Post on 26-Aug-2018
Endogenous Growth:AK Model
Prof. Lutz Hendricks
Econ720
October 24, 2017
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Endogenous Growth
I Why do countries grow?I A question with large welfare consequences.
I We need models where growth is endogenous.I The simplest model is a variation of the Ramsey model.
I Growth can be sustained if the MPK is bounded below.I AK model
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Necessary Conditions for Sustained Growth
I How can growth be sustained without exogenous productivitygrowth?
I A necessary condition: constant returns to thereproducible factors.
I The production functions for inputs that can be accumulatedmust be linear in those inputs.
I Example: In the growth model, K would have to be producedwith a technology that is linear in K
I This motivates a simple class of models in which
1. only K can be produced and2. the production function is AK.
I This can be thought of as a reduced form for more complexmodels (we’ll see examples).
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Solow AK modelTo see what is required for endogenous growth, consider the Solowmodel:
g(k) = s f (k)/k− (n+δ ) (1)
Positive long-run growth requires: As k→ ∞ it is the case that
f (k)/k > n+δ (2)
L’Hopital’s rule implies (if f ′ has a limit):
lim f (k)/k = lim f ′(k) (3)
Sustained growth therefore requires:
limk→∞
f ′(k)> n+δ (4)
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Necessary Conditions for Sustained Growth
I This argument is more general than the Solow model.I It does not matter how s is determined.
I If limk→∞ f ′(k) exists, the production function has asymptoticconstant returns to scale.
f (k)→ Ak+B (5)
I It is fine to have diminishing returns for finite k.
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Examples
1. f (k) = Ak+Bkα with 0 < α < 1
1.1 f (k)/k→ A
2. CES production function with high elasticity of substitution:
F (K,L) =[µKθ +(1−µ)Lθ
]1/θ
(6)
2.1 f (k) =[µkθ +1−µ
]1/θ
2.2 Elasticity of substitution: ε = (1−θ)−1.2.3 If θ > 0 [ε > 1], f (k)/k→ µ1/θ .
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AK Solow Model
I In the Solow model, assume f (k) = Ak.I Law of motion:
g(k) = sA−n−δ (7)
I Changes in parameters alter the growth rate of k.I The model does not have any transitional dynamics: k always
grows at rate sA−n−δ .
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AK Solow Model
I It is not necessary to have constant returns in all sectors of theeconomy.
I Imagine that c is produced from k with diminishing returns toscale: c = [(1− s)Ak]ϕ with ϕ < 1.
I The law of motion for k is unchanged (so is the balancedgrowth rate of k).
I This model still has a balanced growth path with a strictlypositive growth rate, but not c and k grow at differentconstant rates:
g(c) = ϕ g(k) (8)
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AK Neoclassical Growth Model
AK neoclassical growth model
This model adds optimizing consumers to the Ak model.Households maximize ∫
∞
t=0e−(ρ−n) t u(ct)dt (9)
subject to the flow budget constraint
k = (r−n)k− c (10)
There is no labor income because in the Ak world all income goesto capital.
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AK neoclassical growth model
For balanced growth we need
u(c) = c1−σ/(1−σ) (11)
The optimality conditions are the same as in the Cass-Koopmansmodel:
g(c) = (r−ρ)/σ
and the transversality condition (assuming constant r)
limt→∞
kt e−(r−n) t = 0 (12)
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Firms
Firms maximize period profits.The first-order condition is r = A−δ .
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Equilibrium
An allocation: c(t) ,k (t).A price system: r (t).These satisfy:
1. Household: Euler, budget constraint (TVC).2. Firm: 1 foc.3. Market clearing:
k = Ak− (n+δ )k− c (13)
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Summary
Simplify into a pair of differential equations:
k = (A−δ −n)k− c (14)g(c) = (A−δ −ρ)/σ (15)
Boundary conditions: k0 given and the TVC.
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Bounded utility
We need restrictions on the parameters that ensure bounded utility.Lifetime utility is∫
∞
0e−(ρ−n)t [c0 eg(c)t]1−σ dt/(1−σ) (16)
Boundedness then requires that n−ρ +(1−σ)g(c)< 0.Instantaneous utility cannot grow faster than the discount factor(ρ−n).
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Transitional dynamics
This model has no transitional dynamics.Consumption growth is obviously constant over time.To show that g(k) is constant: we need to solve for k (t) in closedform.
Details
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Summary
The AK model has a very simple equilibrium.
1. The saving rate is constant.2. All growth rates are constant.
This is very convenient, but also very limiting in many applications.
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How to think about AK models?
In the data, there is at least one non-reproducible factor: labor.Do models with constant returns to reproducible factors makesense?
The best way of thinking about AK models:
I a reduced form for a model with multiple factorsI there may be transition dynamics, but it does not matter if
you are interested in long-run issuesI there may be fixed factors, but it does not matter if there are
constant returns to reproducible factors.
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Examples: AK as reduced form
1. Human capital: F (K,hL) with K and h reproducible.2. Externalities:
2.1 Romer (1986). For the firm F (ki, liK) = K1−α kαi lθi
2.2 Firms take K as given - diminishing returns to ki.2.3 In equilibrium: K = ∑ki - constant returns to scale to K.
3. Increasing returns to scale at the firm level: y = Akα l1−α
3.1 A can be produced somehow - R&D.3.2 Need imperfect competition.
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Example: Lucas (1988)
Example: Lucas (1988)
A classic endogenous growth paper.Growth is due to human capital accumulation.The model has an AK reduced form.
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Model: Lucas (1988)Demographics:
I A representative, infinitely lived household.
Preferences: ∫∞
0e−ρtu(ct)dt (17)
u(c) = c1−σ/(1−σ) (18)
Technology:
k+ c = f (k,h, l)−δk (19)h = e(k,h, l)−δh (20)
where f (k,h, l) = kα(lh)1−α and e(k,h, l) = B(1− l)h
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Lucas (1988): Balanced growth rates
Law of motion for h:
g(h) = B(1− l)−δ (21)
Law of motion for k:
g(k)+ c/k = (lh/k)1−α −δ (22)
Therefore:g(c) = g(k) = g(h) (23)
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Lucas (1988): Optimality
Current value Hamiltonian:
H = u(c)+λ [e(k,h,u)−δh]+µ [f (k,h,u)−δk− c] (24)
FOCs:
∂H/∂c = u′(c)−µ = 0 (25)∂H/∂u = λel+µfl = 0 (26)
ρλ − λ = λ [eh−δ ]+µfh (27)ρµ− µ = µ [fk−δ ]+λek (28)
Major simplification from ek = 0.
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OptimalityEuler equation (using ek = 0)
g(c) =fk−δ −ρ
σ(29)
From FOC for h:
−g(λ ) = eh−δ −ρ +µ/λ fh (30)
FOC for l:µ
λfh =−
elfhfl
= (Bh)lh= Bl (31)
Substitute into FOC for h:
−g(λ ) = B−δ −ρ (32)
This is an exogenous constant!25 / 35
Balanced growthConstant fk requires constant k/h.Then
µ
λ=
Bhfl
=Bh
(1−α)(k/h)α l1−α(33)
requires constant µ/λ .Then g(uc) =−g(µ) =−g(λ ) = B−δ −ρ .This determines the interest rate:
r = fk−δ = B−δ (34)
The balanced growth rate is determined by the linear human capitaltechnology:
g(c) =B−δ −ρ
σ(35)
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Intuition
I The household has 2 assets: k and h.I One asset has a constant rate of return:
I give up 1 unit of time to gain a fixed increment of futureincome
I regardless of current values of k and h.
I This pins down the interest rate on the other asset by noarbitrage.
I All of this has implicitly assumed an interior solution!
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Summary
Sustained growth requires that inputs are produced with constantreturns to reproducible inputs.Then the model is (at least asymptotically) of the AK form:K = AK.The AK model is a reduced form of something more interesting.
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Reading
I Acemoglu (2009), ch. 11.I Krueger, "Macroeconomic Theory," ch. 9.I Krusell (2014), ch. 8.I Barro and Martin (1995), ch. 1.3, 4.1, 4.2, 4.5.I Jones and Manuelli (1990)I Lucas (1988).
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Digression: Solving for k(t) I
I Law of motion:
kt = (A−δ −n)kt− c0 exp(
A−δ −ρ
σt)
(36)
I Solution to x = ax−b(t) is
xt = x0eat− eat∫ t
0e−asb(s)ds (37)
I To verify:
xt = ax0eat−aeat∫ t
0e−asb(s)ds− eate−atb(t) (38)
= axt−b(t) (39)
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Digression: Solving for k(t) III Define
a = r−n = A−δ −n > 0 (40)
b = gc =A−δ −ρ
σ> 0 (41)
I Then
kt = k0 exp(at)− exp(at)∫ t
0c0 exp([−a+b]s)ds (42)
I Note: ∫ t
0ezsds =
ezt−1z
(43)
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Digression: Solving for k(t) IIII Therefore:
kt = k0eat− c0
b−aeat[e(b−a)t−1
](44)
=
[k0 +
c0
b−a
]eat− c0
b−aebt (45)
I Now we show that g(k) is constant: kt = k0ebt.I Transversality:
limt−∞
e(r−n)tkt = 0 (46)
I Note that a = r−n = A−ρ−n.I If b > a: g(k)→ b > a and TVC is violated.I So we need b < a.
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Digression: Solving for k(t) IV
I With b < a capital grows at rate a, unless the term in bracketsis 0:
k0 +c0
b−a= 0 (47)
I If g(k) = a, then g(e−(r−n)tkt
)= 0 - because a = r−n.
I That violates TVC.
I The only value of c0 consistent with TVC is the one that setsthe term in brackets to 0.
I It implies that k always grows at rate b.
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Saving rateI We can solve for c/k and the saving rate.
g(k)−g(c) = [A−δ −n− c/k]− (A−δ −ρ)/σ = 0
c/k = A−δ −n− (A−δ −ρ)/σ
I And the gross savings rate is
s = (K +δ K)/AK
= [g(K)+δ ]/A
= [g(c)+n+δ ]/A
= [(A−δ −ρ)/σ +n+δ ]/A
I The savings rate is high, if (σ ,ρ or A) are low, or if n is high.
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References I
Acemoglu, D. (2009): Introduction to modern economic growth,MIT Press.
Barro, R. and S.-i. Martin (1995): “X., 1995. Economic growth,”Boston, MA.
Jones, L. E. and R. Manuelli (1990): “A Convex Model ofEquilibrium Growth: Theory and Policy Implications,” Journal ofPolitical Economy, 1008–1038.
Krusell, P. (2014): “Real Macroeconomic Theory,” Unpublished.
Lucas, R. E. (1988): “On the mechanics of economic development,”Journal of monetary economics, 22, 3–42.
Romer, P. M. (1986): “Increasing returns and long-run growth,”The journal of political economy, 1002–1037.
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