Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

fiction:

(14.231)

ecehold grows, the budget

iumeraire commodity (so the optimal per capita ►n (14.232) (plus an NPG- - ion that results from this

(14.233)

-n blueprints) is given by !nal goods market clearing

r

(14.234)

the fact that 4(t) = LR (t)

ading from the simplified dynamic adjustment here l by the augmented model. 'le proportions of labour I LR/L) are both constant,

nd y(t) Y(t)/L(t) are all d by rewriting (14.228) in

p

(14.235)

-t). By differentiating the that LR /LR = nL we obtain

(14.236)

cing symmetry) we find

L (where Lx = kxNX).

Chapter 14: Theories of Economic Growth

From this last expression we find:

Yy = YY

= (11 - 1)YN.

(14.237)

Finally, from the final goods market clearing condition (14.202) we find the growth rate for (per capita) consumption:

Yc = Yc - /IL = yy .

(14.238)

We reach a rather striking conclusion. By modifying the R&D equation suggested by Jones (namely (14.228)) instead of the standard one (14.193) we have managed to eliminate the scale effect altogether (compare (14.211) and (14.236)). The economy still grows and innovation continues to take place in the modified model but growth is exogenous, i.e. it is explained by the rate of population growth just as in the good old Solow model! With a stable population (ni, = 0) innovation ceases in the long run because as N(t) rises over time, more and more labour has to be devoted to the R&D sector to sustain a given rate of innovation.

14.7 Punchlines

We started this chapter by presenting some of the most important stylized facts about growth as they were presented four decades ago by Kaldor. These are: (i) output per worker shows continuous growth, (ii) the capital-output ratio is constant, (iii) labour and capital receive constant shares of total income, and (iv) the rate of productivity growth differs across countries. Together these stylized facts also explain that (v) capital per worker grows continuously and that (vi) the rate of return on capital is steady.

Next we presented the neoclassical growth model as it was developed by Solow and Swan in the mid-1950s. The key elements of this model are the neoclassical production function, featuring substitutability between capital and labour, and the "Keynesian" savings function according to which households save a constant fraction of their income. Although the Solow-Swan model is able to explain all of Kaldor's stylized facts, some economists are disturbed by its prediction that long-run growth is determined entirely by exogenous factors, such as the rate of population growth and the rate of labour-augmenting technological progress. For this reason the Solow-Swan model is often referred to as an "exogenous" growth model. The model is inconsistent with Ricardian equivalence. Further important features of the model are that it allows for the possibility of oversaving (dynamic inefficiency) and that it is consistent with the conditional convergence hypothesis according to which similar countries converge. The standard Solow-Swan model predicts too high a convergence speed but this counterfactual prediction of the model is easily fixed by incorporating human capital into the model.

473

The Foundation of Modern Macroeconomics

Several extensions and applications of the Solow-Swan model are discussed. In the most important extension, the ad hoc savings function is endogenized by introducing infinitely lived optimizing consumers into the model. This approach, which was pioneered by Ramsey more than seven decades ago, precludes the possibility of oversaving and implies the validity of Ricardian equivalence. (Ricardian equivalence fails, even with infinitely lived agents, if population growth consists of the arrival of disconnected generations.) The growth properties of the Ramsey model are very similar to those of the Solow-Swan model.

The final section of the chapter deals with the recent literature on so-called "endogenous" growth. Three major approaches can be distinguished in this literature. The so-called "capital-fundamentalist" models generate perpetual growth by abandoning one of the key elements of the Solow-Swan model, namely the assumption that the average product of capital goes to zero as the capital stock gets very large. If it is easy to substitute labour for capital then the average product of capital reaches a finite limiting value. It is possible to produce without any labour at all and long-run growth depends, among other things, on the savings rate. Similar results are obtained for the AK-model in which labour plays no role at all and production features constant returns to a broad measure of capital.

The second major approach in the endogenous growth literature emphasizes the purposeful accumulation of human capital as the engine of growth. This approach was pioneered by Uzawa in the mid-1960s and further developed by Lucas. The model features infinitely lived households and technology exhibits constant returns to scale in capital and effective labour. The rate of growth in human capital depends on the fraction of time households spend on educational purposes. Even without population growth, consumption, human and physical capital, and output all grow at the same exponential rate.

The third group of studies in the field of endogenous growth is based on the notion that research and development (R&D) activities by firms constitute the engine of growth in the economy. Studies in this vein abandon the assumption of perfect competition and instead analyse monopolistically competitive firms. We present a very simple model (without physical and human capital) in which the R&D sector produces blueprints for new differentiated inputs. In the intermediate goods sector there are many monopolistically competitive firms which each hold a blueprint telling them how to produce their own, slightly unique, input variety. Production of final goods features returns to specialization, i.e. a broader range of differentiated inputs raises productivity because a more roundabout production process can be adopted. The model features a constant rate of innovation which depends positively on the monopoly markup and the scale of the economy. The scale effect is a problematic feature of many R&D based models because it is easily falsified empirically. Elimination of the scale effect is possible but renders the rate of innovation proportional to the rate of population growth, just as in the standard Solow-Swan model.

Further Reading

The first wave of gross (1971), and Hacche (197 (1965), and Koopman , Arrow (1962), Uzawa t

Recent textbooks on and Howitt (1998), (1991), van de Klundc Manuelli (1997). Key ILI and Aghion and Howitt Schumpeter (1934). 1

Lucas (1990a) studies c of the recent empirical gr Rebelo (1993), Munig,: and Bond, Wang, and *; . 4 There is a large literat 1965), Sidrauski (1967►, 1 ature on public investr Uzawa (1988), Aschauc.: (1994), Turnovsky (1996, On the scale effect, see see Griliches (2000). IL -%

integration.

Appendix

In this appendix we shoe step we postulate a trial

k(t) — k* :7,

q(t) — q* ]—[ -

[

where JTki and 70 (i = 1, are, respectively, the ,• :1

eliminate the effects of z

[ 7112	[ 0
irq2

By differentiating (A 14.1

[k (t) ]=

4(t) Xq

474

- nodel are discussed. In is endogenized by intro- ' . This approach, which dudes the possibility of

. t Ricardian equivalence consists of the arrival Ramsey model are very

literature on so-called Dished in this literaperpetual growth by iel, namely the assumpcapital stock gets very erage product of capital )ut any labour at all and ;s rate. Similar results le at all and production

terature emphasizes the 7rowth. This approach loped by Lucas. The thibits constant returns man capital depends mrposes. Even without 3 11, and output all grow

I

!rowth is based on the firms constitute the andon the assumption r competitive firms. We capital) in which the its. In the intermediate -ms which each hold tly unique, input vari- -i , i.e. a broader range oundabout production e of innovation which of the economy. The dels because it is easily but renders the rate just as in the standard

Chapter 14: Theories of Economic Growth

Further Reading

The first wave of growth theory is well surveyed by Burmeister and Dobell (1970), Hamberg (1971), and Hacche (1979). Key contributions to the Ramsey model are Ramsey (1928), Cass (1965), and Koopmans (1965, 1967). Important early papers on endogenous growth are by Arrow (1962), Uzawa (1965), Sheshinski (1967), Shell (1967), and Conlisk (1969).

Recent textbooks on economic growth include Barro and Sala-i-Martin (1995), Aghion and Howitt (1998), Jones (1998), and Gylfason (1999). Good survey articles are Stern (1991), van de Klundert and Smulders (1992), van der Ploeg and Tang (1992), and Jones and Manuelli (1997). Key references to the R&D literature are Grossman and Helpman (1991) and Aghion and Howitt (1998). The classic source on the idea of creative destruction is Schumpeter (1934).

Lucas (1990a) studies capital taxation in a growth model. Temple (1999) presents a survey of the recent empirical growth literature. On the issue of transitional dynamics, see King and Rebelo (1993), Mulligan and Sala-i-Martin (1993), Xie (1994), Benhabib and Perli (1994), and Bond, Wang, and Yip (1996).

There is a large literature on money and growth. The key references are Tobin (1955, 1965), Sidrauski (1967), Fischer (1979), and Ireland (1994). Key contributions to the literature on public investment include Barro (1981, 1990), Aschauer and Greenwood (1985), Uzawa (1988), Aschauer (1988, 1989), Baxter and King (1993), Glomm and Ravikumar (1994), Turnovsky (1996), Turnovsky and Fisher (1995), and Fisher and Turnovsky (1998). On the scale effect, see Young (1998) and Segerstrom (1998). On R&D and education, see Griliches (2000). Rivera-Batiz and Romer (1991) study the growth effects of economic integration.

Appendix

In this appendix we show how the expressions in (14.99)—(14.100) are derived. In the first step we postulate a trial solution for capital and Tobin's q:

[	—	k*	=[ 7rkl	e-A 1 t + [ 7k2	eA2 t (A14.1)
	k(t)
	q(t) — q*		7ro	Thq2

where 7rki and 70 (i = 1, 2) are coefficients to be determined and where —A.1 < 0 and X2 > 0 are, respectively, the stable (negative) and unstable (positive) characteristic root of Ai. To eliminate the effects of the unstable root we must set:

	[ irk2	(A14.2)
	Trq2
	By differentiating (A14.1) with respect to time and noting (A14.2) we obtain:

[ k(t)		(A14.3)
4(0	= —A1 [ Trkl
	7rql
		475

The Foundation of Modern Macroeconomics

where we have also used the fact that k* = = 0 (constant steady state). By substituting (A14.1)-(A14.3) into (14.97) we obtain:

[	811)	—612	-	(A14.4)
	-621	+ 622) [ Trql

where 64 represents element (1,1) of the Jacobian matrix A. Since -X1 is an eigenvalue of Ai, the matrix on the left-hand side of (A14.4) is singular so either row of (A14.4) can be used to solve n-q i in terms of Ytk l . Noting that 811 = 0 we obtain from the first row:

	=	A.1	(A14.5)
		°12

Next we exploit the fact that the capital stock is predetermined, i.e. its value at time t = 0, denoted by /co, is given. Substituting this initial condition in the first equation of (A14.1) and noting (A14.2) we obtain:

k(0) - k* = 1(0 k* = 7rki.

(A14.6)

The second equation of (A14.1) in combination with (A14.2) and (A14.5)-(A14.6) yields the solution for Tobin's q on the saddle path:

q(0) - q* =	- (—)[ko - .	(A14.7)
	312

By substituting (A14.2), (A14.6)-(A14.7) into (A14.1) the expressions in (14.99)-(14.100) are obtained.

The solution method used here is valid provided the forcing term of the dynamical system is time invariant. This covers both the transition path of an economy which starts outside the steady state and the adjustment path following an unanticipated and permanent shock to the investment subsidy (both are discussed in section 5.6 above). In the Mathematical Appendix we present a solution method which can handle more general shock patterns.

Real Busi

The purpose of this cha

1.To introduce an ei study the effects c

2.To turn the exten model by assuma

3.To analyse the z . response function

4.To study the qu,► can be made to fi

5.To briefly discuss to improve the

15.1 Introductio

In this chapter we stuff new classical econon early 1970s. The first equilibrium framewoi were made by Hall recently by Baxter ant The second the: :- fluctuations. PiontLi. approach were made t

Prescott (1986).

476

it steady state). By substituting

(A14.4)

. Since —Ai is an eigenvalue of either row of (A14.4) can be n from the first row:

I

(A14.5)

"mined, i.e. its value at time /talon in the first equation of

(A14.6)

2 and (A14.5)—(A14.6) yields

(A14.7)

^ressions in (14.99)—(14.100)

term of the dynamical system

-nnomy which starts outside rated and permanent shock

above). In the Mathematical , re general shock patterns.

15

Real Business Cycles

The purpose of this chapter is to achieve the following aims:

1.To introduce an endogenous labour supply decision into the Ramsey model and to study the effects of fiscal policy, both theoretically and quantitatively;

2.To turn the extended Ramsey model into a prototypical real business cycle (RBC) model by assuming that technology is stochastic;

3.To analyse the theoretical properties of the RBC model by means of its impulseresponse functions;

4.To study the quantitative performance of the RBC model by showing how well it can be made to fit real world data;

5.To briefly discuss some of the extensions that have been proposed in recent years to improve the model's empirical performance.

15.1 Introduction

In this chapter we study two major themes which were pursued by predominantly new classical economists in the wake of the rational expectations revolution of the early 1970s. The first theme studies the effects of fiscal policy in an optimizing equilibrium framework. Pioneering contributions to this branch of the literature were made by Hall (1971, 1980), Barro (1981), and Aschauer (1988), and more recently by Baxter and King (1993).

The second theme concerns the general equilibrium approach to economic fluctuations. Pioneering contributions to this so-called real business cycle (RBC) approach were made by Kydland and Prescott (1982), Long and Plosser (1983), and Prescott (1986).

(15.1) instantaneous utility

The Foundation of Modern Macroeconomics

15.2 Extending the Ramsey Model

In this section we extend the deterministic Ramsey model (see Chapter 14) by endogenizing the labour supply decision of households. In the model a representative household makes optimal decisions regarding present and future consumption, labour supply, and saving. The representative firm hires the factors of production from the household sector and produces output. The government levies taxes and consumes goods. All agents in the economy operate under perfect foresight. The model can be used to study how the economy reacts to shocks in government spending. Throughout the chapter we abstract from population growth.

15.2.1 Households

The representative agent makes a dynamically optimal decision regarding consumption of goods and leisure both for the present and for the indefinite future. The agent has a time endowment of unity which is allocated over labour and leisure. The agent is infinitely lived and lifetime utility in period t, A(t), is given by the discounted integral of present and future utility:

A (t) f (13.(t)eP(t- T) dr,

where p is the pure rate of time preference (p > 0), and CO is

(or felicity) in period t. Following Campbell (1994, p. 482) and King and Rebelo (1999, p. 954) we assume that the felicity function takes the following form:

	(NT) Ec log C + (1 - cc)	L(r-)]1-1/al- - 1	aL > 0,	(15.2)
		1 - 1/aL,

where C(t) is consumption and 1 - L(r) is leisure. The felicity function is convenient to work with because it nests the two most commonly used specifications in the RBC literature as special cases. The first special case assumes that leisure, like consumption, enters felicity in a logarithmic fashion, i.e. a t, = 1 in (15.2). The second special case is based on the assumption that leisure enters felicity linearly, i.e.

Do in (15.2). The main emphasis in this chapter will focus on the logarithmic case. (In section 15.5.4 below we present the linear case.)

The agent's dynamic budget identity is:

A(r) r(r)A(t) + W(t)L(t) T(r) - C(T),

(15.3)

where r(7-) is the real rate of interest, A(r) is real financial assets, W (r) is the real wage rate, L(r) is labour supply, T(r) is real lump-sum taxes, and C(r) is consumption of a

homogeneous good. A dot above a variable designates the derivative with respect to time, i.e. A(r) dA(t)/dt. As it stands, equation (15.3) simply says that the income

from assets and labour is taxes, or saved. Provide problem defined so far is amount (A(/) -x), s _,ve in a state of utmost h i.e. C(r) +00 and L(r ► now to make for intere.- integrating (15.3):

A(t) = f [C(t) -

where R(t, r) ft r(s) ds is motivate why the term in zero. It is not in the age:- the term in square bracke wish to 'die' heavily inL: term cannot be negative vanishes, i.e. the agent - no-Ponzi-game (NPG)

lim A(t)e-R(tir) = 0

When (15.5) is substituted is obtained. It says that period t (left-hand side u. of consumption over aft of (15.4)). Hence, an .41 eventually consume less ti

The household choose , to maximize lifetime uti. NPG condition (15.5), tal Hamiltonian for this pro:

7-1(r) cc log C(T) +

(r)[r(r)A0

Equation (15.4) is derived

p( r) - r(r)A(r)} CR(' '') =

dr [A(r)e-R('') ] =

where we have used the fact tl integrating over the interval [t

478

odd (see Chapter 14) by In the model a representa- t and future consumption, s the factors of production wernment levies taxes and

r perfect foresight. The to shocks in government lu'ition growth.

vision regarding consumpndefinite future. The agent your and leisure. The agent given by the discounted

(15.1)

r) is instantaneous utility

.82) and King and Rebelo

• e following form:

t

> 0, (15.2)

'elicity function is conve- -- 'v used specifications in assumes that leisure, like = 1 in (15.2). The secers felicity linearly, i.e. focus on the logarithmic

(15.3)

ets, W(r) is the real wage r) is consumption of a -ivative with respect to ', Iv says that the income

Chapter 15: Real Business Cycles

from assets and labour is either consumed, paid to the government in the form of taxes, or saved. Provided the agent has free access to the capital market, the choice problem defined so far is not meaningful: the agent can simply borrow an infinite amount (A(r) -> -oo), service the debt with further borrowings (A(r) < 0), and live in a state of utmost bliss (presumably that would mean "all fun and no work", i.e. C(r) -> +oo and L(r) = 0). Obviously, something is missing in the story up to now to make for interesting macroeconomics. The key to the puzzle is obtained by integrating (15.3):

A(t) f [C(r) - W(r)L(r) + T(r)] e-R(t'r) dr + [ lim A(r)e-R(t ') ] ,

(15.4)

where R(t, r) ft r(s) ds is a discounting factor. 1 A heuristic argument can be used to motivate why the term in square brackets on the right-hand side of (15.4) should be zero. It is not in the agent's interest to 'die' with a positive wealth position. Hence, the term in square brackets cannot be positive. Similarly, although the agent may wish to 'die' heavily indebted, the capital market will not allow this. Hence, the term cannot be negative either. The only possibility that remains is that the term vanishes, i.e. the agent remains solvent. This condition is often referred to as the no-Ponzi-game (NPG) condition (Blanchard and Fischer, 1989, p. 49):

lim A(r)e-R(tir) = 0 (NPG condition).	(15.5)
"C"

When (15.5) is substituted in (15.4), the household intertemporal budget constraint is obtained. It says that the value of financial assets that the agent possesses in period t (left-hand side of (15.4)) equals the present discounted value of the excess of consumption over after-tax labour income (first term on the right-hand side of (15.4)). Hence, an agent who has negative assets (i.e. debt) in period t must eventually consume less than his after-tax labour income at some time in the future.

The household chooses paths for consumption, labour supply, and assets in order to maximize lifetime utility (15.1) subject to the budget identity (15.3), and the NPG condition (15.5), taking as given the initial level of assets. The current-value Hamiltonian for this problem is:

7-1(r) cc log C(r) + (1 -	[ [1 - L(t-)] 1-1 /aL - 1
	1 - 1/aL
+ ,a(r)[r(r)A(r) + W(r) - T(r) - C(r) - W(r) [1 - L(r)]1,		(15.6)
1 Equation (15.4) is derived as follows. We multiply (15.3) by e-R(t 'T) and find:
- r(r)A(r)ie-R(t' r) = [W(r)L(r) — T(r) — C(r)]
dr [A(r)e-R( 1 = [W(r)L(r) — T(r) — C(r)] e-R(t ,		(a)

where we have used the fact that dR(t, r)/ dr r(r). By bringing dr to the right-hand side of (a) and integrating over the interval [t, oo) we obtain (15.4).

479

The Foundation of Modern Macroeconomics

where kt(T) is the co-state variable. The interesting first-order necessary conditions are:2

Ec
C(T) = P (t),	(15.7)

1 - EC

[1 — L(t)] liaL au(T)W (T)'

p - r(r).

µ(t)=

By using (15.7) in (15.8)-(15.9) we can eliminate ,u(T) and order conditions as follows:

	C(r)	(1 - cc ) = W(r),
	[1 - L(T)] 1/aL
		Cc

C(r)

C(r) r(r) - p.

(15.8)

(15.9)

WO and write the first-

(15.10)

(15.11)

Equation (15.10) requires the marginal rate of substitution between leisure and consumption to be equated to the wage rate in each period. This is essentially a static decision which is made in each period. According to (15.10) labour supply depends negatively on consumption and positively on the real wage. 3 The dynamic part of the solution is contained in (15.11) which is the consumption Euler equation. If the real interest rate exceeds (falls short of) the pure rate of time preference, the household chooses an upward (downward) sloping consumption profile over time (see Chapter 14 for further details).

15.2.2 Firms

Perfectly competitive firms produce a homogeneous good by using capital and labour. Since there are constant returns to scale to the production factors, there is no need to distinguish individual firms and we can make use of the notion of a "representative firm". In the interest of simplicity, the production function features the following Cobb-Douglas form:

Y(T) = F (K(r), L(r)) -= ZoL(T)EL K(T) i-EL (15.12)

where Zo is an index of general productivity, Y(r) is aggregate output, and K(r) and L(r) are, respectively, the amounts of capital and labour used in production.

2 The first-order conditions are 87-1/ax = 0 for the control variables (x a {C, 1 — L}) and —87-i/ax =

-for the state variable (x = A).

3 By differentiating (15.10) we obtain:

dLdW dC

=0-04, W T ],

where wa (1 — L)/L. Hence, the wage elasticity of labour supply is given by aLwa• For a given (on, this elasticity rises with aL•

The stockmarket value

flows:

I

V(t) = f [Y(t ) - 1

where R(t , t) is the th.), investment:

K(r) = I (r) - SK(r),

where S is the depreciati( (15.13) subject to the mulation of the firm's capital stock at will, i.e. surprising, therefore, t.. to hire is essentially a su and capital hold:

aY (r) = MO, -

aL(T) ON1

In view of the fact that t the production funct: and the stockmarket di stock, i.e. V(t) = K(t).

15.2.3 Equilibrium

Output can be used fe - purposes. Hence, the

Y(r) = C(r) + /(r

Finally, the model is L ply states that public representative househo

G(r) = T(r).

15.3 The Unit-el

In the previous secth librium model of

480

r necessary conditions

(15.7)

(15.8)

(15.9)

rr 1 r ) and write the first-

(15.10)

(15.11)

etween leisure and con- s is essentially a static

.1bour supply depends

'.3 The dynamic part of

,)n Euler equation. If

)f time preference, the -tion profile over time

I by using capital and xluction factors, there use of the notion of a ction function features

(15.12)

e output, and K(r) and i in production.

C, 1 — L}) and —87-i/ax =

by aLwa. For a given 0LL,

Chapter 15: Real Business Cycles

The stockmarket value of the firm is given by the discounted value of its cash flows:

V(t) =f [Y(r) - W(r)L(r) - I (r)] e -R" dr ,

(15.13)

where R(t, r) is the discounting factor (defined below (15.4)) and I (r) is gross investment:

k(r) = I (r) - SK(r),

(15.14)

where S is the depreciation rate of capital. The firm maximizes its stockmarket value (15.13) subject to the capital accumulation constraint (15.14). Implicit in the formulation of the firm's choice set is the notion that the firm can vary its desired capital stock at will, i.e. there are no adjustment costs on investment. It is not very surprising, therefore, that the firm's decision about how much labour and capital to hire is essentially a static one. Hence, the familiar marginal conditions for labour and capital hold:

	a Y (r)	aY (r)	(15.15)
	aL(T)	= W (T), 0K(r) = (T) + S.

In view of the fact that both factors are paid their respective marginal products, and the production function exhibits constant returns to scale, excess profits are zero and the stockmarket value of the firm is equal to the replacement value of its capital stock, i.e. V(t) = K(t).

15.2.3 Equilibrium

Output can be used for private consumption, public consumption, or for investment purposes. Hence, the condition for goods market equilibrium is:

Y(r) = C(r) + I (r) + G(r).

(15.16)

Finally, the model is completed by the government budget restriction which simply states that public consumption is paid for by lump-sum taxes levied on the representative household:

G(r) = T (r).

(15.17)

15.3 The Unit-elastic Model

In the previous section we have constructed a fairly simple dynamic general equilibrium model of the closed economy. In section 15.5 this model will be used to

481

(T1.4)-(T1.5)

The Foundation of Modern Macroeconomics

Table 15.1. The unit-elastic model
K(t)=1(t)-8K(t)		(T1.1)
E(t)
C(t) = r(t) - p		(T1.2,
G(t) = T(t)		(T1.3)
W(t) = EL (-YT		(T1.4,
L(t)
r(t) 8 = (1 - EL ) (	Y(t)\	(T1.5)
	K(t))
Y(t) = C(t) 1(t) G(t)		(T1.6)
W(t) [1 - L(t)] = ( 1	EC) c(t)	(T1.7)
\ EC
Y(t) = ZoL(t)"K(t)l-E1-		(T1.8)

Definitions: Y real national income, C private consumption, L employment, K capital stock, I gross investment, G public consumption, W real wage rate, r real interest rate, EC taste parameter for consumption, T lump- sum taxes, p pure rate of time preference, B depreciation rate of capital, EL efficiency parameter of labour.

simulate the effects of stochastic productivity shocks. In this section, however, we demonstrate some of the theoretical properties of a special case of the model in which the substitution elasticity for labour supply equals unity, i.e. cr.', = 1. This unit-elastic version of the model is often used in the RBC literature (see e.g. Baxter and King, 1993 and King and Rebelo, 1999) because it is relatively easy to analyse and yet does quite a decent job when confronted with the data (see below). For convenience, the complete unit-elastic model has been summarized in Table 15.1.

Equations (T1.1), (T1.2), (T1.3), (T1.6), and (T1.8) restate, respectively, equations (15.14), (15.11), ( 15.17) (15.16), and (15.12). Equation (T1.7) is obtained by setting oL = 1 in (15.10). Finally, are obtained by using (15.12) in (15.15).

In the appendix to this chapter we present the full derivation of the phase diagram for the unit-elastic model. The derivation proceeds under the assumption that the output share of government consumption is held constant. The phase diagram is presented graphically in Figure 15.1. The K = 0 line represents combinations in (C, K) space for which net investment is zero. For each (C, K) combination there exists a unique equilibrium employment level. Indeed, for points near the origin employment is low whilst for points near the horizontal intercept (KK ) employment is close to its upper limit of unity. The golden-rule capital stock is KGR and the associated consumption level is CGR (see point A). For points above (below) the K = 0 line, consumption is too high (low) and net investment is negative (positive). These dynamic effects have been illustrated with horizontal arrows in Figure 15.1.

C

Cc

CGR

0

Figure 151

The C = 0 line rep. over time, i.e. for whiLi

interest rate depends constant returns to sL

labour ratio and thus (from (T1.7)) that the so. The C = 0 line stock, an increase (d, and equilibrium empl

and the rate of intere! the C = 0 line. This h, It follows from Figu point E0 . The arrow c , associated with

sloping.

15.4 Fiscal Pori(

In this section we dem elastic model develop for the numerical

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