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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