techno-
logical progress!
3.2.8 Future Directions
Two-Sector Models
c + is = zA1kαe
1e kαs
1s l1−αe−αs
1 ,
ie = zqA2k
βe
2e k
βs
2s l
1−βe−βs
2 , (3.20)
γp = γ(αe+αs−βe−βs)/(1+αe−βe)
y γ−1/(1+αe−βe)
q . (3.21)
Equation (3.21) holds irregardless of whether there is endogenous or exogenous growth
in the model. It obtains from two facts: First, the marginal product of capital should
be equalized across sectors so that
αezA1kαe−1
1e kαs
1s l1−αe−αs
1 = pzqβeA2k
βe−1
2e k
βs
2s l
1−βe−βs
2 .
70

Diﬀerence in Capital Share Parameters Across Sectors Maximum Labor Share
Total Equipment Structures in Equipment Sector
(βe + βs) − (αe + αs) βe − αe βs − αs max(1 − βe − βs)
0.10 0.94 -0.84 0.06
0.35 0.80 -0.45 0.20
0.65 0.63 0.02 0.35
0.90 0.49 0.41 0.10
Table 3.1: Structure of Production
Second, in balanced growth k1s and k2s grow at the same rate, γy, while k1e and k2e
grow at the higher rate, γy/γp, a fact that follows from the condition that the return
on structures should be equalized across sectors. Equation (3.21) implies in the absence
of investment-specifc technological progress
(αe + αs) − (βe + βs)
1 + αe − βe
=
ln γp
ln γy
= −1.76. (3.22)
As Table 1 shows, there is no reasonable calibration of the two-sector model that can
generate the stylized facts governing postwar U.S. growth.
Human Capital Accumulation
h0
2 = H(e2)h2, with H0
> 0 and H00
≤ 0,
71

ie = zA2k
βe
2e k
βs
2s (h2l2)1−βe−βs . (3.23)
Observe that the above model is similar to the two-sector model with investment-speciﬁc
technological progress, as can be seen by setting q = h
1−βe−βs
2 .
3.2.9 Solow (1957) versus Solow (1960)
The classic Solow (1960) paper showed how a simple vintage capital model, where new
and improved capital goods come on line each period, could be aggregated into the
standard neoclassical model. He made a distinction between economic and physical
depreciation. This distinction will be developed now.
Consider the following transformed version of the model developed above.
c + ie = ezekαe
e l1−αe
, (3.24)
ek0
e = (1 − eδe)eke + ie, (3.25)
where
ek0
e = k0
e/q,
72

1 − eδe = (1 − δe)(q−1/q), (3.26)
and
ez = z(q−1)αe
.
Equations (8.3) and (3.25) appear as the conventional neoclassical growth
model with neutral technological progress. There is one important modiﬁcation. Ob-
serve that the capital stock is now measured (at market value) in terms of consumption.
The relative price of capital is always one. Under this measurement scheme, a unit of
new capital can be interpreted as being q/q−1 times more productive than a unit of old
capital. Therefore, when new capital comes on line the market value of the old capital
stock is reduced by a factor of q−1/q. Hence, eδe represents the rate of economic, as op-
posed to physical, δe, depreciation. This is an important distinction between this model
and the conventional neoclassical growth model. In a world with investment-speciﬁc
technological progress the rate of economic depreciation will exceed the rate of physical
depreciation due to the fact that this form of technological progress obsoletes the old
capital stock. For example, imagine a world where q has remained forever constant
in value and where the physical depreciation rate on capital is 10%. Now, suppose q
suddenly doubles, in a once-and-for-all manner, due to the invention of a new, more
productive, type of capital good. What happens to the worth of old capital? After
73

production in the current period, only 90% of the old capital stock will remain due to
physical depreciation. But its market will value has also now fallen, in a once-and-for-
all fashion, by 50% due to introduction of new capital goods. Thus, the old capital
stock will be worth 45% (= 90%×50%) of its old value. Therefore, the combined effect
of physical depreciation and obsolesence has been to reduce the market value of the
old capital stock by 55% (= 100% − 45%), which is the rate of economic depreciation.
In the period where the investment-specific technological progress occurred, the rate of
economic depreciation exceeds the physical one by 45 percentage points.
Clearly, either framework could be used for growth accounting. In a world
with perfect data they would yield exactly the same results. The framework adopted in
the text connects directly with Gordon’s measurement of durable goods prices. That is,
q/q−1 can be identified from Gordon’s prices series, p, using the relationship q = 1/p.
Using the framework presented in here, the rate of investment-specific technological
progress could be measured by examining the wedge between the (gross) rates of eco-
nomic and physical depreciation, or from the relationship q/q−1 = (1 − δe)/(1 − eδe).
Similarly, as with the formulation used in the text, this framework speaks a word of cau-
tion for conventional growth accounting, which normalizes the relative price of capital
to be one: failure to distinguish between economic and physical depreciation will cause
investment-specific technological progress to appear as neutral technological progress.
74

3.3 Malthus to Solow
3.3.1 Introduction
• Model the transition from a world with stagnant living standards to a world with
rising living standards.1
• Preindustrial era uses a land-intensive technology, dubbed the Malthus technol-
ogy. Land is in ﬁxed supply.
• Modern era also uses a constant-returns-to-scale technology employing just capital
labor. This is labeled the Solow technology.
• Both technologies are always available. At low levels of development it pays only
to use the Malthus technology. As the economy develops it becomes proﬁtable to
use the Solow technological. The Malthus technology fades away asymptotically.
3.3.2 Facts
England, 1275-1800
• Real wages are roughly constant for a long period of time.
• When population fell, in the Black Death, real wages rose. This is in accord with
Malthusian theory. Here wages adjust to limit the size of the population.
1
This section is based on Hansen and Prescott (forth.).
75

Figure 3.8:
• Malthusian theory predicts that population and land rents will rise and fall to-
gether. Then did over this period.
England 1800-1989
• Population growth did not lead to falling real wages as Malthusian theory predicts.
• It’s hard to see a relationship between population growth and labor productivity.
76

The Solow model doesn’t predict one.
• The value of farmland to GDP fell.
3.3.3 The Model
• A Diamond overlapping generations model.
• Malthus Technology
ym = amkφ
mnµ
ml1−φ−µ
m
• Solow Technology
ys = askθ
sn1−θ
s .
• Resource Constraint
c + k0
= ym + ys.
Firms’ Problems
Firms solve the following maximization problems:
max{amkφ
mnµ
ml1−φ−µ
m − wnm − rkkm − rllm}, (3.27)
and
max{askθ
sn1−θ
s − wns − rkks}, (3.28)
where w is the wage rate and rk and rl are the rental on capital and land.
77

Household’s Problem
Each solves the following maximization problem
max{ln c1,t + β ln c2,t+1}, (3.29)
subject to their budget constraints
c1t + kt+1 + qtlt+1 = wt,
and
c2t+1 = rk,t+1kt+1 + (rl,t+1 + qt+1)lt+1.
Here qt is the period-t price of land.
Demographics
Population growth is simply given by
nt+1 = G(c1,t)nt.
Equilibrium
1. Firms solve problems (3.27) and (3.28).
2. The households solve problem (3.29).
3. All markets clear implying
78

(a)
km + ks = n−1k,
(b)
nm + ns = n,
(c)
n−1l = 1,
(d)
nc1 + n−1c2 + nk0
= ym + ys.
Malthus versus Solow
The cost function for the Solow sector is
Cs(w, rk, ys) = min
km,lm
{rkks + wls : ys = askθ
s n1−θ
s } = a−1
s θ−θ
(1 − θ)−(1−θ)
rθ
kw1−θ
ys.
Hence, marginal cost is
a−1
s θ−θ
(1 − θ)−(1−θ)
rθ
kw1−θ
.
79

Here marginal cost is constant. The cost function for the Malthus sector (holding land
ﬁxed at unity) is
Cm(w, rk, ym) = min
km,nm
{rkm + wnm : ym = amkφ
mnµ
ml1−φ−µ
m and lm = 1}
= a−1/(φ+µ)
m [(
φ
µ
)µ/(φ+µ)
+ (
φ
µ
)−φ/(φ+µ)
]r
φ/(φ+µ)
k wµ/(φ+µ)
y1/(φ+µ)
m ,
so that marginal cost will be
1
(φ + µ)
a−1/(φ+µ)
m [(
φ
µ
)µ/(φ+µ)
+ (
φ
µ
)−φ/(φ+µ)
]r
φ/(φ+µ)
k wµ/(φ+µ)
y1/(φ+µ)−1
m .
Here, marginal cost is increasing and convex. Observe that marginal cost goes to zero
as output goes to zero.
The Solow sector will not operate when
a−1
s θ−θ
(1 − θ)−(1−θ)
rθ
kw1−θ
>
1
(φ + µ)
a−1/(φ+µ)
m [(
φ
µ
)µ/(φ+µ)
+ (
φ
µ
)−φ/(φ+µ)
]
×r
φ/(φ+µ)
k wµ/(φ+µ)
y1/(φ+µ)−1
m .
That is, the Solow sector will not operate at any aggregate output levels, ys, where the
Solow sector has higher marginal cost. Both sectors will operate only when
a−1
s θ−θ
(1 − θ)−(1−θ)
rθ
kw1−θ
=
1
(φ + µ)
a−1/(φ+µ)
m [(
φ
µ
)µ/(φ+µ)
+ (
φ
µ
)−φ/(φ+µ)
]
×r
φ/(φ+µ)
k wµ/(φ+µ)
y1/(φ+µ)−1
m .
The Malthus sector will always operate since, as was mentioned, its marginal cost goes
80

ym, ys
MarginalCost
Malthus + Solow
MCs
Malthus
ym
MCm
Figure 3.9: The Solow Adoption Point
to zero as output goes to zero. Figure 1 shows the adoption point at a given set of
factor prices.
Lemma 22 The Solow technology is not used if
as < (
rk
θ
)θ
(
w
1 − θ
)1−θ
.
Proof. In the Solow sector θ = rkks/ys and 1 − θ = wns/ys. Therefore, proﬁts can be
written as
as(
θys
rk
)θ
(
(1 − θ)ys
w
)1−θ
−
θys
ks
ks −
(1 − θ)ys
ns
ns.
For any ys > 0 proﬁts will be negative whenever
as < (
rk
θ
)θ
(
w
1 − θ
)1−θ
.
81

For alternative proof, suppose that the statement in the lemma holds and that
the Solow sector operates. The first-order conditions to (3.28) imply that
ns
ks
= (
rk
θ
)/(
w
1 − θ
).
Therefore, from the first-order condition for capital
asθ(
ks
ns
)θ−1
= asθ(
rk
θ
)1−θ
(
w
1 − θ
)θ−1
= rk.
This implies that
as = (
rk
θ
)θ
(
w
1 − θ
)1−θ
,
a contradiction.
3.3.4 Calibration
Demographics
G(ct) =



γ
1/(1−µ−φ)
m (2 − c1
c1m
) + 2( c1
c1m
− 1), for c1 < 2c1m,
2 − c1−2c1m
16c1m
, 2c1m ≤ c1 ≤ 18c1m,
1, for c1 > 18.
This function is plotted in the figure below.
Parameter Values
82

Figure 3.10: Demographics
Parameter Value Comment
γm 1.032 Growth in Malthus Era
γs 1.518 Post GDP Growth
φ 0.1 Capital’s Share of Income, Malthus
µ 0.6 Labor’s share, both technologies
θ 0.4 Capital’s Share, Solow
β 1.0 Discount factor
3.3.5 Results
• Figure shows that declining share of inputs devoted to the Malthus sector over
time.
• Figure shows the rising wages and population growth as the economy moves to
the Solow epoch.
83

Figure 3.11:
• Figure shows the declining value of land relative GDP as the Malthus sector dies
out.
3.4 Problems
1. Show that if 1
F
∂F
∂t
is independent of K and L then F must have the form displayed
by (3.2).
84

2. Household Production: Think about the following model of economic growth.
There is a representative agent whose utility is given by
∞X
t=0
βt
(ln ct + ln ht),
where ct is the consumption of market-produced goods and ht is the production
of home produced ones. Market goods are produced according to the following
production function
y = kα
t n1−α
t ,
where kt is the stock of business capital and nt is the amount of time worked
in the market. Market output can be used for consumption, ct, investment in
business capital, it, and investment in household capital, xt. That is, the resource
constraint reads
ct + it + xt = yt.
The law of motion for business capital is
kt+1 = (1 − δ)kt + qtit,
where qt represents investment-speciﬁc technological progress in the production
of business capital. Household goods are produced according to
ht = dθ
t (1 − nt)1−θ
85

where dt is the stock of household capital. It evolves according to
dt+1 = (1 − δ)dt + λtxt,
where λt represents investment-speciﬁc technological progress in the household
sector.
(a) Formulate the dynamic programming problem facing a representative house-
hold.
(b) Let γq = qt+1/qt and γλ = λt+1/λt. Conjecture what a balanced growth path
will look like.
(c) Check whether or not a balanced growth path exists?
3. Consider a representative agent who maximizes his lifetime utility as given by
∞X
t=0
βt
ct,
subject to
ct + kt+1 − kt = yt,
where ct, kt, and yt represent period-t consumption capital and output. Notice
that utility is linear. The agent inelastically supplies s units of skilled labor and
u units of unskilled labor for production. Final output can be produced by one
of following two technologies:
yt = φ[θkρ
t + (1 − θ)sρ
]α/ρ
u(1−α)
,
86

or
yt = ψt[θkρ
t + (1 − θ)sρ
]α/ρ
u1−α
.
The ﬁrst production function represents a primitive technology, the second a more
advanced technology. Capital and skilled labor are complementary in production
in the sense that ρ < 0. At each point in time the agent can pick one of the
technologies to use. Now, let ψt follow a law of motion of the form: ψt = ψ−1/gt
,
where φ > ψ − 1, φ < ψ, and g > 1.
(a) Formulate and solve the representative agent’s dynamic programming prob-
lem.
(b) Compute the skill premium, or the ratio of skilled to unskilled wage rate, as
function of kt/st and u/s.
(c) Characterize the transitional dynamics for the economy understudy. Specif-
ically, analyze how income and the skill premium behave over time. How
are the dynamics aﬀected by g? Explain how this could be a model of an
industrial revolution.
4. The Last Century: Consider the following two-period, two-sector overlapping
generations model. There are two sectors: agriculture and manufacturing. Agri-
cultural goods are used just for consumption and sell at price p. Manufacturing
87

output is used for both consumption and capital goods. Manufactured goods are
produced in line with the Cobb-Douglas production technology
oc = zkκ
c l1−κ
c ,
where oc denotes output, z is total factor productivity, and kc and lc are the
inputs of capital and labor. The production of agricultural goods is also governed
the Cobb-Douglas production function
oa = xkλ
a l1−λ
a ,
where oa is output, x is total factor productivity, and ka and la are the inputs of
capital and labor. Capital and labor ﬂows freely across the two sectors. Aggregate
capital accumulation is governed by the law of motion
k0
= δk + i,
where i represents aggregate investment, k denotes the current capital stock, and
δ adjusts for depreciation. A prime attached to a variable denotes its value next
period. Agents work and have children when young. They earn the wage w per
unit of labor supplied. Each child raised by a young agent takes τ units of time.
Old agents are retired and live solely oﬀ of their savings. Denote the gross interest
rate between today and tomorrow by r0
. A young individual has preferences of
88

the following form
α
a1−ω
− 1
1 − ω
+ ψ ln(c) + βα
a01−ω
− 1
1 − ω
+ βψ ln(c0
) + (1 + β)ξ
[(n − n)w0
]1−ρ
− 1
1 − ρ
,
where the a’s denotes his consumption of the agricultural good, the c’s represent
his consumption of a manufactured good, and n is the number of his kids. Note
that the wage earned by his oﬀspring, w0
(the quality of his child’s life), also enters
his utility function.
(a) Formulate the optimization problem facing a young agent.
(b) Formulate the model’s general equilibrium.
(c) Suppose that x and z are constant over time. Suppose that ω = 1 and that
ρ = 1. How will the model behave?
(d) Let x and z grow over time at a constant rate. Suppose that ω = 1 and that
ρ = 1. Demonstrate analytically how will the model behave?
(e) Let x and z grow over time at a constant rate. Suppose that ω = 1 and that
ρ > 1. Demonstrate analytically how will the model behave? What is the
intuition for your result?
(f) Let x and z grow over time. Suppose that ω > 1 and that ρ = 1. Demon-
strate analytically how will the model behave? What is the intuition for
your result?
89

(g) Let x and z grow over time. Suppose that ω > 1 and that ρ > 1. How will
the model behave? What is the intuition for your result? How does this
relate to U.S. economic history?
90

Chapter 4
Asset Pricing
4.1 Lucas (1978) Trees
4.1.1 Environment
Tastes
E[
∞X
t=0
βt
U(ct)],
Technology
y = δ,
where
δ0
∼ D(δ0
|δ).
91

Each agent can decide how many shares of the tree, s, that he wants to own.
Let the price of a share be p.
4.1.2 Dynamic Programming Problem
V (s, δ) = max
c,s0
{U(c) + βE[V (s0
, δ0
)]},
subject to
c + ps0
= sδ + ps.
The ﬁrst-order condition is
U1(c)p = βE[V1(s0
, δ0
)]
= β{U1(c0
)[δ0
+ p0
].
4.1.3 Equilibrium
In equilibrium
s = 1,
and
c = y = δ.
92

Therefore
U1(δ)p = βE{U1(δ0
)[δ0
+ p0
]}.
Rewrite this to get the following formula for p = P(δ0
):
P(δ) = βE{
U1(δ0
)
U1(δ)
[δ0
+ P(δ0
)]}.
This implies that
P(δ0
) = βE{
U1(δ00
)
U1(δ0
)
[P(δ00
) + δ00
]},
so that
P(δ) = βE{
U1(δ0
)
U1(δ)
[δ0
+ βE{
U1(δ00
)
U1(δ0
)
[P(δ00
) + δ00
]}]}
= E[β
U1(δ0
)
U1(δ)
δ0
+ β2 U1(δ0
)
U1(δ)
U1(δ00
)
U1(δ0
)
δ00
+ β3 U1(δ0
)
U1(δ)
U1(δ00
)
U1(δ0
)
P(δ00
)]
= E{
∞X
j=1
βj
[
j
Y
s=1
U1(δt+s)
U1(δt+s−1)
]δt+j}
= E{
∞X
j=1
βj U1(δt+j)
U1(δt)
δt+j}.
Example 1
Let U(c) = ln c so that U1(δ) = 1/δ. Then, P(δ) = [β/(1 − β)]δ.
93

4.2 Complete Markets à la Arrow, Debreu and McKen-
zie
Now, suppose that agents can trade in contingent claims. Specifically, let q(δ0
) be the
price of a claim that pays off one unit of consumption next period should the event δ0
occur next period. Let a(δ0
) be the number of such claims that the agent purchases.
The goal is to price a stock or a bond using the price of a contingent claim. That
is, contingent claims are the atoms that can be used to create various more elaborate
financial molecules.1
4.2.1 Dynamic Programming Problem
J(a(δ), δ) = max
c,a(δ0
)
{U(c) + β
Z
J(a(δ0
), δ0
)f(δ0
|δ)dδ},
subject to
c +
Z
q(δ0
)a(δ0
)dδ0
= δ + a(δ).
The first-order condition is
U1(c)q(δ0
) = βJ1(a(δ0
), δ0
)f(δ0
|δ).
1
In classic research, Lucas (1982) has shown how a variety of financial assets can be priced using
contingent claims.
94

It is easy to calculate that
J1(a(δ), δ) = U1(c),
so that the above ﬁrst-order condition can be rewritten as
U1(c)q(δ0
) = βU1(c0
)f(δ0
|δ).
4.2.2 Equilibrium
In equilibrium a(δ) = a(δ0
) = 0, which implies that c = δ0
. Hence, the price of a claim
is
q(δ0
) = Q(δ0
|δ) =
βU1(δ0
)
U1(δ)
f(δ0
|δ).
4.2.3 Asset Pricing
Imagine buying a portfolio of claims that pays oﬀ δ0
+ p0
units of consumption next
period. This portfolio would cost
Z
q(δ0
)[δ0
+ p0
]dδ0
=
Z
βU1(δ0
)
U1(δ)
[δ0
+ p0
]f(δ0
|δ)dδ0
= βE{
U1(δ0
)
U1(δ)
[δ0
+ p0
]} = p.
By arbitrage this is the cost of share. Therefore,
95

p =
Z
q(δ0
)[δ0
+ p0
]dδ0
=
Z
q(δ0
){δ0
+
p0
z }| {Z
q(δ00
)[δ00
+ p00
]dδ00
}dδ0
=
Z
q(δ0
){δ0
+
Z
q(δ00
){δ00
+
Z
q(δ000
){δ000
+ p000
}dδ000
}dδ00
}dδ0
=
Z
q(δ0
)δ0
dδ0
| {z }
one period ahead
+
Z
q(δ0
)[
Z
q(δ00
)δ00
dδ00
]dδ0
| {z }
two periods ahead
+
Z
q(δ0
)[
Z
q(δ00
)[
Z
q(δ000
)δ000
dδ000
]dδ00
]dδ0
| {z }
three periods ahead
+ ...
Here
q(δ0
)q(δ00
)q(δ000
) = Q(δ0
|δ)Q(δ00
|δ0
)Q(δ000
|δ00
),
could be thought of as the price for a claim to one unit of consumptions three periods
ahead should the event (δ0
, δ00
, δ000
) occur. The cost of purchasing the dividend stream
three periods ahead, or δ000
, would then be
R R R
Q(δ0
|δ)Q(δ00
|δ0
)Q(δ000
|δ00
)δ000
dδ00
dδ0
. This
is the third term in the above expression.
4.3 The Equity Premium: A Puzzle
4.3.1 The Problem à la Mehra and Prescott (1985)
• Facts:
— From 1889-1978 the average return on equity from the Standard and Poor
500 index as 7%.
96

— The average yield on short term debt was less than 1%.
— Can such a diﬀerential be explained in a frictionless Arrow-Debreu-McKenzie
economy?
• Finding: For the class of economies studied the average real return on equity is
at a maximum 0.4 percentage points higher than on short-term debt.
• Puzzle: To get a low risk free interest rate in a growing economy you need a high
elasticity of intertemporal substitution. To get a large equity premium, you need
a high coeﬃcient of intertemporal substitution. But one is the reciprocal of the
other.
4.3.2 The Environment
Tastes
U(c, α) =
c1−α
− 1
1 − α
, 0 < α < ∞.
Endowments –n-state Markov chain in growth rates.
y0
= x0
y,
where x ∈ {λ1, ..., λn} and
φij = Pr[xt+1 = λj|xt = λi].
97

4.3.3 Asset Pricing
pt = βE{
U1(yt+1)
U1(yt)
[ys + pt+1}. (4.1)
or
pt = E{
∞X
s=t+1
βs−t U1(ys)
U1(yt)
ys}.
Since U1(y) = y−α
then
pt = P(yt, xt) = E[
∞X
s=t+1
βs−t yα
t
yα
s
ys|xt, yt].
Note that (yt, xt) are legitimate state variables for the pricing function since ys =
yt · xt+1 · · · xs. Clearly, then P(y, x) is homogeneous of degree one in y. From (4.1)
P(y, i) = β
nX
j=1
φij(λjy)−α
[yλj + P(λjy, j)]yα
. (4.2)
Now, using the fact that P(y, i) is homogeneous of degree one in y, conjecture a solution
of the form
P(y, i) = wiy,
where the constant wi will have to be determined. Substituting this solution into (4.2)
yields
wi = β
nX
j=1
φijλ1−α
j (1 + wj), for i = 1, 2, · · · , n. (4.3)
98

Therefore,
w = βΛw + γ,
where
w =








w1
...
wn








, Λ =








φ11λ1−α
1 · · · φ1nλ1−α
n
...
...
φn1λ1−α
1 · · · φnnλ1−α
n








, γ =








β
P
j φ1jλ1−α
j
...
β
P
j φnjλ1−α
j








.
Thus,
w = [I − βΛ]−1
γ,
assuming that |I − βΛ| 6= 0.
What is the expected return from holding equity. The realized return, rij,
from moving from state (y, i) to (λjy, j) is
rij =
P(λjy, j) + λjy − P(y, i)
P(y, i)
=
λj(wj + 1)
wi
− 1.
Expected Returns, Conditional on State: The expected return on equity, conditional on
that the current state is i, is
Ri =
nX
j=1
φijrij.
Next consider the price of one-period discount bond in state i, or pf
i . Clearly,
pf
i = Pf
(c, i) =
βE[U1(λjy)]
U1(y)
=
β
Pn
j=1 φijU1(λjy)
U1(y)
= β
nX
j=1
φijλ−α
j .
99

The return on this risk free asset is
Rf
i = 1/pf
− 1.
Expected Returns, Unconditional: To calculate the expected return on either
equity or bonds one needs to know the unconditional probability of being in a particular
state, say i. This comes from the matrix equation
π = πΦ,
where π = (π1, ..., πn) and Φ = [φij]. Therefore, the unconditional return on equity and
bonds is
Re
=
X
πiRe
i ,
and
Rf
=
X
πiRf
i .
The risk premium is Re
− Rf
.
4.3.4 Findings
Two-State Markov Chain
λ1 = 1 +
Growth
z}|{
µ + δ,
100

λ2 = 1 + µ − δ|{z}
St. Dev.
,
φ11 = φ22 ≡ φ
|{z}
Autocorrelation: 2φ−1
and φ12 = φ21 = (1 − φ).
Calibration
For the U.S. economy the mean growth rate in consumption was 0.018. Its
standard deviation and autocorrelation were 0.036 and -0.14. Matching these facts
necessitated setting µ = 0.018, δ = 0.036, and φ = 0.43.
Now, clearly 0 < β < 1, and let 0 < α < 10. Let
X = {(α, β) : 0 < β < 1, 0 < α < 10, and |I − βΛ| 6= 0}.
This deﬁnes two functions, so to speak, where Rf
= R(α, β) and Re
− Rf
= P(α, β).
As can be seen the model can’t simultaneously generate an equity premium of 6.98%
and risk-free return of 0.8%.
4.3.5 Conclusions
• Within the context of a frictionless Arrow-Debreu-McKenzie world it is diﬃcult
to rationalize why the average return on equity was so high while the risk-free
return was so low.
101

Figure 4.1: Equity Premium and Risk-free Rate Combinations. Source: Mehra and
Prescott (1985), pg. 155.
4.4 Problems
1. A Technological Revolution: Imagine the following version of the Lucas tree
economy. The economy is populated by many inﬁnitely-lived identical agents,
and equally many inﬁnitely-lived trees. An agent’s lifetime utility is given by
P∞
t=0 βt
U(ct). A tree – the only source of production – yields a perfectly fore-
seen dividend, yt, each period. Now, assume that the economy has been riding
along in deterministic bliss. Then, unexpectedly, news arrives at t = 0 that a
fraction x of existing trees will die at the beginning of date T (before dividends
are paid). They will be replaced, instantaneously, by equally many new, better
trees, each yielding 1 + z units of output, where z > 0. The lifetime of each tree
102

(either T or ∞) is also announced at date zero. The new trees will not trade on
the stock market until date T, when their ownership is allocated equally among
agents. No technology shocks are expected to occur ever again. Compute the
time path of the stock market from date zero on. (Assume that stocks are traded
at the beginning of each period before dividends are paid.)
2. Consider the Lucas tree model. Show that the price of a share is increasing in the
current level of dividends for the two cases outlined below. [Hint: In answering
the above the questions it may pay formulate the asset pricing equation in terms
of the function G(δ) ≡ U1(δ)P(δ).]
(a) Dividends are independently and identically distribution over time. To get
the desired result, what restriction do you need to impose on the coefficient
of relative risk aversion, −cU11/U1?
(b) Dividends are serially correlated over time, where the distribution function
D is increasing in δ in the sense of first-order stochastic dominance. You
may want to impose your restriction again.
3. q-theory. Imagine a firm with the production function
yt = ztkt,
where z is total factor productivity. The variable zt+1 evolves according to fol-
103

lowing stationary Markov process:
zt+1 ∼ Z(zt+1|zt).
The firm accumulates capital, k, according to the standard law of motion
kt+1 = (1 − δ)kt + xt, (4.4)
where x is investment. There are internal adjustment cost for installing capital.
Period-t adjustment costs are given by
C(
xt
kt
)kt.
This function is assumed to be convex in xt. Let vt denote the stock-market value
of the firm at the end of period t — after period-t dividends have been paid out
— and rt+1 be the gross interest rate between period t and t + 1. (Assume that
the firm pays out its profits each period.) Let qt be the Lagrange multiplier asso-
ciated with the constraint (4.4). Suppose that the firm’s investment and capital
stock are unobservable. Some of it may represent investment in intangibles, such
as knowledge or organizational structure. The size of this capital stock can be
inferred, however.
(a) Lemma: Equality of Marginal and Average q — Hayashi (1982): The value
of the firm, vt, is the product of the shadow price of capital, qt, and the stock
of capital, kt+1. Prove this lemma.
104

(b) Lemma: Quantity Revelation — Hall (2000): Given some starting value for
the capital stock, k0, the time path of the capital stock, {kt+1}T
t=0, can be
inferred from the time path for the stock-market value of the ﬁrm, {vt}T−1
t=1 .
Supply the formal proof.
(c) What is the importance of Hall’s theorem for the IT revolution and recent
stock market experience?
4. In the Arrow-Debreau-McKenzie framework use contingent claims to price:
(a) A discount bond that pays one unit of consumption next period.
(b) An option to buy a tree next period at strike price, σ.
5. (Arrow-Debreu-McKenzie): Consider a one-period agricultural economy made
up of two types of agents, namely greens and reds. There are a large number of
agents of each type. The number of greens equals the number of reds. Each agent
supplies one unit of labor to his farm. A green agent earns g on his farm while a
red agent will get r. Now, g and r are nonnegative random variables distributed
as follows:
g = w + α + ι,
and
r = w + α − ι,
105

where α ∼ A(α), with continuous density a(α), and ι ∼ I(ι), with continuous
density, i(ι). An agent’s utility function is given by U(c), where c is his consump-
tion of the agricultural product. Last, suppose that there is a contingent claims
market in the economy. Trades on this market are effected at the beginning of
period before the shocks are realized.
(a) Set up the optimization problems for green and red agents.
(b) Formulate the model’s general equilibrium.
(c) Characterize an agent’s consumption in the above economy. How will con-
tingent claims be priced?
(d) Suppose that agents live forever. Let their lifetime utility function read
P∞
t=1 βt−1
U(ct). Now, suppose that α0
∼ A(α0
|α), with continuous density a(α0
|α),
and ι ∼ I(ι0
|ι), with continuous density i(ι0
|ι). For the first period α and
ι are drawn from the stationary distributions A(α) and I(ι) defined by
A(α0
) =
R
A(α0
|x)A1(x)dx and I(ι0
) =
R
I(ι0
|x)I1(x)dx. Will the story
change?
6. Does equation (4.3) define a contraction mapping? What are the economic and
mathematical issues here?
106

Chapter 5
Output Effects of Government
Purchases
5.1 The effects of Temporary versus Permanent changes
in Government Spending à al Barro (1987) and
Hall (1980)
• Hall (1980) argued that temporary changes in government spending should have
a larger effect than persistent ones. Barro (1987) states that these multipliers will
have a value less than one. Their argument is portrayed in Figure 1. Aggregate
demand slopes downwards since a fall in the interest rate will stimulate consump-
107

tion and investment spending. Aggregate supply slopes upwards since a rise in
the interest rate entices more labor eﬀort due to the intertemporal substitution
eﬀect on labor supply.
— A temporary change in government moves out the demand curve. This is due
to permanent income theory that supposedly implied consumption should
fall by less than government spending rises. To see this, assume constant
interest rate, r. The permanent income theory would say that consumption
in period t, c(t), should be something like
c(t) = {β
Z ∞
0
[y(t + s) − τ(t + s)] exp[
Z s
0
r(t + v)dv]ds + r(t)a(t)},
where β is the agent’s rate of time preference, y(t + s) − τ(t + s) represents
period-(t+s) income minus lump-sum taxes, r(t+v) is the period-(t+v) real
interest rate, and a(t) is his time t holdings of assets. Essentially, an agent
consumes the constant fraction, β, of his lifetime wealth each period. Here β
could be thought of as the interest rate on his permanent income. Then the
individual consumes interest on his permanent income each period. Now, the
government’s budget constraint requires
R ∞
0
[g(t + s) − τ(t + s)] exp[
R s
0
r(t +
v)dv]ds = 0 so that
c(t) = β
Z ∞
0
[y(t + s) − g(t + s)] exp[
Z s
0
r(t + v)dv]ds.
108

y
AD
AD'
AS
y'
r
r'
∆gT
Figure 5.1: The Effects of Government Spending on Output
A one-shot change in government spending will have no impact on permanent
income and hence aggregate consumption.
— The rise in interest rates, caused by the excess demand, stimulates aggregate
output due to the intertemporal substitution effect on labor supply.
— A permanent change in government spending has no effect on aggregate
demand. Consumption falls by the same amount as the rise in government
spending.
• Barro (1981,1987) felt that a temporary change in government spending will have
a large effect on the interest rate while a permanent change will have no effect.
This argument is shown in Figure 2, using a two-period endowment model.1
Along
1
The argument follows from Figure 1 as well.
109

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