Solution to Homework 2 - Exogeneous Growth Models

ECO-3211 Macroeconomia Aplicada (Applied Macroeconomics)∗

Question 1: Solow Model with a Fixed Factor

1. The law of motion for capital in the Solow economy is given by

K(t) = sY (t)− δK(t) .

Let k = K/L be the capital per worker. Then substituting for output per worker,

y = Y/L, using the aggregate production function, in the law of motion above gives

k(t)

k(t)=

K(t)

K(t)= sk(t)α−1z1−α−β − δ ,

where z = Z/L is land per worker. Note that k(t)/k(t) = K(t)/K(t) because there

is no population growth. Then, in steady-state k(t) = 0, which gives the following

expression for the steady-state k and y:

k∗ =

(sz1−α−β

δ

)1/(1−α)

,

y∗ =(sδ

)α/(1−α)

z(1−α−β)/(1−α) .

The steady-state is unique and stable because for k > k∗, sk(t)α−1z1−α−β − δ < 0 (the

first term is decreasing in k), and for k < k∗, sk(t)α−1z1−α−β − δ > 0. In other words,

k(t) > 0 for k ∈ (0, k∗) and k(t) < 0 for k ∈ (k∗,∞). Thus, if the economy starts from

an initial capital stock k(0) < k∗, it will experience in an increase in k till it reaches k∗,

and if the the economy starts from an initial capital stock k(0) > k∗, it will experience

in a decrease in k till it reaches k∗.

∗Rahul Giri. Contact Address: Centro de Investigacion Economica, Instituto Tecnologico Autonomo de Mexico (ITAM).

E-mail: rahul.giri@itam.mx

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2. Now, with population growth land per worker, z = Z/L is going to decline at the rate

at which population is growing, i.e. z(t) = −nz(t), which implies that z(t) = z(0)e−nt.

Therefore, limt→∞ z(t) = 0. Since

y(t) = k(t)αz(t)1−α−β ,

limt→∞ z(t) = 0 implies that limt→∞ y(t) = 0, which in turn implies that limt→∞ k(t) =

0. Thus, with population growth, there is only one stead-state, which is given by

k∗ = z∗ = y∗ = 0. Intuitively, growth in population reduces the amount of land per

worker, thereby reducing the output per worker, which ultimately tends to zero since

land is an essential factor of production. Since the per capita variables tend to zero, it

must be that aggregate variables, K and Y , grow over time, though at a rate slower

than the population growth rate.

The return to land is given by

pz(t) = (1− α− β)L(t)βK(t)αZ−α−β .

Since L grows at rate n, K also grows, at a rate slower than n, and Z is constant, it

implies that limt→∞ pz(t) = ∞. The intuition is that as population grows land becomes

a scarce factor, and since it is essential for production, the return to land goes to ∞.

The wage rate is given by

w(t) = βL(t)β−1K(t)αZ1−α−β = βk(t)αz(t)1−α−β .

Since limt→∞ k(t) = limt→∞ z(t) = 0, wage rate tends to zero as t → ∞. Intuitively,

as labor becomes more and more abundant, it marginal product falls, and its falls to

zero as t → ∞.

3. The results we found in the previous part hold even when s = 1, i.e. individuals save

all their income. Thus, endogenizing savings rate is not going to offset the effect of

diminishing returns. However, if the growth rate of population was made to depend on

the output per capita, then we could have a steady-state with positive y and k. The

idea is that higher output per capita implies a healthier population, and hence people

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live longer and have more children. This is the idea proposed by Malthus.

L(t)

L(t)= n(y(t)) ,

where n′(y) > 0, limy→∞ n(y) = n > 0 and limy→∞ n(y) = n < 0. Thus, population

growth is positive function of output per capita. If output per capita is very high

population growth is also very high, If output per capita is very low, population growth

may be negative, implying population may shrink. There exists a y = y∗ such that

n(y∗) = 0. Such a formulation is also used in Parente and Prescott (2005).

Question 2: Solow Model with a Different Price of Investment

1. Suppose the population growth rate is n. Define capital per worker as k = K/L,

and therefore output per worker based on our general production function is given by

f(k) = F (k, 1). The law of motion for k is

k(t) = sq(t)f (k(t))− (δ + n)k(t) .

Suppose there exists a BGP (balanced growth path) in which k grows at rate gk ≥ 0.

Then k(t) = k(0)egkt. Furthermore, q(t) = q(0)eγKt. Using these in the law of motion

for k implies

gkk(0)egkt = sq(0)eγKtf

(k(0)egkt

)− (δ + n)k(0)egkt ,

which simplifies to

k(0)

sq(0)(gk + δ + n)e(gk−γK)t = f

(k(0)egkt

).

If gk = 0, then as t → ∞ the left hand side goes to zero whereas the right hand

side is constant. Therefore, it must be that gk is positive. But with positive gk the

argument of f(.), on the right hand side, goes to infinity. Thus we can solve for f(k)

for any k ∈ [k(0),∞). For any k, it is true that k = k(0)egkt, which implies that

t = ln(k/k(0))/gk. Substituting this in the previous equation gives

f(k) =

[k(0)

sq(0)(gk + δ + n)

1

(k(0))(gk−γK)/gk

]k(gk−γK)/gk , ∀ k ∈ [k(0),∞) .

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The term in the big square brackets is a constant. Denote it by C. Then

f(k) = Ck(gk−γK)/gk , ∀ k ∈ [k(0),∞) ,

which implies that

F (K,L) = Lf(k) = CK(gk−γK)/gkLγK/gk ,

which is a Cobb-Douglas production function. The intuition here is related to Uzawa’s

theorem which states that for the existence of a BGP, technological progress must

be labor augmenting asymptotically. In our model, here, technological progress is

capital-augmenting or Hicks-neutral. Thus, there is an inconsistency. But, when the

production function is Cobb-Douglas, its property of unitary elasticity of substitution

between factors ensures that all kinds of technological progress are equivalent. Thus, in

this model only the Cobb-Douglas production function is consistent with the existence

of a BGP.

2. With a Cobb-Douglas production function, the law of motion for k is given by

k(t)

k(t)= sq(t)k(t)α−1 − (δ + n) .

On a BGP the left hand side is constant. This implies that the right hand side is also

constant, i.e. q(t)k(t)α−1 is not growing, which means that

d (q(t)k(t)α−1) /dt

q(t)k(t)α−1= 0 ,

⇒q(t)

q(t)+ (α− 1)

k(t)

k(t)= 0 ,

⇒ gk =k(t)

k(t)=

q(t)/q(t)

1− α=

γK1− α

.

Then, since y(t) = k(t)α, on the BGP

gy =y(t)

y(t)= α

k(t)

k(t)=

α

1− αγK .

Substituting the expression for gk in the original law of motion for k gives

γK1− α

+ δ + n = sq(t)k(t)α−1 ,

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⇒ k∗ =k(t)

q(t)1/(1−α)=

[s

γK1−α

+ δ + n

]1/(1−α)

(1)

This is the steady-state value of the normalized capital-labor ratio. Note that, off the

steady-state, the growth rate of k is given by

g(k(t)

)=

k(t)

k(t)−

γK1− α

,

⇒ g(k(t)

)= sq(t)k(t)α−1 − (δ + n)−

γK1− α

,

⇒ g(k(t)

)= sq(t)

[k(t)q(t)1/(1−α)

]α−1

− (δ + n)−γK

1− α,

⇒ g(k(t)

)≡ sk(t)α−1 − δ − n−

γK1− α

.

In steady-state g(k∗)= 0. Since g

(k)is decreasing in k it implies that

g(k(t)

)> 0 if k(t) < k∗ ,

g(k(t)

)< 0 if k(t) > k∗ .

This means that the steady-state is globally stable.

3. On a BGP, the K/Y ratio must be constant. But in this case gk = γK/(1 − α) and

gy = αgk, which means that aggregate output Y grows at slower rate than aggregate

capital K. Therefore, K/Y is not constant, which violates the Kaldor facts, and hence

the definition of BGP. But, in this model the price of consumption good is not the

same as the price of the investment good. Therefore, K and Y are not expressed in

the same units. Instead of considering K/Y we should consider (K/q)/Y since q is the

inverse of the price of capital in terms of the final good. The growth rate of K/q is

d (K(t)/q(t)) /dt

K(t)/q(t)=

K(t)

K(t)−

q(t)

q(t)= gk + n− γK ,

⇒d (K(t)/q(t)) /dt

K(t)/q(t)=

α

1− αγK + n = gy + n =

Y (t)

Y (t).

Hence K(t)/q(t) grows at the same rate as Y (t), thereby implying that (K/q)/Y is

constant on this BGP.

Question 3: Neoclassical Growth Model with Subsistence

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1. The utility function captures the idea of a subsistence level of consumption, γ. γ is

the minimum level of consumption that consumer must have every period.

2. Define effective capital-labor ratio as k = K/AL. Then, a competitive equilibrium

consists of allocations of consumption and effective capital-labor ratio [c(t), k(t)]∞t=0 and

of sequences of wages and interest rates [w(t), r(t)]∞t=0 such that consumers maximize

utility taking prices as given, firms maximize profits taking prices as given and markets

clear.

3. Maximizing the current value Hamiltonian for the consumer yields the the consumption

Euler equation.C(t)

C(t)=

1

ǫu (C(t))(r(t)− ρ) ,

where ǫu (C(t)) is the inverse of the intertemporal elasticity of substitution, and is

given by

ǫu (C(t)) = −u′′ (C(t))C(t)

u′ (C(t))= θ

C(t)

C(t)− γ.

Notice that if γ = 0 then ǫu (C(t)) = θ, which is the usual case with CRRA pref-

erences. Thus, with a subsistence level of consumption, the intertemporal elasticity

of substitution is no longer constant; instead it depends on the level of consumption.

The interest rate is given by the marginal product of capital net of depreciation, i.e.

r(t) = FK (K(t), A(t)L(t)) − δ = f ′ (k(t)) − δ. Therefore, the consumption Euler

equation becomesC(t)

C(t)=

1

ǫu (C(t))(f ′ (k(t))− δ − ρ) .

The law of motion for capital per unit of effective labor, k, is given by

k(t) = f (k(t))−C(t)

A(t)− (δ + g)k(t) .

If we define consumption in units of effective labor, c = CL/AL = C/A, then the law

of motion for capital and the consumption Euler equation are given by

k(t) = f (k(t))− c(t)− (δ + g)k(t) .

c(t)

c(t)=

1

ǫu (c(t))(f ′ (k(t))− δ − ρ)− g ,

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where

ǫu (c(t)) = θc(t)

c(t)− γA(t)

.

Notice the dependence of the intertemporal elasticity of substitution on the level of

technology in period t. Thus, even though we have expressed our system of differential

equations in terms of effective labor, there is still explicit dependence on A(t). Again

it is due to γ > 0.

The consumption Euler equation and the law of motion for capital along with evolution

of A(t) and the transversality condition characterize the equilibrium. The equation for

the evolution of A(t) and the transversality condition is given by

A(t) = gA(t) ,

limt→∞

e−ρtµ(t)k(t) = 0 .

The factor prices are given by

r(t) = f ′ (k(t))− δ ,

w(t) = f (k(t))− f ′ (k(t)) k(t) .

Suppose there exists a BGP for k(t) = k∗. Then it must be that k(t) = 0, so that

f (k∗)− (δ + g)k∗ = c(t) =C(t)

A(t).

Since the left hand side is constant, it implies that C(t)/A(t) has to be constant, or that

consumption per capita, C(t), grows at rate g. Then the consumption Euler equation

would imply that

g =1

θ

C(t)− γ

C(t)(f ′ (k∗)− δ − ρ) .

But this leads to a contradiction since the left hand side is constant but the right hand

side changes over time as C(t) grows at rate g. Thus, the BGP does not exist in this

economy. The intuition is the consumer’s intertemporal elasticity of substitution is not

constant. Instead it is decreasing in C(t), which means the consumption growth will

be increasing in the level of consumption.

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4. The first order condition from the Hamiltonian gives

µ(t) [f ′ (k(t))− δ − g] = ρµ(t)− µ(t) ,

⇒ µ(t) = − [f ′ (k(t))− δ − g − ρ]µ(t) ,

⇒ µ(t) = µ(0) exp

(−

∫ t

0

(f ′ (k(s))− δ − g − ρ) ds

).

Substituting this in the transversality condition gives

limt→∞

µ(0) exp

(−

∫ t

0

(f ′ (k(s))− δ − g) ds

)k(t) = 0 .

Now, asymptotically limt→∞ ǫu (C(t)) = θ. This implies that

limt→∞

C(t)

C(t)= g =

1

θ(f ′ (k∗)− δ − ρ) .

Using this equation to substitute for f ′ (k∗) in the transversality condition yields

µ(0)k∗ limt→∞

exp [((1− θ)g − ρ)t] = 0 ,

which can be satisfied only if ρ > (1− θ)g. This is the required parametric restriction

for the transversality condition to be satisfied.

5. From the previous part it is clear that this economy has a BGP asymptotically, and

this BGP is exactly that of the economy with γ = 0. Since there is no population

growth both, aggregate and per capita variables, grow at rate g. To see this we go

back to the consumption Euler equation in terms of effective consumption.

c(t)

c(t)=

1

ǫu (c(t))(f ′ (k(t))− δ − ρ) ,

where

ǫu (c(t)) = θc(t)

c(t)− γA(t)

.

Note that asA(t) grows over time, the term γ/A(t) goes to zero, and hence limt→∞ ǫu (c(t)) =

θ, which is the case with standard CRRA preferences.

However, along a transition path the dynamics are different. To see this, define x(t) =

C(t)− γ and x(t) = x(t)/A(t). Then the consumption Euler equation in terms of this

redefined variable becomes

dx(t)/dt

x(t)=

1

θ(f ′ (k(t))− δ − ρ− θg) ,

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which is identical to the one we see in case of our model with standard CRRA prefer-

ences1. However, the law of motion for capital is no longer the same.

k(t) = f (k(t))− x(t)− (δ + g)k(t)− A(t)−1γ .

This equation has an extra term, A(t)−1γ, which is absent in the law of motion of our

standard model. Therefore, the k(t) = 0 locus is given by

x(t) = f (k(t))− (δ + g)k(t)− A(t)−1γ .

This locus shifts up over time as A(t)−1γ decreases with the growth in A(t). As a result

asymptotically this economy moves to the steady state of the economy with γ = 0.

But, at every t, as the k(t) = 0 locus shifts up, so does the saddle path. Also, in every

period t the saddle path is stable.

Question 4: Neoclassical Model with Overlapping Generations

1. A competitive equilibrium is a sequence of aggregate capital and consumption [K(t), C1(t), C2(t)]∞t=0

and sequence of prices [w(t), R(t)]∞t=0 such that households maximize utility given

prices, and firms maximize profits given prices and markets clear. This translates

into

maxC1(t),C2(t+1)

logC1(t) + β logC2(t+ 1) ,

s.t. C1(t) + S(t) ≤ w(t) ,

and C2(t+ 1) ≤ R(t+ 1)S(t) ,

1lnx(t) = ln(C(t) − γ). Taking the derivative w.r.t. time gives us x(t)x(t) = C(t)

C(t)−γC(t)C(t) . Substituting for

C(t)C(t) , multiplying and dividing the right hand side by θ and using the expression for ǫu(C(t)) gives:

x(t)

x(t)=

1

θ(f ′ (k(t))− δ − ρ) .

Since x(t) = x(t)/A(t), it implies that dx(t)/dtx(t) = x(t)

x(t) −A(t)A(t) . Since technology grows at rate g we get

dx(t)/dt

x(t)=

1

θ(f ′ (k(t))− δ − ρ− θg) .

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and

R(t) = αA(t)K(t)α−1L(t)1−α ,

w(t) = (1− α)A(t)K(t)αL(t)−α ,

and finally markets must clear

K(t+ 1) = L(t)S(t) .

2. Define k ≡ K/L, c = C/L, s = S/L as per capita variables and L(t) is the size of

generation t. Then R(t) = αA(t)k(t)α−1, w(t) = (1 − α)A(t)k(t)α and k(t + 1) =

s(t)/(1 + n). Following the class notes we have that

c1(t) =1

1 + βw(t), and s(t) =

β

1 + βw(t) ,

and

k(t+ 1) =β(1− α)A(t)k(t)α

(1 + n)(1 + β).

Define capital per unit of effective labor as k = k/A1/(1−α). Then

k(t+ 1) =β(1− α)

(k(t)

)α

(1 + n)(1 + β)(1 + g)1/(1−α).

Then in steady-state k(t+ 1) = k(t) = k∗, which is given by

k∗ =

[β(1− α)

(1 + n)(1 + β)(1 + g)1/(1−α)

]1/(1−α)

.

In the steady-state the capital labor ratio k grows by a factor of (1 + g)1/(1−α). The

steady state is stable - you can check that off the steady-state(k(t+ 1)/k(t)

)− 1 > 0

if k(t) < k∗ and(k(t+ 1)/k(t)

)− 1 < 0 if k(t) > k∗.

Substituting k = kA1/(1−α) in the equation for interest rate gives us the steady-state

interest rate.

R∗ = α(k∗)α−1

=α(1 + n)(1 + β)(1 + g)1/(1−α)

β(1− α).

Thus, the interest rate is constant in steady-state. Similarly, the wage rate is given by

wt = (1− α)(k∗)α

A(t)1/(1−α) ,

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which implies that wage rate grows by the same factor as the capital labor ratio.

Similarly, consumption per capita of each generation is

c1(t) =1

1 + βw(t), and c2(t) =

β

1 + βw(t)R∗ .

Therefore, in steady-state the consumption per capita of the young and the old grows

by the same factor as capital labor ratio. Finally output per capita is given by

y(t) = A(t)k(t)α =(k∗)α

A(t)1/(1−α) ,

which means that output also grows by the same factor as capital labor ratio.

3. Given a A(0) and k(0), a higher g implies higher k at every t ≥ 2. Also, an increase in

g reduces the steady-state effective capital labor ratio, k∗. It increases w(t) at every

t ≥ 2 since w(t) is positive function of k and A. Lastly, R, c1 and c2 are all higher at

every t ≥ 2.

4. If β′ > β then β′/(1 + β′) > β/(1 + β). Thus, if β increases at t = 1 then k(t) will be

higher for every t ≥ 2. Intuitively, a higher β implies that consumer attaches higher

weight to future consumption, and therefore it induces individuals to save more thereby

increasing capital labor ratio at every t. A higher β also means a higher steady-state

effective capital labor ratio, k∗. Substituting the expression for k∗ in the expression

for c1(t) gives

c1(t) = A(t)1/(1−α)Ωβα/(1−α)(1 + β)−1/(1−α) ,

where Ω is constant that does not depend on β. The expression shows that the effect

of an increase in β is ambiguous. The intuition is that a higher β induces individuals

to save more and therefore reduces c1(t), but higher savings imply higher capital stock

and higher wages which increases c1(t). The net effect is ambiguous. Similar exercise

for c2(t) gives

c2(t) = A(t)1/(1−α)Ω

[β

1 + β

]α/(1−α)

.

This expression shows that c2(t) increases due to a higher β. This is due to higher

savings, which is what the old consume. This is despite the decrease in R(t + 1) as a

result of higher β (due to higher savings). Wage rate increases due to higher capital.

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