Introduction: How to think about economies at the macro level?

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Introduction: How to think about economies at the macro level?
Introduction:
How to think about economies at the macro
level?
Assumptions regarding the decision mechanisms:
• Centralized vs. Decentralized
• Dictatorial vs. Choice
• Homogenous vs. Heterogenous agents.
• Social Choice: Aggregation of individual choices. Social welfare functions and social outcomes.
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Assumptions regarding issues which affect behavior:
• Rational vs. Adaptive Expectations.
• Perfect Foresight vs Limited Foresight
• Full Information vs. Partial Information
• Behavioral parameters: Patience, Savings Behaviour
(as in Solow)
• Herd Behavior.
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Types of Preferences.
• Intertemporally separable vs. Non-separable
• Risk Aversion (Constant Relative Risk Aversion or
Absolute Risk Aversion)
• Hierarchic
• Care about children’s welfare
• Care about others well-being?
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Assumptions regarding the market structures
and public goods:
• Labor Markets ( monopoly, monopolistic, monopsony, perfect competition, etc.)
• Product Markets (similar to labor markets)
• Technology
• Public Goods
• Knowledge
• Rivalry, Excludability
Other Assumptions regarding the systems:
• Open vs. Closed.
• Input and Output Economies
• Interrelatedness of Markets. Contagion.
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History of Economic Growth as a Discipline
Some of the important contributions
• Adam Smith (1776)
• Thomas Malthus ( 1798)
• David Ricardo (1817)
• Roy Harrod (1939)
• Evsey Domar (1946)
• Frank Ramsey (1928) — Start of the modern growth
theory
• Allyn Young (1928)
• Frank Knight (1944)
• Joseph Schumpeter (1934)
• Robert Solow (1956)
• Kenneth Arrow (1960)
• Paul Romer (1986)
• Gene Grossman (1990) and Elhanan Helpman (1990)
• Phillipe Aghion and Peter Howitt (1992)
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• Frank Caselli (2000)
• Daron Acemoglu (2003)
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This courses focuses on the post 1950 period and tries
to explain economic growth using neoclassical concepts.
• Ramsey (1928)
Introduction of consumer optimization using intertemporally seperable utility function.
• Harrod(1939) and Domar (1946)
Keynesian Analysis. Assume little substitutability between capital and labor.
• Robert Solow (1956)
Predicts conditional convergence ( The lower the starting level of real income relative to steady state position
the higher the growth rate.
The steady state level depend on the savings rate,
growth rate of population and the production function
characteristics. Later empirical work show steady state
level also depend on
i) initial human capital level
ii) government policies
Predicts that per capita income growth should eventually go to zero. (similar to Malthus and Ricardo) In
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conflict with stylized facts.
Assumes a neoclassical production function, i.e. constant returns to scale, diminishing returns to each input
and positive and smooth elasticity between the inputs.
(more on this later)
• Cass and Koopmans (1965)
- incorporates the consumer optimization ( in the sense
of rational behaviour) into the neoclassical growth model
and thereby the endogenous determination of savings
rate.
The equilibrium can then be supported by a decentralized, competitive framework as in the neoclassical tradition.
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• 1965-1985
Lack of empirical evidence - death of the growth theory
Focus on short term fluctuations.
Rational Expectations (Lucas, 1976)
Paradigm Business cycle models ( Kydland and Prescott,
1982).
General Equilibrium Modeling of Business Cycle Theory
Introduction of Dynamic Stochastic General Equilibrium Models
• Romer (1986)
Endogenous growth model
Long-term growth rate is not exogenously determined
by the rate of technological progress as in Solow but
rather endogenously determined within the model.
Why do growth rates do not diminish? Because returns to investment in capital goods (especially human
capital) do not diminish. There are spillovers and external benefits to all producers from newly invented technologies.
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• Romer (1990), Aghion and Howitt(1992), Grossman and Helpman (1991)
Technological advance results from R&D and is awarded
monopoly rights.
There is positive growth as long as there are technological advances
New technologies are either process innovations or product innovations.
The resulting level of growth is not necessarly Pareto
optimal because of the distortions created by monopoly
rights.
Long term growth rate depends on government actions, policies such as
infrastructure
protection of property rights
labor market regulations
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• Aghion et al.(2002)
The role of competition in creating new ideas
• Acemoglu(2002), Johnson and Robinson(2006)
and Acemoglu (2005)
Political Economy of Growth
Institutions As the Fundamental Cause of Long-Run
Growth.
Other Important Contributions
• Income Distribution and Growth
Alesina and Rodrik (1994)
Persson and Tabellini (1994)
• Endogenizing Fertility Choice
Barro and Becker (1989)
Parents and Children are linked through altruism.
Role of Intergenerational transfers
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• Environment and Economic Growth
The Green Solow Model
Sustainable Growth
Stokey(1998)
Brock and Taylor (2007)
• Some Empirical Episodes
1st Industrial Revolution
Great Depression
2nd Industrial Revolution
Oil crisis and the productivity slowdown
3rd Industrial Revolution
2nd Great Depression (?)
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Some Empirical Facts (Barro and Sala-i Martin
2002)
In 2000, GDP per capita in the United States was
$32500 (valued at 1995 $ prices). whereas it was $9000 in
Mexico, $4000 in China, $2500 in India, and only $1000
in Nigeria (all figures adjusted for purchasing power parity).
Is catching-up with the leaders possible? Small differences in growth rates over long periods of time can make
huge differences in final outcomes.
Example: US per-capita GDP grew by a factor ∼ 10
from 1870 to 2000: In 1995 prices, it was $3300 in 1870
and $32500 in 2000. Average growth rate was ∼ 1.75%.
If US had grown with 0.75% (like India,Pakistan, or
the Philippines), its GDP would be $8700 in 1990 (i.e.,
∼ 1 /4 of the actual one, similar to Mexico, less than
Portugal or Greece).
If US had grown with 2.75% (like Japan or Taiwan),
its GDP would be $112000 in 1990 (i.e., 3.5 times the
actual one).
Let y0 be the real GDP per capital at year 0, yT the
real GDP per capita at year T , and x the average annual
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growth rate over that period. Then, yT = (1 + x)T y0.
Taking logs,
ln yT -ln y0 = T ln(1+x) ≈ T x, or equivalenty x ≈ (ln
yT -ln y0) /T .
In 2000, the richest country was Luxembourg, with
$44000 GDP per person. The United States came second, with $32500. The G7 and most of the OECD countries ranked in the top 25 positions, together with Singapore, Hong Kong, Taiwan.. Most African countries, on
the other hand, fell in the bottom 25 of the distribution.
Tanzania was the poorest country, with only $570 per
person—that is, less than 2% of the income in the United
States or Luxemburg.
In 1960, on the other hand, the richest country then
was Switzerland, with $15000; the United States was
again second, with $13000, and the poorest country was
again Tanzania, with $450.
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• Kaldor’s (1963) Stylized Facts:
1. Per capita output grows over time and its growth
rate does not diminish
2. Physical capital per worker grows over time.
3. The rate of return to capital is nearly constant
4. The ratio of physical capital to output is nearly
constant.
5. The shares of labor and physical capital in national
income are nearly constant.
6. The growth rate of output per worker differs substantially across countries.
6 fits the cross-country data
1,2,4 and 5 fit well with long term data for developed
countries. Evidence by Maddison (82) and Jorgenson et
al. (1974)
3 does not fit for USA or East asian Economies
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Solow Model
The technology for producing the good is given by
Yt = F (K (t) , L(t), T (t))
(1)
where F : R3++ → R+ is a (stationary) production
function. We assume that F is continuous and twice
differentiable.
K(t) :Durable physical inputs. Produced by the above
function. Subject to rivalry
L(t) : Labor L(t). Inputs associated with human body.
Number of Workers and the amount they work, physical
strength, skill, health. Subject to rivalry.
T (t) : Blueprint or the formula. It is non-rival. Can be
excludable or non-excludable. Therefore it is not necessarily a public good. But public services that are nonrival
can be included in this function.
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We say that the technology is “neoclassical” if F satisfies the following properties
1. Constant returns to scale (CRS), or linear homogeneity:
F (λK, λL, T ) = λF (K, L, T ), ∀λ > 0.
Homogeneity of degree one. Does not apply to T following the "replication" argument. (it is non-rival.)
2. Positive and diminishing marginal products:
FK (K, L, T ) > 0, FL (K, L, T ) > 0
FKK (K, L, T ) < 0, FLL (K, L, T ) < 0.
where F x ≡ ∂F /∂x and F xz ≡ ∂ 2F/(∂x∂z) f or x,
z ∈ {K, L}.
3. Inada conditions:
limFK = limFL = ∞
K→0
L→0
.
limFK = limFL = 0
K→∞
L→∞
By implication, F satisfies
Y = F (K, L, T ) = FK (K, L, T )K + FL (K, L, T )L
or
1 = εK + εL
where
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εK ≡
∂F K
∂K F
and εL ≡
∂F L
∂L F
are capital elasticity of
output and labor elasticity of output, respectively.
Also,FK and FL are homogeneous of degree zero, meaning that the marginal products depend only on the ratio
K/L.
And, FKL > 0, meaning that capital and labor are
complementary.
4. Finally, all inputs are essential: F (0, L, T )
F (K, 0, T ) = F (K, L, 0) = 0.
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=
"Per capita" variables and intensive forms:
Let y ≡ Y/L (output per worker) and k ≡ K/L.
(capital per worker) Since Y
letting λ =
= F (K, L, T ) is CRS,
1
L
gives us
λY = λF (K, L, T ) = F (λK, λL, T ) = F (k, 1, T )
hence
y ≡ F (k, 1, T ) = f (k) (production function in intensive form , no “scale effects”)
In this form, production per person is determined by
the amount of phyical capital each person owns or has
access to.
If k is constant, having more or less workers does not
affect per capita ouput. (no “scale effects”)
By definition of f and the properties of F , we have
f (0) = 0,
f ′ (k) > 0
′′
f (k) < 0
lim f ′ (k) = ∞
k→0
′′
lim f (k) = 0
k→∞
Marginal Products in Intensive Form
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Since Y = Lf (k)
∂Lf (k)
∂K
∂Y
∂K
=
= L L1 f ′ (k) = f ′ (k)
∂Y
∂L
= f (k) − kf ′ (k)
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Example: Cobb-Douglas
Y = AK αL1−α
where A > 0 is the level of technology, α is a constant
with 0 < α < 1
α
f (k) = A K
= Akα
L
Does CD production fucntion satisy neoclassical prop-
erties
1) Y = AK αL1−α is CRS
2) positive and diminishing marginal products
f ′ (k) = Aαk α−1 > 0
f ′′ (k) = Aα(α − 1)k α−2 < 0
3) Inada conditions
lim Aαk α−1 = ∞
k→0
lim Aαk α−1 = 0
k→∞
4. inputs are essential:
f (0) = 0
In a competitive economy with a Cobb-Douglas type
production, capital and labor are each paid their marginal products such that
R = f ′ (k) = Aαk α−1
and
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w = f (k) − kf ′ (k) = (1 − α)Akα
The capital share of income is
capital income
total income
=
Rk
f (k)
= α
and the labor share of income is given by:
labor income
total income
=
w
f (k)
= 1−α
Thus, in a competitive economy with with a CobbDouglas type production factor income shares are constant (independent of k)
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Centralized Dictatorial Allocation
i) The Model in Discrete Time
Time is discrete,t ∈ {0, 1, 2, ...}. You can think of
the period as a year, as a generation, or as any other
arbitrary length of time.
The economy is an isolated island. Many households
live in this island. There are no markets and production
is centralized. There is a benevolent dictator, or social
planner, who governs all economic and social affairs
There is one good, which is produced with two factors
of production, capital and labor, and which can be either
consumed in the same period, or invested as capital for
the next period.
The investment good can be used either as consumption or as inputs to produce more investment goods. (e.g.
farm animals ) or to replace old depreciated capital.
Households are each endowed with one unit of labor,
which they supply inelasticly to the social planner. The
social planner uses the entire labor force together with
the accumulated aggregate capital stock to produce the
one good of the economy.
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In each period, the social planner saves a constant fraction s ∈ (0, 1) of contemporaneous output, to be added
to the economy’s capital stock, and distributes the remaining fraction uniformly across the households of the
economy
In what follows, we let Lt denote the number of households (and the size of the labor force) in period t, Kt
aggregate capital stock in the beginning of period t, Yt
aggregate output in period t, Ct aggregate consumption
in period t, and It aggregate investment in period t. The
corresponding lower-case variables represent per-capita
measures: kt = Kt /Lt , yt = Yt /Lt , it = It /Lt , and
ct = Ct /Lt .
The sum of aggregate consumption and aggregate investment can not exceed aggregate output. That is, the
social planner faces the following resource constraint:
Ct + It Yt
(2)
Equivalently, in per-capita terms:ct + it yt
We assume that population growth is n ≥ 0 per period: Lt = (1 + n)Lt−1 = (1 + n)t L0
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We normalize L0 = 1.
Suppose that existing capital depreciates over time at
a fixed rate δ ∈ [0, 1]. The capital stock in the beginning of next period is given by the non-depreciated part
of current-period capital, plus contemporaneous investment. That is, the law of motion for capital is
Kt+1 = (1 − δ)Kt + It
(3)
Equivalently, in per-capita terms:
Kt+1
Lt
t
= (1 − δ) K
Lt +
It
Lt
since
Lt+1 = (1 + n)Lt
(1+n)Kt+1
Lt+1
= (1 + n)kt+1 = (1 − δ)kt + it
(1 + n)kt+1 = (1 − δ)kt + it
kt+1 = (1 − δ)kt + it − nkt+1
Assuming nkt+1 ∼ nkt since n is small we can approximately write the above as kt+1 ∼ (1 − δ −
n)kt + it
kt+1 = kt − (δ + n)kt + it
The sum δ+n can thus be interpreted as the “effective”
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depreciation rate of per-capita capital. (Remark: This
approximation becomes exact in the continuous-time version of the model.)
The Dynamics of Capital and Consumption
In most of the growth models that we will examine
in this class, the key of the analysis will be to derive a
dynamic system that characterizes the evolution of aggregate consumption and capital in the economy; that is,
a system of difference equations in Ct and Kt (or ct and
kt ). This system is very simple in the case of the Solow
model.
Combining the law of motion for capital , 3, the resource constraint 2, and the technology 1, we derive the
difference equation for the capital stock:
Kt+1 − Kt F (Kt , Lt ) − δKt − Ct
That is, the change in the capital stock is given by
aggregate output, minus capital depreciation, minus aggregate consumption
In capita terms:
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kt+1 − kt f (kt ) − (δ + n)kt − ct.
Feasible and “Optimal” Allocations
Definition: A feasible allocation is any sequence {ct , kt }∞
t=0 ∈
R2 that satisfies the resource constraint
kt+1 f (kt ) + (1 − δ − n)kt − ct .
(4)
The set of feasible allocations represents the ”choice
set” for the social planner. The planner then uses some
choice rule to select one of the many feasible allocations.
We assume here that the dictator follows a simple ruleof-thumb.
Definition: A “Solow-optimal” centralized allocation
is any feasible allocation that satisfies the resource constraint with equality and
ct = (1 − s)f (kt )
(5)
f or some s ∈ (0, 1).
4 and 5 completely describes the system dynamics.
Proposition: Given any initial point k0 > 0, the dynamics of the dictatorial economy are given by the path
{kt }∞
t=0 such that
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kt+1 = G(kt ) for all t ≥ 0, where
G(kt ) = sf (kt ) + (1 − δ − n)kt
Equivalently, the growth rate is given by
γ(kt ) =
kt+1 −kt
kt
= sϕ(kt ) − (δ + n)
where
ϕ(kt ) = f (kt )/kt .
Remark. Think of G more generally as a function that
tells you what is the state of the economy tomorrow as
a function of the state today. Here and in the simple
Ramsey model, the state is simply kt . When we introduce productivity shocks, the state is (kt , At ). When we
introduce multiple types of capital, the state is the vector
of capital stocks. And with incomplete markets, the state
is the whole distribution of wealth in the cross-section of
agents.
Definition: A steady state of the economy is defined
as any level k ∗ such that, if the economy starts with
k0 = k ∗, then kt = k ∗ for all t ≥ 1. That is, a steady
state is any fixed point k ∗ of G in (6), i.e.k ∗ = G(k ∗ ).
Equivalently, a steady state is any fixed point (c∗ , k∗ ) of
the system (4)-(5).
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A trivial steady state is c = k = 0 : There is
no capital, no output, and no consumption. This would
not be a steady state if f (0) > 0. We are interested for
steady states at which capital, output and consumption
are all positive and finite. We can easily show:
P roposition : Suppose δ + n < 1 and s ∈ (0, 1). A
steady state (c∗ , k∗ ) ∈ (0, ∞)2 for the dictatorial economy exists and is unique.
k ∗ and y ∗ increase with s and decrease with δ and n,
whereas c∗ is non-monotonic with s and decreases with
δ and n. Finally, y ∗ /k∗ = (δ + n)/s.
P roof :. k ∗ is a steady state if and only if it solves
0 = sf (k ∗ ) − (δ + n)k ∗
Equivalently k ∗ solves
ϕ(k ∗ ) =
where ϕ(k) ≡
f (k)
k .
δ+n
s
(6)
The function ϕ gives the output-
to-capital ratio in the economy. The properties of f imply that ϕ is continuous and strictly decreasing, with
ϕ′ (k) =
f ′ (k)k−f (k)
k2
= − FkL2 < 0,
ϕ(0) = f ′ (0) = ∞
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and ϕ(∞) = f ′ (∞) = 0
where the latter follow from L’Hospital’s rule. This
implies that equation (6) has a unique solution:
k ∗ = ϕ−1
δ+n s
Since ϕ′ < 0, k ∗ is a decreasing function of (δ + n)/s.
Transitional Dynamics
The above characterized the (unique) steady state of
the economy. Naturally, we are interested to know whether
the economy will converge to the steady state if it starts
away from it. Another way to ask the same question is
whether the economy will eventually return to the steady
state after an exogenous shock perturbs the economy and
moves away from the steady state.
The following uses the properties of G to establish
that, in the Solow model, convergence to the steady is
always ensured and is monotonic:
P roposition. Given any initial k0 ∈ (0, ∞), the dictatorial economy converges asymptotically to the steady
state. The transition is monotonic. The growth rate is
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positive and decreases over time towards zero if k0 < k ∗;
it is negative and increases over time towards zero if
k0 > k ∗ .
P roof . From the properties of f, G′ (k) = sf ′ (k)+(1−
δ −n) > 0 and G′′ (k) = sf ′′ (k) < 0. That is, G is strictly
increasing and strictly concave. Moreover, G(0) = 0 and
G(k∗ ) = k ∗. It follows that G(k) > k for all k < k ∗
and G(k) < k for all k > k ∗ . Therefore G has a unique
fixed point. It follows that kt < kt+1 < k ∗ whenever
kt ∈ (0, k ∗ ) and therefore the sequence {kt }∞
t=0 is strictly
increasing if k0 < k ∗ . By monotonicity, kt converges
ˆ
∗
ˆ
asymptotically to some k · k . By continuity of G, k
ˆ
ˆ
ˆ
must satisfy k = G(k), that is k must be a f ixed point
of G. But we already proved that G has a unique fixed
ˆ
point, which proves that k = k∗ . A symmetric argument
applies when k0 > k ∗
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kt +1
State Transition or the Policy Rule
in the Solow Model kt +1 = G (kt )
kt +1 = kt
G ( kt )
k0
k1
k 2 k3 k *
kt
Macro Lecture Notes– Ozan Hatipoglu
Next, consider the growth rate of the capital stock.
This is given by
kt+1 −kt
kt
= sϕ(kt ) − (δ + n) = γ(kt )
Note that γ(kt ) = 0 if f kt = k ∗ , γ(kt ) > 0 if f kt < k ∗,
and γ(kt ) < 0 if f kt > k ∗ . Moreover, by diminishing
returns, γ′(kt ) = sϕ′ (kt ) < 0. It follows that γ(kt ) >
γ(kt+1 ) > γ(k∗) = 0 whenever kt ∈ (0, k ∗) and γ(kt )
< γ(kt+1) < γ(k ∗ ) = 0 whenever kt ∈ (k∗, ∞). This
proves that γ t is positive and decreases towards zero if
k0 < k ∗ and it is negative and increases towards zero if
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k0 > k ∗ .
γ ( kt )
Behavior of the Growth rate in the
Solow Model
kt
k*
− (δ + n)
Macro Lecture Notes– Ozan Hatipoglu
Golden Rule of Capital Accumulation:
Denote the steady state level of consumption as c∗,
then one might ask the question what are the parameters that maximize steady state level of consumption, c∗.
Since we have perfectly competitive markets with a single good, c∗max amounts to welfare maximizing level of
consumption.
At the steady state we have:
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c∗ = (1 − s)f (k∗ )
since at the steady state sf (k ∗ ) − (δ + n)k ∗ = 0
c∗ = (1 − s)f (k∗ ) = f (k ∗ ) − (δ + n)k ∗
since k ∗ is a function of the parameters as given in
(6)
ϕ(k ∗ ) =
δ+n
s
=
f(k∗ )
k∗
and ϕ(k ∗ ) has a unique solution as we proved earlier,
c∗ is also a function of the parameters.
dc∗
ds
∗
∗
dk
′ ∗
= f ′ (k ∗(s)) dk
ds − (δ + n) ds = [f (k (s)) − (δ +
∗
n)] dk
ds = 0
Since k ∗ = ϕ−1
δ+n s
AND ϕ′ (k ∗ ) < 0, k ∗ is a de-
creasing function of (δ + n)/s , therefore
dk∗
ds
>0
It must be the case that [f ′ (k ∗(s)) − (δ + n)] = 0
∗
(s)) = (δ + n)
f ′ (kgold
(Golden Rule of Capital Ac-
cumulation)
and
∗
∗
c∗gold = f (kgold
) − (δ + n)kgold
Discussion Questions:
1) Is there a "best" savings rate s that the policy maker
can choose in the Solow Model. Explain in detail..
∗
2) Calculate for kgold
for the CD production function.
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Productivity (or Taste) Shocks
The Solow model can be interpreted also as a primitive Real Business Cycle (RBC) model. We can use the
model to predict the response of the economy to productivity, taste, or policy shocks.
Yt = At F (Kt , Lt )
yt = At f (kt ), where At denotes total factor productivity.
Consider a permanent negative shock in A. The G(kt )
and γ(kt ) functions shift down. The economy transits
slowly from the old steady state to the new, lower steady
state.
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kt +1
Negative Productivity Shock in the
Solow Model
kt +1 = kt
G (kt )
k1
k 2 k3 k *
kt
If instead the shock is transitory, the shift in G(kt ) and
γ(kt ) is also temporary. Initially, capital and output fall
towards the low steady state. But when productivity
reverts to the initial level, capital and output start to
grow back towards the old high steady state.
The effect of a productivity shock on kt and yt is illustrated in the figure below The solid lines correspond to
a transitory shock, whereas the dashed lines correspond
to a permanent shock.
36
kt
Effectof a Negative Productivity Shock
in the Solow Model
Transitory
Permanent
t
t
yt
Transitory
t0
t
t1
Permanent
Taste shocks: Consider a temporary fall in the saving rate s. The γ(kt ) function shifts down for a while,
and then return to its initial position. What are the
transitional dynamics? What if instead the fall in s is
permanent?
37
Unproductive Government Spending
Let us now introduce a government in the competitive market economy. The government spends resources
without contributing to production or capital accumulation. The resource constraint of the economy now becomes
ct + gt + it = yt = f (kt ),
where gt denotes government consumption. The latter
is financed with proportional income taxation: gt = τ yt
(balanced budget)
Disposable income for the representative household is
(1 − τ )yt . We continue to assume agents consume a fraction s of disposable income:
it = s(yt − gt ).
Combining the above, we conclude that the dynamics
of capital are now given by
γt =
kt+1 −kt
kt
= s(1 − τ )ϕ(kt ) − (δ + n)
where ϕ(k) ≡ f (k)/k. Given s and kt , the growth rate
γ t decreases with τ
A steady state exists for any τ ∈ [0, 1) and is given by
δ+n
∗
−1
k =ϕ
s(1−τ)
Given s, k∗ decreases with τ .
38
Policy Shocks: Consider a temporary shock in government consumption. What are the transitional dynamics
Suppose now that production is given by yt = f (kt , gt ) =
ktαgtβ , where α > 0, β > 0, and α + β < 1. In this form,
government spending can , for example, be interpreted
as infrastructure or other productive services.
The resource constraint is ct + gt + it = yt = f (kt , gt )
Government spending is financed with proportional income taxation and private consumption is a fraction 1−s
of disposable income:
gt = τ yt ,
ct = (1 − s)(yt − gt )
it = s(yt − gt ).
Substituting gt = τ yt into yt = k αg β and solving for yt
yt = k α (τ yt )β
yt1−β = k ατ β
α
β
yt = kt(1−β) τ (1−β)
α
(1−β)
or yt = Akt
β
where A = τ (1−β)
We conclude that the dynamics and the steady state
are given by
39
γt =
and
k∗ =
kt+1 −kt
kt
α−(1−β)
(1−β)
= s(1 − τ )kt
β
(1−β)
s(1−τ)τ
(δ+n)
β
τ (1−β) − (δ + n)
(1−β)
(1−β)−α
Question: Show that the more productive government
services are, the higher their “optimal” provision.
Proof:
c = (1−s)∗(1−τ )∗f (k)
(1)
At the steady state
kt+1 − kt = s ∗ (1 − τ ) ∗ f (k) − (δ + n) ∗ k = 0
(2)
Substituting in (1)
c = (1−τ )∗f (k ∗)−(δ+n)∗k ∗
(3)
where k ∗ is the steady state level of per capita capital.
Note that with productive spending
yt = ktα ∗ (τ yt )β which implies
α/1−β
yt = τ β/1−β ∗ kt
= f (kt )
(4)
Substituting (4) in (2) one can find the steady state
level of k as
k∗ = (
1−β
δ+n
) α−(1−β)
s∗(1−τ)∗τ β/1−β
40
(5)
Substituting (5) and (4) together in (1) (can also be
solved by substituting in (3))
β
c∗ = (1−s)∗(1−τ )∗τ 1−β ∗(
α
δ+n
) α−(1−β)
s∗(1−τ )∗τ β/1−β
collecting terms yields
β−1
α
α−(1−β) ∗ (1 − τ ) ∗ τ β/1−β α+β−1
c∗ = (1−s)∗( δ+n
)
s
Optimal provision maximizes consumption at the steady
state
∂c∗
∂τ
(1 − τ ) ∗ τ
β−1
α+β−1
= A∗
−α
β/1−β α+β−1
=0
β
2β−1
β
1−β
1−β
∗ −τ
∗
+ (1 − τ ) 1−β ∗ τ
(6)
α
α−(1−β)
where A = (1 − s) ∗ ( δ+n
s )
Note
∂c∗
∂τ
= 0 whenever τ ∗ = 0, τ ∗ = 1 and for some
τ ∈ (0, 1) which can be found by equating the second
term in (6) to zero.
β
β
−τ 1−β + (1 − τ ) 1−β
∗τ
2β−1
1−β
=0
β
divide both sides by τ 1−β and arrange
β
(1 − τ ) 1−β
∗ τ −1 = 1
=⇒
(1−τ )
τ
=
1−β
β
q.e.d.
41
Back to Solow Model in Continuous Version:
Capital Accumulation:
·
K=
dK
dt
·
K(t) = I(t) − δK(t) = sF (K(t), L(t), T (t)) − δK(t)
·
K(t)
L
= sf (k(t)) − δk(t)
Ignore the time subscripts
·
Define k =
d(K/L)
dt
·
=
·
LK−K L
L2
·
=
K
L
·
L
−K
LL =
·
K
L
− nk
·
where n =
L
L
Therefore
·
K(t)
L
·
= k + nk
and
·
k = sf (k)−(n+δ)k (Fundamental Differential Equation of Solow-Swan Model)
Compare with the approximation in the discrete time
version
kt+1 − kt ∼ sf (kt ) − (δ + n)kt
42
With Competitive Markets...
Asset accumulation
Suppose households own assets which deliver a rate
of return r(t) (interest rate received on loans, bank deposits, other financial assets) and labor is paid wage w(t).
The total income received by the household is then given
by
r(t) × assets + w(t)L(t)
The total number of assets then accumulate according
to (ignore time subscripts)
d(assets)
dt
= r(t) × assets + w(t)L(t) − C
where C is total consumption.
assets
L
·
a = d((assets)/L)
dt
Let a =
Define
·
(assets) L
L
L
=
d(assets)
dt
L
·
=
L d(assets)
−(assets)L
dt
L2
=
d(assets)
dt
L
−
− na
Therefore
·
a = (ra + w) − c − na
Firm’s problem:
43
(13)
Firm’s hire capital and labor to produce output. Let
the rental rate R be the rental price for a unit of capital
services and δ rhe consatnt depreciation rate. The net
rate of return for a household is then R − δ
for a unit of capital. Since capitals and loans are perfect substitutes r = R − δ.
The firm’s net receipts
π = F (K, L)−RK −wL = F (K, L)−(r + δ) K −wL
Since F is neoclassical
π = L [f (k) − (r + δ) k − w]
(14)
For a given L the firm chooses k to max profits such
that we have the following first order condition (FOC)
f ′ (k) = (r + δ)
(15)
Note that the resulting profit is either zero, positive or
negative depending on w. But if positives are positive,
then the firm would choose k = ∞ and if it is negative
44
then the firm would choose k = 0. Therefore in equilibrium w must be such that
π = L [f (k) − (r + δ) k − w] = 0
Hence,
w = f (k) − kf ′ (k)
(16)
Another point of view:
We also see that factor prices are equal to marginal
products therefore it must be the case that profits are
zero that is total factor payments exhaust the total output.
Equilibrium:
i) Capital and labor markets clear
Capital markets clear: i.e. all borrowing and lending
must cancel out
a=k
r = f ′ (k) − δ
and
w = f (k) − kf ′(k)
45
Substituting in 13
·
k = f (k) − c − (n + δ)k
(17)
Labor markets clear: Labor supplied=Labor demanded
Since labor is supplied inelastically the eq is determined by the demand side.
ii) Households are "Solow-optimal"
c = (1 − s)f (k)
Therefore 17 can be rewritten as
·
k = sf (k) − (n + δ)k
(18)
Which is exactly the same as the dictatorial version.
iii) Firms max profits by choosing a K/L ratio.
Already shown above.
46
Example: Cobb Douglas Production function
F (K, L) = AK αL1−α
Steady State Level of k
·
k = sf (k) − (n + δ)k = 0
sA (k ∗)α = (n + δ)k ∗
Therefore
1
k ∗ = [sA/(n + δ)] 1−α
(19)
Steady State Level of y
α
1
α
1
1−α
y = A [sA/(n + δ)]
= A 1−α [s/(n + δ)] 1−α
∗
The time path of capital given k(0) is given by
·
k = sAkα − (n + δ)k
We can solve for the exact time path of k by
rewriting the above as
·
kk −α + (n + δ)k 1−α = sA
Substituting v = k 1−α
47
(20)
·
·
v = (1 − α)k−α k
Therefore
·
v + (1 − α)(n + δ)v = (1 − α)sA
(21)
is a first order linear differential equation with a constant coefficient (n + δ) . The solution is given by
1−α
sA
sA
1−α
v=k
= (n+δ) + [k(0)]
− (n+δ) e−(1−α)(n+δ)t
P roof :
·
v + (1 − α)(n + δ)v = (1 − α)sA
·
e(1−α)(n+δ)t v + (1 − α)(n + δ)v dt = e(1−α)(n+δ)t (1−α)sAdt
(22)
Let B = e(1−α)(n+δ)t v + b0 where b0 is a constant then
dB
dt
·
= e(1−α)(n+δ)t v +e(1−α)(n+δ)t v.(1 − α)(n + δ) =
(1−α)(n+δ)t ·
=e
v + (1 − α)(n + δ)v
dB
(1−α)(n+δ)t ·
B = dt dt = e
v + (1 − α)(n + δ)v dt =
e(1−α)(n+δ)t v + b0
The
solution to the right hand side of 22
sA (1−α)(n+δ)t
e(1−α)(n+δ)t (1 − α)sAdt = (n+δ)
e
+ b1
Combining left handside and the right handside
48
e(1−α)(n+δ)t v =
v(t) =
sA
(n+δ)
sA (1−α)(n+δ)t
(n+δ) e
+ b1 − b0
+ (b1 − b0 )e−(1−α)(n+δ)t
How do we determine b1 − b0? Using the steady state
condition for k
v(0) = k(0)1−α =
sA
(n+δ)
(b1 − b0 ) = k(0)1−α −
Therefore
v(t) =
sA
(n+δ)
+ (b1 − b0)
sA
(n+δ)
1−α
+ k(0)
−
sA
(n+δ)
e−(1−α)(n+δ)t
The gap between k 1−α and its steady state value vanishes exactly at the constant rate (1 − α)(n + δ).
49
Convergence
Absolute Convergence
Consider the fundamental Solow equation
·
k = sf (k) − (n + δ)k
(23)
The growth rate of capital per capita is given by
·
k
γ k = = sf (k)/k − (n + δ)
k
(24)
and the derivative of the growth rate of capital per
capita with respect to capital per capita
·
∂(γ k )
∂k
=
·
γy
∂( kk )
∂k
=
s[f ′ (k)− f (k)
k ]
k
·
y
f ′ (k)k
=
=
=
y
f (k)
<0
kf ′ (k)
f (k)
·
·
k
=
k
(25)
k
= [CapitalShare] = [CapitalShare] [sf (k)/k − (n + (1)
δ)]
k
·
′′ ·
∂( yy )
∂(γ y )
f (k)k k (n+δ)f ′ (k)
=
=
[1−CapitalShare] <
∂k
∂k
f (k)
k−
f(k)
·
0 if
k
k
≥ 0 (since CapitalShare < 1)
Therefore The Solow Model predicts absolute conver50
gence
Poor countries tend to grow faster than the rich countries. Empirical Support: Mixed overall. Better support
within homogeneous group of countries.
51
Conditional Convergence
Consider the s.s. condition
sf (k ∗ ) = (n + δ)k ∗
∗
k
s = (n + δ) f (k
∗)
substituting in 24
·
k
k
∗
k
= (n + δ) f(k
∗ ) f (k)/k − (n + δ)
·
k
f (k)/k
−1
= (n + δ)
k
f (k ∗)/k ∗
(26)
A reduction in k increases the average product of capital and increases
·
·
k
k.
k
k
But a lower k increases
more if k
is relatively lower compared to k∗ .
How? Suppose two countries, A and B, have same initial levels of capital stock k(0), if k ∗ is lower for country
A than country B, then country A will grow slower, because the term
f(k)/k
f(k∗ )/k∗
for country A will be lower.
Ex: In the case of a CD p.f.
we have
52
·
k
= (n + δ)
k
53
k
k∗
α−1
−1
(27)
Proposition 1 Absolute Convergence and a Decrease in
the Dispersion of incomes are not equivalent (OPTIONAL)
Proof. Consider N countries. If there is absolute convergence their income process can be approximated by
log(yit ) = a + (1 − b) log(yi,t−1) + uit
(28)
where a and b are constants with 0 < b < 1and uit
is a disturbance term. Since b > 0, this model implies
yit
absolute convergence. log( yi,t−1
) is inversely related to
log(yi,t−1)
Consider the dispersion (or inequality) of per capita
log incomes
Dt =
N
i=1 (log(yi,t )
− µ)2
Using 28
Dt = (1 − b)2Dt−1 + σ 2u
which has a steady state at D ∗ =
54
σ2u
1−(1−b)2
Hence the steady state falls with the strength of the
convergence effect b but rises with the variance of the
disturbance term
The observed evolution of D can be written as
Dt = D ∗ +(1−b)2(Dt−1 −D ∗ ) = D∗ +(1−b)2t (D0 −D ∗)
Where D0 is dipersion at time 0. Since 0 < b < 1 D
monotonically approaches to its steady state value and
Dt , therefore Dt rises if D0 < D∗ and vice versa.
Even though b > 0, Dt rises or falls depending on the
initial condition, q.e.d.
55
More Dynamics with Cobb-Douglas Production
Function and Uzawa’s Theorem
F (K, L, A) = AK αL1−α
0<α<1
Elasticity of substitution δ = −
FK
FL
=
αAK α−1 L1−α
(1−α)AK α L−α
∂ log [F K /F L ]
∂ log (K/L)
−1
= αL/(1 − α)K
α
) − log ( K
log ( FFKL ) = log ( 1−α
L)
∂ log [F K /F L ]
∂ log (K/L)
= −1
⇒
δ=1
Share of capital in national income
αK (t) =
R(t)K(t)
Y (t)
=
FK K
Y (t)
=
αAK α−1 L1−α K
AK α L1−α
=α
(con-
stant)
CES Production Function
Y = AH (t)[φ(AK (t)K(t))
σ−1
σ
+(1 − φ)(AL(t)L(t))
σ−1
σ
σ
σ−1
]
where AH , AK , AL > 0 are three different types of
technological change. φ ∈ (0, 1) is a distribution
parameter referring to the importance of capital in the
final production. And σis the elasticity of substitution.
σ ∈ [0, ∞]
56
σ−1
σ
−1/σ
since FFKL = φAK σ−1K
(1−φ)AL σ L−1/σ
−1
∂ log [F K /F L ]
− ∂ log (K/L)
= δ so δ is indeed the elasticity of
substitution.
σ → 1 Y (t) = AH AφK AL1−φ K φ L1−φ
CD
σ → ∞ Y (t) = φAH AK K + (1 − σ)AH AL
linear
σ→0
Leon-
Y (t) = AH min{φAK K; (1 − φ)AL L}
tieff
Hirofumi Uzawa (1961):
Thm: Let Y (t) = F˜ (K(t), L(t), A(t))
where F : R2 xA → R+ and
˜
A(t)
∈ A represents
technology at time t (where A is an arbitrary set, e.g., a
subset of Rn for some N ∈ N). Let F˜ exhibit CRS.
Let
˙
K(t)
= Y (t) − C(t) − ∂K(t)
L(t) = ent L(0)
and ∃T < ∞ s.t. ∀t ≥ T
˙
K/K
= gk > 0
˙
C/C
= gc > 0 then
(i) gy = gk = gc
57
Y˙ (t)/Y (t) = gy > 0
(ii) For any t ≥ T ∃F : R2+ → R+ homogenous of
degree 1 in its two arguments
s.t.
Y (t) = F (K(t), A(t)L(t))
where A(t) ∈ R+ and
˙
A(t)
A(t)
= g = gy − n
Proof:
(Part 1) By Hypothesis for t ≥ T
Y (t) = egY (t−T ) Y (T ) ,
K(t) = egK (t−T )K(T )
L(t) = en(t−T )L(T )
since K˙ = gK K
the aggregate resource constraint implies
(gK + δ)K(t) = Y (t) − C(t)
Dividing both sides by egK (t−T )
(gK + δ)K(t) = e(gY −gK )(t−T )Y (T ) − e(gC −gK )(t−T )C(T )
for all t ≥ T
Differentiation wrt time
(gY −gK )e(gY −gK )(t−T ) Y (T )−(gC −gK )e(gC −gK )(t−T )C(T ) =
0
This equation holds if any of the following conditions
hold
58
(i) gY = gK = gC
(ii) gY = gC and Y (T ) = (T )
(iii)gY = gK and C(T ) = 0
(iv)gC = gK and Y (T ) = 0
ii), iii), iv) are contradictions because gK > 0 gC > 0
implies C(T ) > 0 K(T ) and hence
Y (T ) > C(t) Y (t) > 0 therefore (i) must hold.
(Part 2) for any t ≥ T
e
−g Y (t−T )
−g K (t−T )
Y (t) = F e
−n(t−T )
K(t), e
˜
L(t), A(T )
Multiply both sides by egY (t−T ) and using the CRS
property of F
(t−T )(g Y −gK )
(t−T )(g Y −n)
˜ )
Y (t) = F e
K(t), e
L(t), A(T
From part (1)
gY = gK therefore
˜ )]
Y (t) = F [K(t), e(t−T )(gY −n)L(t), A(T
Since the above is true for all t ≥ T and
F is homogenous of degree 1 in K and L ∃ a function
F s.t.
F = (K(t), e(gY −n)t L(t))
59
Let A = e(gY −n)t
˙
A(t)
and Y = F (K(t), A(t)L(t)) where A(t)
= gY −n q.e.d.
Extension of Uzawa’s Theorem:(or corollary)
Corollary: Given the assumptions of the previous thm,
the technological progress can be represented as Harrodneutral(labor augmenting).
Some notes:
-The thm simply states that after T, technology should
be labor augmenting to have constant growth rates of
output/capital and consumption.
-The technological change does not have to be labor
augmenting all the time(only after T)
capital augmenting technological change is also feasible before T.
Articles:
Solow(1956)
Swan (1956)
Solow(1970)
Phelps(1966)
Piketty and Sacz(2003)
60
Uzawa(1961)
Jones and Scrimgeour(2006)
Schlicht(2006)
61
Technological Progress
Some Definitions
Definition 2 A capital saving technological progress (or
invention) allows producers to produce the same amount
with relatively less capital input.
Definition 3 A labor saving technological progress (or
invention) allows producers to produce the same amount
with relatively less labor input.
A neutral technological progress allows producers to
produce more with same capital labor ratio ( do not save
relatively more of either input)
i) "Hicks neutral" : Ratio of marginal products remain
the same for a given capital labor ratio. Hicks neutrality
implies the production function can be written as:
Y = T (t)F (K, L)
ii) "Harrod neutral": relative input shares K.Fk /LFL remain
the same for a given capital output ratio
62
Harrod neutrality implies the production function can
be written as:
Y = F [K, LT (t)] (labor-augmenting form)
·
where T (t) is the index of the technology and T (t) > 0
labor-augmenting: it raises output in the same way
as an increase in the stock of labor.
iii) "Solow neutral": relative input shares LFL/K.Fk remain
the same for a given labor output ratio
Solow neutrality implies the production function can
be written as:
Y = F [KT (t), L] (capital-augmenting form)
63
Solow Model with labor augmenting technological progress
Suppose the technology T (t) grows at rate x
·
K(t) = I(t)−δK(t) = sF (K(t), T (t)L(t))−δK(t) (29)
Dividing by L(t)
·
·
k = sF (k, T (t))−(n+δ)k and
k
k
(t))
= s F (k,T
−(n+δ)
k
The average product of per capita capital
F (k,T (t))
k
now
increases over time because the T (t) grows at a rate x.
Steady state growth
· ∗ rate: By definition the steady
state growth rate kk
is constant
∗
s F (k k,T∗ (t)) − (n + δ) =constant
Since F is CRS,
F (k,T (t))
k
= F (1, T k(t) )This implies that
T (t) and k grow at the same rate x, because s,n and δ
are constants
· ∗
k
=x
k
Moreover since y = F (k, T (t)) = kF (1, T k(t) )
64
· ∗
y
=x
y
and c = (1 − s)y,
·
c
c
=
(30)
·
(1−s)y
(1−s)y
Transitional Dynamics
=x
Define: effective amount of labor=physical quantity of
∧
labor× efficiency of labor = L × T (t) ≡ L
∧
k=
K
LT (t)
∧
Y
LT (t)
y=
=
k
T (t) =
∧
capital per unit of effective labor.
∧
= F (k, 1) = f (k) = output per unit effective
labor
We can rewrite
·
K = sF (K, T (t)L) − δK
divide both sides by T (t)L
·
∧
∧
K
= sf (k) − δ k
T (t)L
·
∧
k=
· K
T (t)L
·
=
·
Therefore
·
·
LT (t)K−K(LT (t)+LT (t))
T (t)2 L2
K
T (t)L
·
∧
∧
∧
= k + kn + kx
substituting in (31)
65
(31)
·
=
K
T (t)L
∧
∧
− kn − kx
·
∧
∧
∧
k = sf (k) − (x + n + δ) k
(32)
and
·
∧
k
∧
k
∧
=s
f (k)
∧
− (x + n + δ)
(32)
k
where x + n + δ is the effective depreciation rate
The effective per capita capital depreciates at the rate
x+n+δ
66
Macro Lecture Notes– Ozan Hatipoglu
γ ( kt )
Behavior of the growth rate in the
Solow Model with labor augmenting
technological progress (1)
kt
k
*
− (δ + n)
Macro Lecture Notes– Ozan Hatipoglu
∧
γ (k t )
Behavior of the growth rate in the
Solow Model with labor augmenting
technological progress (2)
(δ + n + x )
∧
s
f (kt )
∧
kt
∧
kt
∧*
k
67
Speed of Convergence:
The speed of convergence is given by
·
∧
∂( k∧ )
k
β=−
∧
(33)
∂ log k
For the CD production fucntion
·
∧
k
∧
k
∧
= sA(k)−(1−α) − (x + n + δ)
or
·
∧
∧
k
= sAe−(1−α) log(k) − (x + n + δ)
∧
k
∧
β = (1 − α)sA(k)−(1−α)
(declines monotonically)
Near the steady state
∧
sA(k)−(1−α) = (x + n + δ)
β ∗ = (1 − α)(x + n + δ)
68
(34)
HomeW ork :
Suppose that in every period a economy neeeds to pay
4% of its output to the rest of the world as interest payments on past government debt. Let the government
collect each period the funds for the repayment of this
debt.
Assume there is a labor augmenting technological progress
Let x = 0.03, n = 0.01 and δ = 0.05, A(0) = 1
α = 0.36 s = 0.3
1) Using the Solow Model find the steady state growth
rate of this economy if
the government collects the funds as
i) lump-sum taxes
ii) capital taxation
and compare i) ii) to the economy without any debt.
Use pencil and matlab.
2) Let s vary and find the golden rule of savings under
i) and ii)
69
ST OCHAST IC SOLOW M ODEL with Hicks Neutral Technological Progress
Alternative formulations
_
1) At = ψ A + (1 − ψ)At−1 + εt
where
0<ψ<1
and
_
A>0
Here the distribution of εt needs to be bounded from
_
below by −ψ A otherwise technology might become negative with some probability
_
2) At = Aeεt
the distribution of εt needs not to be bounded. Rewriting
_
ln At = ln A + εt
Kt+1
Lt
t
= (1 − δ) K
Lt +
It
Lt
since
_
(1 + n)kt+1 = (1 − δ)kt + sAeεt f (kt )
kt+1 = (1 − δ)kt + it − nkt+1
See solowstochastic.m for an example of 2)
70
Matlab Code:
% This simulates the stochastic solow model % % Ozan
Hatipoglu 2010%
clear all;
% define parameters%
a_bar= 1; % Technology
n=0.01; % population growth
alfa=0.36; % capital share
s= 0.2; % savings rate
delta=0.1; % depreciation rate
%%Initializations%%
A=[]; %Technology
K=[]; %Capital Stock Per Capita
Y=[]; %Output Per Capita
C=[]; %Consumption Per Capita
G=[]; % Growth Rates
%%Periods%%
p=50;
%%Shocks%%
epsilon= normrnd(0,0.1,p,1);
%%Initial Levels%%
71
K(1,1)=0.00001;
A(1,1)=a_bar;
Y(1,1)=s*A(1,1)*(K(1,1)^alfa);
G(1,1)=0;
%%Model%%
for i=1:p;
A(i+1)=a_bar*(exp(1).^epsilon(i)); % Technology evolution
K(i+1)=(((1-delta)*K(i))+(s*A(i)*(K(i)^alfa)))/(1+n);
% Capital Stock evolution
Y(i+1)=A(i+1)*K(i+1)^alfa; % Income evolution
C(i)=Y(i)-s*Y(i); % Consumption evolution
G(i+1)=((Y(i+1)-Y(i))/Y(i));
end
% This is just to fit the growth rates into the same
graph
for i=1:2
G(i)=1
end
%% Graphs %%
subplot(4,1,1),plot(K),ylabel(’Capital’);
72
subplot(4,1,2),plot(C),ylabel(’Consumption’);
subplot(4,1,3),plot(Y),ylabel(’Output’);
subplot(4,1,4),plot(G),ylabel(’Growth Rate (%)’);
Homework:
Rewrite solowstochastic.m for 1).
73
LOG-LINEARIZATION OF THE SOLOW MODEL
_
∼
X t = ln(Xt ) − ln(X)
_
_
where X is the time t value of the variable X and X
is its steady state value.
∼
Xt is the log-deviation from the steady state. Rewriting:
_
∼
Xt = XeXt
First order Taylor approximation:
∼
∼
∼
∼ ∼
∼
Xt Xt =0 ∼
Xt =0
+ 1e1! Xt = 1 + Xt
eX t ≈ eXt Similarly
∼
∼
∼
∼
eX t+aY t ≈ 1 + Xt + aYt
∼ ∼
Xt Yt
≈0
∼ Xt+1
≈ aEt [Xt+1] + constant
Et ae
_
(1 + n)kt+1 = (1 − δ)kt + sAeεt f (kt )
since
_
∼
k t+1
kt+1 = k e
_
∼
k t+1
(1 + n)k e
_
∼
e
_
εt
_α
∼
αk t
= (1 − δ)k e + sAe k e
Since
Xt
∼
kt
∼
≈ 1 + Xt
74
_
_
∼
∼
_
∼
∼
_α
(1 + n)k (1 + k t+1) = (1 − δ)k (1 + k t ) + sA(1 + εt )k
(1 +
∼
αk t )
At the nonstochastic steady state k t+1 = k t = εt = 0
_
_
_ _α
(1 + n)k = (1 − δ)k + sAk
subtracting from above
_ ∼
(1 + n)k kt+1
_ _α
∼
sAk αεt k t
= (1 −
_ ∼
δ)k k t
_
_α
+ sAεt k
_ _α ∼
+sAk αk t
+
Since
∼
εt k t ≈ 0
_ ∼
_ ∼
_
_α
_ _α ∼
(1 + n)k k t+1 = (1 − δ)k k t + sAεt k +sAk αk t
∼
k t+1
_ _ α−1
=
((1−δ)+sAk
(1+n)
α)
∼
kt
_ _ α−1
+
sAk
(1+n)
or
∼
k t+1
∼
= M k t + N εt
which is a first order linear stochastic difference equation.
_ _ α−1
M=
((1−δ)+sAk
(1+n)
α)
=
(1−δ)
(1+n)
but
_
_
sf (k) = (n + δ)k
_α−1
k
=
(n+δ)
_
As
75
_ _ α−1
+
sAk
α
(1+n)
(1−δ)
(1+n)
M=
+
_ _ α−1
sAk
(1+n)
N=
α(n+δ)
(1+n)
=
<1
(n+δ)
(1+n)
∼
for k t+1 by
We can solve
∞
∼
k t+1 = N
M i εt−i
recursive substitution
i=0
Variance of the Capital Stock Around steady
state:
∼
var(k t+1)
=
∼
∼
E(k t+1 kt+1 )
= E(N
2
∞
i
M εt−i
i=0
for i = j E(εt−i εt−j ) = 0
∞
M i εt−i )
i=0
Therefore
∼
var(k t+1) = N 2
∞
i=0
N2
1−M 2 var(εt )
M 2i E(ε2t−i ) = N 2
∞
M 2i var(ε) =
i=0
Variance of the Income Around steady state:
Using the similar log-linearization for output equation
_
yt = Aeεt f (kt )
_
∼
_
_α
∼
y (1 + y t ) = A(1 + εt )k (1 + αk t )
Removing the stationary state
_∼
yyt
_ _α
_ _α ∼
_ _α
∼
= Ak + Ak αk t + Ak αεt k t
Since
∼
εt k t ≈ 0
and
76
__
_
y = Ak
we can rewrite the above
∼
yt
=
∼
αk t
+ εt
and
∼
∼
var(y t ) = α2 var(kt ) + var(εt )
and
∼
2
N
var(y t ) = (α2 1−M
2 + 1)var(εt )
77
What does the data say about convergence?
Consider the benchmark case with
x = 0.02, n = 0.01 and δ = 0.05 (for US)
where x is the long term growth rate of GDP/ per
capita
β ∗ = (1 − α)(x + n + δ) = (1 − α)(0.08) depends on
α
Suppose α = 1/3(, based on data) then β ∗ = 5.6%
(half life of 12.5 years)
But the data says that β ∗ ≃ 2 − 3% which implies
α = 3/4 (too high for physical capital)
-A broader definition of capital is needed to reconcile
theory with the facts
78
Extended Solow Model with human capital
α
η
Y = AK H [T (t)L]
·
∧
·
∧
1−α−η
∧
∧α ∧ η
andy = Ak h
∧
∧
k + h = sAk h − (x + n + δ) k + h
∧α ∧ η
(35)
(36)
It must be the case that returns to each type of capital
are equal.
∧
∧
k
h
α ∧y − δ = η y∧ − δ
∧
∧
and h = αη k
Using in (36)
·
∧
∼ ∧ α+η
k = sAk
∼
− (δ + n + x) where A =constant
β ∗ = (1 − α − η)(x + n + δ)
Now with α = 1/3(, based on data) then β ∗ = 2.1%
( a better match)
79
What’s Wrong with Neoclassical Theory??
-Does not explain long-term consistent per capita growth
rates .
-Can not maintain pefect competition assumption when
technological progress is not exogenous.
The AK Model
Y = AK
·
k
k
= sA − (n + δ) > 0 for all k if sA > (n + δ)
does not exhibit conditional convergence.. How?
How about
Y = AK + BK α L1−α whereA > 0, B > 0 and 0 < α <
1
Constant Elasticity of Substitution (CES) Production Functions
ψ
ψ
y = F (K, L) = A a(bK) + (1 − a) [(1 − b)L]
ψ1
(37)
0<a<1
80
0<b<1
and
ψ<1
The elasticity of substitution is a measure of the curvature of the isoquants where the slope of an isoquant is
given by
−1
∂(slope) L/K
1
=
∂(L/K) Slope
1−ψ
−1
∂(log(FK /FL ))
1
or − ∂(log(K/L))
= 1−ψ
(38)
Properties of CES production function
1) The elasticity of substitution between capital and
1
labor, 1−ψ
,is constant
2)CRS for all values of ψ
3) As ψ → −∞, the production function approaches
Y = min [bK, (1 − b)L] , As ψ → 0, Y = (constant)K a L1−a
(CD)
For ψ = 1, Y = abK + (1 − a)(1 − b)L (linear) so
that K and L are perfect substitutes (infinite elasticity
of substitution)
Cobb-Douglas
81
Proof: Take log of Y and apply L’Hospital’s rule.
i.e.find limψ→0 [log Y ]
]=limψ→0 [log A+ ψ1
ψ
ψ
log a(bK) + (1 − a) [(1 − b)L]
limψ→0 [log Y
ψ
1
ψ
log A + limψ→0 ψ log a(bK) + (1 − a) [(1 − b)L]
=
{a(bK)ψ log(bK)+(1−a)[(1−b)L]ψ log[(1−b)L]}
log A+limψ→0
=
{a(bK)ψ +(1−a)[(1−b)L]ψ }
log(A) + {a log(bK) + (1 − a) log [(1 − b)L]}
Transitional Dynamics with CES production function
·
k
k
= s f(k)
k − (n + δ)
Note that for CES
1−ψ
f ′(k) = Aabψ abψ + (1 − a)(1 − b)ψ k −ψ ψ
and
f (k)
k
ψ
ψ
= A ab + (1 − a) (1 − b) k
i) 0 < ψ < 1
limk→∞ (f (k)) = limk→∞ f(k)
k
limk→0 (f ′ (k)) = limk→0 f(k)
k
′
−ψ
ψ1
1
= Aba ψ > 0
=∞
Therefore, CES can exhibit can generate endogenous
growth for 0 < ψ < 1, if savings rates are high enough.
1
ψ
or in general sAba > (n + δ)
82
]=
1
If sAba ψ < (n + δ), we have neoclassical dynamics
1
Macro Lecture Notes– Ozan Hatipoglu
ψ
CES Model with (0 < ψ < 1) and sAba > δ + n
γk > 0
s
f (k )
k
1
sAbaψ
(δ + n)
kt
ii) ψ < 0
′
f(k)
k
limk→∞ (f (k)) = limk→∞
=0
1
ψ < ∞
limk→0 (f ′ (k)) = limk→0 f(k)
=
Aba
k
No endogenous growth, negative growth rates are pos-
sible for low values of s
1
If sAba ψ < (n + δ), we have negative growth
1
ψ
If sAba > (n + δ), we have neoclassical dynamics
83
1
CES Model with
Macro Lecture Notes– Ozan Hatipoglu
(ψ < 0) and sAbaψ < δ + n
(δ + n )
1
sAbaψ
γk < 0
s
f (k )
k
kt
84
Poverty Traps
Macro Lecture Notes– Ozan Hatipoglu
kt +1
Poverty Traps in the Solow Model
kt +1 = kt
G( kt )
k*
*
ktrap
kt
for some range of k, the average product of capital is
increasing in k
- non-constant savings rates
- increasing returns with learning by doing and spillovers
A simple model
Suppose the country has access to two technologies
yA = Ak α
yB = Bk α − b
85
where B>A. To employ yB gov’t has to incur a setup
cost of b per worker.
Under what conditions will the government incur b?
yA · yB
or Ak α · Bk α − b
∼
∼
and k ≥ k where k =
·
α
k
k
= s Akk − (n + δ)
k
k
= s Bkk −b − (n + δ)
·
b
B−A
α
86
α1
Neoclassical Critique of Neoclassical Paradigm
87
ENDOGENIZING SAVINGS RATE: CONSUMER OPTIMIZATION
- Role of Consumer Behavior on the economic dynamics
- Introduce consumer incentives amd see how they are
affected by endogenous factors such as interest rates or
exogenos policy tools such as tax rates, etc.
-Original ideas by Ramsey (1928), Cass and Coopmans
(1965)
The Model:
Households:
- Similar to Solow people except they make their consumption decisions according to an objective function.
- Infinitely lived, altruistic and identical.
- grow at rate n, where n is a net effect of fertility and
mortality
L(t) = ent L(0) where L(0) can be normalized to one.
C(t) = total consumption.
c(t) =
C(t)
L(t)
is the con-
sumption per individual
Each household tries to maximize overall Utility U
88
∞
U = U (c(t)ent e−ρt dt
(39)
0
where u′ (c) > 0, u′′ (c) < 0. Concavity implies people
prefer to smooth their consumption. They prefer a uniform pattern in consumption over a volatile one.
The concavity assumption is the determinant of household consumption behavior: They will tend to borrow
when income is low and save when income is high.
Moreover:
Inada conditions hold
limc→0 u′ (c) = ∞
limc→∞ u′ (c) = 0
ρ > 0 is the rate of time preference. Later utils are
valued less. We assume individuals discount their own
utility at a constant rate but one might also distinguish
the rate at different points in one’s own life
from the rate across generations.
(i.e. ρ =. ρ(t)).
Or one might assume time preference increases with the
number of children such that ρ =. ρ(n))
Households own assets and supply labor similar to
89
Solow-continuous version model such that (13) holds
·
a = (ra + w) − c − na
(40)
Moreover, net debts in the economy are zero in eq
because it is a closed economy.
Return to capital and assets are the same, r, since they
are perfect substitutes.
No Ponzi game:
Suppose some households can borrow an unlimited
amount at the ongoing interest rate, then they might
pursue a Ponzi game (chain letter game).
1) Borrow today to finance current consumption
2) Borrow tomorrow to roll over the prinicipal and pay
all the interest.
In this game, debt grows forever at rate r. Since no
principal is paid, today’s added consumption is effectively free.
To prevent this game we assume credit markets impose
a constraint on the amount of household’s borrowing.
Present value of assets must asymptotically non-negative.
90
lim
t→∞
t
a(t) exp −
[r(v) − n] dv
≥0
(41)
0
where exp [] = e[]
In the long-run a household’s debt per person (negative a(t))can not grow as fast as r(t) − n.
Formal description of the optimization problem
∞
max U (c(t)ent e−ρt dt subject to
{c(t)}
0
·
i ) a = (ra + w) − c − na
t
ii) limt→∞ a(t) exp − 0 [r(v) − n] dv
≥0
iii) a(0) given
iv) c(t) ≥ 0
Because of Inada conditions iv) will never bind.
First Order Conditions
The present-value Hamiltonian
H = u(c(t))e−(ρ−n)t + v(t) {w(t) + [r(t) − n] a(t) − c(t)}
(42)
91
v(t) is the shadow price of income. It represents the
value of a unit increase in income received at time t in
units of utils at time 0.
(Some times, alternatively we denote v(t) = λve−(ρ−n)t .
In this case v(t) I represents the value of a unit increase
in income received at time t in units of utils at time t.)
It depends on time because a household faces a continuum of constraints for each instant.
∂H
= 0 → v = u′ (c)e−(ρ−n)t
∂c
(43)
∂H
·
→ v = −(r − n)v
∂a
(44)
·
v=−
Transversality condition:
lim v(t)a(t) = 0
t→∞
Differentiate 43 wr.t. time
·
·
v = u′′ (c)ce−(ρ−n)t − (ρ − n)u′ (c)e−(ρ−n)t
Substituting in 44
92
(45)
·
−(r − n)v = u′′ (c)ce−(ρ−n)t − (ρ − n)u′ (c)e−(ρ−n)t
Using (43)
·
−(r−n)u′ (c)e−(ρ−n)t = u′′ (c)ce−(ρ−n)t −(ρ−n)u′ (c)e−(ρ−n)t
·
−ru′ (c) = u′′ (c)c − ρu′ (c)
r =ρ−
u′′ (c)c
u′ (c)
·
c
c
(46)
Rate of return to savings= Rate of return to consumption
′′
·
u (c)c
c
r =ρ+ − ′
c
u (c)
Elasticity of
consumption
(i.e. the percentage change in marginal utility with
respect to a percentage change in consumption:
′
′
(c)
− du (c)/u
)
dc/c
The higher the elasticity, the higher the premium required by the households to change their consumption
levels.
93
·
ex: let
c
c
> 0 consumption is low relative to tomorrow.
In this case for a given level of consumption today r
increases with the growth rate of consumption.
Note that Elasticity of consumption=1/ intertemporal
elasticity of substitution
′
(c)
intertemporal elasticity of substitution σ = − uu′′ (c)c
c(t1 )/c(t2 )
−u′ (c(t1 ))/u′ (c(t2 ))
Note that σ =
d[u′ (c(t1 ))/u′ (c(t2 ))]
d[c(t1 )/c(t2 )]
′
(c)
limt2→t1 σ = − uu′′ (c)c
·
To find a steady state in which r and cc are constant in
46 it must be the case that the elsaticity is asymptotically
constant. One utility form that satisfies this condition is
the CIES utility function
c1−θ − 1
u(c) =
1−θ
(47)
where θ > 0 so that the elasticity of marginal utility
equals the constant −θ. The elasticity of substitiution is
σ = 1θ . As θ increases people are less willing to accept deviations from a consant consumption path. Substituting
in 46
94
·
c 1
= (r − ρ)
c θ
95
(48)