Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

E [0,

returns to scale (in these v is also perfectly compet- --1 ' leprints of new varieties ds sector is populated by a lriety of the differentiated petition (see Chapter 13 for

is given by the following

► 1, (14.187)

:ist at time t, Xi(t) is vari- - constant the number of Lout in (14.187), i.e. con- 87) implies that, provided hasized by Ethier (1982). iame amount is used of all

ibrium discussed below), e final goods sector will be

esents the total amount of low). Ceteris paribus Lx (t),

4 intermediate inputs pro-

. -).rieties, producers in the od of production and thus

I nimizes its costs and sets ,rage) cost of production:

p	(14.188)

zing demand for input j

(14.189)

'icily of the demand for

iummation sign replaced by an nted before time t. Following ee Romer (1987) and Grossman

Chapter 14: Theories of Economic Growth

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, variety X1(t). The operating profit of firm j is defined as follows:

Ill (t)-= Pi (t)Xj (t) — W(t)Li(t), (14.190)

where W(t) is the wage rate (common to all firms in the economy as labour is perfectly mobile) and Li (t) is the amount of labour used by firm j. Firm chooses its output level, Xi (t), given the demand for its output (14.189), the production function Xj(t) = (1/kx )Lj (t), and taking the actions of all other producers in the intermediate goods sector as given. As is familiar from the detailed discussion in Chapter 13, the optimal choice of the firm is to set price according to a fixed markup over marginal production cost:

Pi (t) = ,uW(t)kx,

(14.191)

where It thus represents the constant markup. Since all active firms in the intermediate sector possess the same technology and face the same input price and markup, they all choose the same amount of output and charge the same price. Hence, from here on we can suppress the firm subscript, as X i(t) = j((t), Pi (t) = P(t), and Ili(t) = WO for j N(t)], and let the barred variables denote the choices of the representative firm in the intermediate sector. By substituting (14.191) into (14.190) and invoking the symmetry results we obtain the following expression for the profit of a representative firm in the intermediate goods sector:

(t) = [P(t) — W(t)kx] X(t) = — 1 P(t)X(t). (14.192)

In the R&D sector competitive firms use labour (researchers) to produce new blueprints. Since N(t) is the stock of existing blueprints, its time rate of change, N(t), represents the new blueprints. It is assumed, following Benassy (1998) that the production function for new blueprints is given by:

N(t) = (1/kR)N(t)LR (t), (14.193)

where LR(t) is the amount of labour employed in the R&D sector and 1/kR is a productivity parameter. By employing more labour in the R&D sector, more new blueprints are produced per unit of time. Furthermore, equation (14.193) incorporates the assumption, due to Romer (1990), that the stock of existing blueprints positively affects the productivity of researchers. As Romer puts it, "Mlle engineer working today is more productive because he or she can take advantage of all the additional knowledge accumulated as design problems were solved during the last 100 years" (1990, pp. S83-84). Since the R&D sector is competitive the price of a

463

464

(x E {K, H}).

Nx — pt-„

The Foundation of Modern Macroeconomics

new blueprint, PN (t), is equal to the marginal cost of producing it:

k RW	(14.194)
PN(t) = N(t) •

It remains to describe the optimal behaviour of the representative, infinitely lived, household. This household has a utility function as in (14.168) and faces the following budget identity:

PY(t)CM PN(t)N(t) = W(t)L N(t)II(t),

(14.195)

where L is the exogenous supply of labour of the household. Total spending on consumption goods plus investment in new blueprints (left-hand side) equals total labour income plus the total profits the household receives from firms in the differentiated sector (right-hand side). By using the price of final output as the numeraire (Py(t) = 1) we obtain the household budget identity in real terms,

C(t) + PN(t)N(t) = W(t)L N(t)f1(t).

The current-value Hamiltonian associated with the representative household's decision problem is given by:

7-1(t) =	— 1	+ AN(0	[W(t)L N(t)t1(t) — CM] ,	(14.196)
	1 — 1/a		PN ( t)

where ,uN(t) is the co-state variable for N(t). The first-order necessary conditions are. 40

	C(t) —l ia = IJN(t)		(14.197)
		PN(t)'
	µN(t)	fl(t)	(14.198)
	µN (t)	P PN(t)•

By combining these two expressions we obtain the conventional consumption Euler equation:

C(t)
C(t) = [r(t) — p] ,	(14.199)
where r(t) is the rate of return on blueprints:
r(t) = n(o+N(t)	(14.200)
PN

The return on blueprints is the dividend plus the capital gain expressed in terms of the purchase price of the blueprint.

40 The first-order conditions are arc/ax = 0 for the control variables (x E {C,LE}) and —87-(/ax = for the state variables

The model is clog clears provided oL.

Y(t) = C(t).

The labour market t the sum of labour do Since Lx(t) = kxN11 equilibrium cone

N(t) L —

N(t)

where we assume ii does not absorb all strictly positive).

Growth

We are now in a pc the solution appro intermediate resu..

II(t) = (p — 1,

PN

PN(t)

N (t) = ( 77 — 21

I

C(t) = N(t)

Equation (14.203) monopoly markt., sector. It is obtaine symmetry results. is proportional to obtained by using symmetry. This respect to time to condition in the •

In the second si,

Yc(t) = a --i

Yc(t) = (1 —

L — L

YIN; (t) =	kR

troducing it:

(14.194)

le representative, infinitely on as in (14.168) and faces

(14.195)

usehold. Total spending on left-hand side) equals total ceives from firms in the nice of final output as the ' - .2t identity in real terms,

representative household's

(14.196)

order necessary conditions

(14.197)

(14.198)

ntional consumption Euler

(14.200)

gain expressed in terms of

E LED and —87-t/8x =

Chapter 14: Theories of Economic Growth

The model is closed by two market clearing conditions. The final goods market c' jars provided output equals consumption:

Y(t) = C(t).

(14.201)

The labour market equilibrium condition requires the total supply of labour to equal the sum of labour demand in the intermediate and R&D sectors, i.e. Lx (t)+L R = L. Since Lx (t) = k x N (t)X (t) and LR (t) = kRN(t)/N(t) we can rewrite this labour market equilibrium condition as:

	(t) L — kx N (t)X (t)		(14.202)
	N (t)	kR

where we assume implicitly that the differentiated sector is not too large and thus does not absorb all available labour (i.e. the numerator on the right-hand side is strictly positive).

Growth

We are now in a position to determine the growth rate in the economy. We follow the solution approach of Benassy (1998). In the first step, we note a number of intermediate results:

Mt)	= (it 1) kxN(t)X(t)		(14.203)
PN (t)
PN(t)	(10)		(14.204)
T, (0	=( ri 2)	N(t)
i'N (t)
C (t) = N (t) N (t)X (t)			(14.205)

Equation (14.203) expressed the real dividend rate on blueprints in terms of the monopoly markup (A) and the total amount of labour absorbed by the final goods sector. It is obtained by using (14.194) and (14.191) in (14.192) and imposing the symmetry results. Equation (14.204) shows that the capital gains rate on blueprints is proportional to the growth rate of varieties, i.e. the rate of innovation. It is obtained by using (14.191) and (14.188) in (14.194), setting Py (t) = 1 and imposing symmetry. This yields PN(t) = [kR/(ukx)1N(t) 17-2 which can be differentiated with respect to time to obtain (14.204). Finally, (14.205) is the goods market clearing condition in the symmetric equilibrium.

In the second step we write the dynamics of the model as follows:
Yc =a[( 11-1- )Lx	7	(14.206)
	+ ( 1 — 2)YN (t) — 19]
kR
Yc(t) = (ri — 1)YN(t) + Lx (t),		(14.207)
L — Lx (t)		(14.208)
YN(t) =
kR
		465

a < 1,

The Foundation of Modern Macroeconomics

where we use the conventional notation for growth rates, i.e. y, ic(t)/x(t). Equation (14.206) is the consumption Euler equation. It is obtained by combining (14.199)-(14.200) and (14.203)-(14.204) and noting that Lx(t) = kxN(t)X(t). Equation (14.207) is the time derivative of (14.205) and (14.208) is a rewritten version of (14.202).

In the third step we eliminate yN(t) and yc(t) from (14.206)-(14.208) to derive a single differential equation for Lx(t):

Lx(t) = [ala + (1 — )(ri -1)]Lx(t)	[ q 1 + a(2 -	- ap. (14.209)
kRa	kR

The crucial thing to note about this expression is that the coefficient for Lx(t) on the right-hand side is positive, i.e. (14.209) is an unstable differential equation.41 This, of course, means that the only economically sensible solution is such that Lx(t) jumps immediately to its steady-state value:

	[77 - 1 + a(2- 77)]L apkR		(14.210)
	a P., + (1 — a)(ri•	- 1)

Since there is no transitional dynamics in Lx(t) (and thus Lx (t) = 0 for all t) the same hold for the growth rates of the number of varieties and consumption. Indeed, by using (14.210) in (14.206) and (14.208) we obtain:

YN =	au( 1 (1,/kR(77)	yc = yy = i)yN,
	+( 1— a )	(14.211)

where the sign follows from our assumption made in the text below equation (14.202). This expression generalizes the results of Grossman and Helpman (1991, p. 59), Benassy (1998, p. 66), and de Groot and Nahuis (1998, p. 293) to the case of a non-unitary elasticity of intertemporal substitution. Like these authors, we find that the rate of innovation increases with the monopoly markup (ayN /ap, > 0) and the size of the labour force (ayN > 0) and decreases with the rate of time preference (ayN /ap < 0). The partial equilibrium effect for the intertemporal substitution elasticity is:

	aYN _ —1))/N	(14.212)
	aa — a [a + (1 - a)(r) -1)J
		•

Provided the returns to specialization are operative (so that ri > 1), an increase in the willingness of the representative household to substitute consumption across time raises the rate of innovation (ayN /aa > 0). As is evident from (14.211), the growth rate in consumption and output also depends critically on whether or not the technology in the final goods sector is characterized by returns from specialization.

41 Recall that p, > 1 and > 1. For low values of the intertemporal substitution elasticity, 0 <

it follows immediately that the coefficient is positive. If a > 1 we write the numerator of the coefficient as a (,u —1) + -1+ a (2 — ti). This expression is positive provided a mild sufficient condition on holds, i.e. 77 < 2.

Efficiency

One of the classic decentralized marl whether the markt we follow the usua to the decentralize

As is pointed ( quite a lot easier th up front and work and labour used in mize lifetime util it (14.202) and (14.2 find that the Hami

[N(t)r -

7-i(t) =

where AN(t) is the (

(teN(t)) = — kR kx

I

itN(t) =

By combining the straightforward s .

Lx(t) = 1-

Lx(t) kR

Provided there are right-hand side of and the socially u, 0):

LX = - 01.

where the superset itly that LX is fea

466

I

i.e. y, x(t)/x(t). Equais obtained by combining

L x (t) = kxN(t)X(t). Equa-

,.208) is a rewritten version

4.206)-(14.208) to derive a

- ►i) L - a p.

(14.209)

the coefficient for Lx(t) on ble differential equation.' hie solution is such that

(14.210)

bus Lx(t) = 0 for all t) the and consumption. Indeed,

(14.211)

Chapter 14: Theories of Economic Growth

Efficiency

One of the classic questions in economics concerns the welfare properties of the decentralized market equilibrium. In the context of the model we wish to know whether the market rate of innovation is too high or too low. To study this problem we follow the usual procedure by computing the social optimum and comparing it to the decentralized market equilibrium.

As is pointed out by Benassy (1998, p. 66), computation of the social optimum is quite a lot easier than that of the market solution because we can impose symmetry up front and work in terms of aggregates like consumption, the number of firms, and labour used in the intermediate sector. The social planner is assumed to maximize lifetime utility of the representative agent (14.168), subject to the constraints (14.202) and (14.205). By using Lx(t) = kxN(t)X(t) in the various expressions we find that the Hamiltonian for the social welfare programme is given by:

l1(t)	=	[N(tr i ki-c	1 Lx(01 11a —	Lx(t)		(14.213)
					,
		1	— 1/0-	,LN(t)N(t) L - kR

where 1,1 N (0 is the co-state variable for N(t). The first-order necessary conditions are:

AN(0)	N(t)n-2		(14.214)
	1	/ a '

( kR kx[N(t)71-11q,1Lx(td

the text below equation		(11	- Mx(t)N(ty	-2	[L
					AN(t)	- Lx(t) 1	(14.215)
- -ian and Helpman (1991,	µN(t) = PAN(t)					R
kx[N(t)n-lkjilLx(t)]1'
1998, p. 293) to the case of
ke these authors, we find	By combining these two expressions we obtain (after a number of tedious but
markup (ayN/ap, > 0) and
pith the rate of time prefer-	straightforward steps) a differential equation in Lx(t):
-- tertemporal substitution	Lx(t) = rl 11 Lx(t) [(77 - 1)(1 -				a p.
							(14.216)
	Lx(t)	kR	kR

(14.212)

it n > 1), an increase in the consumption across time ‘-,-)m (14.211), the growth v on whether or not the !turns from specialization.

bbstitution elasticity, 0 < a < 1, numerator of the coefficient cient condition on holds,

Provided there are returns to specialization (7/ > 1), the coefficient for Lx(t) on the right-hand side of (14.216) is positive so that the differential equation is unstable and the socially optimal solution is to jump immediately to the steady state (Lx (t) = 0):

LX = (1 - 01, ±	PkR	(14.217)
	-1

where the superscript "SO" denotes the socially optimal value and we assume implicitly that Lv is feasible (positive). The socially optimal rate of innovation associated

467

N(t),-1 (Lx(t)/kx)

The Foundation of Modern Macroeconomics with (14.217) is:

SO	> 0	Yc° =	=(q - 1) YN	(14.218)
yN = kR) a
q 1

The striking conclusion that can be drawn from (14.218) is that the socially optimal rate of innovation does not depend on the markup (A) at all but rather on the parameter regulating the returns to specialization (q). This result is obvious when you think of it—in the symmetric equilibrium (14.187) collapses to Y(t) =

from which we see that the social return to research depends critically on q — 1 (136nassy, 1998, p. 67).

We can now compare the socially optimal and market rate of innovation (given, respectively, in (14.211) and (14.218)) and answer our question regarding the welfare properties of the decentralized market equilibrium. To keep things simple we set a = 1 (logarithmic felicity) for which case yN and )4° are:

(bt — 1)(L/kR) p	so _ (71 – 1) (L IkR) – p		•	(14.219)
YN	Y N	q — 1
tt

These expressions can be used to derive the following result:

[ Aso _ yN. =	L p [it – (9 — 1)	(14.220)

No general conclusion can be drawn from (14.220) and both yN > yN (underinvestment in R&D) and nS°i < yN (overinvestment in R&D) are distinct possibilities as is the knife-edge case for which the parameters are such that the market yields the correct amount of investment in R&D (yV = yN ).42 The literature tends to stress the underinvestment case but that result is not robust as it is based on the implicit assumption that the markup equals the returns to specialization parameter. Indeed, for that special case, ri r, and (14.220) reduces to:

[ yAr	= L p 1	,	(14.221)
		rN •

Hence, if q = and yN > 0 the "traditional" result obtains and the market yields too little R&D and the innovation growth rate is too low (Benassy, 1998, p. 68; de Groot and Nahuis, 1998, p. 294). 43

42 Recall that kt> 1 and 1 < q< 2 so that the term in square brackets on the right-hand side of (14.220) is positive.

43 The example in this paragraph serves to demonstrate that, even though the standard Dixit-Stiglitz preferences (for which = r) are convenient to work with, they are restrictive and may impose too much structure. Ethier (1982) stresses the need to distinguish 77 and p,. Weitzman (1994) provides some micro-foundations for assuming and it to be different. Broer and Heijdra (2001) study diversity and markup effects in a traditional growth model with capital accumulation.

Human and ph)

Romer (1990) ex and by assum I, .4 sector and in the (14.187) is m° L..

Y(t) L,

where L and

in the final good stock is constant.

both in the final

H = Hy (t) + 1

I

A further notable mulation the tech parameterizes the

efficiency parame

Any output of depreciating) "gel

capital is:

K(t) = Y(t)-

where C(t) is aggi lifetime utility fu: assets to smooth c of return, r(t), the

General capital (from the represt ers in order to

in the differential assumes that the ( cialized capital go can be converted defines the follo,

K(t) (1 / kA)

where the left-h,: the right-hand sector.

468

K(t).

1 — aH — aL).

(14.218)

t the socially optimal at rather on the parambvious when you think

= N(t)1-i (Lx(t)/kx)

'Ids critically on 77 — 1

of innovation (given, Hn regarding the welzep things simple we

(14.219)

(14.220)

:h > yN (underin-

.e distinct possibilities that the market yields

• nature tends to stress based on the implicit parameter. Indeed,

(14.221)

Ind the market yields nassy, 1998, p. 68; de

the right-hand side of

the standard Dixit-Stiglitz live and may impose too in (1994) provides some 01) study diversity and

Chapter 14: Theories of Economic Growth

Human and physical capital

Romer (1990) extends the R&D model by recognizing physical capital accumulation and by assuming that a given stock of human capital is used in both the final goods sector and in the R&D sector. In his model the production function for final output (14.187) is modified to:

[N(t)

Y(t) F_- - Hy(t)".1_,(4 f Xj(t) 1-"H' di , 0 < al', aL, al/ + al, < 1, (14.222)

0

where L and Hy (t) are, respectively, the amounts of labour and human capital used in the final goods sector. Labour is only used in the final goods sector and its total stock is constant. The total stock of human capital, H, is also constant, but it is used both in the final goods sector (Hy (t)) and in the R&D sector (HR (t)):

H = Hy (t) HR(t). (14.223)

A further notable difference between (14.187) and (14.222) is that in the latter formulation the technology coefficient, 1 —aH—aL, performs no less than three roles: it parameterizes the returns to specialization, the monopoly markup, and the capital efficiency parameter (i.e. 1/77 = 1/it =

Any output of final goods which is not consumed is added to the stock of (nondepreciating) "general" capital, Hence, the accumulation identity for general capital is:

K(t) = Y(t) c(t), (14.224)

where C(t) is aggregate consumption. The representative household maximizes its lifetime utility function (14.168) using both general capital and new technology as assets to smooth consumption over time. Since these assets attract the same net rate of return, r(t), the consumption Euler equation is still as given by (14.199).

General capital (or "cumulative foregone output" as Romer calls it) is rented (from the representative household) by the monopolistically competitive producers in order to produce units of the differentiated input. The production function in the differentiated sector is KA° = (1 Ikx)X j (t). Romer (1990, p. S80) furthermore assumes that the differentiated inputs, Xj (t), are durable and non-depreciating "specialized capital goods". Capital is "putty-putty" in the sense that specialized capital can be converted back into general capital if the need arises (1990, p. S86). Romer defines the following accounting measure for general capital:

N(t)	(14.225)
K(t) (1 / kx)f X j(t) dj,
13

where the left-hand side of (14.225) denotes the total stock of general capital and the right-hand side is the total amount of general capital used in the differentiated sector.

469

YK = YN ,

LR (t)

The Foundation of Modern Macroeconomics

The R&D sector uses human capital, HR (t), in order to produce new designs. Hence, equation (14.193) is replaced by:

N(t) = (1/kR)N (t)HR(t).

(14.226)

Apart from the fact that raw labour (LR (t)) appears in (14.193) and human capital appears in (14.226), the two expressions for the R&D technology are essentially the same. The engine of growth in both cases is furnished by the fact that the number of designs, N(t), appears linearly in both expressions (see also Romer, 1990, p. S84).

On the balanced growth path consumption, final output, general capital, and the number of designs all grow at the same exponential rate, i.e. yc = yy =

where yN is given by (Romer, 1990, p. S92):

H/kR — pA	A =	aH	(14.227)
YN = 1 + A/a		(1 — aH — aL)(aH + ceL)'

where it is assumed that this growth rate is positive (HR (t) > 0 in the balanced growth path). Comparing (14.211) and (14.227) reveals a number of differences and similarities. Both models give rise to similar effects on the rate of innovation of preference and technology parameters like s, p, and kR . An important difference between the two models concerns the scale factor. In (14.211) the stock of raw labour (L) determines the growth rate whereas in (14.227) it is the stock of human capital (H) which determines the rate of innovation. This key difference is, of course, directly attributable to the different specifications of the R&D sector (namely (14.193) and (14.226)).

Scale effect

In the previous subsections we have developed two R&D-type growth models which have in common the prediction that the scale of an economy is an important determinant of that economy's balanced growth rate in the economy. This so-called scale effect is in fact a common feature of many important R&D growth models such as Grossman and Helpman (1991) and Aghion and Howitt (1992). In a recent paper, Jones (1995) has argued that the prediction of scale effects is easily falsified empirically. In the US, for example, the amount of labour employed in R&D activities grew from 160,000 in 1950 to about 1,000,000 in 1988 whereas total factor productivity growth stayed the same (or even declined somewhat) during that period (Jones, 1995, p. 762). Similar data can be quoted for other industrialized countries such as France, West Germany, and Japan. On the basis of the empirical evidence, Jones concludes that "the assumption embedded in the R&D equation that the growth rate of the economy is proportional to the level of resources devoted to R&D is obviously false" (1995, p. 762).

Jones suggests 1 refutation, it ship(

N(t) = (1/k,

where cap R&D sector. In eqi diminishing re t take LR (t) as give function is linear a more general ), via the stock of it the standard Ilt\ with an exponent generalized R&D (14.193) as a spe, and the R&D equi We now dem( general specifica t4 section, the act the determinatiL„ R&D model (with (14.228) is used L. by assuming nonThe key ingrec output is as givt: the specialization

N,;,

Y(t) [fo

The simplificatit • tions (14.188)—( 1- replaced by (14.2"

PN(t) = kRi%A;(04

The price of a rh. in our first R&D r in the R&D sect labour is now pose The representati• .

470

0

D produce new designs.

(14.226)

193) and human capital iology are essentially the e fact that the number lso Romer, 1990, p. S84). , general capital, and the i.e. yc = YY = YK = YN,

(14.227)

'41 > 0 in the balanced a number of differences n the rate of innovation An important differIn (14.211) the stock of 4.227) it is the stock of ion. This key difference ons of the R&D sector

pe growth models which ny is an important determy. This so-called scale ) growth models such as 1992). In a recent paper, easily falsified empiri- ed R&D activities grew total factor productivity [ring that period (Jones, llized countries such as rnpirical evidence, Jones quation that the growth -- ces devoted to R&D is

Chapter 14: Theories of Economic Growth

Jones suggests that, since the R&D equation is clearly the cause of the empirical refutation, it should be replaced by the following specification:

N(t) = (1/kAR(ON(t)01 [LR(t)] h 1 , 0 <

02 1,

(14.228)

where LR(t) captures an external effect due to unintended duplication of work in the R&D sector. In equilibrium LR(t) = LR(t), and the production of new designs features diminishing returns to labour provided 0 2 < 1. Individual R&D firms, however, take LR (t) as given and operate under the assumption that the R&D production function is linear in LR(t). Apart from a duplication externality, (14.228) also features a more general specification of the knowledge externality which operates across time via the stock of invented product varieties. Indeed, whereas N(t) enters linearly in the standard R&D equation (14.193), it features in the augmented R&D equation with an exponent 0 1 which may or may not equal unity. An attractive feature of the generalized R&D equation (14.228) is that it contains the standard R&D equation (14.193) as a special case. Indeed, if 0 1 = 02 = 1 the duplication externality is absent and the R&D equation is linear in N(t) and the two expressions coincide.

We now demonstrate the implications for economic growth of adopting the more general specification of the R&D function. As we have seen in the previous subsection, the accumulation of physical capital does not play an essential role in the determination of the steady-state growth rate. For that reason we use our first R&D model (without physical capital) and derive the steady-state growth rate when (14.228) is used as the R&D function. We augment our simple R&D model, however, by assuming non-zero population growth.

The key ingredients of the model are as follows. The production function for final output is as given in (14.187) except that we follow convention by assuming that the specialization parameter equals the markup (1) = ,u):

N(t)	(14.229)
Y(t) -=[f Xi (t) 11A dji , ,u >

The simplifications that result from assuming ri = are easily incorporated in equations (14.188)—(14.189). Equations (14.190)—(14.192) are unchanged, (14.193) is replaced by (14.228), and (14.194) is replaced by:

PN(t) — kR W(t)	[I,R(t)]1-h	(14.230)
N(04, 1

The price of a new design is equal to the private marginal cost of producing. As in our first R&D model, labour is used in both the intermediate goods sector and in the R&D sector. In contrast to what was assumed in that model, the stock of labour is now postulated to grow at a constant exponential rate, i.e. L(t)/L(t) = nL. The representative household is assumed to care about its per capita consumption,

471

LR = LR:

(14.234)

LR(t) = LR(t)

The Foundation of Modern Macroeconomics

c(t) C(t)/L(t), and has the following lifetime utility function:

	A(0) =	fa°	c(t)	1-1/a 1 ]	edt.	(14.231)
		0	L 1	- 1/a

Finally, since the number of family members of the household grows, the budget identity for the household is changed from (14.195) to:

L(t)c(t) + PN(t)N(t) = W(t)L(t) + N(t)I1(t),

(14.232)

where we once again assume that final output is the numeraire commodity (so that Py(t) = 1). The representative household chooses the optimal per capita consumption path in order to maximize (14.231) subject to (14.232) (plus an NPG- no-Ponzi-game—condition). The consumption Euler equation that results from this choice problem is given by:

c(t) = a [r(t) -	(p + nL)]	(14.233)
c(t)

where the rate of interest (r(t), representing the yield on blueprints) is given by (14.200). The remaining equations of the model are the final goods market clearing condition (14.201) and the labour market condition:

1/02

L(t) = Lx(t) LR(t) = Lx(t) -F[kRN(01-01 (Mt)]N(t)

where the second equality uses (14.228) and incorporates the fact that in equilibrium.

Although we could, in principle, retrace the steps leading from the simplified model to the expression in (14.211), we skip the details of dynamic adjustment here and simply compute the steady-state growth rates implied by the augmented model. We are looking for a balanced growth path in which (a) the proportions of labour going into the intermediate and R&D sectors (Lx/L and LR/L) are both constant, and (b) the proportional rates of growth in N(t), c(t), and y(t) Y(t)/L(t) are all constant. The steady-state innovation rate is easily found by rewriting (14.228) in steady-state format and substituting

[YN =1 N	= (1/kR)N	4'1-1 42 .	(14.235)
—

The left-hand side of (14.235) is constant (as yN is constant). By differentiating the left-hand side of (14.235) with respect to time and noting that LR /LR = nL we obtain 0 = (01 - 1)YN + 0211L which can be solved for yN:

02111,	(14.236)
YN = 1 - •

By using the steady-state version of (14.229) (and imposing symmetry) we find Y = MLA which can be rewritten as Y/L = (1/kx)NA -1 Lx /L (where Lx = kxNX).

From this

YY = YY

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472