Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

effective and money is not it hinges on a highly elastic by the empirical evidence. mes that there are convex h, the individual firm tries ble to its "ideal" price path Dsts. The presence of adjust- ; a weighted average of last explicitly forward looking. h thus provides a microeco-

. 1 ips curve of Friedman and

e pricing friction is stochasirm which may be a "green

-obabilities are the same for a green light can change its od but must maintain that theory differs substantially is level, the two approaches tic pricing equation.

w Keynesian economics. Also - or overviews of new Keyneion for the multiplier, see Ng 1987), Dixon (1987), Mankiw and Moutos (1992), Dixon Ligthart, and van der Ploeg competitive equilibrium, see ctre (1993), and Matsuyama

1.

emberg, and Summers (1986), 43 ). Levy et al. (1997) present chains. For the envelope :carve, see Ball, Mankiw, and :e stickiness is widely used in °6, 1999), Clarida, Gall, and I Woodford (1999), and Yun

solistic competition it may I here is a large literature on

Chapter 13: New Keynesian Economics

multiple equilibria and coordination failures. See Diamond (1982, 1984a,b), Howitt (1985), Shleifer (1986), Diamond and Fudenberg (1989, 1991), Cooper and John (1988), Weil (1989a), and Benhabib and Farmer (1994). An excellent survey of some of this literature is presented by Cooper (1999). The classic source on multiple equilibria and animal spirits is

Keynes (1937).

403

14.1 Stylized Facts of Economic Growth

The neoclassical growth me Swan (1956). The central production function (wh, in a very general form as:

According to Kaldor (1961, pp. 178-179), a satisfactory theory of economic growth should be able to explain the following six "stylized facts" by which we mean results that are broadly observable in most capitalist countries.

(SF1) (*) Output per worker shows continuing growth "with no tendency for a falling rate of growth of productivity".

(SF2) Capital per worker shows continuing growth. (SF3) The rate of return on capital is steady.

(SF4) (*) The capital-output ratio is steady.

(SF5) (*) Labour and capital receive constant shares of total income.

Y(t) = F [K(t), L(t), tj ,

where t is the time ind, to indicate that the tech.. the assumption of perfectll production function must ( of technology (P1): 1

F RK(t), AL(t), t] =

I See the Intermezzo on prods

conomic growth? ' . 7ed facts?

to the Solow—Swan model? )del based on dynamically

various traditional growth

(SF11) Both skilled and unskilled workers tend to migrate towards high-income countries.

Although we shall have very little to say about the last three stylized facts, the other facts will be referred to regularly.

!ory of economic growth ly which we mean results

ith no tendency for a

income.

The neoclassical growth model was developed independently by Solow (1956) and Swan (1956). The central element of their theory is the notion of an aggregate production function (which has been used throughout the book). It can be written in a very general form as:

Y(t) = F [K(t), L(t), t] , (14.1)

where t is the time index which appears separately in the production function to indicate that the technology itself may not be constant over time. We retain the assumption of perfectly competitive behaviour of firms which implies that the production function must obey constant returns to scale. We label this first property of technology (P1): 1

1 See the Intermezzo on production theory in Chapter 4 above.

405

The Foundation of Modern Macroeconomics

It is assumed that the household sector as a whole (or the representative household) consumes a constant fraction of output and saves the rest. Aggregate saving in the economy is then:

where s is the constant propensity to save which is assumed to be exogenously given. In a closed economy, output is exhausted by household consumption C(t) and investment I(t):

where we have assumed that government consumption is zero for now. Aggregate gross investment is the sum of replacement investment, SK(t) (where 8 is the constant depreciation rate), and the net addition to the capital stock, K(t):

I(t) = SK(t) K(t). (14.4)

We assume that labour supply is exogenous but that the population grows as a whole at a constant exponential rate nL:

where we can normalize L(0) = 1.

14.2.1 No technological progress

We first look at the case for which technology itself is time-invariant, so that the production function (14.1) has no separate time index:

In addition to linear homogeneity (property (P1)), the production function features positive but diminishing marginal products to both factors:

A more controversial assumption, but one we will make nevertheless, is that F(.) obeys the so-called Inada conditions (after Inada (1963)) which ensure that it has nice curvature properties around the origin (with K or L equal to zero) and in the limit (with K or L approaching infinity): 2

The model consists of tity, S(t) I (t). Becau , state in levels of output, suring all variables in p k(t) K(t)/L(t), etc. Th equation in the per capi

k(t) = sf (k(t)) –

where f (k(t)) is the inte made of the linear horI

f (k(t)) F [K(t)/L.

We can obtain insight i diagram for k(t)—see 1 .t sents the amount of invl each existing worker N• the line features the gn s, is constant by assu: the intensive-form pro,. happens for k(t) = 0 and

f' (k(t)) FK [k(t),

about which the Inada at the origin, is conca accumulated. Hence

It follows in a straight From any initial posi:...

Eo. In the steady state c implies that the capital i.e. k(t)/K(t) = L(t)/1_ , steady-state output per Hence, output itself al5, and since the savings ra investment. In the bal.

111

representative household) t_ Aggregate saving in the

(14.2)

umed to be exogenously ce hold consumption C(t)

(14.3)

is zero for now. Aggre-

---nt, 8K(t) (where 8 is the apital stock, k(t):

(14.4) pulation grows as a whole

(14.5)

.,, e-invariant, so that the

(14.6)

auction function features s:

(P2)

^ evertheless, is that F(.) which ensure that it has !qual to zero) and in the

(P3)

1 innocuous and actually

least. The question "Where t with by theologians than by dy of theology.

Chapter 14: Theories of Economic Growth

The model consists of equations (14.2)—(14.5) plus the savings-investment identity, S(t) I (t). Because the labour force grows, it is impossible to attain a steady state in levels of output, capital, etc., but this problem is easily remedied by measuring all variables in per capita or intensive form, i.e. we define y(t) Y (t)/ L(t), k(t) K(t)/ L(t), etc. The model can then be condensed into a single differential equation in the per capita capital stock:

where f (k(t)) is the intensive form of the production function and use has been made of the linear homogeneity property (P1):

We can obtain insight into the properties of the model by working with a phase diagram for k(t)—see Figure 14.1. In that figure, the straight line (8 + nL)k(t) represents the amount of investment required to replace worn-out capital and to endow each existing worker with the same amount of capital. Since the work force grows, the line features the growth rate of the labour force, nL. Since the savings rate, s, is constant by assumption, the per capita saving curve has the same shape as the intensive-form production function. To draw this curve we need to know what happens for k(t) = 0 and k(t) oo. We obtain from (14.8):

about which the Inada conditions (P3) say all we need to know: f (k(t)) is vertical at the origin, is concave, and flattens out as more and more capital per worker is accumulated. Hence f (k(t)) and sf (k(t)) are as drawn in Figure 14.1. 3

It follows in a straightforward fashion from the diagram that the model is stable. From any initial position k(t) will converge to the unique equilibrium at point E0 . In the steady state capital per worker is constant and equal to k(t) = k*. This implies that the capital stock itself must grow at the same rate as the work force, i.e. K(t)/K(t) = L(t)/ L(t) = nL . The intensive-form production function says that steady-state output per worker, y*, satisfies y* = and is thus also constant. Hence, output itself also grows at the same rate as the work force, i.e. Y(t)/Y(t) = nL, and since the savings rate is constant, the same holds for the levels of saving and investment. In the balanced growth path we thus have:

Since the rate of population growth is exogenous, the long-run growth rate of the economy is exogenously determined and thus cannot be influenced by government

3 Barro and Sala-i-Martin (1995, p. 52) show that both inputs are essential if the properties (P1)—(P3) are satisfied. Hence, F(0, L) = F (K , 0) = f (0) = 0.

407

The Foundation of Modern Macroeconomics

s f (k (t))

+ n ,) k (t)

Figure 14. The Solow-Swan model

policy or household behaviour. For example, an increase in the savings rate rotates the savings function counter-clockwise and gives rise to a higher steady-state capitallabour ratio but it does not affect the rate of economic growth along the balanced growth path.

Before turning to a detailed examination of the properties of the Solow-Swan model we first expand the model by re-introducing technological change into the production function.

14.2.2 Technological progress

Technical change can be embodied or disembodied (see Burmeister and Dobell, 1970, ch. 3). Embodied technical change is only relevant to newly acquired and installed equipment or workers and therefore does not affect the productivity of existing production factors. Disembodied technical progress takes place if, independent of changes in the production factors, isoquants of the production function shift inwards as time progresses (Burmeister and Dobell, 1970, p. 66). Reasons for this inward shift may be improvements in techniques or organization which increase the productivity of new and old factors alike. We focus on disembodied technical progress in the first part of this chapter but will return to examples of embodied technical progress later on.

We can represent different cases of factor-augmenting disembodied technical change by writing the production function (14.1) in the following form:

Y(t) = F [AK (t)K(t), AL(OL(t)i

where AK(t) and AL (t) 4 tive capital" and "effec augmenting if

> 0, and equally 1 Three different con,

the literature (BurmeisL 33). Technological ch. constant over time for a is constant over time foi share is constant over (14.11), the three cases AL (t) --a 1.

Of course, for the C neutrality are indistingL

Y(t) = [AK (t)K

= K(t)" [A L ( t

= A(t)K(t)

For non-Cobb-Douglas ferent implications for t show, for example, tha menting) for the mod, steady state we must th: for forms of technologi' shares approaches zero have balanced growth a between capital and labs remainder of the disco. holds.

The production fungi1

Y(t) = F [K(t), N (t)]

where N (t) measures the that technical progress

A (t) = nA, A(t)

A(t)

Since the labour force i the effective labour force

By measuring out; it and k(t) K(t)/N(t),

(6 + n L ) k (t)

f (k (t))

sf (k (t))

k (t)

in the savings rate rotates " ; 7her steady-state capital- _ )wth along the balanced

?erties of the Solow-Swan knological change into the

rmeister and Dobell, 1970, sly acquired and installed e productivity of existing yes place if, independent --Auction function shift ), p. 66). Reasons for this

nization which increase disembodied technical to examples of embodied

g disembodied technical ',, llowing form:

Chapter 14: Theories of Economic Growth

where AK(t) and AL(t) only depend on time, and AK(t)K(t) and AL (t)L(t) are "effective capital" and "effective labour" respectively. Technical progress is purely labour augmenting if AK (t) 0 and AL (t) > 0, purely capital augmenting if AL (t) 0 and AK(t) > 0, and equally capital and labour augmenting if AK (t) AL (t) > 0.

Three different concepts of neutrality in the process of technical advance exist in the literature (Burmeister and Dobell, 1970, p. 75; Barro and Sala-i-Martin, 1995, p. 33). Technological change is (a) if the relative input share FKK/FLL is constant over time for a given capital-output ratio, K/Y, (b) Hicks neutral if this share is constant over time for a given capital-labour ratio, K/L, and (c) Solow neutral if this share is constant over time for a given labour-output ratio, L/Y. In terms of equation (14.11), the three cases correspond to, respectively, AK (t) 1, AK (t) AL (t), and AL(t) 1.

Of course, for the Cobb-Douglas production function the three concepts of neutrality are indistinguishable, since:

Y(t) = [AK(OK(t)r

For non-Cobb-Douglas cases, however, the different neutrality concepts have different implications for balanced growth. Barro and Sala-i-Martin (1995, pp. 54-55) show, for example, that technical progress must be Harrod neutral (labour augmenting) for the model to have a steady state with a constant growth rate. In a steady state we must have a constant capital-output ratio and it can be shown that for forms of technological progress that are not Harrod neutral, one of the factor shares approaches zero if the capital-output ratio is to be constant. So if we wish to have balanced growth and be able to consider a non-unitary substitution elasticity between capital and labour, we must assume Harrod-neutral technical progress. The remainder of the discussion in this section will thus assume that Harrod neutrality holds.

The production function is written as:

where N(t) measures the effective amount of labour (N(t) A(t)L(t)) and we assume that technical progress occurs at a constant exponential rate:

Since the labour force itself grows exponentially at a constant rate nL (see (14.5)), the effective labour force grows at a constant exponential rate nr, +

By measuring output and capital per unit of effective labour, i.e.

The Foundation of Modern Macroeconomics

above, the fundamental differential equation for k(t) is obtained:

k(t) = sf(k(t)) — (6 + nL + nA)k(t)• (14.15)

In the steady state, k* = sy* / (6 + nt + nA), so that output and the capital stock grow at the same rate as the effective labour input. Hence, equation (14.10) is changed to:

Hence, exactly the same qualitative conclusions are obtained as in the model without technological advance. Long-term balanced growth merely depends on the exogenous factors nL and nA.

14.3 Properties of the Solow—Swan Model

In this section we study the most important properties of the Solow—Swan model. In particular, we look at (a) the golden rule and the issue of over-saving, (b) the transitional dynamics implied by the model as well as the concept of absolute versus conditional convergence, and (c) the speed of dynamic adjustment.

14.3.1 The golden rule of capital accumulation

One of the implications of the model developed thus far is that, even though longterm balanced growth is exogenous (and equal to n + nA), the levels of output, capital, and consumption are critically affected by the level of the savings rate. In other words, even though s does not affect long-term growth it does affect the path along which the economy grows. This prompts the issue concerning the relative welfare ranking for these different paths. To the extent that the policy maker can affect s, he/she can also select the path on which the economy finds itself. We first consider steady-state paths.

In the steady state, equation (14.15) implies a unique implicit relationship between the savings rate and the equilibrium capital-labour ratio which can be written as:

k* = k*(s), (14.17)

with dk* /ds = y* /[8 + n — sf/(k*)] > 0. Suppose that the policy maker is interested in steady-state per capita consumption and, to keep things simple, assume that there is no technical progress (i.e. nA = 0 and n = nL). Consumption per capita can then

be written

c(s) = 11

which in I- tion and the c(s) for differ stock per 1‘ its maximun

I dc(s) = I

ds

In terms of Fi the slope of I

function. In

I

f' [k* (s

The golden and Dobell The produce( to f' — b , N,

.gilding an (:) the rates of r

)Ids.

Note that tl

GR =

S

Lquation ( 14. share of c,,

rule sa the capital in(

We are nu. economy dyn off (ands(

in i ,8ure 14.2 economy is at

> sGR ), duricult to sib per capita c( shows that a I state from L :le aid of ..b.

ained:

(14.15)

d the capital stock grow ition (14.10) is changed

riA. (14.16)

d as in the model with- merely depends on the

the Solow—Swan model. of over-saving, (b) the mcept of absolute versus

Istment.

'tat, even though long- /1 4 ), the levels of output, of the savings rate. In th it does affect the path roncerning the relative the policy maker can )my finds itself. We first

le implicit relationship Th. ir ratio which can be

(14.17)

y maker is interested in 1e, assume that there ion per capita can then

Chapter 14: Theories of Economic Growth

be written as:

which in Figure 14.1 represents the vertical distance between the production function and the required-replacement line in the steady state. In Figure 14.2 we plot c(s) for different savings rates. Any output not needed to replace the existing capital stock per worker in the steady state can be consumed. Per capita consumption is at its maximum if the savings rate satisfies dc(s)I ds = 0, or:

In terms of Figure 14.1, per capita consumption is at its maximum at point A where the slope of the production function equals the slope of the required-replacement function. In view of (14.19), the golden rule savings rate,

The golden rule savings rate is associated with point E1 in Figure 14.2. Burmeister and Dobell (1970, pp. 52-53) provide the intuition behind the result in (14.20). The produced asset (the physical capital stock) yields an own-rate of return equal to f' — 8, whereas the non-produced primary good (labour) can be interpreted as yielding an own-rate of return nL = n. Intuitively, the efficient outcome occurs if the rates of return on the two assets are equalized, i.e. if the equality in (14.20) holds.

Note that the expression in (14.20) can be rewritten as:

Equation (14.21) shows that the golden rule savings rate should be equated to the share of capital income in national income (which itself in general depends on the golden rule savings rate). In the Cobb—Douglas case, with f = k(t)" , a represents the capital income share so that the golden rule savings rate equals SGR = a.

We are now in a position to discuss the concept of dynamic inefficiency. We call an economy dynamically inefficient if it is possible to make everybody at least as well off (and some strictly better off) by reducing the capital stock. Consider the situation in Figure 14.2, and assume that the actual steady-state savings rate is so so that the economy is at point Eo. Since this savings rate exceeds the golden rule savings rate (so sGR\) per capita consumption is lower that under the golden rule. It is not difficult to show that point E0 is dynamically inefficient in the sense that higher per capita consumption can be attained by reducing the savings rate. Figure 14.2 shows that a reduction in the savings rate from so to SGR would move the steady state from E0 to E 1 and lead to higher per capita steady-state consumption. With the aid of Figure 14.3 we can figure out what happens to per capita consumption

411

The Foundation of Modern Macroeconomics

c [k* (s)]

c [k* (s)] = f (k* (s)) — (6+ n) k" (s)

Figure 14.2. Per capita consumption and the savings rate

during the transitional phase. The economy is initially at point E 0 and the initial steady-state capital-labour ratio is 4. A reduction in the savings rate (from s o to sGR) rotates the per capita consumption schedule in a counter-clockwise fashion and the economy jumps from Eo to A at impact. Since the transition towards the golden-rule capital-labour ratio kGR is stable, the economy moves from A to the new steady-state point E 1 as k(t) falls towards kGR during transition. Hence, as a result of the decrease in the savings rate, consumption is higher than it would have been, both during transition and in the new steady state, i.e. the reduction in s is thus Pareto-improving. As a result, we can conclude that savings rates exceeding SGR are dynamically inefficient.

The same conclusion does not hold if the savings rate falls short of SGR as the Pareto-optimality property cannot be demonstrated unambiguously. Consider an economy in which the savings rate is too low, i.e. s1 < SGR . In terms of Figures 14.2 and 14.3, the economy is initially at point £2. An increase in the savings rate from s1 to SGR still leads to an increase in steady-state per capita consumption. During transition, however, per capita consumption will have to fall before it can settle at its higher steady-state level prescribed by the golden rule. In terms of Figure 14.3, at impact the economy jumps from E2 to B as the savings rate is increased. During part of the transition consumption is lower than it would have been in the absence of the shock. Since we have no welfare function to evaluate the uneven path of per capita consumption we cannot determine whether the increase in s is Pareto-improving in this case.

c (t)

k (t)

Figure 14

its goldei 1

14.3.2 Transitional

1

Up to now attention the model with exogc is given in (14.15). By from (14.15):

yk(t) sf (k(t))/k(1

+ nA . difference between thi 14.4 for that matter, i faster than countries v should converge!

Note that the gro1.

where cox (t) r(k( t For a Cobb—Douglas pi does not hold if the s unity.

4 The Inada conditions ei