Endogenous Growth: AK Model

Prof. Lutz Hendricks

Econ720

October 14, 2020

1 / 35 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

2 / 35 Necessary Conditions for Sustained Growth

I How can growth be sustained without exogenous productivity growth? I A necessary condition: constant returns to the reproducible factors. I The production functions for inputs that can be accumulated must be linear in those inputs. I Example: In the growth model, K would have to be produced with a technology that is linear in K I This motivates a simple class of models in which 1. only K can be produced and 2. the production function is AK. I This can be thought of as a reduced form for more complex models (we’ll see examples).

3 / 35 Solow AK model To see what is required for endogenous growth, consider the Solow model: g(k) = sf (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 0 has a limit): lim f (k)/k = lim f 0(k) (3)

Sustained growth therefore requires:

lim f 0(k) > n + δ (4) k→∞

4 / 35 Necessary Conditions for Sustained Growth

I This argument is more general than the Solow model. I It does not matter how s is determined. 0 I If limk→∞ f (k) exists, the production function has asymptotic constant returns to scale.

f (k) → Ak + B (5)

I It is fine to have diminishing returns for finite k.

5 / 35 Examples

f (k) = Ak + Bkα (6)

I 0 < α < 1 I f (k)/k → A as k → ∞

CES production function with high elasticity of substitution:

h i1/θ F (K,L) = µKθ + (1 − µ)Lθ (7)

 1/θ I f (k) = µkθ + 1 − µ I Elasticity of substitution: ε = (1 − θ)−1. I If θ > 0 [ε > 1], f (k)/k → µ1/θ .

6 / 35 AK Solow Model

I In the Solow model, assume f (k) = Ak. I Law of motion: g(k) = sA − n − δ (8)

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.

7 / 35 AK Solow Model

I It is not necessary to have constant returns in all sectors of the economy. I Imagine that c is produced from k with diminishing returns to scale: c = [(1 − s)Ak]ϕ with ϕ < 1. I The law of motion for k is unchanged (so is the balanced growth rate of k). I This model still has a balanced growth path with a strictly positive growth rate, but not c and k grow at different constant rates: g(c) = ϕ g(k) (9)

8 / 35 AK Neoclassical Growth Model AK neoclassical growth model

This model adds optimizing consumers to the Ak model. Households maximize Z ∞ −(ρ−n)t e u(ct)dt (10) t=0

subject to the flow budget constraint

k˙ = (r − n)k − c (11)

There is no labor income because in the Ak world all income goes to capital.

10 / 35 AK neoclassical growth model

For balanced growth we need

u(c) = c1−σ /(1 − σ) (12)

The optimality conditions are the same as in the Cass-Koopmans model: g(c) = (r − ρ)/σ

and the transversality condition (assuming constant r)

−(r−n)t lim kt e = 0 (13) t→∞

11 / 35 Firms

Firms maximize period profits. The first-order condition is r = A − δ .

12 / 35 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 (14)

13 / 35 Summary

Simplify into a pair of differential equations:

k˙ = (A − δ − n)k − c (15) g(c) = (A − δ − ρ)/σ (16)

Boundary conditions: k0 given and the TVC. Of course, we could have simply taken the equilibrium of the standard growth model and replaced f 0 (k) = A and f (k) = Ak.

14 / 35 Bounded utility

We need restrictions on the parameters that ensure bounded utility. Lifetime utility is

Z ∞ −(ρ−n)t g(c)t 1−σ e [c0 e ] dt/(1 − σ) (17) 0

Boundedness then requires that n − ρ + (1 − σ)g(c) < 0. Instantaneous utility cannot grow faster than the discount factor (ρ − n).

15 / 35 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 closed form.

Details

16 / 35 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.

17 / 35 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 make sense? The best way of thinking about AK models: I a reduced form for a model with multiple factors I there may be transition dynamics, but it does not matter if you are interested in long-run issues I there may be fixed factors, but it does not matter if there are constant returns to reproducible factors.

18 / 35 Examples: AK as reduced form

1. Human capital: F (K,hL) with K and h reproducible. 2. Externalities: 1−α α θ 2.1 Romer (1986). For the firm F (ki,liK) = K ki li 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.

19 / 35 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.

21 / 35 Model: Lucas (1988) Demographics: I A representative, infinitely lived household. Preferences: Z ∞ −ρt e u(ct)dt (18) 0 u(c) = c1−σ /(1 − σ) (19)

Technology:

k˙ + c = f (k,h,l) − δk (20) h˙ = e(k,h,l) − δh (21)

where f (k,h,l) = kα (lh)1−α and e(k,h,l) = B(1 − l)h

22 / 35 Lucas (1988): Balanced growth rates

Law of motion for h:

g(h) = B(1 − l) − δ (22)

Law of motion for k:

g(k) + c/k = (lh/k)1−α − δ (23)

Therefore: g(c) = g(k) = g(h) (24)

23 / 35 Lucas (1988): Optimality

Current value Hamiltonian:

H = u(c) + λ [e(k,h,l) − δh] + µ [f (k,h,l) − δk − c] (25)

FOCs:

∂H/∂c = u0(c) − µ = 0 (26)

∂H/∂u = λel+µfl = 0 (27) ˙ ρλ − λ = λ [eh − δ] + µfh (28)

ρµ − µ˙ = µ [fk − δ] + λek (29)

Major simplification from ek = 0.

24 / 35 Optimality Euler equation (using ek = 0) f − δ − ρ g(c) = k (30) σ

From FOC for h:

−g(λ) = eh − δ − ρ + µ/λfh (31)

FOC for u: µ elfh l fh = − = (Bh) = Bl (32) λ fu h

Substitute into FOC for h: −g(λ) = B − δ − ρ (33)

This is an exogenous constant! 25 / 35 Balanced growth

Constant fk requires constant k/h. Then µ Bh Bh = = (34) α 1−α λ fl (1 − α)(k/h) l requires constant µ/λ.

Then g(uc) = −g(µ) = −g(λ) = B − δ − ρ. This determines the interest rate:

r = fk − δ = B − δ (35)

The balanced growth rate is determined by the linear human capital technology: B − δ − ρ g(c) = (36) σ

26 / 35 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 future income I regardless of current values of k and h. I This pins down the interest rate on the other asset by no arbitrage. I All of this has implicitly assumed an interior solution!

27 / 35 Summary

Sustained growth requires that inputs are produced with constant returns 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.

28 / 35 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).

29 / 35 Digression: Solving for k(t)I

I Law of motion: A − δ − ρ  k˙ = (A − δ − n)k − c exp t (37) t t 0 σ

I Solution to x˙ = ax − b(t) is Z t at at −as xt = x0e − e e b(s)ds (38) 0

I To verify: Z t at at −as at −at x˙t = ax0e − ae e b(s)ds − e e b(t) (39) 0 = axt − b(t) (40)

30 / 35 Digression: Solving for k(t)II I Define a = r − n = A − δ − n > 0 (41)

A − δ − ρ b = g = > 0 (42) c σ

I Then Z t kt = k0 exp(at) − exp(at) c0 exp([−a + b]s)ds (43) 0

I Note: Z t ezt − 1 ezsds = (44) 0 z

31 / 35 Digression: Solving for k(t)III

I Therefore: c h i k = k eat − 0 eat e(b−a)t − 1 (45) t 0 b − a  c  c = k + 0 eat − 0 ebt (46) 0 b − a b − a

bt I Now we show that g(k) is constant: kt = k0e . I Transversality: (r−n)t lime kt = 0 (47) t−∞

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.

32 / 35 Digression: Solving for k(t)IV

I With b < a capital grows at rate a, unless the term in brackets is 0: c k + 0 = 0 (48) 0 b − a

−(r−n)t
I If g(k) = a, then g e kt = 0 - because a = r − n. I That violates TVC.

I The only value of c0 consistent with TVC is the one that sets the term in brackets to 0. I It implies that k always grows at rate b.

33 / 35 Saving rate

I 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.

34 / 35 ReferencesI

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 of Equilibrium Growth: Theory and Policy Implications,” Journal of Political 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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