1. No category

Ozan Hatipoglu
Macro Lecture Notes
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.
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.
Ozan Hatipoglu
Macro Lecture Notes
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? ,
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
Ozan Hatipoglu
Macro Lecture Notes
Other Assumptions regarding the systems:
Open vs. Closed.
Input and Output Economies
Interrelatedness of Markets. Contagion.
Ozan Hatipoglu
Macro Lecture Notes
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)
Ozan Hatipoglu
Macro Lecture Notes
Elhanan Helpman (1990)
Frank Caselli (2000)
Daron Acemoglu (2003)
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.
Ozan Hatipoglu
Macro Lecture Notes
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
con ict 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
Ozan Hatipoglu
Macro Lecture Notes
and thereby the endogenous determination of savings rate.
The equilibrium can then be supported by a decentralized, competitive framework as in the neoclassical tradition.
1965-1985
Lack of empirical evidence - death of the growth theory
Focus on short term uctuations.
Rational Expectations
Paradigm Business cycle models.
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
Ozan Hatipoglu
Macro Lecture Notes
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.
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
Ozan Hatipoglu
Macro Lecture Notes
rights.
Long term growth rate depends on government actions,
policies such as
infrastructure
protection of property rights
labor market regulations
Aghion et al.(2002)
The role of competition in creating new ideas
Acemoglu, Johnson andRobinson(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)
Ozan Hatipoglu
Macro Lecture Notes
Persson and Tabellini (1994)
Endogenizing Fertility Choice
Barro and Becker (1989)
Parents and Children are linked through altruism.
Role of Intergenerational transfers
Some Empirical Episodes
1st Industrial Revolution
Great Depression
2nd Industrial Revolution
Oil crisis and the productivity slowdown
3rd Industrial Revolution
Ozan Hatipoglu
Macro Lecture Notes
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 Mex-
ico, $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
Ozan Hatipoglu
Macro Lecture Notes
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
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
Ozan Hatipoglu
Macro Lecture Notes
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.
Ozan Hatipoglu
Macro Lecture Notes
• 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
Ozan Hatipoglu
Macro Lecture Notes
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.
Ozan Hatipoglu
Macro Lecture Notes
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
Ozan Hatipoglu
Macro Lecture Notes
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
∂F K and ε ≡ ∂F L are capital elasticity of outεK ≡ ∂K
L
F
∂L F
put 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.
Ozan Hatipoglu
Macro Lecture Notes
"Per capita" variables and intensive forms:
Let y ≡ Y /L (output per worker) and k ≡ K/L. (capital
per worker) Since Y = F (K, L, T ) is CRS, letting λ =
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 0(k) > 0
Ozan Hatipoglu
Macro Lecture Notes
00
f (k) < 0
lim f 0(k) = ∞
k→0
00
lim f (k) = 0
k→∞
Marginal Products in Intensive Form
Since Y = Lf (k)
∂Y = ∂Lf(k) = L 1 f 0 (k) = f 0 (k)
L
∂K
∂K
∂Y = f (k) − kf 0 (k)
∂L
Ozan Hatipoglu
Macro Lecture Notes
Example: Cobb-Douglas
Y = AK αL1−α
where A > 0 is the level of technology, α is a constant
with 0 < α < 1
³ ´α
f (k) = A K
L
= Ak α
Does CD production fucntion satisy neoclassical properties
1) Y = AK αL1−α is CRS
2) positive and diminishing marginal products
f 0(k) = Aαk α−1 > 0
f 00(k) = Aα(α − 1)k α−2 < 0
3) Inada conditions
lim Aαkα−1 = ∞
k→0
lim Aαk α−1 = 0
k→∞
4. inputs are essential:
f (0) = 0
Ozan Hatipoglu
Macro Lecture Notes
In a competitive economy with a Cobb-Douglas type
production, capital and labor are each paid their marginal
products such that
R = f 0(k) = Aαk α−1
and
w = f (k) − kf 0(k) = (1 − α)Akα
The capital share of income is
capital income
Rk
total income = f (k) = α
and the labor share of income is given by:
labor income = w = 1 − α
total income
f (k)
Thus, in a competitive economy with with a Cobb-Douglas
type production factor income shares are constant (independent
of k)
Ozan Hatipoglu
Macro Lecture Notes
Centralized Dictatorial Allocations
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.
Ozan Hatipoglu
Macro Lecture Notes
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.
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
Ozan Hatipoglu
Macro Lecture Notes
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)tL0
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.
Ozan Hatipoglu
Macro Lecture Notes
That is, the law of motion for capital is
Kt+1 = (1 − δ)Kt + It
(3)
Equivalently, in per-capita terms:
Kt+1
Kt
It
Lt = (1 − δ) Lt + Lt
since
Lt+1 = (1 + n)Lt
(1+n)Kt+1
= (1 + n)kt+1 = (1 − δ)kt + it
Lt+1
(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
The sum δ + n can thus be interpreted as the “effective”
depreciation rate of per-capita capital. (Remark: This ap-
Ozan Hatipoglu
Macro Lecture Notes
proximation 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
Ozan Hatipoglu
Macro Lecture Notes
That is, the change in the capital stock is given by aggregate output, minus capital depreciation, minus aggregate consumption
In capita terms:
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.
Ozan Hatipoglu
Macro Lecture Notes
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
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+1k−kt = sϕ(kt) − (δ + n)
t
where
Ozan Hatipoglu
Macro Lecture Notes
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).
Ozan Hatipoglu
Macro Lecture Notes
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
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
Ozan Hatipoglu
Macro Lecture Notes
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 ϕ0(k) =
f 0 (k)k−f(k)
FL
=
−
2
k
k 2 < 0,
ϕ(0) = f 0(0) = ∞
and ϕ(∞) = f 0(∞) = 0
Ozan Hatipoglu
Macro Lecture Notes
where the latter follow from L’Hospital’s rule. This implies that equation (6) has a unique solution:
k ∗ = ϕ−1
³
δ+n
s
´
Since ϕ0 < 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:
Ozan Hatipoglu
Macro Lecture Notes
P roposition. Given any initial k0 ∈ (0, ∞), the dictator-
ial economy converges asymptotically to the steady state.
The transition is monotonic. The growth rate is 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, G0(k) = sf 0(k) + (1 −
δ − n) > 0 and G00(k) = sf 00(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 ∗. It follows that kt <
kt+1 < k ∗ whenever kt ∈ (0, k ∗) and therefore the se∗
quence {kt}∞
t=0is strictly increasing if k0 < k . By monotonicˆ
ity, kt converges asymptotically to some k ≤ k∗. By conˆ
ˆ
ˆ
ˆ
tinuity of G, k must satisfy k = G(k), that is k must be
a f ixed point of G. But we already proved that G has a
ˆ
unique fixed point, which proves that k = k∗. A symmet-
Ozan Hatipoglu
Macro Lecture Notes
ric argument applies when k0 > k∗
Next, consider the growth rate of the capital stock. This
is given by
kt+1 −kt
= sϕ(kt) − (δ + n) = γ(kt)
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, γ0(kt) = sϕ0(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 k0 > k∗.
Ozan Hatipoglu
Macro Lecture Notes
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 consump-
Ozan Hatipoglu
Macro Lecture Notes
tion.
At the steady state we have:
c∗ = (1 − s)f (k ∗)
since at the steady state sf (k∗) − (δ + n)k ∗ = 0 at the
steady state
c∗ = (1 − s)f (k ∗) = f (k ∗) − (δ + n)k ∗
since k∗ is a function of the parameters as given in (6)
f (k ∗ )
ϕ(k∗) = δ+n
=
s
k∗
and ϕ(k∗) has a unique solution as we proved earlier, c∗
is also a function of the parameters.
dc∗ = f 0 (k ∗ (s)) dk∗ −(δ+n) dk ∗ = [f 0 (k ∗ (s))−(δ+n)] dk∗ = 0
ds
ds
ds
ds
Since
k ∗ = ϕ−1
³
δ+n
s
´
AND ϕ0(k∗) < 0, k∗ is a de-
creasing function of (δ + n)/s , therefore
dk ∗ > 0
ds
It must be the case that [f 0(k∗(s)) − (δ + n)] = 0
∗ (s)) = (δ + n)
f 0(kgold
mulation)
(Golden Rule of Capital Accu-
Ozan Hatipoglu
Macro Lecture Notes
and
∗ ) − (δ + n)k ∗
c∗gold = f (kgold
gold
Discussion Questions:
1) Is there a "best" savings rate s that the policy maker
can choose in the Solow Model. Explain in detail..
∗
for the CD production function.
2) Calculate for kgold
Ozan Hatipoglu
Macro Lecture Notes
Decentralized Market Allocation:
Households are dynasties, living an infinite amount of
time. We index households by j ∈ [0, 1], having normalized L0 = 1.
The number of heads in every household grow at a constant rate n ≥ 0. Therefore, the size of the population in
period t is Lt = (1 + n)t and the number of persons in
each household in period t is also Lt.
We write cjt , ktj , bjt , ijt for the per-head variables for
household j.
Each person in a household is endowed with one unit of
labor in every period, which he supplies inelastically in a
competitive labor market for the contemporaneous wage
j
wt. Household j is also endowed with initial capital k0 .
Capital in household j accumulates according to
j
j
j
(1 + n)kt+1 = (1 − δ)kt + it ,
Ozan Hatipoglu
Macro Lecture Notes
which we approximate by
j+1
kt
j
j
= (1 − δ − n)kt +it
Households rent the capital they own to firms in a competitive market for a (gross) rental rate Rt.
The household may also hold stocks of some firms in
the economy. Let πjt be the dividends (firm’s profits) that
household j receive in period t. It is without any loss
of generality to assume that there is no trade of stocks
(because the value of stocks will be zero in equilibrium).
We thus assume that household j holds a fixed fraction aj
of the aggregate index of stocks in the economy, so that
j
π t = aj Πt,
R
where Πt are aggregate profits and aj dj = 1
The household uses its income to finance either consumption or investment in new capital:
j
j
j
ct + it = yt .
Total per-head income for household j in period t is
Ozan Hatipoglu
Macro Lecture Notes
simply
j
j
j
yt = wt + Rtkt + πt .
Combining, we can write the budget constraint of household j in period t as
j
j
j
j
ct + it = wt + Rtkt + π t
Finally, the consumption and investment behavior of
household is a simplistic linear rule. They save fraction s
and consume the rest:
j
j
ct = (1 − s)yt
and
j
j
it = syt .
Ozan Hatipoglu
Macro Lecture Notes
Firms
There is an arbitrary number Mt of firms in period t, indexed by m ∈ [0, Mt]. Firms employ labor and rent capital
in competitive labor and capital markets, have access to
the same neoclassical technology, and produce a homogeneous good that they sell competitively to the households
in the economy.
Let Ktm and Lm
t denote the amount of capital and labor
that firm m employs in period t. Then, the profits of that
firm in period t are given by
m m
m
m
Πm
t = F (Kt , Lt ) − Rt Kt − wt Lt .
The firms seek to maximize profits. The FOCs for an
interior solution require
FK (Ktm, Lm
t ) = Rt .
FL(Ktm, Lm
t ) = wt .
Remember that the marginal products are homogenous
Ozan Hatipoglu
Macro Lecture Notes
of degree zero; that is, they depend only on the capitallabor ratio. In particular, FK is a decreasing function
m m
of Ktm/Lm
t and FLis an increasing function of Kt /Lt
. Each of the above conditions thus pins down a unique
capital-labor ratio Ktm/Lm
t . For an interior solution to
the firms’ problem to exist, it must be that Rt and wt are
consistent, that is, they imply the same Km/Lm . This is
the case if and only if there is some Xt ∈ (0, ∞) such that
Rt = f 0(Xt)
(7)
wt = f (Xt) − Xtf 0(Xt)
(8)
and
where f (k) ≡ F (k, 1); this follows from the proper-
Ozan Hatipoglu
Macro Lecture Notes
ties FK (K, L) = f 0(K/L) and FL(K, L) = f (K/L) −
f 0(K/L)(K/L), which we established earlier. That is, (wt, Rt)must
satisfy wt = W (Rt)· where
−1
−1
W (r) ≡ f (f 0 (r)) − rf 0 (r).
If (7) and (8) are satisfied, the FOCs reduce to Ktm/Lm
t
= Xt, or
Ktm = XtLm
t .
That is, the FOCs pin down the capital-labor ratio for
m
each firm (Ktm/Lm
t ), but not the size of the firm (Lt
). Moreover, Ktm = XtLm
t imply all firms use the same
capital-labor ratio
.
(7) and (8) imply
RtXt + wt = f (Xt).
it follows that
RtKtm + wtLm
= (RtXt + wt)Lm
= f (Xt)Lm
=
t
t
t
Ozan Hatipoglu
Macro Lecture Notes
F (Ktm, Lm
t ),
and therefore
Πm = Lm[f (Xt) − RtXt − wt] = 0.
That is, when (7) and (8) are satisfied, the maximal
profits that any firm makes are exactly zero, and these
profits are attained for any firm size as long as the capitallabor ratio is optimal. If instead (7) and (8) were violated, then either RtXt + wt < f (Xt), in which case the
firm could make infinite profits, or RtXt + wt > f (Xt),
in which case operating a firm of any positive size would
generate strictly negative profits.
Market Clearing
The capital market clears if and only if
R Mt
R
mdm = 1 (1+ n)t k j dj
K
t
t
0
0
or
R Mt
0
Ktmdm = Kt
Ozan Hatipoglu
Macro Lecture Notes
R Lt j
where Kt = 0 kt dj is the aggregate capital stock in the
economy.
The labor market, on the other hand, clears if and only
if
R Mt m
R1
t
L
dm
=
t
0
0 (1+ n) dj
or
R Mt m
0 Lt dm = Lt
Ozan Hatipoglu
Macro Lecture Notes
General Equilibrium
1.Def inition
The definition of a general equilibrium is more
meaningful when households optimize their behavior (maximize
utility) rather than being automata (mechanically save a
constant fraction of income). Nonetheless, it is always
important to have clear in mind what is the definition
of equilibrium in any model.
For the decentralized version of the Solow model:
An equilibrium of the economy is an allocation
j
j
j
{(kt , ct , it )j∈[0,1],
j
∞
∞
(Ktm, Lm
t , )m∈[0,Mt] }t==0 , a distribution of profits {(π t )j∈[0,1] }t=0 ,
and a price path {Rt, wt}∞
t=0 such that
j
j
j
j
∞
∞
(i) Given {Rt, wt}∞
t=0and {(π t )j∈[0,1] }t=0, the path {(kt , ct , it )j∈[0,1] }t
is consistent with the behavior of household j for every j .
(ii) (Ktm,Lm
t ) maximizes firm profits, for every m and
Ozan Hatipoglu
Macro Lecture Notes
t.
(iii) The capital and labor markets clear in every period
2. Characterization
For any initial positions (k0j ), j ∈ (0, 1), an equilibrium
exists. The allocation of production across firms is indeterminate, but the equilibrium is unique with regard to
aggregates and household allocations.
i) The capital-labor ratio in the economy is given by
{kt}∞
t=0such that, for all t ≥ 0,
kt+1 = G(kt)
(9)
R1 j
with k0= 0 k0 dj given and with G(kt) ≡ sf (kt) + (1 −
δ − n)kt.
ii) Equilibrium growth is
Ozan Hatipoglu
Macro Lecture Notes
− kt
k
γ t = t+1
= γ(kt),
kt
(10)
where γ(kt) = sϕ(kt) − (δ + n) and ϕ(kt) = f (kt)/kt.
iii) Finally, equilibrium prices are given by
Rt = r(kt) = f 0(kt),
(11)
wt = w(kt) ≡ f (kt) − f 0(kt)kt
(12)
and
P roof :
We first characterize the aggregate equilibrium, assuming it exists. Using Ktm = XtLm
t , we can write the aggreR Mt
R
mdm = X Mt Lmdm.
K
t 0
t
t
0
R
From the labor market clearing condition 0Mt Lm
t dm =
R Mt m
Lt, combining, we get 0 Kt dm = XtLt, and substitut-
gate demand for capital as
Ozan Hatipoglu
Macro Lecture Notes
ing in the capital market clearing condition, we conclude
R
j
XtLt = Kt, where Kt = 0Lt kt dj denotes the aggregate
capital stock. Equivalently, letting kt = Kt/Lt denote the
capital-labor ratio in the economy, we have Xt = kt.
That is, all firms use the same capital-labor ratio as the
aggregate of the economy.
Proof of iii)
Substituting Xt = kt into (7) and (8) we infer that equilibrium prices are given by
Rt = r(kt) ≡ f 0(kt) = FK (kt, 1)
wt = w(kt) ≡ f (kt) − f 0(kt)kt = FL(kt, 1)
Proof of i)
Adding up the individual capital accumulation rules
i
R j
R h
j
j
kt+1dj =
(1 − δ − n)kt + it dj
we get the capital accumulation rule for the aggregate
of the economy. In per-capita terms,
Ozan Hatipoglu
Macro Lecture Notes
kt+1 = (1 − δ − n)kt + it
Adding up cjt = (1 − s)ytj and ijt = sytj across
households, we similarly infer
it = syt = sf (kt).
Combining, we conclude
kt+1 = sf (kt) + (1 − δ − n)kt = G(kt),
which is exactly the same as in the centralized allocation
Proof of ii) Trivial
Note that r0(kt) = f 00(kt) = FKK < 0 and w0(k) =
−f 00(kt)kt = FLK > 0. That is, the interest rate is a de-
creasing function of the capital-labor ratio and the wage
rate is an increasing function of the capital-labor ratio.
The first property represents diminishing returns, the second represents the complementarity of capital and labor.
Ozan Hatipoglu
Macro Lecture Notes
Q: Show that the resource constraint of this economy
canbe written as ct + it = f (kt).
Adding up the budget constraints of the households, we
get
R j
Ct + It = RtKt + wtLt + π t dj
R1 j
R1 j
where Ct = 0 ct dj and It = 0 it dj . Aggregate divR
R
idends must equal aggregate profits, 01 πjt dj = 0Mt Πm
t
R1 j
dj. Since profits for each firm, Πm
are
zero,
t
0 π t dj = 0,
implying
Ct + It = RtKt + wtL
Equivalently, in per-capita terms,
ct + it = Rtkt + wt.
Rtkt + wt = yt = f (kt) (no-profit condition)
We conclude that the household budgets imply
ct + it = f (kt)
which is simply the resource constraint of the economy.
Ozan Hatipoglu
Macro Lecture Notes
2) Existence and Uniqueness
Finally, existence and uniqueness is now trivial. (9)
maps any kt ∈ (0, ∞)to a unique kt+1 ∈ (0, ∞). Similarly, (11) and (12) map any kt ∈ (0, ∞) to unique Rt, wt ∈
R1 j
(0, ∞). Therefore, given any initial k0 = 0 k0 dj, there ex-
∞
∞
ist unique paths {kt}∞
t=0and {Rt , wt}t=0. Given {Rt , wt}t=0
the allocation {ktj , cjt , ijt } for any household j is then uniquely
determined by
i) ktj+1 = (1 − δ − n)ktj +ijt
ii) ytj = wt + Rtktj + π jt
iii) cjt = (1 − s)ytj and ijt = sytj .
Finally, any allocation(Ktm, Lm
t ), m ∈ [0, Mt]of production across firms in period t is consistent with equilibrium
as long as Ktm = ktLm
t .
P roposition : The aggregate and per-capita allocations
in the competitive market economy coincide with those
Ozan Hatipoglu
Macro Lecture Notes
in the dictatorial economy.
We can thus immediately translate the steady state and
the transitional dynamics of the centralized plan to the
steady state and the transitional dynamics of the decentralized market allocations.
Remark: This example is just a prelude to the first and
second welfare theorems, which we will have once we
replace the “rule-of-thumb” behavior of the households
with optimizing behavior given a preference ordering over
different consumption paths: in the neoclassical growth
model, Pareto efficient and competitive equilibrium allocations coincide.
Ozan Hatipoglu
Macro Lecture Notes
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 = AtF (Kt, Lt)
yt = Atf (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.
Ozan Hatipoglu
Macro Lecture Notes
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
Ozan Hatipoglu
Macro Lecture Notes
a permanent shock.
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?
Ozan Hatipoglu
Macro Lecture Notes
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 frac-
tion s of disposable income:
it = s(yt − gt).
Combining the above, we conclude that the dynamics
of capital are now given by
γ t = kt+1k−kt = s(1 − τ )ϕ(kt) − (δ + n)
t
where ϕ(k) ≡ f (k)/k. Given s and kt, the growth rate
Ozan Hatipoglu
Macro Lecture Notes
γ t decreases with τ
A steady state exists for any τ ∈ [0, 1) and is given by
k ∗ = ϕ−1
³
δ+n
s(1−τ )
´
Given s, k∗ decreases with τ .
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) =
β
ktα gt , 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)
Ozan Hatipoglu
Macro Lecture Notes
it = s(yt − gt).
Substituting gt = τ yt into yt = kαgβ and solving for yt
yt = k α (τ yt)β
1−β
yt
= kατ β
α
(1−β)
yt = kt
τ
β
(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
α−(1−β)
β
kt+1 −kt
(1−β)
(1−β)
γt = k
= s(1 − τ )kt
τ
− (δ + n)
t
and
k∗ =
Ã
β
(1−β)
s(1−τ )τ
(δ+n)
!
(1−β)
(1−β)−α
The tax rate which maximizes either k∗
Result: The more productive government services are,
the higher their “optimal” provision.
Ozan Hatipoglu
Macro Lecture Notes
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
·
·
·
·
·
·
d(K/L)
L = K − K L = K − nk
k = dt = LK−K
L
LL
L
L2
where n =
·
L
L
Therefore
·
·
K(t)
L = k + nk
and
·
k = sf (k) − (n + δ)k (Fundamental Differential Equa-
tion of Solow-Swan Model)
Ozan Hatipoglu
Macro Lecture Notes
Compare with the approximation in the discrete time
version
kt+1 − kt ∼ sf (kt) − (δ + n)kt
Ozan Hatipoglu
Macro Lecture Notes
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)
= r(t) × assets + w(t)L(t) − C
dt
where C is total consumption.
Let a =
Define
·
assets
L
·
·
L d(assets)
−(assets)
L
d((assets)/L)
dt
a=
=
=
L2
dt
(assets) L
L
L =
d(assets)
dt
Therefore
L
− na
d(assets)
dt
L
−
Ozan Hatipoglu
Macro Lecture Notes
·
a = (ra + w) − c − na
(13)
Firm’s problem:
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
Ozan Hatipoglu
Macro Lecture Notes
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 0(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
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 0(k)
(16)
Ozan Hatipoglu
Macro Lecture Notes
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 0(k) − δ
and
w = f (k) − kf 0(k)
Substituting in 13
·
k = f (k) − c − (n + δ)k
(17)
Ozan Hatipoglu
Macro Lecture Notes
Labor markets clear: Labor supplied=Labor demanded
Since labor is supplied inelatically 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.
Ozan Hatipoglu
Macro Lecture Notes
Example: Cobb Douglas Production function
F (K, L) = AK α L1−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
iα
h
1
1
α
∗
1−α
1−α
1−α
y = A [sA/(n + δ)]
= A [s/(n + δ)]
The time path of capital given k(0) is given by
·
k = sAkα − (n + δ)k
(20)
Ozan Hatipoglu
Macro Lecture Notes
We can solve for the exact time path of k by
rewriting the above as
·
kk −α + (n + δ)k1−α = sA
Substituting v = k1−α
·
−α
v = (1 − α)k 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
v = k 1−α =
P roof :
·
sA +
(n+δ)
n
o
1−α
sA
[k(0)]
− (n+δ) e−(1−α)(n+δ)t
v + (1 − α)(n + δ)v = (1 − α)sA
Ozan Hatipoglu
Macro Lecture Notes
Z
Z
h·
i
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)tv + b0 where b0 is a constant then
dB = e(1−α)(n+δ)t v v· +e(1−α)(n+δ)t v.(1 − α)(n + δ) =
dt
i
h
(1−α)(n+δ)t ·
=e
v + (1 − α)(n + δ)v
Z
h·
i
R dB
(1−α)(n+δ)t
B = dt dt = e
v + (1 − α)(n + δ)v dt =
e(1−α)(n+δ)tv + b0
The solution to the right hand side of 22
Z
sA e(1−α)(n+δ)t + b
e(1−α)(n+δ)t(1 − α)sAdt = (n+δ)
1
Combining left handside and the right handside
sA e(1−α)(n+δ)t + b − b
e(1−α)(n+δ)tv = (n+δ)
1
0
sA + (b − b )e−(1−α)(n+δ)t
v(t) = (n+δ)
1
0
How do we determine b1 − b0? Using the steady state
condition for k
Ozan Hatipoglu
Macro Lecture Notes
sA + (b − b )
v(0) = k(0)1−α = (n+δ)
1
0
sA
(b1 − b0) = k(0)1−α − (n+δ)
Therefore
sA +
v(t) = (n+δ)
n
o
sA
1−α
k(0)
− (n+δ) e−(1−α)(n+δ)t
q.e.d
The gap between k1−α and its steady state value vanishes exactly at the constant rate (1 − α)(n + δ).
Ozan Hatipoglu
Macro Lecture Notes
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
·
γk =
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
h
i
f (k)
0
s f (k)− k
∂(γ k )
=
=
<0
∂k
∂k
k
·
k
∂( k )
·
∙ 0 ¸·
·
kf (k) k
y f 00(k)k
=
=
γy = =
y
f (k)
f (k) k
·
= [CapitalShare]
(25)
k
= [CapitalShare] [sf (k)/k − (n +(1)
δ)]
k
Ozan Hatipoglu
Macro Lecture Notes
= sf 0(k) − (n + δ) [CapitalShare]
·
i·
h 00
∂( yy )
∂(γ y )
f (k)k k (n+δ)f 0 (k)
∂k = ∂k = f (k) k − f (k) [1−CapitalShare] <
0 if
·
k ≥0
k
(since CapitalShare < 1)
Therefore The Solow Model predicts absolute convergence
Poor countries tend to grow faster than the rich countries
Ozan Hatipoglu
Macro Lecture Notes
Conditional Convergence
Consider the s.s. condition
sf (k ∗) = (n + δ)k ∗
∗
k
s = (n + δ) f (k
∗)
substituting in 24
·
k = (n + δ) k ∗ [f (k)/k − (n + δ)]
k
f (k ∗ )
∙
¸
f (k)/k
k
= (n + δ)
−1
k
f (k ∗)/k ∗
·
(26)
A reduction in k increases the average product of capital and increases
·
k.
k
But a lower k increases
·
k
k
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, be-
Ozan Hatipoglu
Macro Lecture Notes
cause 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
·
k
= (n + δ)
k
"µ
k
k∗
¶α−1
#
−1
(27)
Proposition: Absolute Convergence and a Decrease in
the Dispersion of incomes are not equivalent
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
Ozan Hatipoglu
Macro Lecture Notes
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 =
PN
2
i=1(log(yi,t ) − μ)
Using 28
Dt = (1 − b)2Dt−1 + σ 2u
which has a steady state at D∗ =
σ 2u
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
Ozan Hatipoglu
Macro Lecture Notes
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.
Ozan Hatipoglu
Macro Lecture Notes
Technological Progress
Some Definitions
A capital saving technological progress (or invention)
allows producers to produce the same amount with relatively less capital input.
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:
Ozan Hatipoglu
Macro Lecture Notes
Y = T (t)F (K, L)
ii) "Harrod neutral": relative input shares K.Fk /LFLremain
the same for a given capital output ratio
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)
Ozan Hatipoglu
Macro Lecture Notes
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)
·
·
F (k,T (t))
k = sF (k, T (t)) − (n + δ)k and kk = s
− (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
s
µ · ¶∗
k
k
is constant
F (k ∗ ,T (t))
− (n + δ) =constant
k∗
Since F is CRS,
F (k,T (t))
T (t)
=
F
(1,
k
k )This
implies that
Ozan Hatipoglu
Macro Lecture Notes
T (t) and k grow at the same rate x, because s,n and δ are
constants
µ · ¶∗
k
k
=x
Moreover since y = F (k, T (t)) = kF (1, T k(t) )
à · !∗
y
=x
y
and c = (1 − s)y,
³·´
c
c
=
Transitional Dynamics
µ
·
(1−s)y
(1−s)y
(30)
¶
=x
Define: effective amount of labor=physical quantity of
∧
labor× efficiency of labor = L × T (t) ≡ L
∧
k = LTK(t) = T k(t) = capital per unit of effective labor.
∧
∧
Y
y = LT (t) = F (k, 1) = f (k)
∧
labor
We can rewrite
= output per unit effective
Ozan Hatipoglu
Macro Lecture Notes
·
divide both sides by T (t)L
K = sF (K, T (t)L) − δK
·
∧
∧
K
= sf (k) − δ k
T (t)L
·
∧
k=
∧
kx
T (t)L
³ · ´
K
T (t)L
Therefore
·
(31)
·
·
·
∧
LT (t)K−K(LT (t)+LT (t))
K − kn −
=
=
2
2
T (t) L
T (t)L T (t)L
·
·
∧
∧
∧
K = k + kn + kx
T (t)L
T (t)L
T (t)L
substituting in (31)
·
∧
∧
∧
k = sf (k) − (x + n + δ) k
(32)
and
·
∧
∧
f (k)
=
s
− (x + n + δ)
∧
∧
k
k
k
(32)
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Macro Lecture Notes
where x + n + δ is the effective depreciation rate
The effective per capita capital depreciates at the rate
x+n+δ
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)
*
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Macro Lecture Notes
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
Speed of Convergence:
The speed of convergence is given by
·
∧
β=−
∂( k∧ )
k
∧
∂ log k
For the CD production fucntion
(33)
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Macro Lecture Notes
·
∧
∧
k = sA(k)−(1−α) − (x + n + δ)
∧
k
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)
(34)
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
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Macro Lecture Notes
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
Ozan Hatipoglu
Macro Lecture Notes
Extended Solow Model with human capital
∧α ∧η
∧
1−α−η
α
η
Y = AK H [T (t)L]
andy = Ak h
·
∧
·
∧
µ
¶
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.
y
∧
y
∧
k
h
α∧ − δ = η ∧ − δ
∧
∧
η
and h = α 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)
Ozan Hatipoglu
Macro Lecture Notes
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 = sA − (n + δ) > 0
k
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
Ozan Hatipoglu
Macro Lecture Notes
o1
n
ψ ψ
ψ
y = F (K, L) = A a(bK) + (1 − a) [(1 − b)L]
(37)
0<a<1
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
dL = − ∂F/∂K
dK
∂F/∂L
∙
¸
1
∂(slope) L/K −1
=
∂(L/K) Slope
1−ψ
(38)
Properties of CES production function
1) The elasticity of substitution between capital and
Ozan Hatipoglu
Macro Lecture Notes
1 ,is constant
labor, 1−ψ
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 aL1−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)
Proof: Take log of Y and apply L’Hospital’s rule. i.e.find
limψ→0 [log Y ]
Transitional Dynamics with CES production function
·
k = s f (k) − (n + δ)
k
k
Note that for CES
h
f 0(k) = Aabψ abψ + (1 − a)(1 − b)ψ k −ψ
i 1−ψ
ψ
Ozan Hatipoglu
Macro Lecture Notes
and
h
f (k)
ψ + (1 − a) (1 − b)ψ k −ψ
=
A
ab
k
i1
ψ
i) 0 < ψ < 1
h
i
1
£ 0 ¤
f (k)
limk→∞ f (k) = limk→∞ k = Aba ψ > 0
h
i
£ 0 ¤
f (k)
limk→0 f (k) = limk→0 k = ∞
Therefore, CES can exhibit can generate endogenous
growth for 0 < ψ < 1, if savings rates are high enough.
1
or in general sAba > (n + δ)
ψ
1
If sAba < (n + δ), we have neoclassical dynamics
ψ
Ozan Hatipoglu
Macro Lecture Notes
1
Macro Lecture Notes– Ozan Hatipoglu
CES Model with (0 < ψ < 1) and sAba > δ + n
ψ
γk > 0
s
f (k )
k
1
sAbaψ
(δ + n)
kt
ii) ψ < 0
h
i
£ 0 ¤
f (k)
limk→∞ f (k) = limk→∞ k = 0
h
i
1
£ 0 ¤
f (k)
limk→0 f (k) = limk→0 k = Aba ψ < ∞
No endogenous growth, negative growth rates are pos-
sible for low values of s
1
If sAba ψ < (n + δ), we have negative growth
Ozan Hatipoglu
Macro Lecture Notes
1
If sAba > (n + δ), we have neoclassical dynamics
ψ
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
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Macro Lecture Notes
Poverty Traps
Macro Lecture Notes– Ozan Hatipoglu
kt +1
Poverty Traps in the Solow Model
kt +1 = kt
G ( kt )
k
*
trap
k*
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
Ozan Hatipoglu
Macro Lecture Notes
yA = Ak α
yB = Bk α − b
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 Akα ≤ Bkα − b
∼
and k ≥ k where
∼
b
k = B−A
·
k = s Ak α − (n + δ)
k
k
·
h
k = s Bk α −b − (n + δ)
k
k
i1
α
Ozan Hatipoglu
Macro Lecture Notes
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
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Macro Lecture Notes
L(t) = entL(0) where L(0) can be normalized to one.
C(t)
C(t) = total consumption. c(t) = L(t) is the consump-
tion per individual
Each household tries to maximize overall Utility U
Z∞
U = U (c(t)ente−ρtdt
(39)
0
where u0(c) > 0, u00(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 u0(c) = ∞
Ozan Hatipoglu
Macro Lecture Notes
limc→∞ u0(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 Solowcontinuous 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
Ozan Hatipoglu
Macro Lecture Notes
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.
Ozan Hatipoglu
Macro Lecture Notes
¸¾
½
∙ Z t
lim a(t) exp −
[r(v) − n] dv
≥0
t→∞
(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
Z∞
max U(c(t)ente−ρtdt subject to
{c(t)}
·
0
i ) a = (ra + w) − c − na
io
n
h R
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
Ozan Hatipoglu
Macro Lecture Notes
H = u(c(t))e−(ρ−n)t + v(t) {w(t) + [r(t) − n] a(t) − c(t)}
(42)
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 = u0(c)e−(ρ−n)t
∂c
(43)
∂H
·
→ v = −(r − n)v
∂a
(44)
·
v=−
Ozan Hatipoglu
Macro Lecture Notes
Transversality condition:
lim v(t)a(t) = 0
t→∞
(45)
Differentiate 43 wr.t. time
·
·
v = u00(c)ce−(ρ−n)t − (ρ − n)u0(c)e−(ρ−n)t
Substituting in 44
·
−(r − n)v = u00(c)ce−(ρ−n)t − (ρ − n)u0(c)e−(ρ−n)t
Using (43)
·
−(r−n)u0(c)e−(ρ−n)t = u00(c)ce−(ρ−n)t−(ρ−n)u0(c)e−(ρ−n)t
·
−ru0(c) = u00(c)c − ρu0(c)
¸·
∙ 00
u (c)c c
r =ρ−
u0(c) c
(46)
Rate of return to savings= Rate of return to consumption
Ozan Hatipoglu
Macro Lecture Notes
¸ ·
∙ 00
u (c)c
c
r =ρ+ − 0
c
u (c)
{z
}
|
Elasticity of
consumption
The higher the elasticity, the higher the premium required by the households to change their consumption
levels.
ex: let
·
c > 0
c
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
0
(c)
intertemporal elasticity of substitution=− uu00(c)c