ECON 582: The Neoclassical

Growth Model (Chapter 8,

Acemoglu)

Instructor: Dmytro Hryshko

1 / 25

Consider the neoclassical economy without populationgrowth and technological progress. The optimal growthproblem is then

max[k(t),c(t)]∞t=0

∫ ∞0

exp(−ρt)u(c(t))dt

s.t.

k(t) = f(k(t))− δk(t)− c(t),

k(0) > 0, k(t) ≥ 0 for all t.The current-value Hamiltonian can be written as

H(k, c, µ) = u(c(t)) + µ(t) [f(k(t))− δk(t)− c(t)] ,

with state variable k, control variable c, and current-valuecostate variable µ.

2 / 25

Theorem 7.13 Suppose the problem of maximizing (7.60)subject to (7.61) and (7.62). Let H(t, x, y, µ) be thecurrent-value Hamiltonian given by (7.64). Then except atpoints of discontinuity of y(t), the optimal control pair(y(t), x(t)) satisfies the following necessary conditions:

Hy(t, x(t), y(t), µ(t)) = 0 for all t ∈ R+ (7.65)

ρµ(t)− µ(t) = Hx(t, x(t), y(t), µ(t)) for all t ∈ R+ (7.66)

x(t) = Hµ(t, x(t), y(t), µ(t)) for all t ∈ R+, x(0) = x0, limt→∞

b(t)x(t) ≥ x1(7.67)

limt→∞

[exp(−ρt)µ(t)x(t)] = 0. (7.69)

where (7.69) is the transversality condition.

3 / 25

The economy consists of a set of identical households;population within each household grows at the rate n,L(0) = 1:

L(t) = exp(nt).

Labor supply is inelastic. Per-capita utility u(c(t));household utility at t is L(t)u(c(t)) = exp(nt)u(c(t)). Thenthe objective function of each household at time t = 0 canbe written as ∫ ∞

0

exp(−(ρ− n)t)u(c(t))dt (8.3)

4 / 25

Assumption 3 (Neoclassical preferences): u(c(t)) isstrictly increasing, concave, and twice differentiable.

Assumption 4′ (Discounting): ρ > n. (Ensures that, inthe model without growth, discounted utility is finite.)

Assume the economy without technological progress;production function Y (t) = F (K(t), L(t)).

y(t) ≡ Y (t)

L(t)= F

(K(t)

L(t), 1

)≡ f(k(t)).

Competitive factor markets imply

R(t) = f ′(k(t)) (8.5)

w(t) = f(k(t))− k(t)f ′(k(t)). (8.6)

5 / 25

Note thatY = F (K,L) = Lf( k︸︷︷︸

=KL

).

You can show (8.5) by differentiating Lf(k) with respect toK

R =∂

∂K[Lf(k)] = Lf ′(k)

∂

∂K

[K

L

]= f ′(k)

and (8.6) by differentiating Lf(k) with respect to L:

w =∂

∂L[Lf(k)] = f(k)− Lf ′(k)

K

L2= f(k)− kf ′(k).

6 / 25

The law of motion for the total household assets is

A(t) = r(t)A(t) + w(t)L(t)− c(t)L(t).

Define per-capita assets as a(t) ≡ A(t)L(t)

. Note that

a(t) = A(t)L(t)−A(t)L(t)L(t)2

= A(t)L(t)− a(t)n. Thus,

a(t) = (r(t)− n)a(t) + w(t)− c(t). (8.8)

Market-clearing implies

a(t) = k(t). (8.9)

The no-Ponzi condition:

limt→∞

[a(t) exp

(−∫ t

0

(r(s)− n)ds)

)]= 0. (8.16)

7 / 25

Definition of equilibrium

Definition 8.1A competitive equilibrium of the neoclassical growth modelconsists of paths of consumption, capital stock, wage rates,and rental rates of capital, [C(t), K(t), w(t), R(t)]∞t=0, suchthat:

the representative household maximizes its utility giveninitial asset holdings (capital stock) K(0) > 0 andtaking the time path of factor prices [w(t), R(t)]∞t=0 asgiven;

firms maximize profits taking the time path of factorprices as given; and

factor prices are such that all markets clear.

8 / 25

Household maximization

The current-value Hamiltonian:

H(t, a, c, µ) = u(c(t)) + µ(t) [w(t) + (r(t)− n)a(t)− c(t)] .

Applying Theorem 7.13,

Hc = u′(c(t))− µ(t) = 0 (8.17)

Ha = µ(t)(r(t)− n) = (ρ− n)µ(t)− µ(t) (8.18)

limt→∞

[exp (−(ρ− n)t)µ(t)a(t)] = 0. (8.19)

9 / 25

(8.18) implies:

µ(t)

µ(t)= −(r(t)− ρ). (8.20)

Differentiating (8.17) wrt t,

u′′(c(t))c(t) = µ(t),

and dividing by µ(t),[u′′(c(t))c(t)

u′(c(t))

]︸ ︷︷ ︸≡−εu(c(t))

c(t)

c(t)=µ(t)

µ(t).

Using (8.20),c(t)

c(t)=

1

εu(c(t))(r(t)− ρ). (8.22)

10 / 25

From (8.22), consumption grows if the discount rate isbelow the rate of return on assets; the speed of growth isrelated to the elasticity of marginal utility of consumption.(8.20) implies

µ(t) = µ(0) exp

(−∫ t

0

(r(s)− ρ)ds

)= u′(c(0)) exp

(−∫ t

0

(r(s)− ρ)ds

). (8.24)

The transversality condition (8.19) can be expressed as

limt→∞

exp (−(ρ− n)t)︸ ︷︷ ︸=∫ t0 (n−ρ)ds

a(t)u′(c(0)) exp

(−∫ t

0

(r(s)− ρ)ds

) = 0

limt→∞

[a(t) exp

(−∫ t

0

(r(s)− n)ds

)]= 0. (8.25)

11 / 25

Thus, the transversality condition implies the no-Ponzicondition (8.16). Since in equilibrium a(t) = k(t), it can bewritten as

limt→∞

[k(t) exp

(−∫ t

0

(r(s)− n)ds

)]= 0. (8.26)

Thus, the discounted market value of capital in the very farfuture should equal zero.

Any pair (k(t), c(t)) that satisfies (8.22) and (8.25)corresponds to a competitive equilibrium.Since r(t) = R(t)− δ, (8.22) implies

c(t)

c(t)=

1

εu(c(t))(f ′(k(t))− δ − ρ). (8.28)

12 / 25

Steady-state equilibrium

Requires that c(t) = 0. From (8.28),

f ′(k∗) = δ + ρ. (8.35)

(8.35) corresponds to the modified golden rule. Togetherwith Assumption 4′ this implies

r∗ = f ′(k∗)− δ = ρ > n. (8.36)

SS consumption is

c∗ = f(k∗)− (n+ δ)k∗.

13 / 25

Redefine the production function as f(k) = Af(k), where Ais a shift parameter. Additional comparative statics resultsare

∂k∗(A, ρ, n, δ)

∂ρ< 0,

∂c∗(A, ρ, n, δ)

∂ρ< 0,

∂k∗(A, ρ, n, δ)

∂n= 0,

∂c∗(A, ρ, n, δ)

∂n< 0.

A lower discount rate implies greater patience and thusgreater savings. In the model with no technologicalprogress, the savings rate in SS is

s∗ =(n+ δ)k∗

f(k∗). (8.38)

14 / 25

New results relative to the Solow model

Comparative statics with respect to ρ (replacescomparative statics wrt exogenously given s in theSolow model)

SS values k∗ and c∗ do not depend on theinstantaneous utility function u(·) as the SS isdetermined by the modified golden rule

The form of u(·) affects transitional dynamics but notthe SS

k∗ is not affected by n (depends on the wayintertemporal discounting is introduced)

15 / 25

Proposition 8.4

In the neoclassical growth model with no technologicalprogress, with Assumptions 1–4′, there exists a uniqueequilibrium path starting from any k(0) > 0 and convergingmonotonically to the unique steady-state (k∗, c∗) with k∗

given by (8.35). This equilibrium path is identical to theunique optimal growth path.

16 / 25

Transitional dynamics in the neoclassical growth

modelIntroduction to Modern Economic Growth

c(t)

kgold0k(t)

k(0)

c’(0)

c’’(0)

c(t)=0

k(t)=0

k*

c(0)

c*

k

Figure 8.1. Transitional dynamics in the baseline neoclassicalgrowth model.

Therefore, the correct notion of stability in models with state and control variables

is one in which the dimension of the stable curve (manifold) is the same as that of

the state variables, requiring the control variables jump on to this curve.

Fortunately, the economic forces are such that the correct notion of stability is

guaranteed and indeed there will be a one-dimensional manifold of stable solutions

tending to the unique steady state. There are two ways of seeing this. The first one

simply involves studying the above system diagrammatically. This is done in Figure

8.1.

The vertical line is the locus of points where c = 0. The reason why the c = 0

locus is just a vertical line is that in view of the consumer Euler equation (8.25), only

the unique level of k∗ given by (8.21) can keep per capita consumption constant.

The inverse U-shaped curve is the locus of points where k = 0 in (8.24). The

intersection of these two loci defined the steady state. The shape of the k = 0 locus

388

Source: Acemoglu. Introduction to Modern Economic Growth.17 / 25

Technological change and the canonical

neoclassical model

The production function takes the form

Y (t) = F (K(t), A(t)L(t)),

A(t) = A(0) exp(gt). Kaldor facts: constant capital-outputratio, constant growth in output and constant capital sharein income. Since αK(t) = R(t)K(t)

Y (t), this implies that R(t)

and therefore r(t) should be constant.

For consumption to grow at a constant rate, εu(c(t)) shouldbe constant as well.

18 / 25

Balanced growth in the neoclassical model requires thatasymptotically all technological change is purelylabor-augmenting and the elasticity of intertemporalsubstitution tends to a constant εu.

19 / 25

It can be shown that the intertemporal elasticity ofsubstitution is the inverse of the elasticity of marginalutility.

The Arrow-Pratt coefficient of relative risk aversion for atwice differentiable concave utility function u(c) is definedas

< = −u′′(c)c

u′(c).

The Constant Relative Risk Aversion utility function(CRRA) satisfies the property that < is constant:

u(c) =c1−θ − 1

1− θif θ 6= 1, θ ≥ 0

= log c if θ = 1.

20 / 25

The Euler equation of the representative household is

c(t)

c(t)=

1

θ(r(t)− δ) . (8.50)

Consumption per capita will be growing; define

c(t) =C(t)

A(t)L(t)=

c(t)

A(t),

which is constant along the BGP. Then,

dc(t)/dt

c(t)=

1

θ(r(t)− ρ− θg) .

21 / 25

The law of motion of k(t) = K(t)A(t)L(t)

:

k(t) = f(k(t))− c(t)− (n+ g + δ)k(t). (8.51)

The transversality condition:

limt→∞

[k(t) exp

(−∫ t

0

[f ′(k(s))− g − δ − n]ds

)]= 0.

(8.52)Since c(t) remains constant on a BGP,

f ′(k∗) = ρ+ δ + θg. (8.53)

The equation uniquely determines the SS value of k∗.

22 / 25

The transversality condition can be written as

limt→∞

[k(t) exp

(−∫ t

0

[ρ− (1− θ)g − n]ds

)]= 0,

This is ensured if the following holds:

Assumption 4 (discounting with technologicalprogress): ρ− n > (1− θ)g.

This is equivalent to r∗ > n+ g.

The neoclassical growth model, like the Solow model,endogenizes the capital-labor ratio, but not the growth rateof the economy. The advantage of the neoclassical growthmodel is that the capital-labor ratio, normalized outputand consumption are determined by the preferences of theindividuals rather than by an exogenously given saving rate.

23 / 25

The role of policySuppose the returns on capital net of depreciation aretaxed at the rate τ and the proceeds are redistributedlump-sum back to households. The relevant equations are

k(t) = f(k(t))− c(t)− (n+ g + δ)k(t)

dc(t)/dt

c(t)=

1

θ(r(t)− ρ− θg)

=1

θ((1− τ)(f ′(k(t))− δ)− ρ− θg)

On a BGP,

f ′(k∗) = δ +ρ+ θg

1− τ. (8.57)

Thus, a higher tax rate τ reduces k∗ and income per capita.24 / 25

Dynamic response of capital and consumption to

a decline in the tax rate from τ to τ ′ < τIntroduction to Modern Economic Growth

c(t)

kgold0k(t)

k*

c(t)=0

k(t)=0

k**

c**

c*

k

Figure 8.2. The dynamic response of capital and consumption to adecline in the discount rate from ρ to ρ0 < ρ.

8.8. The Role of Policy

In the above model, the rate of growth of per capita consumption and output

per worker (per capita) are determined exogenously, by the growth rate of labor-

augmenting technological progress. The level of income, on the other hand, depends

on preferences, in particular, on the intertemporal elasticity of substitution, 1/θ,

the discount rate, ρ, the depreciation rate, δ, the population growth rate, n, and

naturally the form of the production function f (·).If we were to go back to the proximate causes of differences in income per capita of

economic growth across countries, this model would give us a way of understanding

those differences only in terms of preference and technology parameters. However, as

already discussed in Chapter 4, and we would also like to link the proximate causes

of economic growth to potential fundamental causes. The intertemporal elasticity

of substitution and the discount rate can be viewed as potential determinants of

400

Source: Acemoglu. Introduction to Modern Economic Growth.25 / 25