程序代写代做代考 Lecture 1: Solow Growth Model
Lecture 1: Solow Growth Model
Patrick Macnamara
ECON60111: Macroeconomic Analysis University of Manchester
Fall 2020
Introduction and Stylized Facts
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Introduction
• Large growth of real income over the past couple of centuries
• In the US and Western Europe, real incomes today are between 5 and 20 times larger than a century ago, and between 15 and 100 times larger than two centuries ago.
• A lot of variation in worldwide growth over time
• Low growth over the millenia before the industrial revolution;
• Average growth rates in industrialized countries increasing
over 18th, 19th, and 20th centuries;
• Yet, since 1970s, fall in growth in the US and other
industrialized countries
• Large differences in growth across countries
• Some examples of growth miracles (Japan and East Asian countries) and growth disasters (Argentina, sub-Saharan Africa)
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Introduction
• To summarize: vast differences in standards of living over time and across countries.
⇒ enormous implications for human welfare
• Differences in income associated with large differences in nutrition, literacy, infant mortality, life expectancy
• Therefore, this week and the next, we will focus on growth • This week: Solow growth model
• Next week: Ramsey-Cass-Koopmans model
• Main conclusions of Solow growth model:
• Capital accumulation cannot account for either vast growth
over time of output per person, or the vast differences of
output per person across countries
• Other potential sources of differences in real incomes are
exogenous (e.g., technological progress) or not modeled (e.g., positive externalities from capital).
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Kaldor Facts
Stylized facts of long-run growth:1
1. Output per worker grows at a roughly constant rate
2. Capital per worker grows at a roughly constant rate
3. Capital/output ratio is roughly constant
4. Shares of income accruing to capital and labor are roughly constant over time
5. Rate of return to capital is constant
1See pages 591-592 in Kaldor, Nicholas (1957). “A Model of Economic Growth”. The Economic Journal, 67(268): 591-624. link
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The Solow Model: Assumptions
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Why Build a Model?
• The purpose of a model is not to be realistic.
• Making a model more realistic may make it too complicated
to understand.
• The model’s purpose is to isolate the key assumptions which provide insights about particular features of the world.
• Ideally, the simplifying assumptions should not cause the model to give incorrect answers. Nevertheless, if this isn’t the case, the model could still be a useful reference point.
“All theory depends on assumptions which are not quite true. That is what makes it theory. The art of successful theorizing is to make the inevitable simplifying assumptions in such a way that the final results are not very sensitive.” Solow (1956, pg. 65)2
2 Link
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Production
• Model is set in continuous time. • 4 variables, function of time t:
Y(t) = output, K(t) = capital, L(t) = labor,
A(t) = “knowledge”, “effectiveness of labor” or “technology”
• Aggregate production function:
Y(t) = F(K(t),A(t)L(t))
• F exhibits Constant Returns to Scale (CRS). • AL is effective labor input.
• technological progress is labor-augmenting.3 • Other inputs (like land) are neglected.
3Specifying technology this way ensures steady-state K/Y is constant.
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Production Function: Intensive Form
y≡AL=F AL,1 ≡f(k).
Output per unit of effective labor (y) is a function of
capital per unit of effective labor k ≡ K AL
• y = f (k) assumed to satisfy
f (0) = 0
Note that ∂F(K,AL) = f ′(k) f ′(k) > 0, ∂K
f ′′(k) < 0, and the Inada conditions:
lim f ′(k) = ∞, k→0
lim f′(k)=0. k→∞
• Given CRS: intensive form of production function YK
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Evolution of Inputs
• Initial level of K, L, and A are taken as given. e.g., fix K(0), L(0) and A(0)
• L and A grow at constant rates:4
d lnA(t) = A ̇ (t) = g dt A(t)
d lnL(t) = L ̇ (t) = n dt L(t)
• Therefore:
lnA(t)=lnA(0)+gt or A(t)=A(0)egt, A(0)>0
lnL(t)=lnL(0)+nt or L(t)=L(0)ent, L(0)>0 4X ̇ (t) is shorthand for dX(t)/dt.
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Capital Accumulation
• Assuming closed economy:
Y (t ) = C (t ) + I (t )
where C is consumption, I is investment.
• Law of motion for capital:
K ̇ ( t ) = I ( t ) − δ K ( t ) = Y ( t ) − C ( t ) − δ K ( t )
I(t) where δ is the capital depreciation rate.5
• Assume a constant saving rate s ∈ (0, 1). Hence:
I(t) = sY(t)
C (t ) = (1 − s )Y (t )
K ̇ (t) = sY (t) − δK(t)
Next week, I & C are endogenous (Ramsey-Cass-Koopmans) 5Assume that n + g + δ > 0.
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The Solow Model: Dynamics
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Dynamics of Effective Capital
• It’s much easier to focus on the dynamics of k rather than K • WehavealreadyspecifiedK ̇(t),L ̇(t)andA ̇(t):
K ̇ (t) = sY (t) − δK(t) L ̇(t) = nL(t)
A ̇ (t) = gA(t)
• Next step: use the equations above to determine k ̇(t)
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Dynamics of Effective Capital
• Use k = K/AL and apply quotient rule (and product rule): k ̇(t)= K ̇(t)A(t)L(t)−K(t)A(t)L ̇(t)+A ̇(t)L(t)
[A(t )L(t )]2
k ̇(t)= K ̇(t) − K(t) L ̇(t)− K(t) A ̇(t)
A(t)L(t) A(t)L(t) L(t) A(t)L(t) A(t)
k(t) n k(t) g k ̇(t)=s Y(t) −δ K(t) −(n+g)k(t)
sY(t)−δK(t)
A(t )L(t )
y(t)=f (k(t))
A(t )L(t )
k(t)
• We get:
k ̇(t) = sf (k(t)) − (n + g + δ)k(t)
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Dynamics of Effective Capital
• Key assumptions: (1) n+g+δ>0
(2) f(0)=0,f′ >0,f′′ <0 (3) Inada conditions
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Dynamics of Effective Capital: Steady State
• Denote steady-state level of effective capital by k∗
• Given assumptions (1)-(3) on previous slide, there exists a
unique k∗ > 0 where k ̇(t) = 0. • k∗ implicitly defined by
sf(k∗) = (n+g+δ)k∗ .
actual investment break-even investment
• If k ̸= k∗, k will converge back towards k∗:
k ̇ < 0 when k > k∗ k ̇ > 0 when k < k∗ k ̇ = 0 when k = k∗
Think about the intuition.
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Balanced Growth Path
• Regardless of starting point, economy converges to a balanced growth path where all variables grow at a constant rate (and k ̇ = 0).
• By assumption, A ̇/A = g and L ̇/L = n,
⇒ growth rate of effective labor (AL) is n + g.6
• SinceK =ALk andk ̇ =0insteadystate, ⇒ growth rate of K in steady state is n + g.
• y = f (k) and k ̇ = 0 ⇒ y ̇ = 0. Then using Y = ALy
⇒ growth rate of Y in steady state is n + g.7
6If X ̇ /X = gX and Y ̇ /Y = gY , it is straightforward to show that
̇ ̇
(XY)/(XY)=gX +gY and(X/Y)/(X/Y)=gX −gY.
7Y =F(K,AL)exhibitsCRS.Sincebothinputsgrowattheraten+g, Y growsatn+g.
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Balanced Growth Path
• Growth rate of K is n + g and growth rate of L is n: ⇒ growth rate of K/L in steady state is g
• Growth rate of Y is n + g and growth rate of L is n: ⇒ growth rate of Y /L in steady state is g
• Growth rate of K is n+g and growth rate of Y is n+g:
⇒ growth rate of K/Y in steady state is 0 matching a stylized fact!
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The Solow Model: Change in Saving Rate
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Increase in Saving Rate
• Increase in steady-state capital per effective worker
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Increase in Saving Rate
• k ̇ > 0 on impact, k ̇ gradually falls back to zero
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Increase in Saving Rate
• Permanent increase in s leads to …
• temporary increase in growth rate of Y /L • increase in level of Y /L
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Increase in Saving Rate: Summary So Far
Permanent increase in saving rate s leads to …
• increase in steady state k∗
• temporary increase in k ̇ as economy converges to new balanced growth path
• temporary increase in growth rate of Y /L above g • Increase in level of Y /L
What’s the effect on consumption?
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Increase in Saving Rate: Consumption
• Individual welfare depends on consumption, not output. ⇒ we are more interested in effect on consumption
• Consumption per effective labor:
c = (1 − s)f (k)
• On impact, when s increases, c must fall (because k is fixed). • As the economy converges to the new steady state,
k increases, increasing c.
Yet, it’s ambiguous whether c∗ is higher/lower than c∗
NEW
c∗ ≡ (1 − s )f (k∗ ) ⪋ (1 − s OLD OLD OLD
OLD NEW NEW
NEW
)f (k∗ ) ≡ c∗
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Increase in Saving Rate: Consumption
• Steady state c∗:
c∗ =f(k∗)−sf(k∗)=f(k∗)−(n+g+δ)k∗
∗ =sf(k )
• Effect of change in saving rate:8
∂c∗ ′∗ ∂k∗
∂s =[f (k )−(n+g+δ)] ∂s (1)
• c∗ increasesiff′(k∗)>(n+g+δ) • c∗ decreasesiff′(k∗)<(n+g+δ)
pos.
8Think of k∗ being a function of (s, n, g, δ), k∗(s, n, g, δ), which is implicitly defined by sf (k∗) = (n + g + δ)k∗.
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Golden-Rule
• Value of k∗ which maximizes steady-state c∗ is determined by (2)
At this point, dc∗/ds = 0 in (1) on previous slide
• k∗ defined by (2) is known as the golden-rule level of the
GR
capital stock.
• Saving rate s such that k∗(s) = k∗ GR
golden-rule saving rate.
is known as the
f′(k∗ )=n+g+δ GR
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Golden-Rule: Illustration
GoldHLeoingw-hRSualeviSnagvRinagteRate
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The Solow Model: Quantitative Implications
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Quantitative Implications
• We are not just interested in the model’s qualitative implications, but it’s quantitative predictions.
For example, a higher savings rate increases growth temporarily, but how much and for how long?
• In the next few slides, we’ll focus the quantitative effects of an increase in the saving rate.
• How much does y∗ increase?
• How quick is the convergence to the new steady state?
• Then, we’ll take the model to the data.
• According to the model, what explains income differences
(across countries and over time)?
• Do poor countries tend to grow faster than rich ones, as
predicted by the model?
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Effect on Output
• Long-run effect on y∗ = f (k∗):
∂y∗ ′∗∂k∗
∂s =f(k)∂s
• To determine ∂k∗/∂s, use implicit differentiation:
sf (k∗) = (n + g + δ)k∗
∗ ′∗dk∗ dk∗
f(k )+sf (k )ds =(n+g+δ)ds ∂k∗ f(k∗)
• Therefore:
∂s =(n+g+δ)−sf′(k∗)>0
∂y∗ f ′(k∗)f (k∗)
∂s =(n+g+δ)−sf′(k∗)>0
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Effect on Output
• Convert to elasticity (for easier interpretation):
s ∂y∗ s f′(k∗)f(k∗) s ∂y∗ s f′(k∗
y∗ ∂s = f(k∗)(n+g +δ)−sf′(k∗)y∗ ∂s = f(k∗)(n+g +δ • Use substitution sf (k∗) = (n + g + δ)k∗:
s ∂y∗ (n+g+δ)k∗f′(k∗)/f(k∗)
y∗ ∂s = (n+g +δ)−(n+g +δ)k∗f′(k∗)/f(k∗)
= k∗f ′(k∗)/f (k∗)
1 − k∗f ′(k∗)/f (k∗)
• Re-write:
s ∂y∗ αK(k∗) kf′(k)
y∗ ∂s =1−αK(k∗) where αK(k)≡ f(k) and αK(k) is the elasticity of output w.r.t. capital.9
9Note that, for F(K,AL) = Kα(AL)1−α, αK(k) = α.
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Effect on Output: Numerical Example
• Elasticity of output w.r.t. to the saving rate: s ∂y∗ αK(k∗)
y∗ ∂s =1−αK(k∗) • Usually,αK(k)≈1/3,implying:
s∂y∗ 1 y∗ ∂s ≈2
• 10% increase in saving rate (e.g., from 20% to 22%):
⇒ 5% increase in Y /L in long run (relative to original path)
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Speed of Convergence
• How quickly does k converge to k∗? • Key equations:
k ̇(k)≡sf(k)−(n+g+δ)k AND k ̇(k∗)=0
• First-order Taylor approximation of k ̇(k) around k = k∗:10
∂ k ̇ k ̇(k)≈k ̇(k∗)+ ∂k
(k−k∗)
= 0 + [sf ′(k∗) − (n + g + δ)](k − k∗)
• Usesf(k∗)=(n+g+δ)k∗ again:
k ̇ (k) ≈ [(n + g + δ)k∗f ′(k∗)/f (k∗) − (n + g + δ)] (k − k∗) αK(k∗)
= − (1 − αK (k∗)) (n + g + δ)(k − k∗)
10First order Taylor-series approximation of f(x) around a is 33/40
k=k∗
≈′−
Speed of Convergence
• Summarizing:
k ̇ (k) ≈ − (1 − αK (k∗)) (n + g + δ)(k − k∗) = −λ(k − k∗)
λ
• Re-write as a function of time t:
k ̇ ( t ) ≈ − λ ( k ( t ) − k ∗ )
• Solution to this differential equation:11
k(t) ≈ k∗ + e−λt (k(0) − k∗)
approximately describes the evolution of k(t) near the SS.
• Similarly, it can be shown:
y(t) ≈ y∗ + e−λt (y(0) − y∗) (3)
11Trick: define x(t) ≡ k(t) − k∗ and use the fact that the solution to the differential equation x ̇(t)/x(t) = g is x(t) = x(0)egt.
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Speed of Convergence: Numerical Example
• Supposen+g+δ=0.06peryearandassumeαK =1/3 ⇒ λ = (1 − αK )(n + g + δ) = 0.04 per year
⇒ Every year, the gap y(t) − y∗ is reduced by 4%.
• Supposeatdate0,y(0)−y∗ >0.
• Howmanyyearsuntily(t)−y∗ =0.5[y(0)−y∗]?
From (3) on previous slide: −λt y(t)−y∗
e =y(0)−y∗ =0.5 ⇒ t=ln(0.5)/0.04≈17years Effect on output is slow to occur
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The Solow Model and Key Questions of Growth
• The model matches the stylized facts, in particular that the capital-output ratio remains constant as the economy grows.
• But can the model explain observed income differences (across countries and over time)?
• The Solow model identifies two possible sources of variation:
• capital per worker (K/L)
• effectiveness of labor (A)
(knowledge? technology? skills of labor force? strength of
property rights? etc.)
• Changes in capital per worker (over time and across countries) not large enough to explain the income differences.
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Convergence
• Do poor countries tend to grow faster than rich ones?
• Suppose countries have similar characteristics (e.g., saving rates), but different initial levels of capital.
• Then, Solow growth model predicts that poor countries should catch up to rich ones (i.e., convergence).
• We take a historical perspective of the evidence which informs the debate.
• Basic idea: regress the growth rate over a given period on initial (real) income (per capita).
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Evidence on Convergence: Baumol (1986)
• Baumol (1986) examines convergence from 1870 to 1979 • 16 industrialized countries for which long historical data
series were readily available • Suggests convergence
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Evidence on Convergence: DeLong (1988)
• DeLong considers the richest countries in 1870; includes 7 countries Baumol excluded and drops 1.
• Weaker evidence after correcting for sample selection. • Less evidence after correcting for measurement error.
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Evidence on Convergence
• Shorter time period, but most countries included (Middle Eastern oil producers excluded).
• Little evidence of convergence.
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