14.05 Lecture NotesIntroduction and The Solow ModelGeo.docx

14.05 Lecture Notes
Introduction and The Solow Model
George-Marios Angeletos
MIT Department of Economics
February 20, 2013
1
1 Preliminaries
• In the real world, we observe for each country time series of
macroeconomic variables such as
aggregate output (GDP), consumption, investment, employment,
unemployment, etc. These
are the typical data that concern the macroeconomist.
• We also observe certain patterns (correlations, stylized facts)
either over time or in the cross-
section. For example, here is a pattern in the time-series
dimension: during booms and
recessions, output, consumption, investment and unemployment
all move together in the

same direction. And here is a pattern in the cross-section: richer
countries tend to have more
capital.
• Understanding what lies beneath these patterns and deriving
lessons that can guide policy is
the job of the macroeconomist. But “understanding” for the
formal economist does not mean
just telling a “story” of the short you can find in the financial
news or the blogosphere. It
means to develop a coherent, self-consistent, formal explanation
of all the relevant observed
patterns.
2
14.05 Lecture Notes: The Solow Model
• To this goal, macroeconomists develop and work with
mathematical models. Any such model
abstracts from the infinity of forces that may be at play in the
real world, focuses on a few
forces that are deemed important, and seeks to work out how
these forces contribute towards
generating the observed patterns.
• Any such model thus features abstract concepts that are meant

to mimic certain aspects of
the world. There are “households” and “firms” in our models
that are meant to be proxies
for real-world people and businesses. And they are making
choices whose product at the
aggregate level is some times series for aggregate output,
employment, etc. We thus end up
with a mathematical model that generates the kind of times
series we also observe in the real
world. And by figuring how these times series are generated in
the model, we hope to also
understand some of the forces behind the actual macroeconomic
phenomena.
3
• In this lecture note, we will go over our first, basic,
mathematical model of the macroeconomy:
the Solow model. We are going to use this model extensively to
understand economic growth
over time and in the cross-section of countries. But we are also
going to use it to standard
understanding economic fluctuations and the economic impact
of various policies. All in all,

we will thus see how a very simple—in fact, ridiculously
simple—mathematical model can
give us a lot of insight about how the macroeconomy works.
• On the way, we will also familiarize ourselves with formal
notions that we will use in subsequent
richer models, including the difference (or coincidence)
between market outcomes and socially
optimal outcomes.
• In particular, we will start analyzing the model by pretending
that there is a social planner, or
“benevolent dictator”, that chooses the static and intertemporal
allocation of resources and
dictates these allocations to the households and firms of the
economy. We will later show that
the allocations that prevail in a decentralized competitive
market environment coincide with
the allocations dictated by the social planner (under certain
assumptions).
4
• Be aware of the following. To talk meaningfully of a
benevolent social planner, we need to have
well specified preferences for the households of the economy.
This is not going to be the case

in the Solow model. Nevertheless, we will establish a certain
isomorphism between centralized
and decentralized allocations as a prelude to a similar exercise
that we will undertake in the
Ramsey model, where preferences are going to be well
specified. This isomorphism is going
to be the analogue within the Solow model of an important
principle that you should know
more generally for a wide class of convex economies without
externalities and other market
frictions: for such economies, the two welfare theorems apply,
guaranteeing the set of Pareto
Optimal allocations coincides with the set of Competitive
Equilibria.
5
2 Introduction and stylized facts about growth
• How can countries with low level of GDP per person catch up
with the high levels enjoyed by
the United States or the G7?
• Only by high growth rates sustained for long periods of time.

• Small differences in growth rates over long periods of time
can make huge differences in final
outcomes.
• US per-capita GDP grew by a factor ≈ 10 from 1870 to 2000:
In 1995 prices, it was $3300
in 1870 and $32500 in 2000.1 Average growth rate was ≈
1.75%. If US had grown with .75%
(like India, Pakistan, or the Philippines), its GDP would be only
$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).
1Let y0 be the GDP per capital at year 0, yT the GDP per capita
at year T, and x the average annual growth rate
over that period. Then, yT = (1 + x)
Ty0. Taking logs, we compute ln yT − ln y0 = T ln(1 + x) ≈ Tx,
or equivalenty
x ≈ (ln yT − ln y0)/T.
6
• At a growth rate of 1%, our children will have ≈ 1.4 our
income. At a growth rate of 3%, our
children will have ≈ 2.5 our income. Some East Asian countries
grew by 6% over 1960-1990;
this is a factor of ≈ 6 within just one generation!!!

• Once we appreciate the importance of sustained growth, the
question is natural: What can
do to make growth faster? Equivalently: What are the factors
that explain differences in
economic growth, and how can we control these factors?
• In order to prescribe policies that will promote growth, we
need to understand what are the
determinants of economic growth, as well as what are the
effects of economic growth on social
welfare. That’s exactly where Growth Theory comes into
picture...
2.1 The World Distribution of Income Levels and Growth Rates
• As we mentioned before, in 2000 there were many countries
that had much lower standards
of living than the United States. This fact reflects the high
cross-country dispersion in the
level of income.
7
• Figure 3.1 in the Barro textbook shows the distribution of
GDP per capita in 2000 across
the 147 countries in the Summers and Heston dataset. 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, and Cyprus. 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!
• Figure 3.2 shows the distribution of GDP per capita in 1960
across the 113 countries for which
data are available. 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.
• The cross-country dispersion of income was thus as wide in
1960 as in 2000. Nevertheless,
there were some important movements during this 40-year
period. Argentina, Venezuela,
Uruguay, Israel, and South Africa were in the top 25 in 1960,
but none made it to the top 25
in 2000. On the other hand, China, Indonesia, Nepal, Pakistan,
India, and Bangladesh grew
fast enough to escape the bottom 25 between 1960 and 1970.
These large movements in the

8
distribution of income reflects sustained differences in the rate
of economic growth.
• Figure 3.3 shows the cross-country distribution of the growth
rates between 1960 and 2000.
Just as there is a great dispersion in income levels, there is a
great dispersion in growth rates.
The mean growth rate was 1.8% per annum; that is, the world on
average was twice as rich
in 2000 as in 1960. The United States did slightly better than
the mean. The fastest growing
country was Taiwan, with a annual rate as high as 6%, which
accumulates to a factor of 10
over the 40-year period. The slowest growing country was
Zambia, with an negative rate at
−1.8%; Zambia’s residents show their income shrinking to half
between 1960 and 2000.
• Most East Asian countries (Taiwan, Singapore, South Korea,
Hong Kong, Thailand, China,
and Japan), together with Bostwana (an outlier as compared to
other sub-Saharan African
countries), Cyprus, Romania, and Mauritus, had the most stellar
growth performances; they

were the “growth miracles” of our times. Some OECD countries
(Ireland, Portugal, Spain,
Greece, Luxemburg and Norway) also made it to the top 20 of
the growth-rates chart. On
the other hand, 18 out of the bottom 20 were sub-Saharan
African countries. Other notable
“growth disasters” were Venezuela, Chad and Iraq.
9
2.2 Stylized Facts
The following are stylized facts that should guide us in the
modeling of economic growth (Kaldor,
Kuznets, Romer, Lucas, Barro, Mankiw-Romer-Weil, and
others):
1. In the short run, important fluctuations: Output, employment,
investment, and consumptio
vary a lot across booms and recessions.
2. In the long run, balanced growth: Output per worker and
capital per worker (Y/L and K/L)
grow at roughly constant, and certainly not vanishing, rates.
The capital-to-output ratio

(K/Y ) is nearly constant. The return to capital (r ) is roughly
constant, whereas the wage
rate (w) grows at the same rates as output. And, the income
shares of labor and capital
(wL/Y and rK/Y ) stay roughly constant.
3. Substantial cross-country differences in both income levels
and growth rates.
4. Persistent differences versus conditional convergence.
5. Formal education: Highly correlated with high levels of
income (obviously two-direction
10
causality); together with differences in saving rates can
“explain” a large fraction of the
cross-country differences in output; an important predictor of
high growth performance.
6. R&D and IT: Most powerful engines of growth (but require
high skills at the first place).
7. Government policies: Taxation, infrastructure, inflation, law
enforcement, property rights and
corruption are important determinants of growth performance.

8. Democracy: An inverted U-shaped relation; that is, autarchies
are bad for growht, and democ-
racies are good, but too much democracy can slow down
growth.
9. Openness: International trade and financial integration
promote growth (but not necessarily
if it is between the North and the South).
10. Inequality: The Kunzets curve, namely an inverted U-shaped
relation between income in-
equality and GDP per capita (growth rates as well).
11. Ferility: High fertility rates correlated with levels of income
and low rates of economic growth;
and the process of development follows a Malthus curve,
meaning that fertility rates initially
increase and then fall as the economy develops.
11
12. Financial markets and risk-sharing: Banks, credit, stock
markets, social insurance.
13. Structural transformation:
agriculture→manifacture→services.

14. Urbanization: family production→organized production;
small vilages→big cities; extended
domestic trade.
15. Other institutional and social factors: colonial history,
ethnic heterogeneity, social norms.
The Solow model and its various extensions that we will review
in this course seek to explain how
all the above factors interrelate with the process of economic
growth. Once we understand better
the “mechanics” of economic growth, we will be able, not only
to predict economic performance
for given a set of fundamentals (positive analysis ), but also to
identify what government policies or
socio-economic reforms can promote social welfare in the long
run (normative analysis ).
12
3 The Solow Model: Centralized Allocations
• The goal here is to write a formal model of how the
macroeconomy works.
• To this goal, we shall envision a central planner that takes as
given the production possibilities

of the economy and dictates a certain behavior to the
households of the economy. As noted
earlier, we will later see how the dynamics of this centralized,
planning economy coincide with
the dynamics of a decentralized, market economy.
• The “inputs” (or “assumptions”) of the model are going to be a
certain specification of the
aforementioned production possibilities and behavior.
• The “output” (or “predictions”) of the model will be the
endogenous macroeconomic outcomes
(consumption, saving, output, growth, etc.).
• We will then be able to use this model to understand the
observed macroeconomic phenomena,
as well as to draw policy lessons.
13
3.1 The Economy and the Social Planner
• 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.
• Households are each endowed with one unit of labor, which
they supply inelastically 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.
14
• 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.
15
3.2 Technology and Production Possibilities
• The technology for producing the good is given by
Yt = F(Kt,Lt) (1)
where F : R2+ → R+ is a (stationary) production function. We
assume that F is continuous
and (although not always necessary) twice differentiable.
16
• We say that the technology is “neoclassical ” if F satisfies the
following properties
1. Constant returns to scale (CRS), a.k.a. homogeneity of degree
1 or linear homogeneity:2
F(µK,µL) = µF(K,L), ∀ µ > 0.
2. Positive and diminishing marginal products:

FK(K,L) > 0, FL(K,L) > 0,
FKK(K,L) < 0, FLL(K,L) < 0.
where Fx ≡ ∂F/∂x and Fxz ≡ ∂2F/(∂x∂z) for x,z ∈ {K,L}.
3. Inada conditions:
lim FK = lim FL = ∞,
K→0 L→0
lim FK = lim FL = 0.
K→∞ L→∞
2We say that a function g : Rn+ → R is homogeneous of degree
λ if, for every vector x ∈ Rn+ and every scalar
µ ∈ R λ a1 a2+, g(µx) = µ g(x). E.g., the function g(x) = x1 x2
is homogenous of degree λ = a1 + a2.
17
• By implication of CRS, F satisfies
Y = F(K,L) = FK(K,L)K + FL(K,L)L
That is, total output equals the sum of the inputs times their
marginal products. Equivalently,
we can think of quantities FK(K,L)K and FL(K,L)L as the
contributions of capital and labor

into output.
• Also by CRS, the marginal products F 3K and FL are
homogeneous of degree zero. It follows
that the marginal products depend only on the ratio K/L :
K K
FK(K,L) = FK
(
, 1
)
FL(K,L) = FL
L
(
, 1
L
)
• Finally, it must be that FKL > 0, meaning that capital and
labor are complementary inputs.4
3This is because of the more general property that, if a function
is homogenous of degree λ, then its first derivatives
are homogeneous of degree λ− 1.
4We say that two inputs are complementary if the marginal
product of the one input increases with the level of
the other input.

18
• Technology in intensive (or per-capita) form. Let
Y K
y = and k = .
L L
denote the levels of output and capital per head (or,
equivalently, per worker, or per labor).
Then, by CRS, we have that
y = f(k) (2)
where the function f is defined by
f(k) ≡ F(k, 1).
19
• Example: Cobb-Douglas production function
F(K,L) = AKαL1−α
where α ∈ (0, 1) parameterizes output’s elasticity with respect
to capital and A > 0 parame-

terizes TFP (total factor productivity).
In intensive form,
f(k) = Akα
so that α can also be interpreted as the strength of diminishing
returns: the lower α is, the
more fastly the MPK, f ′(k) = αkα−1, falls with k.
Finally, as we will see soon, α will also coincide with the
income share of labor (that is,
the ratio of wL/Y ) along the competitive equilibrium. This will
give us a direct empirical
counterpart for this theoretical parameter.
20
• Let us now go back to a general specification of the
technology. By the definition of f and
the properties of F, it is easy to show that f satisfies that
following properties:
f(0) = 0,
f ′(k) > 0 > f ′′(k)
lim f ′(k) =
k→0

∞, lim f ′(k) = 0
k→∞
The first property means that output is zero when capital is
zero. The second property means
that the marginal product of capital (MPK) is always positive
and strictly decreasing in the
capital-labor ratio k. The third property means that the MPK is
arbitrarily high when k is
low enough, and converges to zero as k becomes arbitrarily
high.
• Also, it is easy to check that
FK(K,L) = f
′(k) and FL(K,L) = f(k) −f ′(k)k
which gives us the MPK and the MPL in terms of the intensive-
form production function.
• Check Figure 1 for a graphical representation of a typical
function f.
21
22
yt

kt
kt
f’(k)
0
0
f(k)
ytδ
δkt
Graphical Representation of
a Typical Function f.
Figure 1
Image by MIT OpenCourseWare.
3.3 The Resource Constraint
• Remember that there is a single good, which can be either
consumed or invested. Of course,
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 (3)
Equivalently, in per-capita terms:
ct + it ≤ yt (4)
• Suppose that population growth is n ≥ 0 per period. The size
of the labor force then evolves
over time as follows:
L tt = (1 + n)Lt−1 = (1 + n) L0 (5)
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,
23
plus contemporaneous investment. That is, the law of motion for
capital is
Kt+1 = (1 − δ)Kt + It. (6)
Equivalently, in per-capita terms:
(1 + n)kt+1 = (1 − δ)kt + it
We can approximately write the above as

kt+1 ≈ (1 − δ −n)kt + it (7)
The sum δ+n can thus be interpreted as the “effective”
depreciation rate of per-capita capital:
it represents the rate at which the per-capita level of capital will
decay if aggregate saving
(investment) is zero.
(Remark: The above approximation becomes arbitrarily good as
the economy converges to
its steady state. Also, it would have been exact if time was
continuous rather than discrete.)
24
3.4 Consumption/Saving Behavior
• We will later derive consumption/saving choices from proper
micro-foundations (well specified
preferences). For now, we take a short-cut and assume that
consumption is a fixed fraction
(1 −s) of output:
Ct = (1 −s)Yt = (1 −s)F(Kt,Lt) (8)
where s ∈ (0, 1). Equivalently, aggregate saving is given by a
fraction s of GDP.
• Remark: in the textbook, consumption is defined as a fraction

s of GDP net of depreciation.
This makes little difference for all the economic insights we
will deliver, but be aware of this
minor mathematical difference.
25
3.5 The Aggregate Dynamics
• 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 (6), the resource
constraint (3), and the technology
(1), we derive the following dynamic equation for the capital
stock:
Kt+1 −Kt = F(Kt,Lt) − δKt −Ct (9)
That is, the change in the capital stock is given by aggregate
output, minus capital deprecia-
tion, minus aggregate consumption.

26
• Combining conditions (8) and (9), we get a simple difference
equation for the capital stock:
Kt+1 = (1 − δ)Kt + sF(Kt,Lt) (10)
At the same time, the law of motion for labor gives another
difference equation:
Lt+1 = (1 + n)Lt (11)
• Taken together, these two conditions pin down the entire
dynamics of the labor force and
the capital stock of the economy for any arbitrary initial levels
(K0,L0): starting from such
an initial point, we can compute the entire path {Kt,Lt} simply
by iterating on conditions
(10) and (11 ). Once we have this path, it is straightforward to
compute the paths of output,
consumption, and investment simply by using the facts that Yt =
F(Kt,Lt), Ct = (1 − s)Yt
and It = sYt for all t.
27

• We can reach a similar result in per-capita terms. Using (6),
(4) and (2), we get that the
capital-labor ratio satisfies the following difference equation:
kt+1 = (1 − δ −n)kt + sf(kt), (12)
Starting from an arbitrary initial k0, the above condition alone
pins down the entire path
{kt}
∞
t=0 of the capital-labor ratio. Once we have this path, we can
then get the per-capita
levels of income, consumption and investment simply by the
facts that
yt = f(kt), ct = (1 −s)yt, and it = syt ∀ t (13)
• From this point and on, we will analyze the dynamics of the
economy in per capita terms only.
Translating the results to aggregate terms is a straightforward
exercise.
28
• We thus reach the following characterization of the planner’s
allocation for the Solow economy.
Proposition 1 Given any initial point k0 > 0, the dynamics of
the planner’s solution are given by

the path {kt}∞t=0 such that
kt+1 = G(kt), (14)
for all t ≥ 0, where
G(k) ≡ sf(k) + (1 − δ −n)k.
Equivalently, the growth rate of capital is given by
k
γt ≡
t+1 −kt
= γ(kt), (15)
kt
where
γ(k) ≡ sφ(k) − (δ + n), φ(k) ≡ f(k)/k.
• This result is powerful because it permits us to understand the
entire macroeconomic dynamics
simply by studying the properties of the function G (or
equivalently the function γ).
29
3.6 Steady State
• A steady state of the economy is defined as any level k∗ such

that, if the economy starts
k0 = k
∗ , then kt = k
∗ for all t ≥ 1. That is, a steady state is any fixed point k∗ of
(14).
• A trivial steady state is k = 0 : There is no capital, no output,
and no consumption.
would not be a steady state if f(0) > 0. We are interested for
steady states at which capi
output and consumption are all positive (and finite). We can
then easily show the followi
Proposition 2 Suppose δ + n ∈ (0, 1) and s ∈ (0, 1). A steady
state with k∗ > 0 exists an
unique.
with
This
tal,
ng:
d is
30
Proof. k∗ is a steady state if and only if it solves

k∗ = G(k∗ )
Equivalently
sf(k∗ ) = (δ + n)k∗ ,
or
δ + n
φ(k∗ ) = (16)
s
where the function φ gives the output-to-capital ratio in the
economy (equivalently, the average
product of capital):
f(k)
φ(k) ≡ .
k
We infer that characterizing the steady state of the economy
reduces to the simple task of
characterizing the solution to equation (16). To do this, in turn,
we simply need to study the
properties of the function φ.
31

The properties of f, which we studied earlier, imply that φ is
continuous (and twice differen-
tiable), decreasing, and satisfies the Inada conditions at k = 0
and k = ∞:
f ′(k)k
φ′(k) =
−f(k) F
=
k2
− L < 0,
k2
φ(0) = f ′(0) = ∞ and φ(∞) = f ′(∞) = 0,
where the last two properties follow from an application of
L’Hospital’s rule.
The continuity of φ and its limit properties guarantee that
equation (16) has a solution if and
only if δ + n > 0 and s > 0, which we have assumed. The
monotonicity of φ then guarantees that
the solution is unique. We conclude that a steady-state level of
capital exists, is unique, and is
given by
k∗ = φ−1
(
δ + n

s
)
,
which completes the proof.
32
3.7 Comparative statics of the steady state
• We now turn to the comparative statics of the steady state. In
order to study the impact of
the level of technology, we now rewrite the production function
as
Y = AF(K,L)
or, in intensive form,
y = Af(k)
where A is an exogenous scalar parameterizing TFP (total factor
productivity).
Proposition 3 Consider the steady state.
The capital-labor ratio k∗ and the per-capita level of income,
y∗ increase with the saving rate s
and the level of productivity A, and decrease with the
depreciation rate δ and the rate of population

growth n.
The per-capital level of consumption, c∗ , increases with A,
decreases with δ and n, and is non-
monotonic in s.
33
Proof. By the same argument as in the proof of the previous
proposition, the steady-state level
of the capital-labor ratio is given by
k∗ = φ−1
(
δ + n
.
sA
)
Recall that φ is a decreasing function. It follows from the
Implicit Function Theorem that k∗ is
a decreasing function of (δ + n)/(sA), which proves the claims
about the comparative statics of
k∗ . The comparative statics of y∗ then follow directly from the
fact that y∗ = Af(k∗ ) and the

monotonicity of f. Finally, consumption is given by
c∗ = (1 −s)f(k∗ ).
It follows that c∗ increases with A and decreases with δ + n,
but is non-monotonic in s.
34
• Example If the production functions is Cobb-Douglas, namely
y = Af(k) = Akα, then φ(k) ≡
f(k)/k = k−(1−α) and therefore
(
sA
) 1 α
1−α sA 1−α
k∗ = .y∗ = A .
δ + n
(
δ + n
)
Equivalently, in logs,
1 α α
log y∗ = log A + log s
1 α 1 −α

− log(δ + n)
− 1 −α
Note that 1 > 1, which means that, in steady state, output (and
consumption) increases
1−α
more than one-to-one with TFP. This is simply because capital
also increases with TFP, so
that output increases both because of the direct effect of TFP
and because of its indirect
effect through capital accumulation.
35
3.8 Transitional Dynamics
• The preceding analysis has 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 if an exogenous shock perturbs the
economy away from the steady
state. The following propositions uses the properties of the

functions G and γ (defined in
Proposition 1) to establish that, in the Solow model,
convergence to the steady is always
ensured and is indeed monotonic.
36
Proposition 4 Given any initial k0 ∈ (0,∞), the economy
converges asymptotically to the steady
state:
lim kt = k
∗
t→∞
Moreover, the transition is monotonic:
k0 < k
∗ ⇒ k0 < k1 < k2 < k3 < ... < k∗
and
k0 > k
∗ ⇒ k0 > k1 > k2 > k3 > ... > k∗
Finally, the growth rate γt is positive and decreases over time
towards zero if k0 < k
∗ ,while it is

negative and increases over time towards zero if k0 > k
∗ .
37
Proof. 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, G′(0) = ∞, G(∞) = ∞,
G′(∞) = (1 − δ − n) < 1. By definition of k∗ , G(k) = k iff 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 sequence {kt}∞t=0 is strictly increasing if k0
< k∗ . By monotonicity, kt converges
ˆ ˆ ˆ ˆ ˆasymptotically to some k ≤ k∗ . By continuity of G, k
must satisfy k = G(k), that is k must be
a fixed point of G. But we already proved that G has a unique
fixed point, which proves that
k̂ = k∗ . A symmetric argument proves that, when k0 > k
∗ , {kt}∞t=0 is stricttly decreasing and again
converges asymptotically to k∗ . Next, consider the growth rate
of the capital stock. This is given
by
k

≡ t+1γt
−kt
= sφ(kt) − (δ + n)
kt
≡ γ(kt).
Note that γ(k) = 0 iff k = k∗ , γ(k) > 0 iff k < k∗ , and γ(k) < 0
iff k > k∗ . Moreover, by diminishing
returns, γ′(k) = sφ′(k) < 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 γt is positive
and decreases
towards zero if k0 < k
∗ and it is negative and increases towards zero if k0 > k
∗ .
38
• Figure 2 depicts G(k), the relation between kt and kt+1. The
intersection of the graph of G
with the 45o line gives the steady-state capital stock k∗ . The
arrows represent the path {kt}∞t=
for a particular initial k0.
• Figure 3 depicts γ(k), the relation between kt and γt. The

intersection of the graph of γ with
the 45o line gives the steady-state capital stock k∗ .
• The negative slope of the curve in Figure 3 (equivalently, the
monotonic dynamics of the
growth rate stated in the previous proposition) captures the
concept of conditional conver-
gence: if two countries have different levels of economic
development (namely different k0
and y0) but otherwise share the same fundamental
characteristics (namely share the same
technologies, saving rates, depreciation rates, and fertility
rates), then the poorer country will
grow faster than the richer one and will eventually
(asymptotically) catch up with it.
• Discuss local versus global stability: Because φ′(k∗ ) < 0, the
system is locally stable. Because
φ is globally decreasing, the system is globally stable and
transition is monotonic.
39
40
kt+1
kt

k0 k1 k2 k3 k*
45
G(k)
0
0
V V V
Figure 2
41
0 k*
kt
kt+1-kt
kt
(k)γ
(δ+η)
t=γ
-
-
Figure 3

4 The Solow Model: Decentralized Market Allocations
• In the preceding analysis we characterized the centralized
allocations dictated by a certain
social planner. We now characterize the allocations chosen by
the “invisible hand” of a
decentralized competitive equilibrium.
4.1 Households
• 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 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,k
j
t ,b
j
t, i
j
t 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
inelasticly in a competitive labor market for the
contemporaneous wage wt. Household j is
42
also endowed with initial capital k
j
0. Capital in household j accumulates according to
(1 + n)k
j
t+1 = (1 − δ)k
j
t + it,
which we once again approximate by
k
j
t+1 = (1 − δ −n)k
j
t + it. (17)
• We assume that capital is owned directly by the households.
But capital is productive only
within firms. So we also assume that there is a competitive
capital market in which firms rent

the capital from the households so that they can use it as an
input in their production. The
capital market thus takes the form of a rental market and the
per-period rental rate of capital
is denoted by rt.
• Note that this capital market is a rental market for real,
physical capital (machines, buildings),
not for financial contracts (funds). We are abstracting from this
kind of market and also
abstracting from borrowing constrain and any other form of
financial frictions (frictions in
how funds and resources can be channeled from one agent to
another).
43
• The households may also hold stocks of the firms in the
economy. Let πjt be the dividends
(firm profits) that household j receive in period t. As it will
become clear later on, it is without
any loss of generality to assume that there is no trade of stocks.
(This is because the value
of firms stocks will be zero in equilibrium and thus the value of
any stock transactions will

be also zero.) We thus assume that household j holds a fixed
fraction αj of the aggregate
index of stocks in the economy, so that π
j j∫ t = α Πt, where Πt are aggregate profits. Of course,
αjdj = 1.
• Finally, there is also a competitive labor market, in which the
households supply their labor
and the firms are renting this labor to use it in their production.
The wage rate (equivalently,
the rental rate of labor) is denotes by wt.
• Note that both rt and wt are in real terms, not nominal: they
are the rental prices of capital
and labor relative to the price of the consumption good (which
has been normalized to one).
44
• The household uses its income to finance either consumption
or investment in new capital:
c
j
t + i
j
t = y

j
t .
Total per-head income for household j in period t is simply
y
j
t = wt + rtk
j
t + π
j
t . (18)
Combining, we can write the budget constraint of household j in
period t as
c
j
t + i
j
t = wt + rtk
j
t + π
j
t (19)
• Finally, the consumption and investment behavior of each
household is assumed to follow a
simple rule analogous to the one we had assumed for the social
planner. They save fraction s

and consume the rest:
c
j
t = (1 −s)y
j
t and i
j
t = sy
i
t. (20)
45
4.2 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 Kmt and Lmt 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
Πmt = F(K
m
t ,L
m
t ) − rtK
m
t −wtL
m
t .
• The firms seeks to maximize profits. The FOCs for an interior
solution require
FK(K
m
t ,L
m
t ) = rt. (21)
FL(K
m
t ,L
m
t ) = wt. (22)
• Remember that the marginal products are homogenous of
degree zero; that is, they depend
only on the capital-labor ratio. In particular, F is a decreasing
function of KmK t /L

m
t and FL
46
is an increasing function of Kmt /L
m
t . Each of the above conditions thus pins down a unique
capital-labor ratio Km/Lmt 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 K
m
t /L
m
t . This is the case if and
only if there is some Xt ∈ (0,∞) such that
rt = f
′(Xt) (23)
wt = f(Xt) −f ′(Xt)Xt (24)
where f(k) ≡ F(k, 1); this follows from the properties
FK(K,L) = f
′(K/L) and FL(K,L) = f(K/L) −f ′(K/L) · (K/L),

which we established earlier.
• If (23)-(24) are satisfied, the FOCs reduce to Kmt /Lmt = Xt,
or
Km mt = XtLt . (25)
That is, the FOCs pin down the capital labor ratio for each firm
(Kmt /L
m
t ), but not the size
of the firm (Lmt ). Moreover, because all firms have access to
the same technology, they use
exactly the same capital-labor ratio.
47
• Besides, (23)-(24) imply
rtXt + wt = f(Xt). (26)
It follows that
r Km + w Lm = (r X + w )Lm = f(X )Lm = F(Km,Lmt t t t t t t t
t t t t ),
and therefore
Πmt = L
m
t [f(Xt) − rtXt −wt] = 0. (27)

That is, when (23)-(24) 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
capital-labor ratio is optimal If
instead (23)-(24) 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 entail strictly negative profits.
48
4.3 Market Clearing
• The capital market clears if and only if∫ Mt 1
Kmt dm =
0
∫
(1 + n)tk
j
tdj
0

Equivalently, ∫ Mt
Kmt dm = Kt (28)
0
L
where K
t
t ≡
∫
k
j
tdj is the aggregate capital stock in the economy.0
• The labor market, on the other hand, clears if and only if∫ Mt
Lmt dm =
∫ 1
(1 + n)tdj
0 0
Equivalently, ∫ Mt
Lmt dm = Lt (29)
0
where Lt is the size of the labor force in the economy.
49

4.4 General Equilibrium: Definition
• 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, we let:
Definition 5 An equilibrium of the economy is an allocation
{(kj,cj, ij) , (Km,Lmt t t j∈ [0,1] t t )m [0,Mt]}t∞ ,∈ =0
a distribution of profits {(πjt )j [0,1]}, and a price path
{rt,wt}t∞ such∈ =0 that
(i) Given {rt,wt}∞t=0 and {π
j
t}∞t=0, the path {k
j
t ,c
j
t, i
j
t} is consistent with the behavior of household
j, for every j.
(ii) (Kmt ,L
m

t ) maximizes firm profits, for every m and t.
(iii) The capital and labor markets clear in every period
(iv) Aggregate dividends equal aggregate profits.
50
4.5 General Equilibrium: Existence, Uniqueness, and
Characterization
• In the next, we characterize the decentralized equilibrium
allocations:
Proposition 6 For any initial positions (k
j
0)j [0,1], an equilibrium exists. The allocation of produc-∈
tion across firms is indeterminate, but the equilibrium is unique
as regards aggregate and household
allocations. The capital-labor ratio in the economy is given by
{kt}∞t=0 such that
kt+1 = G(kt) (30)
for all t ≥ 0 and k0 =
∫
k
j
0dj historically given, where G(k) ≡ sf(k) + (1 − δ − n)k.
Equilibrium

growth is given by
k
γt ≡
t+1 −kt
= γ(kt), (31)
kt
where γ(k) ≡ sφ(k) − (δ + n), φ(k) ≡ f(k)/k. Finally, equilibrium
prices are given by
rt = r(kt) ≡ f ′(kt), (32)
wt = w(kt) ≡ f(kt) −f ′(kt)kt, (33)
where r′(k) < 0 < w′(k).
51
Proof. We first characterize the equilibrium, assuming it exists.
Using Km mt = XtLt by (25), we can write the aggregate
demand for capital as∫ Mt t
Kmt dm = Xt
∫ M
Lmt dm
0 0

From the labor market clearing condition (29),∫ Mt
Lmt dm = Lt.
0
Combining, we infer ∫ Mt
Kmt dm = XtLt,
0
and substituting in the capital market clearing condition (28),
we conclude
XtLt = Kt,
where Kt ≡
∫ Lt
k
j
0 t
dj denotes the aggregate capital stock.
52
Equivalently, letting kt ≡ Kt/Lt denote the capital-labor ratio in
the economy, we have
Xt = kt. (34)
That is, all firms use the same capital-labor ratio as the

aggregate of the economy.
Substituting (34) into (23) and (24) we infer that equilibrium
prices are given by
rt = r(kt) ≡ f ′(kt) = FK(kt, 1)
wt = w(kt) ≡ f(kt) −f ′(kt)kt = FL(kt, 1)
Note that r′(k) = f ′′(k) = FKK < 0 and w
′(k) = −f ′′(k)k = FLK > 0. That is, the interest
rate is a decreasing function of the capital-labor ratio and the
wage rate is an increasing function
of the capital-labor ratio. The first properties reflects
diminishing returns, the second reflects the
complementarity of capital and labor.
53
Adding up the budget constraints of the households, we get
Ct + It = rtKt + wtL
j
t +
∫
πtdj,

where∫ Ct ≡
∫
c
j
tdj and It ≡
∫
i
j j
tdj. Aggregate dividends must equal aggregate profits, πtdj =
Πmt dj. By (27), profits for each firm are zero. Therefore,
∫
∫
π
j
tdj = 0, implying
Ct + It = Yt = rtKt + wtLt
Equivalently, in per-capita terms,
ct + it = rtkt + wt.
From (26) and (34), or equivalently from (32) and (33), rtkt +
wt = yt = f(kt). We conclude that
the household budgets imply
ct + it = f(kt),
which is simply the resource constraint of the economy.
54

Adding up the individual capital accumulation rules (17), we
get the capital accumulation rule
for the aggregate of the economy. In per-capita terms,
kt+1 = (1 − δ −n)kt + it
Adding up (20) across household, 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.
Finally, existence and uniqueness is now trivial. (30) maps any
kt ∈ (0,∞) to a unique kt+1 ∈
(0,∞). Similarly∫ , (32) and (33) map any kt
j
∈ (0,∞) to unique rt,wt ∈ (0,∞). Therefore, given any
initial k0 = k0dj, there exist unique paths {kt}∞t=0 and
{rt,wt}∞t=0. Given
j
{rt,wt}∞t=0, the allocation
{kt ,c

j
t, i
j
t} for any household j is then uniquely determined by (17), (18),
and (20). Finally, any
allocation (Km,Lmt t )m [0,Mt] of production across firms in
period t is consistent with equilibrium as∈
long as Kmt = ktL
m
t .
55
• An immediate implication is that the decentralized market
economy and the centralized dic-
tatorial economy are isomorphic:
Corollary 7 The aggregate and per-capita allocations in the
competitive market economy coincide
with those in the dictatorial economy.
• Given this isomorphism, we can 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:

Corollary 8 Suppose δ + n ∈ (0, 1) and s ∈ (0, 1). A steady
state (c∗ ,k∗ ) ∈ (0,∞)2 for the
competitive economy exists and is unique, and coincides with
that of the social planner. 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.
Corollary 9 Given any initial k0 ∈ (0,∞), the competitive
economy converges asymptotically to
the steady state. The transition is monotonic. The equilibrium
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
∗ .
56
• The bottom line is that the allocations that characterize the
frictionless competitive equilib-
rium coincide with those that characterize the planner’s
solution. The only extra knowledge
we got by considering the equilibrium is that we found the
prices (wages, rental rates) that
“support” the allocation as a decentralized market outcome, that
is, that make this allocation

individually optimal in the eyes of firms and households. Keep
this in mind: the planner’s
solution is merely an allocation, a market equilibrium is always
a combination of an allocation
and of prices that support this allocation.
• By finding the prices that support the planner’s solution as a
market equilibrium, we can
thus make predictions, not only about the real macroeconomic
quantities (GDP, investment,
consumption, etc) but also about wages, interest rates, and more
generally market prices.
• Finally, remember that all this presumes that we have a
competitive market economy without
externalities and without any kind of friction to drive the
equilibrium away from the planner’s
solution. If we were to allow for, say, externalities or monopoly
power, the equilibrium would
differ from the planner’s solution—and then we could start
making sense of policies that seek
to correct the underlying market inefficiencies. We will
consider such situations in due course.
57

5 The Solow Model: Introducing Shocks and Policies
• The Solow model can be used to understand business cycles
(economic fluctuations).
• To do this, we must first extend the model in a way that it can
accommodate stochasticity
in its equilibrium outcomes. This is done by introducing
exogenous random disturbances in
the primitives of the model (technologies, preferences, etc).
This means that we model the
“deeper origins” of booms and recession as exogenous forces
and then use the model to make
predictions about how the endogenous macroeconomic variables
respond over time to these
exogenous disturbances.
• In the sequel, we do this kind of exercise to predict the
response of the economy to productivity
shocks (changes in the production possibilities of the economy,
taste shocks (changes in the
saving rate), and policy shocks (changes in government
policies).
58

5.1 Productivity (TFP) Shock
• Let us introduce exogenous shocks to the Total Factor
Productivity (TFP) of the economy.
To this goal, we modify the production function as follows:
Yt = AtF(Kt,Lt)
or, in intensive form,
yt = Atf(kt)
where At identifies TFP in period t.
• We thus want to consider the possibility that At varies over
time and to examine how the
economy responds to changes in At, according to our model.
Before we do this, let us first
show that variation in At is not merely a theoretical possibility;
it is an actual fact in US
data.
• To this goal, suppose further that F takes a Cobb-Douglas
form: F(K,L) = KαL1−α. It
follows that
log Yt = log At + α log Kt + (1 −α) log Lt
59

and therefore
∆ log At = ∆ log Yt −α∆ log Kt − (1 −α)∆ log Lt
where ∆Xt ≡ Xt − Xt−1 for any variable X . Note that ∆ log Yt
is the growth rate of
GDP, ∆ log Kt is the change in capital (net investment), and ∆
log Lt is the net change in
employment. For all these variables, we have readily available
data in the US. Furthermore,
under the assumption of perfect competition, wt = AtFL(Kt,Lt).
Given the Cobb-Douglas
specification, this gives
wt = (1 −α)A α αtKt L
−
t = (1 −α)Yt/Lt
and therefore
w
1 − t
Lt
α =
Yt
which means that 1−α coincides with the income share of labor.
In the US data, the income
share of labor is about 70%. It follows that α ≈ .3. We conclude
that the change in TFP can
be computed by using the available macro data along with the
following equation

∆ log At = ∆ log Yt − .3∆ log Kt − .7∆ log Lt
60
If you do this, you get a times series for ∆ log At, the growth
rate in TFP, that looks as in
the following figure.
• There are two notable features in this figure. The first is that
on average ∆At is positive.
This means that on average there is long-run technological
progress: out of the same inputs,
61
we get more and more output as time passes. The second is that
∆At fluctuates a lot around
its trend and tends to be lower during recessions (periods
highlighted by the white areas in
the figure) as opposed to normal times (grey areas in the
figure).
• This second systematic feature, that TFP tends to fall during
recessions, motivates the exercise

we do here. We take for given that At fluctuates over time and
study the model’s predictions
regarding how all other macroeconomic variables (output,
investment, consumption) respond
to such fluctuations in At. We are thus interested to see if the
model makes reasonable and
empirically plausible predictions about the cyclical behavior of
these variables.
• In particular, we know that, in the data, TFP, output,
consumption and investment all fall
during recessions. In the model, we will only assume that TFP
falls during recession. We
then ask whether the model predicts that output, consumption,
and investment must fall in
response to a fall in TFP.
62
• Thus consider a negative shock in TFP. This shock could be
either temporary or permanent.
Also, keep in mind this shock can be interpreted literally as a
change in the know-how of
firms and the talents of people; but it could also be proxy from
changes in the efficiency of the

financial system and more generally in the efficiency of how
resources are used in the economy.
• Recall that the dynamics of capital are given by
kt+1 = G(kt; At) = sAtf(kt) + (1 − δ −n)kt
As a result of the drop in At, the G function shifts down. If the
drop in At is permanent, the
shift in G is also permanent; if the drop in At is transitory, the
shift in G is also transitory. The
same logic applies if we look at the γ function, which gives the
growth rate of the economy.
See Figure 4.
63
• Suppose now that the economy was resting at its steady state
before the drop in At. At the
moment At falls, output falls by exactly the same amount,
because at the moment resources
are fixed and simply TFP has fallen. But the drop in output
leads to a drop in investment,
which in turn leads to lower capital stock in the future. It
follows that after the initial shock
there are further and further reductions in output, due to the

endogenous reduction in the
capital stock. In other words, the endogenous response of
capital amplifies the effects of the
negative TFP shock on output. Furthermore, if at some point the
TFP shock disappears and
At returns to its initial value, output (and by implication
consumption and investment) do
not return immediately to their initial values. Rather, because
the capital stock has been
decreased, it takes time for output to transit back to its original,
pre-recession value. In this
sense, the endogenous response of capital, not only amplifies
the recessionary effects of the
exogenous TFP shock, but it also adds persistence: the effects
of the shock are felt in the
economy long time after the shock has itself gone away.
Equivalently, recoveries take time.
• The aforementioned dynamics are illustrated in Figure 5. The
solid lines represent the response
of the economy to a transitory negative TFP shock, which only
lasts between t1 and t2 in the
figure. The dashed lines show what the response would have
been in the case the shock were
64

permanent, starting at t1 and lasting for ever.
either temporarily or permanently. What are the effects on the
steady state and the transi-
tional dynamics, in either case?
• See Figures 4 and 5 for a graphical representation of the
impact of a (temporary) negative
productivity shock.
• Taste shocks: Consider a temporary fall in the saving rate. The
γ(k) function shifts down for
a while, and then return to its initial position. What the
transitional dynamics?
65
66
kt+1
kt
kt
γt

Figure 4
67
kt
yt
kt
t
t
t1 t2
yt = At f(kt)
Figure 5
5.2 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. It follows that the
dynamics of capital are given
by
kt+1 −kt = f(kt) − (δ + n)kt − ct −gt
• Government spending is financed with proportional income
taxation, at rate τ ≥ 0. The
government thus absorbs a fraction τ of aggregate output:
gt = τyt.
68
• Disposable income for the representative household is (1 −
τ)yt. We continue to assume that
consumption and investment absorb fractions 1 −s and s of
disposable income:
ct = (1 −s)(1 − τ)yt.
• Combining the above, we conclude that the dynamics of
capital are now given by
kt+1

γt =
−kt
= s(1 τ
t
− )φ(kt)
k
− (δ + n).
where φ(k) ≡ f(k)/k. Given s and kt, the growth rate γt decreases
with τ.
• A steady state exists for any τ ∈ [0, 1) and is given by
k∗ = φ−1
(
δ + n
s(1 − τ)
)
.
Given s, k∗ decreases with τ.
• Policy Shocks: Consider a temporary shock in government
consumption. What are the tran-
sitional dynamics?
69

5.3 Productive Government Spending
• Suppose now that production is given by
yt = f(kt,gt) = k
α
t g
β
t ,
where α > 0, β > 0, and α + β < 1. Government spending can
thus be interpreted as
infrastructure or other productive services. The resource
constraint is
ct + gt + it = yt = f(kt,gt).
• We assume again that government spending is financed with
proportional income taxation at
rate τ, and that private consumption and investment are
fractions 1 − s and s of disposable
household income:
gt = τyt.
ct = (1 −s)(1 − τ)yt
it = s(1 − τ)yt
70

• Substituting gt = τyt into yt = kαt g
β
t and solving for yt, we infer
α β
yt = k
1−β τ 1 a bt −β ≡ kt τ
where a ≡ α/(1 − β) and b ≡ β/(1 − β). Note that a > α, reflecting
the complementarity
between government spending and capital.
• We conclude that the growth rate is given by
kt+1
γt =
−kt
= s(1 − τ)τbka−1
k tt
− (δ + n).
The steady state is
1/(1
s(1 τ)τb
−a)
k∗ =

(
−
)
.
δ + n
71
• Consider the rate τ that maximizes either k∗ , or γt for any
given kt. This is given by
d
[(1
dτ
− τ)τb] = 0 ⇔
bτb−1 − (1 + b)τb = 0 ⇔
τ = b/(1 + b) = β.
That is, the “optimal” τ equals the elasticity of production with
respect to government ser-
vices. The more productive government services are, the higher
their optimal provision.
72

6 The Solow Model: Miscellaneous
6.1 The Solow Model in Continuous Time
• Recall that the basic growth equation in the discrete-time
Solow model is
kt+1 −kt
= γ(kt) k
t
≡ sφ( t)
k
− (δ + n).
We would expect a similar condition to hold under continuous
time. We verify this below.
• The resource constraint of the economy is
C + I = Y = F(K,L).
In per-capita terms,
c + i = y = f(k).
• Clearly, these conditions do not depend on whether time is
continuous or discrete. Rather, it
is the law of motions for L and K that slightly change from
discrete to continuous time.

73
• Population growth is now given by
L
̇
= n
L
and the law of motion for aggregate capital is
K
̇ = I − δK
• Let k ≡ K/L. Then,
˙ ˙ ˙k K L
= − .
k K L
Substituting from the above, we infer
k̇ = i− (δ + n)k.
Combining this with
i = sy = sf(k),
we conclude
k̇ = sf(k) − (δ + n)k.
74

• Equivalently, the growth rate of the economy is given by
k̇
= γ(k) ≡ sφ(k) − (δ + n). (35)
k
The function γ(k) thus gives the growth rate of the economy in
the Solow model, whether
time is discrete or continuous.
75
6.2 Mankiw-Romer-Weil: Cross-Country Differences
• The Solow model implies that steady-state capital,
productivity, and income are determined
primarily by technology (f and δ), the national saving rate (s),
and population growth (n).
• Suppose that countries share the same technology in the long
run, but differ in terms of saving
behavior and fertility rates. If the Solow model is correct,
observed cross-country income and
productivity differences should be explained by observed cross-
country differences in s and n,

• Mankiw, Romer and Weil tests this hypothesis against the
data. In it’s simple form, the Solow
model fails to explain the large cross-country dispersion of
income and productivity levels.
• Mankiw, Romer and Weil then consider an extension of the
Solow model, that includes two
types of capital, physical capital (k) and human capital (h). The
idea is to take a broader
perspective on how to map the model to reality.
76
• Output is given by
y = kαhβ,
where α > 0,β > 0, and α + β < 1. The dynamics of capital
accumulation are now given b
k̇ = sky − (δ + n)k
ḣ = shy − (δ + n)h
where sk and sh are the investment rates in physical capital and
human capital, respectivel
The steady-state levels of k,h, and y then depend on both sk and
sh, as well as δ and n.
• Proxying sh by education attainment levels in each country,

Mankiw, Romer and Weil fin
that the Solow model extended for human capital does a pretty
good job in explaining th
cross-country dispersion of output and productivity levels.
y
y.
d
e
77
6.3 Log-linearization and the Convergence Rate
• Define z ≡ ln k − ln k∗ . We can rewrite the growth equation
(35) as
ż = Γ(z),
where
Γ(z) ≡ γ(k∗ ez) ≡ sφ(k∗ ez) − (δ + n)
Note that Γ(z) is defined for all z ∈ R. By definition of k∗ ,
Γ(0) = sφ(k∗ ) − (δ + n) = 0.
Similarly, Γ(z) > 0 for all z < 0 and Γ(z) < 0 for all z > 0.
Finally, Γ′(z) = sφ′(k∗ ez)k∗ ez < 0

for all z ∈ R.
• We next (log)linearize ż = Γ(z) around z = 0 :
ż = Γ(0) + Γ′(0) ·z
or equivalently
ż = λz
where we substituted Γ(0) = 0 and let λ ≡ Γ′(0).
78
• Straightforward algebra gives
Γ′(z) = sφ′(k∗ ez)k∗ ez < 0
f ′(k)k
φ′(k) =
−f(k) f ′(k)k f(k)
= − 1
k2
−
f(k) k2
sf(k∗ ) = (δ + n)k∗
[ ]

We infer
Γ′(0) = −(1 −εK)(δ + n) < 0
where εK ≡ FKK/F = f ′(k)k/f(k) is the elasticity of production
with respect to capital,
evaluated at the steady-state k.
• We conclude that
k̇
= λ ln
k
(
k
k∗
)
where
λ = −(1 −εK)(δ + n) < 0
The quantity λ is called the convergence rate.−
79
• Note that, around the steady state
˙ẏ k

= εK
y
·
k
and
y k
= ε
y∗
K ·
k∗
It follows that
ẏ y
= λ ln
y
(
y∗
)
Thus, −λ is the convergence rate for either capital or output.
• In the Cobb-Douglas case, y = kα, the convergence rate is
simply
−λ = (1 −α)(δ + n),
where α is the income share of capital. Note that as λ → 0 as α
→ 1. That is, convergence

becomes slower and slower as the income share of capital
becomes closer and closer to 1.
Indeed, if it were α = 1, the economy would a balanced growth
path.
80
• In the example with productive government spending, y =
kαgβ = kα/(1−β)τβ/(1−β), we get
−λ =
(
α
1 −
1 −β
)
(δ + n)
The convergence rate thus decreases with β, the productivity of
government services. And
λ → 0 as β → 1 −α.
• Calibration: If α = 35%, n = 3% (= 1% population growth+2%
exogenous technological
process), and δ = 5%, then −λ = 6%. This contradicts the data.
But if α = 70%, then
−λ = 2.4%, which matches the date.

81
6.4 Barro: Conditional Convergence
• Recall the log-linearization of the dynamics around the steady
state:
ẏ y
= λ ln .
y y∗
A similar relation will hold true in the neoclassical growth
model a la Ramsey-Cass-Koopmans.
λ < 0 reflects local diminishing returns. Such local diminishing
returns occur even in
endogenous-growth models. The above thus extends well
beyond the simple Solow model.
• Rewrite the above as
∆ ln y = λ ln y −λ ln y∗
Next, let us proxy the steady state output by a set of country-
specific controls X, which
include s,δ,n,τ etc. That is, let
−λ ln y∗ ≈ β′X.

We conclude
∆ ln y = λ ln y + β′X + error
82
• The above represents a typical “Barro” conditional-
convergence regression: We use cross-
country data to estimate λ (the convergence rate), together with
β (the effects of the saving
rate, education, population growth, policies, etc.) The estimated
convergence rate is about
2% per year.
• Discuss the effects of the other variables (X).
83
6.5 The Golden Rule and Dynamic Inefficiency
• The Golden Rule: Consumption at the steady state is given by
c∗ = (1 −s)f(k∗ ) =
= f(k∗ ) − (δ + n)k∗

Suppose the social planner chooses s so as to maximize c∗ .
Since k∗ is a monotonic function
of s, this is equivalent to choosing k∗ so as to maximize c∗ .
Note that
c∗ = f(k∗ ) − (δ + n)k∗
is strictly concave in k∗ . The FOC is thus both necessary and
sufficient. c∗ is thus maximized
if and only if k∗ = kgold, where kgold solve
f ′(kgold) − δ = n.
Equivalently, s = sgold, where sgold solves
sgold ·φ (kgold) = (δ + n)
The above is called the “golden rule” for savings, after Phelps.
84
• Dynamic Inefficiency: If s > sgold (equivalently, k∗ > kgold),
the economy is dynamically
inefficient: If the saving raised is lowered to s = sgold for all t,
then consumption in all periods
will be higher!
• On the other hand, if s < sgold (equivalently, k∗ > kgold),

then raising s towards sgold will
increase consumption in the long run, but at the cost of lower
consumption in the short run.
Whether such a trade-off between short-run and long-run
consumption is desirable will depend
on how the social planner weight the short run versus the long
run.
• The Modified Golden Rule: In the Ramsey model, this trade-
off will be resolved when k∗
satisfies the
f ′(k∗ ) − δ = n + ρ,
where ρ > 0 measures impatience (ρ will be called “the discount
rate”). The above is called
the “modified golden rule.” Naturally, the distance between the
Ramsey-optimal k∗ and the
golden-rule kgold increase with ρ.
85
• Abel et. al.: Note that the golden rule can be restated as
Ẏ
r − δ = .

Y
˙Dynamic inefficiency occurs when r − δ < Y /Y, dynamic
efficiency is ensured if r − δ >
Ẏ /Y. Abel et al. use this relation to argue that, in reality, there
is no evidence of dynamic
inefficiency.
• Bubbles: If the economy is dynamically inefficient, there is
room for bubbles.
86
6.6 Poverty Traps, Cycles, etc.
• Discuss the case of a general non-concave or non-monotonic
G.
• Multiple steady states; unstable versus stable ones; poverty
traps.
• Local versus global stability; local convergence rate.
• Oscillating dynamics; perpetual cycles.
• See Figures 6 and 7 for examples.
87

88
kt+1
kt
kt
G(k)
G(k)
45o
Kt+1
v v v
v v v v v
v v v
Figure 7
Figure 6
6.7 Introducing Endogenous Growth
• What ensures that the growth rate asymptotes to zero in the

Solow model (and the Ramsey
model as well) is the vanishing marginal product of capital, that
is, the Inada condition
limk f
′(k) = 0.→∞
• Continue to assume that f ′′(k) < 0, so that γ′(k) < 0, but
assume now that limk f ′(k) =→∞
A > 0. This implies also limk φ(k) = A. Then, as k→∞ →∞,
k k
t ≡
t+1
γ
− t
kt
→ sA− (n + δ)
• If sA < (n + δ), then it is like before: The economy converges
to k∗ such that γ(k∗ ) = 0. But
if sA > (n + x + δ), then the economy exhibits diminishing but
not vanishing growth: γt falls
with t, but γt → sA− (n + δ) > 0 as t →∞.
89
• Consider the special case where f(k) = Ak (linear returns to

capital). This is commonly
referred as the AK model. The economy then follows a
balanced-growth path from the very
beginning. Along this path, the growth rate of consumption,
output and capital are all equal
to γ = sA− (n + δ) in all dates.
• Note then that the growth rate depends both on “preferences”
(through s) and on “technol-
ogy” (through A). Hence, the same forces that determined the
long-run level of income in the
Solow model now also determine the long-run rate of growth.
• We will later “endogenize” A in terms of policies, institutions,
markets, etc.
• For example, Romer/Lucas: If we have human capital or
spillover effects,
yt = Atk
α
t h
1−α
t .
If (for reasons that we will study later) it happens that h is
proportional to k, then we get
that y is also proportional to k, much alike in the simple AK
model that we briefly mentioned
earlier (i.e., the version of the Solow model with f(k) = Ak).

90
• In particular, let
k̃t ≡ kt + ht
denote the ”total” capital of the economy and suppose that there
exists a constant λ ∈ (0, 1)
such that
˜ ˜kt = λkt ht = λkt.
The we can write output as
˜ ˜yt = Atkt.
where
Ãt ≡ Atλα(1 −λ)1−α.
represent the effective productivity of the ”total” capital of the
economy. Finally, assuming
that households save a fraction s of their income to ”total”
capital, the growth rate of the
economy is then simply
˜γt = sAt − (δ + n).
91

• Clearly, this is the same as in the simple AK model, except for
the fact that effective produc-
tivity is now endogenous to the allocation of savings between
the two types of capital (that
˜is, At depends on λ).
• Question: what is the λ that maximizes effective productivity
and growth?
92
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1
A Simple Climate-Solow Model for Introducing the Economics

of Climate Change to
Undergraduate Students
Panagiotis Tsigaris1
Professor
Department of Economics
and
Joel Wood
Assistant Professor
Department of Economics
Thompson Rivers University
900 McGill Road
Kamloops, B.C.,
CANADA, V2C 0C8
May 24, 2016

Abstract
In this paper the simplest integrated assessment model is
developed in order to illustrate to
undergraduate students the economic issues associated with
climate change. The growth model
developed in this paper is an extension of the basic Solow
model and includes a simple climate
model. Even though the model is very simple it is very powerful
in its predictions. Students use
the model to explore various scenarios illustrating how
economic activity today will inflict
damages from higher temperatures on future generations. But
students also observe that future
generations will be richer than today’s generation due to
productivity growth and population
stabilization. Hence, the richer future generations will not be as
rich as they would be without
climate change. Since the cost of action is absorbed by the
current generation and the benefits of
action accrue to future generations students can conduct a cost-
benefit analysis and explore the
importance of the discount rate. The appendix provides step-by-
step instructions for students to

setup the model in MS Excel and to conduct simulations.
Keywords: Integrated Assessment Models, Climate Change,
Solow Growth Model, Teaching
Economics.
JEL: A22, O44, Q54.
1 Authors email addresses: [email protected] and
[email protected]
mailto:[email protected]
mailto:[email protected]
2
Introduction
“Greenhouse gas (GHG) emissions are externalities and
represent the biggest market failure
the world has seen” –Sir Nicholas Stern (2007)
Climate change caused by the Greenhouse Gas (GHG) emissions
released by the burning of
fossil fuels and land use changes imposes damages to future

generations.2 GHG emissions trap
heat and affect the future climate resulting in damages from
increased temperatures. For
example, increased temperatures are expected to cause sea level
rise, increased floods, increased
droughts and heat waves, and possibly even increased human
conflict. The current generation
benefits from using fossil fuels, but does not internalize these
external costs. As a result, climate
change is what economists call a negative externality. Covert et
al., (2016) examine historical
data on fossil fuel production and consumption and conclude
that neither supply (e.g., Peak Oil)
nor demand (e.g., development of low-carbon technologies)
factors will sufficiently reduce GHG
emissions. Without government intervention, humans will
overproduce GHG emissions.
The climate change problem is further complicated as being a
global externality rather
than a local one. Even though each nation emits a different
amount of greenhouse gases (GHG),
the marginal impact of a tonne of GHG is independent of where
it is emitted (Stern, 2007);
whereas, the effects of smog in a city are local and

heterogeneous depending on the geography
and demographics of the city. Furthermore, GHGs accumulate in
the atmosphere and stay a long
time, i.e., carbon dioxide has an average atmospheric life of
over a century (Archer et al., 2013).
The impact is persistent and long term, whereas the effects of
smog in a city are relatively
immediate following exposure.
2 For scientific consensus on the issue see Oreskes, (2004).
3
Due to the persistence of GHGs in the atmosphere, the climate
change problem is
characterized by the issue of inter-generational equity: The
current generation is imposing
external costs on future generations and would have to forego
some economic growth to limit
those costs. How the costs of action of the current generation
varies relative to the benefits, in
terms of reduced damages, to future generations depends
heavily on the discount rate used. The

discount rate in turn depends on the social rate of time
preference, risk aversion and, per capita
economic growth. Discounting at normal discount rates does not
put too much value on what
happens 100 or 200 years from now; however, a very low
discount rate, such as that used in the
Stern Review (Stern, 2007), places much more weight on future
damages. Discounting plays a
significant role as to whether it is optimal, from an inter-
generational perspective, to undertake
strong emission reduction action immediately or to start
reducing emissions more slowly and to
follow an increasingly stringent climate policy (a ramp up
climate policy). 3
In addition to the inter-generational equity, climate change is
also characterized by issues
of intra-generational equity. For example, rich nations which
are relatively GHG intensive are
located in temperate climates and have the funds and strong
institutions to more easily adapt to
climate change; whereas, poorer nations, say in sub-Saharan
Africa, are expected to be hit
relatively harder by the negative impacts of higher
temperatures.4

Complicating the problem is the fact that uncertainty and risk
are significant. Damages
from climate change could be potentially large and irreversible
(Weitzman, 2009, 2011).
Furthermore, the continuous disposal of carbon into the
atmosphere, oceans, and land could thus
result in the tragedy of the commons (Broome, 2012). Finally,
reducing GHG emissions can be
characterized as a public good in that the benefits of mitigation
are non-rival and non-
3 This issue will be explored in more detail in section 4.
4 For a critical review of inter-generational and intra-
generational climate justice see Forsyth (2013).
4
exclusionary resulting in a free-rider problem and the under-
provision of mitigation policy. 5 This
free-rider problem can provide insights into the failure of the
Kyoto Protocol and subsequent
annual meetings. It is no wonder that Sir Nicholas Stern
considers this issue the biggest market
failure the world has ever seen.

One of the most common approaches to evaluate the impact of
climate change is to use
an Integrated Assessment Model (IAM). These models integrate
a model of the world economy
with a representation of the global climate system. The models
assess different scenarios from
these complex systems and are used by governments when
evaluating the impact of climate polices
(e.g., estimating the Social Cost of Carbon) and informing the
general public (Schwanitz,
(2013)).6
In spite of these significant issues and all the research being
undertaken to study the
economics of climate change, not much has been formally done
to introduce IAMs to
undergraduate students. Tol (2014) is a notable exception, as a
text on climate economics
suitable for a full course in climate economics with a specific
focus on the IAM at the masters’
or advanced undergraduate levels. Yet there is little available to
introduce undergraduate
students to IAMs for a portion of a climate economics course or
for courses in macroeconomic
growth theory, development economics or environmental

economics. The existing IAMs are
overly complex for teaching the economics of climate change to
undergraduate students. For
example, the Dynamic Integrated Climate Economics (DICE)
model is based on the Ramsey
growth model that many economics students do not encounter
until graduate school.7 Our
5 Recently, Nordhaus (2015) has proposed the formation of
climate clubs to solve the free rider problem.
6 Because of the large amount of uncertainty with respect to
climate change and climate damages Pindyck (2013,
2015) concludes that IAMs are not very useful for guiding
policy;
7 The closest economic models to the one we have constructed
are Nordhaus’ DICE model, Brock and Taylor
(2010), and Taylor (2014). None of these three closely related
works are aimed at educating undergraduate students
about the economics of climate change.
5
approach adjusts the simple Solow growth model that
undergraduate economics students are
familiar with.8 Furthermore, the existing IAMs include a
complex representation of the climate

system that takes a significant amount of time to explain to
undergraduate students. Our model
replaces the complex climate system with a simple linear
relationship between atmospheric
carbon accumulation and expected temperature change
demonstrated by Matthews et al (2012).
This paper is aimed at making the simple IAM model available
to instructors and undergraduate
students in order to explore the economics of climate change.
The model is available in two
possible formats both accompanying this article: an MS Excel
workbook or an R code version.
Throughout the paper figures and key points are provided for
instructors to highlight to students
and to use as starting points to motivate in-class discussion.
The step-by-step instructions to replicate the simple IAM
(included in an accompanying
appendix) and the exercises provided throughout the paper
allow students to learn the economic
issues surrounding climate change in a hands-on way. Learning-
by-doing, rather than watching
only the instructor’s lectures, is a more effective way to absorb
and understand the material
(Findley (2014), Dalton et. al., (2015)). After being exposed to

the topic by the instructor,
students learn more when they can use the simple IAM to make
and graph the projections and to
explain the results. Visually seeing the pattern students created
themselves is a powerful teaching
tool (Watts and Becker, (2008)). Psychological studies show
that visuals improve learning
outcomes and learning-by-doing increases knowledge retention
and also becomes a more
enjoyable experience to students (Vazquez and Chiang, (2014)).
This paper (and accompanying
appendix) guides instructors and students to create visuals of
future trajectories of the standard of
living of the world economy with and without climate change
under different scenarios.
8 In case students are not exposed to the Solow model, more
time can be spent explaining the basics of the Solow
model and the concept of steady state levels.
6
Section 2 describes the basic climate-Solow model for the world
economy. Section 3

alters the model to examine damages which are more severe.
This section provides direction for
instructors to use the model to illustrate the impact of climate
change when damages are more
severe at higher temperatures, and when temperature increases
affect the depreciation of capital
and productivity growth. Section 4 uses the model developed in
section 2 to illustrate the costs
and benefits of emission reductions by conducting a simple
Benefit-Cost Analysis for the 2
degrees target. Finally, concluding remarks and other possible
classroom extensions are
mentioned. The appendix provides step-by-step instructions for
students to create and run the
base case version of the simple IAM outlined in the paper
following an approach similar to
Tebaldi and Elmslie (2010). Students can construct, on their
own, the income per capita
trajectories with and without damages, the Environmental
Kuznets curve, and the time paths of
other variables over 200 years. Exercises are provided
throughout the main text.
2. The Simple Climate-Solow Model

2.1 Economic Growth & Climate Impacts
The economic growth component of the model is a variation on
the standard Solow Growth
model. In the standard undergraduate treatment of the Solow
model, output is produced by the
combination of capital, Kt labor, Lt and technology, At
according to the Cobb-Douglas production
function �� = ��
� ��
1−� , which can be rearranged in terms of output per worker as
�� = ��
� .
This is the standard Solow Growth model that students should
be already familiar with.
For the purposes of studying climate change, the effect of
increased temperatures is added to the
model in a similar way as by Nordhaus (2008) and Fankhauser
and Tol (2005). This is a standard
7
assumption in most IAMS. The production function in the model
is slightly altered to be the

following
�� = ��
�,
where �� = 1/(1 + �1��
�2 ) ≤ 1 is the damage function and Tt is the temperature
anomaly in
year t. The production function looks the same as the standard
Cobb-Douglas production
function, except output per worker is now reduced by increased
temperatures, i.e., the higher is
Tt, the lower is yt ceteris paribus.
The savings rate, s is constant, leading to investment per worker
in period t of ��.
Capital depreciates at a constant rate, ��. To reflect recent UN
population projections that
predict global population will plateau around 10.5 billion, total
population and the labor force
grow at a decreasing rate over time, ��,� = ��,0/(1 + �� )
� determined by the parameter �� > 0
which reduces the degree of population growth over time. The
term gL,0 is the population growth
rate in the base year of 2010. Total factor productivity, At also
grows at a decreasing rate over

time: ��,� = ��,0/(1 + ��)
�.9 This leads to the following difference equation to describe
the
transitional dynamics in the model:
��+1 − �� = �� − (�� + ��,� ) ��.
Given this equation it is easy to show convergence to a balanced
growth stable steady
state capital labour ratio ��,� = [
��
��+��,�
]
1/(1−�)
for a given time period t.10 Due to population
9 The assumption of a declining growth rates of total factor
productivity and population growth as shown above are
also used in Nordhaus (2013). Most undergraduate students will
be familiar with the Solow model with constant
rates of population and technology growth; therefore, the
diminishing growth rates used here may appear more
complicated at first glance to the students. However, this
change has little effect on how an instructor would
traditionally introduce the dynamics of the Solow model.
10 Simulations can also conducted using transitional dynamics

but this is a possible extension. The differences
between the two paths is not significant and this path will
converge to the same unique steady state values when
technology is constant and population growth is constant.
8
growth declining and technology advancing, the balanced
growth steady state capital labor ratio
will increase over time (offset by damages). Along the
balanced growth path, output per worker,
��,� = �� ,�
� grows at a rate dependent on changes in temperature
(outlined in subsection
2.3), the growth rate of total factor productivity, gA,t (which
grows at a declining rate) and the
growth rate of the capital labour ratio which is weighted by the
income share of capital, α. It can
be easily seen that in the absence of climate damages (i.e., ��
= 1), yt grows at a faster rate.
To identify the impact of Business-As-Usual (BAU) in the
model, a simple comparison
of �� = 1 for all t (i.e., no climate damages) to �� < 1 (i.e.,
with climate damages) is required.

This comparison is shown in Figure 1 for the parameter values
displayed in the appendix. The
figure is very useful to highlight to students the central trade-
off involved in the climate change
problem. There are two important aspects of this figure to
highlight. First, that the model,
consistent with other IAMs, predicts that future generations are
better off despite climate
damages. Second, that the climate change problem is
intergenerational in nature; the damages of
climate change, as represented by the wedge between the two
lines, are imposed mainly on
future generations.11 Combined, these two aspects highlight
that the climate change problem can
be encapsulated by the following trade-off: A relatively poorer
current generation is imposing
damages (costs) on relatively richer future generations. This is
of course only true in the base
case of the model, and altering either the damage function or
where damages enter the model can
lead to future generations being made worse off; which is a
useful exercise for instructors to do
for their class using our provided Excel workbook or R code.

11 Damages by 2100 are 5.5 percent (as a % of the income per
person without climate change) and increase to 17
percent by 2200. These damages are within the range found in
the literature (See Tol (2015)).
9
Figure 1: The Solow Model with and without Climate Impacts
Source: Authors’ calculations.
0
10
20
30
40

50
60
70
2000 2050 2100 2150 2200
S
te
a
d
y
S
ta
te
i
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cm
o
e
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e
r
p
e

rs
o
n
(
0
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s
o
f
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5
$
s)
Year
base case damages no damages
Question 1: The base model predicts that future generations will
be worse off because of

climate change but that they will still be richer than the current
generation. What are the
implications for the climate policy decisions being made by
politicians in the current generation?
10
2.2 Carbon Emissions
Carbon emissions, Et are generated in the model by the
production process based on a variable,
y or
clean) the production technology is at
time t. The emissions intensity variable defines how much
emissions are released per unit of
output. Carbon emissions in year t are calculated by multiplying
the emissions intensity in year t
by the output in year t
�� = �� .
where Et is tonnes of carbon released and Yt is total output. For
modelling purposes, emissions
intensity is computed by assuming a level in the base year and
then specifying the growth rate of
emissions intensity over time into the future. Figure 2 shows

that global emissions intensity has
steadily declined between 1950 and 2010. This decline has
occurred for many reasons. Sectors
that have been growing most rapidly, like information
technology or health care, are generally
less energy intensive than the sectors that are growing more
slowly or stagnating. Also, the
advance of technology improves the efficiency of production, so
that it now takes less energy to
produce the same product. There has also been a general shift in
the composition of the sources
of energy away from coal and towards natural gas, nuclear,
hydroelectricity, and others. Future
declines in emissions intensity take the following relationship
��,� = ��,�−1/(1 + �� ),
where ��,� < 0 is the growth rate of emissions intensity
between periods t and t-1 and �� < 0.
The value of emissions intensity in year t can then be calculated
as12
�� = ��−1(1 + ��,� ).
12 Similar assumptions about emissions intensity were made by
Nordhaus (2013). For details see
http://www.econ.yale.edu/~nordhaus/homepage/documents/DIC

E_Manual_103113r2.pdf
http://www.econ.yale.edu/~nordhaus/homepage/documents/DIC
E_Manual_103113r2.pdf
11
This formula can also be expressed in terms of the base year
�� = �0 ∏[1 + ��,0/(1 + �� )
� ]
�=�
�=1
.
This information is provided for the benefit of instructors and
can be given to especially
interested students; however, the important thing to highlight to
students is that emissions
intensity of output is assumed to decline at an increasing rate
into the future (consistent with past
history)
Figure 2. Global Emissions Intensity, 1950-2010
Source: CDIAC, 2015; Maddison Project, 2013; authors’

calculations.
The carbon emissions predicted by the model follow an inverse-
u shape consistent with the
Environmental Kuznets’ Curve hypothesis and are displayed in
Figure 3A and 3B. As income
per capita increases emissions initially increase, peak in the
later part of this century when
income per capita reaches approximately twenty eight thousand
dollars and then emissions start
0.0005
0.0006
0.0007
0.0008
0.0009
0.0010
0.0011
0.0012
1
9
5
0

1
9
5
3
1
9
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6
1
9
5
9
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9
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9
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0
1
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3
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6

1
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1
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2
0
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2
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0
G
lo
b
a
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C
O
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G
lo
b
a
l
G
D

P
Year
12
declining. Along a steady state, emissions initially grow
because output grows faster than the
rate at which intensity falls but after a certain period the latter
becomes stronger than the former
causing emissions to fall. This can be seen as follows (See also
Taylor and Brock for a similar
expression, 2010):
��,� = ��,� + ��,� .
This relationship is important as it indicates to students how
difficult it is to reduce emissions in
an economy that is growing along a steady state due to
population growth, total factor
productivity growth and capital per worker growth.13
This relationship can also be connected to the IPAT equation
when expressed in growth
rates. The IPAT equation is used by the IPCC for setting future
emission targets. It links

environmental impact (I) to population (�� ), affluence (
��
��
) and technology (
��
��
). In our experience,
students find the IPAT equation easy to understand even though
it is an identity.14 The IPAT
equation for carbon emissions is usually expressed as follows:
�� ≡ ��
��
��
��
��
.
Carbon dioxide emissions at time t (i.e., ��) are proportional
to population multiplied by
affluence as measured by output per capita at time t and
technology as measured by carbon
emissions per dollar of output (recall that in the model �� /��
= ��). In growth rates, after
cancelling out the growth of population, this identity becomes:

��,� ≡ ��,� + ��,�
which is identical to the growth rate of emissions from the
model. The difference now is that the
Solow model provides a theory that explains why output grows.
Emissions grow because
13 This is offset partially by the growth rate of the damage that
occurs with increasing temperature.
14 Students can download yearly data from Gapminder.org to
explore this relationship for individual countries.
13
affluence grows along a steady state that in the Solow model is
due to population growth, growth
in total factor productivity and growth of capital per worker
offset by the impact on growth from
damages growing over time. The emissions growth rate is also
affected by the emissions
intensity falling over time (i.e., ��,� < 0). Hence the IPAT
equation in growth rates arises from
the long run properties of the Solow model and can explain why
the model produces an inverse
u-shaped emissions path over time (as displayed in Figure 3A).
At first, −��,� < ��,� but over

time the growth rate of output slows down (due to the assumed
diminishing TFP and population
growth) and eventually −��,� > ��,� producing negative
emissions growth (i.e., ��,� < 0).
Figure 3A. Predicted Global Carbon Emissions, 2010-2200
Source: Authors’ calculations.
0
5
10
15
20
25
2000 2050 2100 2150 2200
C
a
rb
o
n
E
m

is
si
o
n
s
(b
il
li
o
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s
o
f
to
n
n
e
s)
Year
14
Figure 3B. Environmental Kuznet’s Curve

�� ≡ ��
��
��
��
��
to find what it takes in terms of technology to reduce emissions
in 2050 by 50% below 2010 levels with
an assumed population growth of 1.5 percent and growth of
affluence as measured by income per person
by 2.5% per year.
0
5
10
15
20
25

0 10 20 30 40 50 60
C
a
rb
o
n
E
m
is
si
o
n
s
(b
il
li
o
n
s
o
f
to
n

n
e
s)
Income per person with damages (000s of constant $s)
Question 2: Use the IPAT equation
15
2.3 Carbon Accumulation & Temperature Change
One of the aspects that make this model so useful for teaching
is the simplicity of how the
climate system is modelled.15 The simple proportional stable
linear relationship between carbon
accumulation and global temperature change found by Matthews
et al. (2012) is used in the
model. They found that temperature increases by approximately
1.8 Celsius per 1000 billion
tonnes of carbon (i.e., 1000 PgC) emitted with a 95 percent
confidence band between 1 and 2.5
degrees Celsius. This relationship is found to be independent of
both time and the level of
stabilization of atmospheric carbon concentration (i.e., the
emissions scenario). Using this

scientifically based relationship avoids modelling much of the
complexity of the climate system
done by other IAM models.16 The following relationship shows
the cumulative emissions from
pre-industrial levels to 2010. The cumulative emissions from
the pre-industrial levels to 2010
(the base year for the simulations) are labelled as C0, i.e., these
are the sum of past emission
releases. The global temperature change relationship to carbon
accumulation into the future is:
�� = � [�0 + ∑ ��
�
�=1
],
where t ≥ 1. The first term, � �0, is the impact on global
temperature change relative to pre-
industrial levels due to the accumulated carbon emissions that
were released prior to 2010 (i.e.,
there are 530 billion tonnes already accumulated). The second
term, � ∑ ��
�
�=1 , is the impact on
global temperature at any time t in the future due to the

additional emissions accumulated since
15 It is important to give students a basic understanding of the
science of climate change before exposing them to the
modelling of temperature anomaly. Basics understanding of
climate change can be found at the U.S. EPA
http://www.epa.gov/climatechange/basics/ or showing students
the IPCC AR5 short video on the physical science
basis at https://www.youtube.com/watch?v=6yiTZm0y1YA. For
students that want to go beyond the basics on the
science of climate change, Professor Archer’s video lectures are
recommended:
http://forecast.uchicago.edu/lectures.html.
16 This complexity arises because there is uncertainty
associated with the path of carbon emissions towards affecting
the atmospheric concentration level, through carbon sensitivity.
Also there is uncertainty as to the impact of the
concentration level of carbon to temperature anomaly change
via the climate sensitivity parameter.
http://www.epa.gov/climatechange/basics/
https://www.youtube.com/watch?v=6yiTZm0y1YA
http://forecast.uchicago.edu/lectures.html
16

2010. Because of a growing economy, as shown in the previous
section, emissions will continue
to accumulate resulting in a higher temperature change.
Note that the above relationship is independent of the emissions
pathway selected. What
matters in terms of temperature change anomaly is the
cumulative carbon emissions and the
targeted budget. For example, to keep global temperature
anomaly below 2 degrees Celsius
relative to pre-industrial levels then cumulative emissions
should not increase more than
approximately 1110 billion tonnes (i.e., the budget). If they
increase by 470 billion tonnes over
the next 50 years which is within the current BAU pathway (See
Figure 3A and 3B) they will
reach 1000 billion tonnes. This will result in a temperature
increase of 1.8 degrees Celsius
relative to pre-industrial level given that Matthew et al. found �
to be 0.0018 per 1 billion tonnes
of cumulative carbon emitted. Figure 4 shows the path of
cumulative carbon emissions starting
from 530 billion tonnes. Figure 4 also shows the corresponding
temperature increases as well as
the 2degC target. With business as usual, 2 degrees Celsius will

be reached just before 2050 and
surpass 2000 billion tonnes by 2100 leading to a temperature
increase of 4 degrees Celsius which
is considered dangerous climate change.17
17 There is an estimated 6000 PgC that can be accumulated
given the fossil fuels available. Recently, the relationship
has been found to be stable within 5000 PgC (Tokarska et al.
2016).
17
Figure 4. Predicted Cumulative Carbon Emissions and
temperature anomaly
Source: authors’ calculations.
to keep the accumulation of carbon below 1000 billion tones by

2100. Note that each box in Figure 3A is
250 billion tonnes of carbon and that 530 billion tonnes since
2010 have already been accumulated. Can
emissions increase in the short run? Does stabilizing emissions
reduce the concentration? What are the
implications of the alternative paths?
0
1
2
3
4
5
6
7
8
0
500

1000
1500
2000
2500
3000
3500
4000
4500
2010 2060 2110 2160
Te
m
p
e
ra
tu
re
A
n
o
m
a

ly
(
d
e
g
re
e
s
C
e
ls
iu
s)
C
u
m
m
u
la
ti
v
e
C
a

rb
o
n
E
m
is
si
o
n
s
(b
il
li
o
n
s
o
f
to
n
n
e
s)
Year

Cummulative Carbon Emisiosns two degrees Temperature
Anomaly
Question 3: Find different paths, using Figure 3A, in
order
18
3 Additions to the base model for discussion
Damages in the base model enter multiplicatively in the
production function as in Nordhaus
(2013). It is assumed that climate change causes losses to
production in the same period only via
the damage function. Temperature increases are assumed not to
affect the depreciation of
physical capital nor any other form of capital such as
environmental, social and organizational
capital. In addition, climate change is assumed not to impact the
factors of production
individually nor the growth rate of total factor productivity.
Also, the damage function used in
the base model has been calibrated for losses when temperature
increases to 2.5-3 degrees
Celsius but it does not apply for higher temperature changes

which are a real possibility under
BAU (Stern, 2013). Furthermore, catastrophic damages are not
incorporated into the base model
(See Pyndick (2013), Weitzman (2013)). Below some of these
additions are incorporated into the
base model. This enriches the simple model in terms of
illustrating impacts to students. 18
First consider the depreciation rate of physical capital. It is easy
to conceive that
increased temperatures and more severe weather will lead to
capital having a shorter life span. It
was mentioned as a possibility by Fankhauser and Tol (2005)
and by Stern (2013). Recently, it
has been incorporated into the DICE model by Dietz and Stern
(2014) as well as Moore and Diaz
(2015). Climate change can affect the durability and the
longevity of stock of capital, for
example, increased temperatures cause increased frequency of
storms, more extreme weather,
rising sea levels, and many other impacts. Such events can
cause permanent damage to capital
infrastructure. 19 Capital will require more maintenance to keep
it from further wear and tear due

14.05 Lecture NotesIntroduction and The Solow ModelGeo.docx

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