1 Introduction

The growth model developed by Ramsey (1928) and subsequently adapted by

Cass (1965) and Koopmans (1968) is arguably the single most important

theoretical construct in modern macroeconomics. It is the foundation of both

growth theory and business cycle theory. It takes pride of place in every graduate

text in macroeconomics and growth theory.1

The geometry of the dynamics of the Ramsey model is well understood;

every graduate student ritually ponders its phase diagram. There is a paucity of

analytical solutions, however. Long and Plosser (1983) developed a discrete-time

example with log utility, Cobb-Douglas technology, and 100 percent depreciation

which is often used to exposit RBC models.2 It is also possible to derive an

explicit solution when there is a constant gross saving rate [Kurz (1968), Barro,

Mankiw, and Sala-i-Martin (1995), Barro and Sala-i-Martin (2004, Chapter 2),

and Wälde (2005)]. Recently, Mehlum (2005) has provided a closed form solution

for the case of a Leontieff technology. While these are illuminating examples,

each involves some unappealing features. The assumption of 100 depreciation in

Long and Plosser is highly unrealistic, as is the restriction to log preferences. At

any rate, this example only yields a solution in discrete time; it doesn’t work in

continuous time [Chang (1988)] . The constant saving rate formulation requires

the elasticity of intertemporal substitution to be tied to the rates of depreciation

and time preference in an unintuitive way. The Leontieff technology has long

been known to be a razor-edge case. It would be nice to have an example with a

reasonable rate of depreciation, a non-unitary elasticity of intertemporal

substitution, and diminishing returns. Absent such a solution, empirical

implementation of the model has required either a linear approximation around

the steady state [Barro and Sala-i-Martin (2004)] or numerical simulation

[Mulligan and Sala-i-Martin (1991)].

This paper will derive an explicit solution to the canonical version of the

Ramsey model with constant elasticity preferences and Cobb-Douglas

technology. The solution can only be obtained when capital’s share of GDP is

equal to the reciprocal of the intertemporal elasticity of substitution. The

distinguishing feature of this constellation of parameters is that it generates a

1 Some examples are Blanchard and Fischer (1989), Romer (2005), Sargent (1989, 2002),

Turnovsky (1999, 2000), Obstfeld and Rogoff (1999), and Barro and Sala-i-Martin (2004).

2 In this case both the utility function and production function have a unitary elasticity of

substitution. Benhabib and Rustichini (1994) show that solutions can be found for a more general

class of models, where (given CRRA preferences and CES technology) the elasticities of

substitution of the utility function and production function are equal, but not necessarily equal to

one.

linear saddle path. This permits an explicit solution for the dynamics of the

capital stock, along with an exact expression for the speed of convergence.

Although this is admittedly a special case, it provides a concrete example

that makes the dynamic workings of the model transparent. It is not intended to

supplant graphical and numerical analysis, but to serve as a complementary tool

that helps reveal the economic properties of the model. As Boucekkine and RuizTamarit (2004b) put it when motivating the same parametric device

in studying

the Lucas (1988) model, “. . . there is an obvious need to complement this

computational and qualitative literature with full-fledged theoretical proofs . . .”

Two elements of this solution have been anticipated in the literature. First,

what makes the model tractable is the fact that the consumption/capital ratio is

constant when labor’s share equals the intertemporal elasticity. Barro and Sala-iMartin (2004) are aware of this fact in the Ramsey model. Xie

(1991) exploits this

property in the Romer (1986) model, while Xie (1994) and Boucekkine and RuizTamarit (2004a) use it to shed light on the Lucas (1988) model. The

tractability

of this case has also been put to good use in some stochastic growth models. For

example, Chang (1998) solved the closely related “inverse optimal” problem for

this case, deriving the utility function and rate of time preference that generate a

constant equilibrium consumption/capital ratio. More recently, Wälde (2005) has

used this combination of parameters to solve a Poisson model of endogenous

growth cycles. The innovation here is to use the constancy of the

consumption/capital ratio to arrive, via a Bernoulli transformation, at closed form

solutions for the capital stock and the speed of convergence in the non-stochastic

Ramsey model.

Second, the derivation centers around the hypergeometric function.

Boucekkine and Ruiz-Tamarit (2004b) have recently, and independently, applied

the hypergeometric function to the Lucas (1988) model. The recursive structure of

the Lucas model allows them to derive analytical results without restricting the

intertemporal elasticity. The Ramsey model, however, is generally not recursive;

the only case where it is recursive seems to be the example studied here, where

the intertemporal elasticity is equal to the reciprocal of capital’s share.

The outline of the paper is as follows. Section 2 recapitulates the Ramsey

model. Section 3 sketches the solution, leaving the details to an appendix.

Section 4 discusses the properties of the solution. It shows vividly how the

speed of convergence varies with time, a feature of the model that is masked by

estimates of the speed of convergence calculated from a Taylor-series

approximation around the steady state. It also suggests a pedagogical application

of the model: students are often asked to solve the Cobb-Douglas version of the

Solow (1956) model by using a Bernoulli transformation. This model could be

adapted to serve as a complementary example of a tractable solution to the

Ramsey model.

Section 5 extends the model to study two problems that have heretofore

lacked analytical solutions. The first is the dynamic effects of a temporary change

in government expenditures; the second is to the transition dynamics of the Jones

and Manuelli (1990) model.

Section 6 calibrates the model. This is obviously a very special case, so I

do not wish to press the issue. However, I suggest that this calibration may not be

terribly unrealistic. If we adopt a broad notion of capital and embrace a degree of

agnosticism about the elasticity of intertemporal substitution it may be a

reasonable first approximation to the real world.

Section 7 offers some concluding thoughts.

2 The Model

Since the model is so familiar, my description of it will be succinct. Imagine an

economy inhabited by a large number of identical, infinitely-lived consumers.

For simplicity there is neither population growth nor technological growth. I

adopt the functional forms commonly used in empirical, and in many theoretical,

applications: technology is Cobb-Douglas, with a capital share ofα ∈[0,1]; the

rate of depreciationδ ∈[0,1]is constant; there is a constant rate of time preference

ρ ∈[0,∞]and preferences are time-separable, with a constant elasticity of

intertemporal substitution, σ ∈[0,∞].

It is well known that the dynamics of per capita consumption c and capital

k are determined by the following pair of differential equations:

k = k α -δk – c

= σ (αk α -1 – δ – ρ) (1)

(2)

cc

Equation (1) is the resource constraint; Equation (2) is the Euler equation. There

are two boundary conditions: the initial capital stock ko and the transversality

condition

.

lim 0

1

→ ∞

=

–

–

t

e ρtc σ k (3)

For future reference, the steady state is

1

1 –

+

=

α

ρ α

δ

k (4)

( )

.

1

c k k k

α

ρ δ α

= α – δ = + – (5)

The system is saddle-path stable. Given the initial capital stock k0 there is a

unique value of initial consumption c0 that will satisfy the transversality

condition.

3 A Solution

The system of equations (1) – (2) is dauntingly nonlinear. To approach a solution,

define the capital-output ratio (a Bernoulli transformation) by z = k 1-α and the

consumption-capital ratio by x = c k . Using these transformations the system can

be rewritten as

z = (1-α )[1- (δ + x)z] (6)

( )

+ – –

1 x

+ 1 (7)

x z

x

–

= σα δ σ ρσ The Bernoulli transformation converts the capital accumulation equation

[Equation (1)] into the linear, albeit non-autonomous, differential equation in

Equation (6). Despite this simplification, the system still does not allow an

analytical solution. Equation (7) is still non-linear.

Suppose, however, that σ =1 α . In that case, z disappears from equation

(7), and the system becomes recursive: Equation (7) reduces to

( )

,

1

x

xx

+

– +

= –

α

δ α ρ

(8)

while z evolves according to Equation (6), given the forcing process x determined

in Equation (8). This is a simple, autonomous logistic equation.

The solution to Equations (6) and (8), subject to the initial condition

k0 and the transversality condition [Equation (3)] is given by the following result.

PROPOSITION 1: If σ = 1 α the transitional paths of consumption and capital

are

( )

,

1

c k

α

ρ + δ -α

= (9)

-α ( -α -α ) -( -α )ρα+δ -α

= + –

1

1

1

1 1

0

1 t

k k k k e . (10)

Since to some extent the method is the message of this result, I will sketch its

derivation before discussing its properties.

PROOF:

Consider the logistic equation in Equation (8). This has the general

solution

xAe xt

x

x

+

=

1

(11)

where A is an arbitrary constant of integration and ( )

α

ρ +δ -α

=

1

x . Substitute

this back into Equation (6) to arrive at

( )

= – – +

z

1 α 1 δ

+

z

xAe

x

1 xt (12)

In the appendix I demonstrate that the general solution to Equation (12) is:

1 1(1 α,1, ;ω) α 1 (1 α )( δ ) (1 )1 α ,

– – – + –

– + +

=

F d Bx e x t xAe xt

z (13)

2 δ

+

x

where B is another arbitrary constant of integration, d = 1+ (1-α)(δ + x) x ,

,

1 xt

xt

xAe

xAe

+

ω = and

( ) ( )( )

( )

( )( )( )

(1 )(2 ) . . .

1 2 3

1

1

1 2

d d

2 1 1 ,1; ; 1 2 3 +

+ +

– – –

+

+

– –

+

–

– = + ω

α α α

ω

α α

ω

α

α ω

d d d d

F d

(14)

is a hypergeometric function.3

The following lemma pins down the constants A and B .

LEMMA. Given the initial condition

k0 the transversality condition [Equation

(3)] will be satisfied if and only if A = 0 and

.

1

0

1

= –

–

δ

B x z (15)

+

α

x

The proof of this lemma is sufficiently involved to be relegated to the appendix.

The intuition, however, is straightforward. Notice that the transversality

condition (3) can be written as

.

lim 0

→ ∞

=

– –

–

t

e ρt x σ k σ (16)

1

1 1

Consider the solution for x in Equation (11). If A > 0 then x would fall to zero

over time, driving marginal utility and k to infinity. If A < 0 then x would

become negative in finite time. If A = 0 x is constant and, given Equation (10),

Equation (16) is satisfied. Therefore, the transversality condition is satisfied if and

only if A = 0.

Applying this Lemma to Equations (11) and (13) completes the proof:

Since A = 0, Equation (11) implies that x = x,∀t. This yields Equation (9). Next,

evaluating Equation (13) at the values of A and B described in the Lemma yields

( )(x )t

e

x

α δ

δ

– – +

+

1

0

1

. z

=

1 δ (17)

z

x

+ –

+

Using the definition of z we can then back out the solution for the capital stock in

Equation (10).

4 Discussion

Note several interesting features of the equilibrium. First, Equation (9) asserts

that in every period the ratio of consumption to capital must be constant,

3 The function 2 F1(a, b, d;ω)is convergent for arbitrary a,b, and d for real -1 < ω < 1. See

Weisstein (2004).

c k = x = x. In other words, the saddle-path is linear, with a slope equal to x.

This is depicted in Figure 1. This property is now fairly well known, for a range

of models [Chang (1988), Xie (1991, 1994), Barro and Sala-i-Martin (2004),

Boucekkine and Ruiz-Tamarit (2004a, b), Wälde (2005)].

The Phase Diagram

Figure 1

The Phase Diagram

Second, Equation (10) provides an exact solution for the dynamics of the

capital stock. The growth rate of the capital stock is

( )

–

+

=

+

– –

+ –

= = –

–

+

–

– –

– –

1

.

1

1

1 1

0

1 1

0

α

δ α

ρ

α

α α

α α

δ α

ρ

ρ δ

α α

ρ δ

γ

k k

k k e

k k

kk

t

k

(18)

On inspection, the growth rate decreases over time: The growth rate depends upon

the deviation between net output and consumption. However, consumption grows

linearly with the capital stock, while gross output increases at a decreasing rate,

due to diminishing returns.

We can also obtain an exact, analytical calculation of the speed of

convergence to the steady state:

(1 ) .

ln

-1

+

= –

∂ ∂

= –

α

δ α

ρ

α

γ

β

k k

k

k

k (19)

Barro and Sala-i-Martin (2004, chapter 2), in contrast, resort to a calibration of a

linear approximation around the steady state. The accuracy of their estimate

diminishes rapidly the further away capital is from its steady state. The

numerical calculations of Mulligan and Sala-i-Martin (1991) are exact, but lack

the transparency of the solution in Equation (19).

Third, the model yields a clean expression for the dynamics of the gross

saving rate, s =1- c k α . From Equations (9) and (10) it follows that

( ) ( ) ( )

( ) ( ) ,

1

1

1

0

1

1 1

0

1

t

t

s s s e

s k k k e

δ α

ρ

α

δ α

ρ

α

α α α

α

ρ δ α

+

– –

+

– –

– – –

= + –

+ –

+ –

= –

(20)

where s =1- xα (δ + ρ) is the steady state saving rate. The model predicts that, if

k0 < k , the saving rate will decrease monotonically over the transition to the

steady state. The intuition is simple. In general, the increase in the capital stock

exerts offsetting income and substitution effects on saving [Barro and Sala-iMartin (2004, Chapter 2, Section 6.3)]. As interest rates fall the

substitution effect

tends to reduce saving. However, as the gap between current income and

permanent income diminishes, there is less incentive to save. Because people like

intertemporal substitution (σ > 1) in this calibration of the model the substitution

effect is strong, and dominates the income effect: Since the consumption-capital

ratio is constant it follows that s =1- xk 1-α =1-αx R, where R =αk α -1 is the

gross interest rate. The saving rate therefore decreases as the interest rate falls.

This is not consistent with the empirical evidence [Barro and Sala-i-Martin

(2004)], since the saving rate actually increases over the transition.

Finally, the solution suggests a useful pedagogical exercise. It is well

known that the Solow (1956) model with Cobb-Douglas technology is an

autonomous Bernoulli equation [Chiang (1967, p. 454), Obstfeld and Rogoff

(1999), Barro and Sala-i-Martin (2004, p. 44)].4 Students are often asked to solve

4 Barro and Sala-i-Martin (2004, appendix 2.C) attribute the use of this transformation in the

Solow model to Jaume Ventura. However, it was known at least as early as Chiang (1967).

the Solow model explicitly using this “trick.” It is non-trivial to show – using

either the methods here, or those in Barro and Sala-i-Martin (2004, Appendix 2b)

— that the saddle-path is linear when σ =1 α . However, students may as an

exercise be told to assume this result – or tell an intuitive story to motivate it —

and then be invited to solve the rest of the model. Once they have determined the

steady-state capital-output ratio, they can write Equation (6) as an autonomous

Bernoulli equation, which can be solved just like the Solow model.

5 Extensions

The model can be adapted to provide explicit solutions to a range of problems in

growth theory. I will consider two here.

5.a A Temporary Increase in Government Expenditures

A traditional thought exercise [Romer (2005), Turnovsky (1999)] is to trace out

the dynamic effects of a temporary increase in government expenditures in the

Ramsey model. This problem has been addressed graphically [Blanchard and

Fischer (1989), Romer (2005)] and solved using linear approximations

[Turnovsky (1999)]. I provide an explicit solution for the case where σ = 1 α. 5

First we need to incorporate the government into a decentralized version

of the model. Suppose that that government issues debt b , collects lump-sum

taxes τ, and engages in expenditures G. Given an initial debt of b0 and the

interest rate r = αk α -1 – δ , its flow budget constraint is then

b = rb + G -τ. (21)

To obtain an analytical solution I assume that government expenditures are

proportional to the capital stock, G = gk. While this assumption is just a

convenience here, it is standard in endogenous growth models [see Turnovsky

(2000), for example].6

Imagine a temporary increase in g from time zero to time T :

5 It is also possible to consider expected future changes in government expenditure, as well as

temporary and expected future changes in distortionary taxes on capital.

6

The traditional exercise involves an increase in government expenditures that is independent of

the capital stock or output [Blanchard and Fischer (1989), Romer (2005)]. This makes no

substantive difference. Alternatively, the model could be solved by making government

expenditures proportional to output.

g g t T

g t g t T

= < ≥

= ≤ <

2 1

1, 0

(22)

The dynamics of capital and consumption are now given by

, .

, 0

1 2

k k c g k t T

k k k c g k t T

= – – – >

= – – – ≤ ≤

δ

δ

α

α

(23)

= σ (αk α -1 -δ – ρ)

cc

(24)

Define ( )

1 1

1

x – g

+ –

=

α

ρ α δ

,

( )

2 2

1

x – g

+ –

=

α

ρ α δ

and .

∆g = g1 – g 2 In the

appendix I prove

PROPOSITION 2: Assuming that σ =1 α the equilibrium paths of consumption

and capital in response to a temporary increase in government expenditures from

time 0 to time T are:

( )

, .

, 0 ,

1

2

2

1

1

x k t T

k t T

e

x

xg

c

x t T

= >

≤ ≤

∆

–

=

–

(25)

( )

( )

( ) ( )( )

( )

, .

0 ,

1 ,

1

1 ,1, ;

1

1

1

1

1

1

1

1

2

2

2

2 1 1

1

1

1

k e t T

t T

e e

x

g

e

x

e g

x

g

k F d

t

T

x t T x t

x t T

x t T

>

= + + – +

≤ ≤

+ Ω – ∆

∆

∆ –

– –

+

=

–

+

– –

–

–

– – +

–

–

–

–

α

δ α

ρ

α

α

α

α δ

α

ρ δ

α

ρ δ

α

α

ρ δ

α

(26)

where

1 .

1

1 1 ,1, ; 1

2

2

2

0 2 1 1

+ ∆

∆

∆ –

– –

+

Ω = – –

–

–

x T

xT

xT

e

x

g

e

x

e g

x

g

F d

x

z α

δ

(27)

Equation (25) makes the dynamics of consumption per unit capital

transparent: As shown in Figure 2, consumption initially falls, given the initial

capital stock, so that c k drops.7 Then, as the capital stock falls, c k increases

until the economy regains the saddlepath at timeT ; thereafter it remains constant.

Intuitively, people respond to the increase in government expenditures by

reducing consumption. Since the change in government expenditures is

temporary, however, they reduce consumption by less than the increase in

government expenditures. This causes the capital stock to decrease over time, so

that consumption per unit capital has to accelerate as it approaches the saddlepath.

Effects of a Temporary Increase in Government Expenditures Figure 2

Effects of a Temporary Increase in Government Expenditures

7 Figure 2 is identical to Figure 2.9 in Romer (2005) except that the k = 0 locus rotates around the

origin rather than making a parallel shift.

5.b Transition Dynamics in an Endogenous Growth Model

Another illuminating application of the model is to the transitional dynamics of

endogenous growth. This too has been analyzed graphically [Barro and Sala-iMartin (2004)], but never solved explicitly.

I adopt a Cobb-Douglas version of the a-k technology suggested by Jones

and Manuelli (1990): y = k α + ak. Since the marginal product of capital is

bounded away from zero, this is sufficient to induce long run growth. The

dynamics of k and c are now

k = k α + ak -δk – c

= σ (αk α -1 + a -δ – ρ). (28)

(29)

cc

In order to generate an asymptotically balanced growth path I assume that

a > δ + ρ. In order to satisfy the transversality condition I also assume that

ρ > (1-α )(a -δ ).

In the appendix I derive the closed form solution to this model:

PROPOSITION 3: If σ = 1 α the transitional paths of consumption and capital in

the endogenous growth model are

( )( )

,

1

c a k

α

ρ – -δ -α

= (30)

( )

( δ ρ ) α

α α

α

δ ρ

α

δ ρ

α – – –

–

–

= – – – + + – – 1

1

1

10

a t

e

a

k

a

k . (31)

Equation (30) asserts that the consumption/capital ratio is again constant over

time. Equation (31) implies that the growth rate of capital is

( )

(a )t

δ ρ

– –

–

–

1 a k

α

–

+ a

ρ

– – k

10 a

α

δ ρ

– –

=

e

ρ

a

kk

α α

α

δ α

δ ρ

α

δ α

–

– –

–

– –

+

10

. (32)

This makes it clear that the growth rate falls over time, tending asymptotically

to(a – δ – ρ) α .

6 Calibration

We can only solve the model in the special case where the intertemporal elasticity

of substitution σ is equal to the reciprocal of capital’s share of income α. Is this

likely to be a reasonable description of the real world?

At first glance, the answer to this question would seem to be no. The

traditional estimate of physical capital’s share of GDP is about one third [Simon

(1990)]. This would require the intertemporal elasticity of substitution to be as

large as three. However, the intertemporal elasticity of substitution is usually

thought be quite small, certainly less than one [Hall (1988), Attanasio and Weber

(1993), Epstein and Zin (1991), Campbell, Lo, and MacKinlay (1997)].

Recent empirical work casts this conventional wisdom in doubt. Barro and

Sal-i-Martin (2004) point out that if α = .3the Ramsey model does a poor job at

explaining a range of empirical features of growth. However, “all of these

shortcomings are eliminated” [Barro and Sal-i-Martin (2004); see also Mankiw,

Romer, and Weil (1992) and Obstfeld and Rogoff (1994)] if we adopt a broader

notion of capital (so that it includes human as well a physical capital), and

increase capital’s share: Barro and Sala-i-Martin suggest α = .75, Obstfeld and

Rogoff (1994) α = .66. If we set α = .75, we are only restricting the elasticity of

substitution to be about σ =1.33. The last decade has seen more and more

empirical evidence that σ is in fact greater than one, with estimates ranging from

1.5to as much as 3[Bufman and Leiderman (1990), Attanasio and Weber (1989),

Koskievic (1999), Vissing-Jorgensen (2003)]. There is also the interesting

empirical fact that, in a cross section of countries, growth rates are negatively

correlated with production risk [Ramey and Ramey (1995)]. In the canonical

stochastic A-k growth model [Smith (1996)] this can only occur if the

intertemporal elasticity is greater than one.

What of the model’s prediction that the consumption-capital ratio should

be constant? It is a traditional “stylized fact” of growth, enunciated by Kaldor

(1963) and later supported by Maddison (1982), that the ratio of consumption to

physical capital is roughly constant over time. What can we say about the

consumption-capital ratio if we adopt a broader notion of capital, one that that

encompasses human as well as physical capital? Empirical estimates [Jorgenson

and Fraumeni (1989), Lettau and Ludvigson (2004)] suggest that human capital

represents between two thirds and 90 percent of total wealth. In a recent paper

Zhang (2004) establishes that the ratio of consumption to total wealth is roughly

constant. He estimates the steady-state consumption-capital ratio to be .05739, in

annual terms. Now compare this with the predicted consumption-capital ratio of

the model: Following Barro and Sala-i-Martin (2004), set the rate of depreciation

at .05 and the rate of time preference at .02 per year. Let capital’s share be .66 [as

suggested by Obstfeld and Rogoff (1994)], which restricts the intertemporal

elasticity of substitution to the fairly low value of 1.5. The consumption-capital

ratio predicted by Equation (9) is then .055. This matches the data remarkably

well.

In conclusion, if we are willing to embrace a broad notion of capital and

entertain the possibility that the elasticity of substitution may be even a bit larger

than one, this calibration may work fairly well.

7 Conclusion

This paper provides an explicit solution to the Ramsey model when capital’s share

is equal to the reciprocal of the intertemporal elasticity of substitution. This is

admittedly a special case. However, it illuminates the dynamics of the model in a

way that approximations and numerical methods cannot: it helps us to “inspect

the mechanism” [Campbell (1994)]. In principle it could be used as a benchmark

from which to undertake perturbation analyses of less restrictive cases [Judd

(1997)]. In addition, it offers a useful pedagogical exercise, providing a closedform solution to the Ramsey corresponding to the Bernoulli

solution of the Solow

model.

In another paper [Smith (2005)] I derive an explicit solution to a stochastic

version of the Ramsey model. Applying the methods of Merton’s (1975) classic

paper on stochastic growth I can then derive an explicit solution for the ergodic

distribution of the capital stock. The nonlinearities of the model (in both the utility

and production functions) interact with uncertainty — via Jensen’s inequality — to

affect the steady state in complicated and even surprising ways. The presence of

uncertainty makes the “mechanism” much more complicated than generally

appreciated.

Appendix

DERIVATION OF THE GENERAL SOLUTION FOR z

Consider first the homogeneous equation

( )

1

z

= – -α δ + .

1

z

xAe

x

xt

+ (A.1)

This can be integrated to find the complementary solution

z c = x α -1e -(1-α )(x+δ )t (1 + xAe xt )1-α . (A.2)

To find the particular solution to Equation (12), I will use the method of

variation of parameters. Conjecture that the particular solution is

z p = zcΨ, where Ψ is an unknown function of time. Substituting this conjecture

into Equation (12), it follows that

Ψ = 1 – α = (1 – α ) 1-α (1-α )(x+δ )t (1 + xt )α -1.

c

x e xAe

z

(A.3)

Integrating Equation (A.3) leads to1

( )( ) ( )( ) ( )( )

; .

1

1 1 2 1 1 , 1 ,1

–

– +

+

– +

–

+

Ψ = – +

–

x t xAe xt

x

x

x

x

e F

xx

α δ α δ

α

δ

α δ

α

(A.4)

It follows that

( ) ( )( ) ( )( )

; .

1

1 1 2 1 1 , 1 ,1

–

– +

+

– +

–

+

+

=

–

xt

xt

p xAe

x

x

x

x

F

x

xAe

z

α δ α δ

α

δ

α

(A.5)

However, a useful property of the hypergeometric function [Weisstein (2004)] is

that ( ) ( )

–

= – –

–

1

2 1 , , ; 1 2 1 , , ,

z

z

F a b c z z a F a c b c . Therefore Equation (A.4) can

be written as

.

1

1 2 1 1 ,1, ;

z p = x +δ F -α d +xAe xAe xt xt (A.6)

where d = 1+ (1-α)(δ + x) x . To verify that this is a solution; differentiate

Equation (A.6) with respect to time, using the definition of the hypergeometric

function given by Equation (14) of the text:

( )

( )( )

– –

+

–

1 2

1 2

2

α α α

x Ae

xt

xt =

1

z

p

( )

( )( )( )

( )( )

( ) ( )( )

( )

( )( )( )

( )( )

+

+ +

– – –

+

+

– –

+

–

–

+

=

+

+ +

– – –

+

+

+ +

. . .

1 2

1 2 3

3

1

1 2

1 1 2

. . .

1 2

1 2 3

3

1

1

2

2

2

ω

α α α

ω

α α α

ω ω

δ

ω

α α α

ω

δ

x d d d d d d

x

xAe d d d d d d

x

(A.7)

Substitute this back into Equation (12) in the text to find

( ) ( )( )

( )

( )( )( )

( )( )

( ) [ ( )] ( )( )

( )

( )( )( )

( 1)( 2) . . .

1 2 3

1

1 1 2

1 1 1 1

. . .

1 2

1 2 3

3

1

1 2

1 1 2

2 3

2

+

+ +

– – –

+

+

– –

+

–

+

+

+ –

– –

=

+

+ +

– – –

+

+

– –

+

–

–

+

ω

α α α

ω

α α

ω

α

ω δ

δ

α

ω

α α α

ω

α α α

ω ω

δ

x d d d d d d

x

x d d d d d d

x

(A.8)

Collect the terms inω i (i = 0,1,2,…). Some tedious algebra reveals that the

coefficient attached to each ω i in the left hand side of Equation (A.8) equals the

coefficient attached to the corresponding ω i on the right hand side. Therefore

Equation (A.6) solves Equation (14) in the text.

Using Equations (A.2) and (A.6) leads to the general solution

α ( α )( δ ) α ( α )( δ ) ( ) α

δ

x + x 1+xAe

Bx e xAe

– – – + –

+ +

z = 1 2 F1 1- ,1,1+ 1- x + ;xAe xt xt 11 x t 1 xt 1

(A.9)

where B is an arbitrary constant.

PROOF OF THE LEMMA

Using the initial condition k0 and Equations (13) and (14), it follows that

( )

+

–

+

= –

–

xA

xA

F d

x

B A x z

1

1 0 1 2 1 1 α,,1, ;

δ

α

. (A.10)

Now consider the transversality condition in Equation (3) of the text. Recalling

that σ =1 α , use Equations (11) and (13) to write this limit as

( )

( )( )

1 1 1

– – – +

x t

α α δ lim 1 1 ,1; ;

–

xAe

F a d

xt

α

1

+

xAe 2

+

x

δ 1

+

xAe

xt (1 ) .

+

–

Ae

xt

α + xt

→ ∞

– t

xx e

B A

x

e

t

ρ

(A.11)

We need to show that this limit is zero if and only if A = 0.

First, suppose that A = 0. Note that, from Equation (14),

(0) 1 1 .

= –

–

B x z (A.12)

0 +

δ

α

x

Use this fact to evaluate the limit in Equation (A.11) when A = 0 :

( )( )

0

1 1

lim 1 1

0

=

→ ∞

+

+ –

+

– – – – +

t

e

x

k

x

e

ρt α α δ x t

δ δ (A.13)

This establishes sufficiency.

To establish necessity first rewrite Equation (A.11) as

t ( xt ) xt xt ( ) (

xAe e

– + – + x xAe δ → ∞

α ρ

–

xt )[ ( )(x )]t

AB

t

xAe

e xAe F a d

x

x

ρ α δ

+ +

lim + 1+ 2 1 ,1; ;1+ 1 1

(A.14)

Now consider the limit of the second term:

( )( ) [ ( )( )]

.

lim 1 1

→ ∞

+ – + – +

t

xAe e

Ax

B

xt ρ α x δ t

(A.15)

Since ρ + (1-α)(δ + x) > 0 and – ρ + x -(1-α)(δ + x) = 0 ,

( )( ) [ ( )( )] ( )

.

lim 1 1

→ ∞

+ – + – + =

t

xAe e AB A

Ax

B

xt ρ α x δ t

(A.16)

If B(A) ≠ 0 this will be zero if and only if A = 0.

Next consider the first expression in Equation (A.14):

( )

1 .

1 2 1 ,1; ;

++ +

xAe

xAe

e xAe F a d lim

–

ρt xt

xt α

→ ∞

–

t

xx

xt

α

δ (A.17)

First consider the limit of the hypergeometric function. Note that

lim 1 =1.

→ ∞

+

t

xAe

xAe

xt

xt

(A.18)

[Lebedev (1972, p. 244, Equation (9.3.1))] provides a limit result about

hypergeometric functions which then establishes

( ) ( )

( ) ( )

→ ∞

= Θ >

Γ – + Γ –

Γ Γ – +

=

+

t

d d

d d

xAe

xAe

F a d

xt

xt

0.

1 1

2

1

lim 2 1 ,1; ;

α

α

(A.19)

Since the hypergeometric function converges to this finite limit as t → ∞ it follows

that

( )

=

++ +

xAe

xAe

e xAe F a d

xt

t xt

lim

1 2 1 ,1; ;

( )

+ Θ

+

– –

e xAe

t xt

1 .

ρ α

α – xt lim α

→ ∞

→ ∞

– t

xx

t

xx

1

ρ

α

δ δ

(A.20)

Now rewrite this as

( ) ( ) ( ) ( )

→ ∞

t → ∞

lim

Θ

+

+ Θ =

+

–

–

– –

–

e xA

xx

t

e e xA

xx

x t xt x t

lim α ρ α .

α

α

α ρ

α

δ δ (A.21)

This will go to zero if and only if A = 0. This establishes necessity.

Given A = 0, B is given by Equation (A.12). This proves the lemma.

SOLUTION FOR THE MODEL WITH A TEMPORARY INCREASE IN

GOVERNMENT EXPENDITURES

The dynamics of x and z are given by

( )

( )

, ;

1

1 , 0 ,

1 2

g x t T

g x t T

xx

+ + >

– +

= –

+ + ≤ ≤

– +

= –

α

δ α ρ

α

δ α ρ

(A.22)

( )[ ( ) ]

(1 )[1 ( ) ], .

1 1 , 0 ,

2

1

g x z t T

z g x z t T

= – – + + >

= – – + + ≤ ≤

α δ

α δ

(A.23)

First, define xi = +( – ) – g i

α

ρ 1 α δ

, and =1+ (1- )( + ), i =1,2.

x

x

d

i

i

α δ

Let

0.

∆g = g1 – g 2 > Note in passing that

x + g = x + g = x =

1 α

1 2 2 1 .

()ρ +δ -α

Following the derivation of the basic model without government, the general

solutions for x and z are

, ;

1

, 0 ,

1

1 2

2 2

2

1 1

1

t T

x A e

x

t T

x A e

x

x

x t

x t

>

+

=

≤ ≤

+

=

(A.24)

( )( ) ( )

( )( ) (1 ) , .

1

1 1 ,1, ;

1 , 0 ,

1

1 1 ,1, ;

1

2 2

1 1

2 2

2 2

2 2

2 1 2

1

1 1

1 1

1 1

1 1

1 1

2 1 1

21

2

1

1

1

B x e x A e t T

x A e

x A e

F d

x

B x e x A e t T

x A e

x A e

F d

x

z

x t x t

xt

x t

x t x t

x t

x t

+ + >

+

–

+

=

+ + ≤ ≤

+

–

+

=

–

– – – +

– – – + –

α α δ α

α α δ α

α

δ

α

δ

(A.25)

There are four arbitrary constants, A1, A2 , B1, and B2. To pin down these

constants, we have four boundary conditions:

(a)

(b)

(c)

(d) the initial capital stock, k0,

the transversality condition [Equation (3) in the text],

continuity of c at T,

continuity of k at T.

Consider the A’ s first. Using the argument developed in the previous

appendix, the transversality condition (b) requires A2 = 0. Using the continuity

condition (c) along with Equation (A.24) and the fact that A2 = 0, we find

.

1

1 2

x T

e

x x

g

∆ – (A.26) 1

A = –

To find

B1, substitute Equation (A.26) into Equation (A.25) evaluated at t = 0 and

using the initial condition (a), z0 = k01-α :

∆

∆ –

– –

–

F d

z

0 2 1 1

1 1 α,1, ; = – ∆

– –

x T

e

x

g

B x

2

11

1

1 1

α –

xT (A.27)

+

x

δ 1

–

–

xT

e

x

e g

x

g

2

2

1

α

This implies, from Equation (A.25), that for the interval 0 ≤ t ≤ T

( )

( )

( )

( )( )

.

1

1

1

1 1 ,1, ;

1

1 1 ,1, ;

1

1

2

2

2

2

0 2 1 1

2

2

2 1 1

1

1

1

1

x t

x T

x t T

xT

xT

x t T

x t T

e

e

x

e g

x

g

e

x

e g

x

g

F d

x

z

e

x

e g

x

g

F d

x

z

α δ

α

α

δ

α

δ

– – +

–

–

–

–

–

–

–

∆

∆ –

–

∆

∆ –

– –

+

–

+

∆

∆ –

– –

+

=

(A.28)

Note that

( )( )

.

1

1

1

1 1 ,1, ;

1

1 1 ,1, ;

1

1

2

2

2

2

0 2 1 1

2

2

2 1 1

1

x T

xT x T

xT

T

e

e

x

g

x

g

e

x

e g

x

g

F d

x

z

x

g

x

g

F d

x

z

α δ

α

α

α δ

δ

– – +

–

– –

–

∆

–

∆

–

∆

∆ –

– –

+

–

+

∆

∆ –

– –

+

=

(A.29)

Finally, to find B2 substitute Equation (A.29) to the second interval (for t > T ) of

Equation (A.26), evaluated at t = T. Recalling that z =1/(x +δ )this yields

B2 = x21-α (zT – z)e(1-α )(x+δ )T

Substituting this back into Equation (A.26) yields z for t > T :

z z (zT z)e (1)(x)(tT ). (A.30)

= + – –α +δ – (A.31)

SOLUTION FOR THE ENDOGENOUS GROWTH MODEL

Assuming thatσ = 1 α , Equations (28) and (29) in the text imply that the

dynamics of z and x are

z = (1-α )[1+ (a -δ – x)z]

(a )(1 ) x.

= – -δ -σ – ρσ + (A.32)

(A.33)

xx

I assume that a > δ + ρ in order to generate long-term growth.

Begin with the logistic equation (A.33). This has the general solution

,

1 ˆ

ˆ

x x

= (A.34)

xAe xˆt

+

where ( )( )

α

ρ – -δ -α

=

1

xˆ a . The transversality condition will require that xˆ > 0.

Substitute Equation (A.34) back into the linear equation (A.32):

( )

z

+

= – + – – xAe

x

z a

1 ˆ xˆt

ˆ

1 α 1 δ . (A.35)

Following the derivation of Equation (A.9), the general solution to Equation

(A.35) is

α ( α )( δ ) α ( α )( δ ) ( ) α

δ

xˆ + – a x 1+xˆAe xt

2 Bxˆ e 1 xˆAe

–

– – – + –

+ +

z = 1 F1 1- ,1,1+ 1- xˆˆ+ – a ;xˆAe xˆt ˆ11 xˆ a t xt 1

.

(A.36)

The initial condition

z0 requires

( )

1

ˆ

1

= –

–

B A x z

α ,

ˆ

1 ,,1, ˆ;

–

xA

F d

α

1 ˆ

+

xA ˆ

0 2 1

+ –

x a

δ

(A.37)

where ( )( ).

ˆ

1 ˆ

ˆ 1

x

x a

d = + -α +δ – The transversality condition requires

( )

( )( )

ˆ

ˆ

,1; ˆ;

1

ˆ 1

1 1 ˆ

ˆ

ˆ

2 1

– – – + –

x e

B A

xt

x a t

xt

xt

(1 ˆ ) .

+

–

xAe 1 ˆ

+

+

xAe ˆ

+ –

x a

δ 1 ˆ

ˆ

+

Ae

ˆ lim

e –

xt

α xAe

F a d α α

→ ∞

– t

xx

t

α

δ

ρ

(A.38)

Following the reasoning in the previous lemma, the transversality condition will

be satisfied if and only if (1) A = 0 and (1) ρ > (1-α)(a -δ ). The latter condition

implies that xˆ > 0, which yields Equation (30) in the text.

Setting A = 0 , Equation (A.37) yields

.

ˆ

1

ˆ

0

1

+ –

= –

–

x a

B x z

δ

α (A.39)

Substituting this into Equation (A.36) yields

( )( )

( )

( )

.

ˆ

1

ˆ

1

1

0

1 ˆ

0

a t

x a t

e

a

z

a

e

x a

z

x a

z

δ ρ

δ α α

α

δ ρ

α

δ ρ

α

δ δ

– –

–

– – + –

= – – – + + – –

+ –

+ –

+ –

=

(A.40)

Substituting z = k 1-α into Equation (A.40) yields Equation (31) in the text.

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