Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

k(t)

•

r and

(14.151)

first task at hand is Is under the technol-

I nresentative (perfectly ns in order to maximize function (14.151) and e manipulation we find 7 , nal product of capital

subsidy.

(14.152)

ed. As in section 5.1, it ev:

(14.153)

(14.154)

ty (see (14.65)), tc is a -nment (or tax if it is

Chapter 14: Theories of Economic Growth

negative), A(t) represents financial assets, and r(t) is the rate of interest. Using the analytical methods discussed in section 5.1, the representative household's Euler equation can be derived:

Since the model deals with a closed economy and there is no government debt, the only financial asset which can be accumulated consists of company shares, i.e. A(t) = K(t). The key equations of the basic AK growth model have been summarized in Table 14.5.

Equations (T5.1) and (T5.3) have been explained above, and (T5.2) is the capital accumulation identity (14.4) combined with the output constraint (14.3) and the production function (14.151). It is now straightforward to demonstrate the existence of perpetual "endogenous" growth in the model. We focus attention on the case for which both the consumption tax and the investment subsidy are (expected by agents to be) constant over time, i.e. tc (t) = = 0. 32 In that case the interest rate is constant and the growth rate of consumption is fully determined by (T5.1). At the same time, similar arguments to those explained in section 5.1 can be used to show that the propensity to consume out of total wealth is constant also, i.e. it is optimal for the representative household to maintain a constant ratio between consumption and the capital stock. 33 But if C(t)/K(t) is constant, so is I(t)/K(t). Hence, the common growth rate for output, consumption, the capital stock, and investment is given by:

The striking conclusion is that the growth rate of the economy can be permanently affected by the investment subsidy, a result which is impossible in the traditional growth model discussed above. Intuitively, a higher investment subsidy leads to a

Table 14.5. The basic AK growth model

Notes: C(t) is consumption, K(t) is the capital stock, r(t) is the interest rate, tc (t) is the consumption tax, silt) is an investment subsidy, p is the pure rate of time preference, and S is the depreciation rate of capital.

32 The key point to note in Table 14.5 is that the level of the consumption tax does not influence the growth rate as this tax does not distort the intertemporal consumption decision.

33 Since there is no labour in the model, human wealth is zero and the capital stock equals total wealth.

453

The Foundation of Modern Macroeconomics

higher interest rate, a steeper intertemporal consumption profile, and thus a higher rate of capital accumulation in the economy. Furthermore, taste parameters also exert a permanent effect on the growth rate of the economy. Hence, an economy populated by patient households (a low p) or households with a high willingness to substitute consumption intertemporally (a high a), tends to have a high rate of economic growth.

The level of the different variables can be determined by using the initial con-

A number of further properties of the basic AK model must be pointed out. First, the model contains no transitional dynamics. The initial levels of the different macroeconomic variables are tied down by the initial capital stock (see (14.151) and (14.157)), and the rate of growth is constant and the same for all these variables. This result does not hold if the consumption tax or the investment subsidy are timevarying, since in that case the real interest rate will vary over time and the agents will react to this. Second, the equilibrium in the basic AK model is Pareto-efficient in that the market outcome and the central planning solution coincide. Intuitively this result holds because there is no source of market failure in the model (Barro and Sala-i-Martin, 1995, p. 144).

In a recent paper, Barro (1990) has proposed a model in which productive government spending has an effect on the economic growth rate. The production function (14.151) is replaced by:

where G(t) is the flow of public spending. The idea is that productive public spending affects all producers equally, these services are provided free of charge, and there is no congestion effect. Note that (14.158) reinstates diminishing returns to private capital, K(t), because a is less than unity. If somehow the government succeeds in maintaining a constant ratio between its productive spending and the private capital stock, however, the model ends up looking very much like the basic AK model and thus will display endogenous growth.

The government is assumed to finance its spending by means of a tax on output:

where ty is the output tax. The representative firm takes the level of G(t) as given and maximizes the present value of after-tax cash flows:

subject to the capit (14.158), an initial After some manipuiu;

marginal product of c

4

r(t) 3 = (1 — t '.

a

Since nothing is char is still of the form g. computes the margin;

BY (t) = aA ( 6

K(t)

By using (T5.1) in con ernment maintains a rate implied by (this I

'(t) i7 ( Y * = C(t) = Y(;,

=Q [a(1 — t

The striking conclu of diminishing return productive spending. returns to private ca p mulation. It is able to tax rates because the stock.

By using the prod t. (14.159) we can exp., and the output tax r (14.163), we obtain -

y* = a [a (1 — ty

which has been plot:. tax on the rate of ecc we know that a h growth. Second, a 1-1, capital which is good save more. For low

the growth rate increi

h profile, and thus a higher nore, taste parameters also ^ r)my. Hence, an economy :s with a high willingness ends to have a high rate of

d by using the initial con- r..h 'K(0) = y* in (T5.2) and

(14.157)

must be pointed out. First, al levels of the different tal stock (see (14.151) and ame for all these variables. ivestment subsidy are time- F over time and the agents K model is Pareto-efficient lution coincide. Intuitively :lure in the model (Barro

which productive govern- e. The production function

(14.158)

it productive public spend- ed free of charge, and there tinishing returns to private to government succeeds in nding and the private cap- - h like the basic AK model

I

means of a tax on output:

(14.159)

s the level of G(t) as given

(14.160)

Chapter 14: Theories of Economic Growth

subject to the capital accumulation identity (14.4), the production function (14.158), an initial condition on the capital stock, and a transversality condition. After some manipulation we find that the rental rate on capital equals the after-tax marginal product of capital:

Since nothing is changed on the household side of the model, the Euler equation is still of the form given in (T5.1) (with ic (t) = 0 imposed). The representative firm computes the marginal product of capital for a given level of public spending (G(t)):

By using (T5.1) in combination with (14.161)-(14.162) and assuming that the government maintains a constant ty and G(t)/K(t)-ratio, we find the common growth rate implied by (this version of) the Barro model:

C(t) Y(t) 1(t)

Y = = = - --

C(t) Y (t) K(t) I (t)

The striking conclusion is that endogenous growth emerges despite the existence of diminishing returns to private capital! Intuitively, by ever increasing its level of productive spending, the government manages to negate the effect of diminishing returns to private capital that would otherwise result from continuing capital accumulation. It is able to do so without ever-increasing (and thus ultimately infeasible) tax rates because the tax base (gross output) grows at the same rate as the capital stock.

By using the production function (14.158) and the government budget constraint (14.159) we can express the G(t)/K(t)-ratio in terms of the productivity parameter and the output tax rate, i.e. G(t)/K(t) = (tyA) l ia . By substituting this result into (14.163), we obtain an expression linking the rate of growth to the tax rate:

which has been plotted in Figure 14.15. There are two offsetting effects of the output tax on the rate of economic growth. First, via the government budget constraint we know that a higher G(t)/Y(t)-ratio requires a higher tax rate which is bad for growth. Second, a higher G(t)/ Y(t)-ratio also raises the marginal product of private capital which is good for growth as it raises the interest rate and makes households save more. For low initial tax rates, the second effect dominates the first effect and the growth rate increases if the output tax is raised, and vice versa for high tax rates.

455

The Foundation of Modern Macroeconomics

Y*

7MAX

Figure 14.15. Productive government spending and growth

The growth-maximizing tax rate (and share of productive government spending) is obtained by maximizing y* with respect to ty . After some manipulation we obtain:

The interpretation of this result is as follows. The social cost of a unit of government spending is unity and the social benefit is aY(t)/aG(t) = (1—a)Y(t)/G(t) = (1—a)/ty . By equating marginal costs and benefits we obtain the expression in (14.165) (see Barro and Sala-i-Martin, 1995, p. 155).

14.6.2 Human capital formation

In a path-breaking early contribution to the literature, Uzawa (1965) argued that (labour-augmenting) technological progress should not be seen as some kind of "manna from heaven" but instead should be regarded as the outcome of the intentional actions by economic agents employing scarce resources in order to advance the state of technological knowledge. Uzawa (1965) formalized his notions by assuming that all technological knowledge is embodied in labour, i.e. in terms of the aggregate production function (14.11) he sets = 1 for all t and proposes a theory which endogenizes A L (t) (and thus nA in (14.14)). Uzawa postulates the existence of a broadly defined educational sector which uses labour, LE (t), in order to augment the state of knowledge in the economy according to the following

knowledge produt. Li

AL(t) = L

AL (t) L:

where L(t) = LE W -1 production of good that there are now the stock of physiL shows how a benefit special case of a

in (14.53)). One of optimal assignme • the proportion of increase but produc Uzawa's ideas L. extended by Romer section is to discu• that human capita. Lucas (1988) ma

whereas Uzawa tion, health, const adopts a more sh ond, Lucas cites Ra on individual earn Despite the fact th early on in life, thi to knowledge accu the fact that age' L for an octogenarian additional skills is the above consider capital accumuld

H(t) (

H(t) = 111E T.1

where ";frE > 0 is a a curved (rather - representative in LLi

A(0) =

where a is the ir and C(t) is consu'

ty = G(t)/ Y(t)

growth

7—wernment spending) is manipulation we obtain:

I

I

t of a unit of government

-a)Y(t)/G(t) = (1—a)/ty . pression in (14.165) (see

zawa (1965) argued that seen as some kind of outcome of the intenrrces in order to advance alized his notions by in labour, i.e. in terms

• = 1 for all t and pro- 4. 1 4)). Uzawa postulates ch uses labour, LE(t), in ►rding to the following

Chapter 14: Theories of Economic Growth

knowledge production function:

where L(t) = LE(t) Lp(t) is the total labour force, Lp(t) is labour employed in the production of goods, and tP(x) satisfies W'(x) > 0 > W"(x) for 0 < x < 1. It is clear that there are now two stocks that can be accumulated in this economy, namely the stock of physical capital goods (K(t)) and the stock of knowledge (AL (t)). Uzawa shows how a benevolent social planner would optimally choose these stocks for the special case of a linear felicity function (i.e. under the assumption that U[c(t)] = c(t) in (14.53)). One of the trade-offs which the planner must make is of course the optimal assignment of labour to the production and educational sectors. By raising the proportion of workers in the educational sector the growth of knowledge will increase but production of goods (and thus the rate of investment) will decrease.

Uzawa's ideas lay dormant for two decades until they were taken up again and extended by Romer (1986), Lucas (1988, 1990b), and Rebelo (1991). The aim of this section is to discuss (a simplified version of) the Lucas model in order to demonstrate that human capital accumulation can serve as the engine of (endogenous) growth.

Lucas (1988) modifies and extends Uzawa's analysis in various directions. First, whereas Uzawa interprets AL (t) very broadly as consisting of activities like education, health, construction and maintenance of public goods (1965, p. 18), Lucas adopts a more specific interpretation by interpreting AL (t) as human capital. Second, Lucas cites Rosen (1976) whose findings suggests that the empirical evidence on individual earnings is consistent with a linear knowledge production function. Despite the fact that in reality people tend to accumulate human capital mainly early on in life, this does not necessarily imply that there are diminishing returns to knowledge accumulation (as is assumed in (14.166)) but rather may be due to the fact that agents' lives are finite (Lucas, 1988, p. 19). It simply makes no sense for an octogenarian to go to school as the time during which he can cash in on his additional skills is too short for the investment to be worthwhile. On the basis of the above considerations, Lucas adopts the following specification for the human capital accumulation function:

where *E > 0 is a constant. The third modification that Lucas makes is to assume a curved (rather than linear) felicity function. The lifetime utility function for the representative infinitely lived household is thus given by:

where a is the intertemporal substitution elasticity, p is the rate of time preference, and C(t) is consumption. The remainder of the model is fairly standard. To keep

457

The Foundation of Modern Macroeconomics

things simple we abstract from population growth and normalize the size of the population to unity (L(t) = 1). This means that the time constraint can be written as:

Following Lucas we assume that the aggregate production function is CobbDouglas:

where N(t) is effective labour used in goods production, i.e. skill-weighted manhours:34

We are now in a position to solve the model and to demonstrate that it contains a mechanism for endogenous growth. The institutional setting is as follows. Perfectly competitive firms hire capital and labour from the household sector. Households receive rental payments on the two production factors and decide on the optimal accumulation of physical and human capital and the optimal time profile for consumption.

Since technology is linearly homogeneous and competition is perfect it is appropriate to postulate the existence of a representative firm. This firm hires units of labour and capital from the household in order to maximize profit, fl(t)

Y(t) - W (t)Lp(t) - RK (0K(t), subject to the technology (14.170) and the definition of effective labour (14.171). This yields the familiar expressions for the rental rate on capital RK(t) and the wage rate W(t):

Equation (14.172) is the standard condition equating the marginal product of capital to the rental rate. The key thing to note about (14.173) is that, for a given capital- effective-labour ratio, K/N, the wage rate increases as the skill level increases. This gives the household a clear incentive to accumulate human capital.

The representative household chooses sequences for consumption and the stocks of physical and human capital in order to maximize lifetime utility (14.168) subject

34 In adopting (14.169)-(14.171) we have simplified the Lucas model by assuming that the population is constant and that there is no external effect of human capital. See Lucas (1988, p. 18) for the latter effect.

to the time constra the following buc

I(t) C(t) =

where I (t) is gross in sion for the real v, point of the individi (FN ) is taken as give and effective laboui The Hamiltonian

is given by:

H(t) =

C(t) 1- 1

1 - 1

+ AIX

where plat) and first-order necessa.

C(t) -11° =

ilx(t)FN(•) =

1:11<(t)

µK (t) = P

IH(t) = p

11,11(t)

0 = A., t-

where we have us,. conditions (explaii intuition behind tl must on the margii capital accumulat its two uses, nam, p. 21). The intuitio slightly and appe,.. the rate of return o

35 Since, capital ail

is zero (ri(t) = 0).

36 The first-order co fix - Au, for the state N

normalize the size of the

, nstraint can be written as:

(14.169)

ction function is Cobb-

(14.170)

i.e. skill-weighted man-

(14.171)

!monstrate that it contains a Ming is as follows. Perfectly ehold sector. Households )rs and decide on the opti- r' he optimal time profile for

oetition is perfect it is appro- MI. This firm hires units

:o maximize profit, n(t) (14.170) and the defini-

ir expressions for the rental

(14.172)

(14.173)

marginal product of capital is that, for a given capitalle skill level increases. This

an capital.

.1sumption and the stocks me utility (14.168) subject

cil by assuming that the popula- _ See Lucas (1988, p. 18) for the

Chapter 14: Theories of Economic Growth

to the time constraint (14.169), accumulation identities (14.4) and (14.167), and the following budget identity:

where I (t) is gross investment in physical capital and we have substituted the expression for the real wage from (14.173). 35 The crucial thing to note is that, from the point of the individual agent described here, the marginal product of effective labour (FN) is taken as given as it depends on the aggregate ratio between physical capital and effective labour.

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

where plat) and auH(t) are the co-state variables for, respectively, K(t) and H(t). The first-order necessary conditions are: 36

where we have used (14.172) to simplify (14.178) and (14.180) are the transversality conditions (explained in detail by e.g. Benhabib and Perli (1994, p. 117)). The intuition behind these expressions is as follows. First, according to (14.176) goods must on the margin be equally valuable in their two uses, namely consumption and capital accumulation. Similarly, (14.177) says that time must be equally valuable in its two uses, namely the accumulation of physical and human capital (Lucas, 1988, p. 21). The intuition behind (14.178)-(14.179) is best understood by rewriting them slightly and appealing to the fundamental principle of valuation according to which the rate of return on different assets (dividends plus capital gains) must be equalized

35 Since, capital and effective labour receive their respective marginal products, it follows that profit

The Foundation of Modern Macroeconomics

(cf. Miller and Modigliani, 1961, p. 412).

where DK (t) FK [KM , N(0] — 8 is the "dividend" on physical capital, consisting of the net marginal product of physical capital, and DH(t) stands for the "dividend" on human capital. The latter can be written in a number of different (but equivalent) ways:

Recall that µK (t) and p,H (t) are the imputed shadow prices of the two assets owned by the household. According to the fundamental principle of valuation, the rate of return (consisting for each asset of dividends plus capital gains expressed in terms of the value of the asset) must be equalized across assets. This is essentially what (14.181) says in the context of the household's choice regarding physical and human capital. The expressions in (14.182), which are obtained by substituting (14.177) into (14.179), show that the dividend on human capital can be written in terms of the additional wage payments it causes (first equality) or in terms of the increase in the marginal productivity of educational activities it gives rise to (second term).

We now have all the ingredients of the model and proceed to characterize its balanced growth path (BGP). 37 Along the BGP consumption and physical and human capital are all growing at constant exponential growth rates, the fraction of labour used in education is constant, and the shadow prices decline at constant exponential rates. We define the exponential growth rate of a variable along the BGP as

(for x E (K, C, H, Y}). First we note that by differentiating (14.176) with

where we have incorporated the fact that F[.] is homogeneous of degree one (so that FK [.] is homogeneous of degree zero). Equation (14.183) implies that the capital- effective-labour ratio is constant along the BGP, i.e. yi< = yN = YH, where the final equality follows from (14.171) plus the fact that Lp is constant along the BGP. It follows from (14.177) that µH /µK = FN is constant also, so that (14.177)—(14.179) together imply that FK = 8 + E. Using this value for FK in (14.183) we find that Yc = a(E — p) . The macroeconomic resource constraint along the BGP can be written as follows:

Since Y IK = F[1, N IK] is constant along the BGP (as KIN is constant) it follows from (14.184) that C K is constant also. Hence, consumption, human and physical

37 The issue of transitional dynamics is studied by, among others, Mulligan and Sala-i-Martin (1993), Benhabib and Pei-1i (1994), Xie (1994), and Bond et al. (1996).

capital, and outpu.

YK = YY = Yc =

It remains to be cht ally feasible. Accui, (equalling

(LE = 1 and Lp = n

The feasw intertemporal sub

We have thus dern ful accumulation of 1 assumption is neede more complex than ulation growth r1L al have a positive extet duction function in society and as > that the formation 4 households are in L nize the link betwei average economy-•. efficient. Lucas (19 sistent differences in are no barriers to

14.6.3 Endogenoi

In the previous su. capital ("skills") forr growth. In this sut which the purposeft. the key source of grx R&D affects econon. Benassy (1998) by at- setting all saving

There are three p. duces a homogeneo

38 Key contributions Grossman and Helpal

(14.181)

hysical capital, consisting of *ands for the "dividend" on different (but equivalent)

(14.182)

-- es of the two assets owned iple of valuation, the rate of '11 gains expressed in terms - s. This is essentially what 71rding physical and human J by substituting (14.177) it can be written in terms of )r in terms of the increase in :s rise to (second term). Dceed to characterize its bal- - 1 and physical and human rates, the fraction of labour

-I ine at constant exponen-

ivariable along the BGP as ifferentiating (14.176) with

(14.183)

neous of degree one (so that

►implies that the capital-

=YN = YH, where the final

constant along the BGP. It ). so that (14.177)-(14.179) in (14.183) we find that lint along the BGP can be

(14.184)

I

kils1 is constant) it follows `ion, human and physical

;an and Sala-i-Martin (1993),

Chapter 14: Theories of Economic Growth

capital, and output all grow at the same exponential rate:

It remains to be checked that the (common) growth rate given in (14.185) is actually feasible. According to (14.167) the maximum growth rate of human capital (equalling IfrE) occurs if the entire labour stock is devoted to educational activities (LE = 1 and Lp = 0). Hence, the growth rate in (14.185) is feasible if and only if YH < *E. The feasibility requirement thus places an upper limit on the allowable intertemporal substitution elasticity:

We have thus demonstrated that endogenous growth can result from the purposeful accumulation of human capital by maximizing agents. No "manna from heaven" assumption is needed to generate this result. The model studied by Lucas (1988) is more complex than the one studied here because he introduces (exogenous) population growth nL and, more importantly, because he argues that knowledge may have a positive external effect on productivity. Instead of (14.170) he uses the production function Y(t) = N(t) l-aKK(t)a4-1(t)as, where H(t) is the average skill level in society and as > 0. Intuitively, his formulation attempts to capture the notion that the formation of human capital is, in part, a social activity. Since individual households are infinitesimally small (relative to the economy) they will not recognize the link between their own human capital choice and the resulting level of average economy-wide human capital. As a result, the market economy will not be efficient. Lucas (1990b) uses this extended model to explain why there can be persistent differences in the marginal product of capital across countries even if there are no barriers to international capital flows.

14.6.3 Endogenous technology

In the previous subsection we showed that the purposeful accumulation of human capital ("skills") forms the key ingredient of the Uzawa-Lucas theory of economic growth. In this subsection we briefly review a branch of the (huge) literature in which the purposeful conduct of research and development (R&D) activities forms the key source of growth. 38 In order to demonstrate the key mechanism by which R&D affects economic growth we follow Grossman and Helpman (1991, ch. 3) and Benassy (1998) by abstracting from physical and human capital altogether. In such a setting all saving by households is directed towards the creation of new technology.

There are three production sectors in the economy. The final goods sector produces a homogeneous good using varieties of a differentiated intermediate good

38 Key contributions to this literature are Romer (1987, 1990), Aghion and Howitt (1992), and Grossman and Helpman (1991).

461

The Foundation of Modern Macroeconomics

as productive inputs. Production is subject to constant returns to scale (in these inputs) and perfect competition prevails. The R&D sector is also perfectly competitive. In this sector units of labour are used to produce blueprints of new varieties of the differentiated input. Finally, the intermediate goods sector is populated by a large number of small firms, each producing a single variety of the differentiated input, who engage in Chamberlinian monopolistic competition (see Chapter 13 for a detailed account of this market structure).

The production function in the final goods sector is given by the following (generalized) Dixit-Stiglitz (1977) form:

where N(t) is the number of different varieties that exist at time t, Xj(t) is variety j, and /2 and 77 are parameters. 39 Note that, holding constant the number of varieties, doubling all inputs leads to a doubling of output in (14.187), i.e. constant returns to scale prevail. The specification in (14.187) implies that, provided > 1, there are returns to specialization of the form emphasized by Ethier (1982). This can be demonstrated as follows. Suppose that the same amount is used of all inputs (as will indeed be the case in the symmetric equilibrium discussed below), i.e. Xj(t) = X(t) for j E [0, N(t)]. Then total output in the final goods sector will be Y(t) = N (011-1 (Lx(t)/kx), where Lx(t) = kxN(t)X(t) represents the total amount of

labour used up in the intermediate goods sector (see below). Ceteris paribus Lx(t), output in the final goods sector rises with the number of intermediate inputs pro-

vided rj exceeds unity. By having a larger number of varieties, producers in the final goods sector can adopt a more "round-about" method of production and thus produce more.

The representative producer in the final goods sector minimizes its costs and sets the price of final goods equal to the marginal (equals average) cost of production:

where P1(t) is the price of input variety j. The cost-minimizing demand for input j is given by:

where /2/(1 - /2) thus represents the (constant) price elasticity of the demand for variety j.

39 Note that (14.187) is similar to (13.2) in Chapter 13 with the summation sign replaced by an integral sign. Strictly speaking N(t) is now the "measure" of products invented before time t. Following convention we will continue to refer to N(t) as the number of firms. See Romer (1987) and Grossman and Helpman (1991, p. 45) for details.

In the inter! firms which eacl

unique, variety :

I

11(t) =Pi m. (1

where W(t) is tl perfectly mobile its output level. function Xj (t) =- intermediate goo Chapter 13, the ( over marginal

Pi(t) =

where /2 thus red mediate sector markup, they al Hence, from h, and FI1 (t) = II t of the represent 114.190) and inv the profit of a re

1

Mt) = [P(ti

In the R&D s blueprints. Sind N(t), represents the production I

N(t) =

where LR (t) is productivity par blueprints are i rates the assumi positively affect working today additional know

100 years" (199