Bhattacharya R., Majumdar M. Random Dynamical Systems: Theory and Applications

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1.8 Comparative Statics and Dynamics 43

Example 8.2 (Verhulst equation) We return to the Verhulst equation

discussed in Example 2.3. Consider the dynamical system (S,α) where

S =

and α : S → S is deﬁned by

α(x) =

x + θ

, (8.12)

where θ

,θ

are two positive numbers, treated as parameters. Note that

α is increasing:



(x) =

(x + θ

)

> 0 for x ≥ 0.

Also,



(x) =

−2θ

(x + θ

)

< 0 for x ≥ 0.

Verify that

lim

x→0



(x) = (θ

/θ

)

and

lim

x→∞



(x) = 0.

To study the long-run behavior of alternative trajectories, three cases are

distinguished (of which one is left as an exercise):

Case I: θ

<θ

Case II: θ

= θ

Case III: θ

>θ

In Case I, the unique ﬁxed point x

∗

= 0 is locally attracting. But in

this case, α



(0) < 1 and α



(x) < 0 for x ≥ 0 imply that α



(x) < 1 for

all x > 0 (the graph of α is “below” the 45

◦

-line). By the mean value

theorem, for x > 0

α(x) = α(0) + xα



(z), where 0 < z < x

= xa



(z) < x

Hence, from any x > 0,

t+1

= α(x

) < x

Thus, the trajectory τ (x) ={x

} is decreasing. Since x

≥ 0, the se-

quence τ (x) must converge to some ﬁxed point

x of α (by using the

44 Dynamical Systems

continuity of α). Since 0 is the unique ﬁxed point, we conclude the fol-

lowing:

when θ

<θ

, for any x > 0 the trajectory τ (x) is decreasing and

converges to the unique ﬁxed point of α, namely, x

∗

= 0.

(A graphical analysis might be useful to visualize the behavior of

trajectories.)

We now turn to Case III. Here, x

∗

(1)

= 0 is repelling and x

∗

(2)

= θ

− θ

is locally attracting. But by the mean value theorem, for x > 0 and

“sufﬁciently small,” α(x) > x. But this also means that for all x sat-

isfying 0 < x < x

∗

(2)

, it must be that α(x) > x (otherwise, there must be

another ﬁxed point between 0 and x

∗

(2)

, and this leads to a contradiction).

Consider x > x

∗

(2)

. Since α



∗

(2)

) < 1 and α



(x) < 0 for all x, it fol-

lows that α



(x) < 1 for all x > x

∗

(2)

. Write

η(x) = α(x) − x.

Then η



(x) < 0 for all x ≥ x

∗

(2)

and η(x

∗

(2)

) = 0. This implies that

η(x) < 0 for all x > x

∗

(2)

, i.e., α(x) < x for all x > x

∗

(2)

To summarize, we have

α(x) > x for x satisfying 0 < x < x

∗

(2)

α(x) < x for x > x

∗

(2).

We can now adapt the discussion of Example 5.1, to conclude

when θ

>θ

, for any x > 0, τ (x) converges to x

∗

= θ

− θ

. If

0 < x < x

∗

, τ (x) is increasing; if x > x

∗

, τ (x) is decreasing.

Finally, the case θ

= θ

is left as an exercise.

Example 8.3 Here S = R

and α : S → S is deﬁned by

α(x) =

+ θ

, (8.13)

where θ

,θ

are two positive real numbers (parameters).

First, note that α is increasing:



(x) =

2θ

+ θ

)

> 0 for x > 0. (8.14)

As to the ﬁxed points of α, there are three cases:

Case I: If θ

< 2

√

, the only ﬁxed point of α is x

∗

(1)

= 0.

Case II: If θ

= 2

√

, the ﬁxed points of α are x

∗

(1)

= 0, x

∗

(2)

1.8 Comparative Statics and Dynamics 45

Case III: If θ

> 2

√

, the ﬁxed points of α are

∗

(1)

= 0

∗

(2)

= [θ

−



− 4θ

]/2

∗

(3)

= [θ



− 4θ

]/2. (8.15)

Since α



(0) = 0, we see that the ﬁxed point x

∗

(1)

= 0 is locally attracting

(in all cases).

In Case II, the ﬁxed point x

∗

(2)

= θ

/2 satisﬁes α



(

) = 1, i.e., x

∗

(2)

non-hyperbolic.

In Case III, after some simpliﬁcations, we get



∗

(2)

) =

4θ

(θ

−



− 4θ

)



∗

(3)

) =

4θ

(θ



− 4θ

)

. (8.16)

Using the fact that θ

> 2

√

, in this case one can show that





∗

(2)



> 1,





∗

(3)



< 1.

Hence, x

∗

(2)

is repelling while x

∗

(3)

is locally attracting.

We now turn to the global behavior of trajectories and their sensitivity

to changes in θ

and θ

In Case I, by continuity of α



, α



(x) < 1 for all x ∈ (0,

x) for some

x > 0. Hence, α(x) < x for x ∈ (0,

x). But this means that α(x) < x for

all x > 0, otherwise there has to be a ﬁxed point of α,sayz > 0. But this

would contradict the fact that in this case 0 is the unique ﬁxed point.

Since α(x) < x for all x > 0, it follows from each x > 0, the trajectory

τ (x) is decreasing and converges to 0.

In Case II we can similarly prove that for all x ∈ (0, x

∗

(2)

), the trajectory

τ (x) from x is decreasing and converges to 0. Next, one can verify that

for x sufﬁciently large, α(x) < x. But this implies that α(x) < x for all

x > x

∗

(2)

. Otherwise there would have to be another ﬁxed point z of α

such that z > x

∗

(2)

, and this would contradict the fact that x

∗

(2)

is the only

positive ﬁxed point of α. Hence, for each x > x

∗

(2)

, the trajectory τ (x)

from x is decreasing and converges to x

∗

(2)

46 Dynamical Systems

In Case III, arguments similar to those given above show that for

all x ∈ (0, x

∗

(2)

), α(x) < x. Hence, the trajectory τ (x) is monotonically

decreasing and converges to 0. Similarly, for x > x

∗

(3)

, α(x) < x. Hence,

for each x > x

∗

(3)

the trajectory τ (x) is decreasing and converges to x

∗

(3)

Finally, α



∗

(2)

) > 1 implies that for x > x

∗

and x sufﬁciently close

to x

∗

(2)

, α



(x) > 1. By using the mean value theorem we can show that

α(x) > x. But this means that for all x ∈ (x

∗

(2)

, x

∗

(3)

),α(x) > x (other-

wise there must be a ﬁxed point z of α between x

∗

(2)

and x

∗

(3)

, a contra-

diction). Also α



(x) > 0 for x > 0 implies that α(x) ≤ α(x

∗

(3)

) = x

∗

(3)

for

all x ∈ (x

∗

(2)

, x

∗

(3)

). We can use our analysis in Section 1.5 to show that

for each x ∈ (x

∗

(2)

, x

∗

(3)

), the trajectory τ (x) is increasing and converges

to x

∗

(3)

The above discussion suggests that bifurcations occur near non-

hyperbolic ﬁxed and periodic points. This is indeed the only place where

bifurcations of ﬁxed points occur, as the following result on comparative

statics demonstrates.

Proposition 8.1 Let α

be a one-parameter family of functions, and sup-

pose that α

) = x

and α



) = 1. Then there are intervals I about

and N about θ

, and a smooth function p : N → I such that p(θ

) =

and α

(p(θ)) = p(θ). Moreover, α

has no other ﬁxed points in I .

Proof. Consider the function deﬁned by G(x,θ) = α

(x) − x.Byhy-

pothesis, G(x

,θ

) = 0 and

∂G

∂x

,θ

) = α



) − 1 = 0.

By the implicit function theorem, there are intervals I about x

and N

about θ

, and a smooth function p : N → I such that p(θ

) = x

and

G( p(θ),θ) = 0 for all θ ∈ N . Moreover, G(x,θ) = 0 unless x = p(θ).

This concludes the proof.

1.9 Some Applications

1.9.1 The Harrod–Domar Model

We begin with the Harrod–Domar model, which has left an indelible

impact on development planning and has ﬁgured prominently in policy

1.9 Some Applications 47

debates on the appropriate strategy for raising the growth rate of an

economy.

The “income” or output at the “end” of period t, Y

, is determined by

the amount of capital K

at the “beginning” of the period according to a

linear production function:

= β K

, (9.1)

where β>0 is the productivity index (or the “output–capital” ratio).

At the end of period t, a fraction s (0 < s < 1) is saved and invested

to augment the capital stock. Write S

as the saving in period t. The stock

of capital does not depreciate. Hence we have

= sY

, 0 < s < 1, (9.2)

and the capital stock “at the beginning” of period t + 1 is given by

t+1

= K

+ S

= K

+ sY

. (9.3)

From (9.1) and (9.3)

β K

t+1

= β K

+ βsY

or, Y

t+1

= Y

(1 + βs). (9.4)

This law of motion (9.4) describes the evolution of income (output) over

time. Clearly,

t+1

− Y

= sβ. (9.5)

The last equation has ﬁgured prominently in the discussion of the role of

saving and productivity in attaining a high rate of growth (it is assumed

that there is an “unlimited” supply of labor needed as an input in the

production process).

1.9.2 The Solow Model

1.9.2.1

Homogeneous Functions: A Digression

Let F(K , L):

→ R

be a homogeneous function of degree 1, i.e.,

F(λK,λL) = λF(K, L) for all (K , L) ∈

,λ>0. Assume that F is

twice continuously differentiable at (K , L) >> 0, strictly concave, and

> 0, F

< 0, F

< 0at(K, L) >> 0.

48 Dynamical Systems

For L > 0, note F(K , L) = F(K /L , 1)/L. Write k ≡ K /L, and

f (k) ≡ F(K /L, 1)/L. A standard example is the Cobb–Douglas form:

F(K, L) = K

, a > 0, b > 0, a + b = 1.

Then f (k) ≡ k

, and 0 < a < 1.

One can show that



(k) ≡

f (k) = F

at (K , L) >> 0

= f (k) − kf



(k)at(K, L) >> 0

= f



/L < 0at(K, L) >> 0

= [k



(k)]/L < 0at(K, L) >> 0.

1.9.2.2 The Model

There is only one producible commodity that can be either consumed

or used as an input along with labor to produce more of itself. When

consumed, it simply disappears from the scene. Net output at the “end”

of period t, denoted by Y

(≥0), is related to the input of the producible

good K

(called “capital”) and labor L

employed “at the beginning

of ” period t according to the following technological rule (“production

function”):

= F(K

, L

), (9.6)

where K

≥ 0, L

≥ 0.

The fraction of output saved (at the end of period t) is a constant s,so

that total saving S

in period t is given by

= sY

, 0 < s < 1. (9.7)

Equilibrium of saving and investment plans requires

= I

, (9.8)

where I

is the net investment in period t. For the moment, assume that

capital stock does not depreciate over time, so that at the beginning of

period t + 1, the capital stock K

t+1

is given by

t+1

≡ K

+ I

. (9.9)

1.9 Some Applications 49

Suppose that the total supply of labor in period t, denoted by

,is

determined completely exogenously, according to a “natural” law:

(1 + η)

> 0,η>0. (9.10)

Full employment of the labor force requires that

. (9.10



)

Hence, from (9.7), (9.8), and (9.10



), we have

t+1

= K

+ sF(K

Assume that F is homogeneous of degree 1. We then have

t+1

+ sF



, 1



Writing k

≡ K

,weget

t+1

(1 + η) = k

+ sf(k

), (9.11)

where

f (k) ≡ F(K /L, 1).

From (9.11)

t+1

= k

/(1 + η) + sf(k

)/(1 + η)

t+1

= α(k

), (9.12)

where

α(k) ≡ [k/(1 + η)] + s[ f (k)/(1 + η)]. (9.12



)

Equation (9.12) is the fundamental dynamic equation describing the

intertemporal behavior of k

when both the full employment condi-

tion and the condition of short-run savings–investment equilibrium (see

(9.10



) and (9.8)) are satisﬁed. We shall refer to (9.12) as the law of

motion of the Solow model in its reduced form.

Assume that

f (0) = 0, f



(k) > 0, f



(k) < 0 for k > 0

50 Dynamical Systems

and

lim

k↓0



(k) =∞, lim

k↑∞



(k) = 0.

Then, using (9.12



α(0) = 0,



(k) = (1 +η)

−1

[1 + sf



(k)] > 0atk > 0,



(k) = (1 +η)

−1



(k) < 0atk > 0.

Also verify the boundary conditions:

lim

k↓0



(k) = lim

k↓0

[(1 + η)

−1

+ (1 +η)

−1



(k)] =∞,

lim

k↑∞



(k) = (1 +η)

−1

< 1.

By using Propositions 5.1 and 5.3 we get

Proposition 9.1 There is a unique k

∗

> 0 such that

∗

= α(k

∗

);

equivalently,

∗

= [k

∗

/(1 + η)] + s[ f (k

∗

)/1 + η]. (9.13)

If k < k

∗

, the trajectory τ (k) is increasing and converges to k

∗

If k > k

∗

, the trajectory τ (k) is decreasing and converges to k

∗

A few words on comparative statics are in order. Simplifying (9.13)

we get

∗

f (k

∗

). (9.14)

How do changes in the parameters s and η affect the steady state value

∗

? Keep η ﬁxed, and by using the implicit function theorem we get

∂k

∗

∂s



f (k

∗

)



1 −



∗

f (k

∗

)



1.9 Some Applications 51

By our assumptions,

f (k)

> f



(k) for all k > 0.

Hence,

1 >



(k)k

f (k)

for all k > 0.

This leads to

∂k

∗

∂s

> 0,

∂y

∗

∂s

= f



∗

)

∂k

∗

∂s

> 0.

We see that an increase in s increases the steady state per capita capital

and output. A similar analysis of the impact of a change in η on k

∗

and

∗

is left as an exercise.

Exercise 9.1 An extension of the model introduces the possibility of

depreciation in the capital stock in the production process and replaces

Equation (9.9) by

t+1

= (1 − d)K

+ I

, (9.9



)

where d ∈ [0, 1] (the depreciation factor) is the fraction of capital “used

up” in production. In our exposition so far d = 0. The other polar case

is d = 1, where all capital is used up as an input. In this case (9.9) is

replaced by

t+1

= I

. (9.9



)

It is useful to work out the details of the dynamic behavior of the more

general case (9.9



In his article Solow (1956) noted that the “strong stability” result

that he established “is not inevitable” but depends on the structure of

the law of motion α in (9.12) (“on the way I have drawn the productivity

curve”). By considering simple variations of his model, Solow illustrated

some alternative laws of motion that could generate qualitatively different

trajectories.

This point was subsequently elaborated by Day (1982), who stressed

the possibility that with alternative speciﬁcations of the basic ingredients

52 Dynamical Systems

of the Solow model, one could obtain laws of motion that are consistent

with cyclical behavior and Li–Yorke chaos. Day’s examples involved

(a) allowing the fraction of income saved to be a function of output

(i.e., dispensing with the assumption that s is a constant in (9.7));

(b) allowing for a “pollution effect” of increasing per capita capital.

Day’s examples provide the motivation behind one of the examples in

Chapter 3.

Exercise 9.2 Consider the variation of the Solow model in which η = 0

(i.e., labor force is stationary) and (9.9



) holds (i.e., capital depreciates

fully). Then (9.12) is reduced to

t+1

= sf(k

). (9.12



)

In one of Day’s examples (Day 1982), f (k) is speciﬁed as

f (k) =



(m − k)

for 0 ≤ k ≤ m

0 for k ≥ m

(9.D)

where B, m > 0 and β,γ ∈ (0, 1). The term (m − k)

on the right-hand

side of (9.D) attempts to capture the pollution effect. The law of motion

can be explicitly obtained as

t+1

= α(k

) = sBk

(m − k)

for 0 ≤ k ≤ m.

(1) Fix the savings rate s ∈ (0, 1), B ∈ (0,

], and β = γ =

. Ve r-

ify that under these assumptions α(k) = sBk

(m − k)

is a map from

[0, m] into itself. [Hint: Compute k

∗

= arg max

k∈[0,m]

α(k), and let

= α(k

∗

) =

Bsm

. Use this information along with the inequalities im-

plied by the restriction on B to prove the result.]

(2) Now ﬁx B = B

∗

. Verify that k

, the smallest root of

∗

(m − k)

= k

∗

is given by k

m(2 −

√

3), and the following string of inequalities

hold: 0 < k

< k

∗

< k

(i.e., α

) < k

<α(k

) <α

)).

(3) Using (1) and (2) above, apply the Li–Yorke sufﬁciency

theorem.