Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

The Foundation of Modern Macroeconomics

7, k(0)

( 5 + n

sf(k(t))/ k(t)

Figure 14.4. Growth convergence

Take, for example, the CES production function:

where ola, (>0) represents the substitution elasticity between capital and labour. The capital share implied by (14.24) is given by (K (t) EE a[f (k(t))/ k(t)ria -1)/aKL , which thus depends on k(t) according to:

It follows that for 0-KL, > 1 (<1), an increase in k(t) results in an increase (decrease) in the share of capital in national income. By using (14.23) and (14.25) we obtain the result linking output growth to the output level in efficiency units of labour:

For economies with positive growth in k(t) (for which yk(t) > 0) the term in square brackets on the right-hand side of (14.26) is guaranteed to be positive, so that a higher output level in efficiency units of labour is associated with a lower growth rate in output. The same holds for declining economies (for which yk(t) < 0) operating to the right of their steady-state position, provided they are not too far from this steady state (i.e. yk(t) must not be too negative).

This suggests that tht is based on the convergL. countries. We take a grou to the closed economy _. sess the same structural f so that in theory they 1 - hypothesis (ACH) then st countries. Barro and SaL on log y(t) for a

finding a negative effect , i.e. initially rich count: does not seem to hold

This rejection of the A( is refuted because one o results could be false. For poor country, it could act country is from its steau ; country will be growing fi of Barro and Sala-i-Mar . where sp and sR are

and (k1P and (k*)R are t at kP (0) and the rich cu.. (the vertical distance CD A refined test of the • hypothesis (CCH) accorui

E

Figure 14.5.

(5+ n

sf(k(t))1k(t)

k(t)

F KL /(01a-1) <=>.

(14.24)

I

-een capital and labour. The t))1k(t)] (,KL -1)1G-KL which

(14.25)

& in an increase (decrease)

.23) and (14.25) we obtain `ficiency units of labour:

(14.26)

t (t) > 0) the term in square to be positive, so that a LI with a lower growth rate which yk(t) < 0) operating y are not too far from this

Chapter 14: Theories of Economic Growth

This suggests that there is a simple empirical test of the Solow-Swan model which is based on the convergence property of output in a cross-section of many different countries. We take a group of closed economies (since the Solow-Swan model refers to the closed economy) and assume that they are similar in the sense that they possess the same structural parameters, s, n, and 8, and the same production function, so that in theory they have the same steady state. The so-called

hypothesis (ACH) then suggests that poor countries should grow faster than rich countries. Barro and Sala-i-Martin (1995, p. 27) show the results of regressing of yy (t) on log y(t) for a sample of 118 countries. The results are dismal: instead of finding a negative effect as predicted by the ACH, they find a slight positive effect, i.e. initially rich countries grow faster than poor countries. Absolute convergence does not seem to hold and (Romer's) stylized fact (SF7) is verified by the data.

This rejection of the ACH does not necessarily mean that the Solow-Swan model is refuted because one of the identifying assumptions underlying the regression results could be false. For example, if a rich country has a higher savings rate than a poor country, it could actually be further from its (higher) steady state than the poor country is from its steady state. The Solow-Swan model then predicts that the rich country will be growing faster than the poor country, as indeed the empirical results of Barro and Sala-i-Martin (1995) suggest. We demonstrate this result in Figure 14.5 where sp and sR are the savings rates of the poor and the rich country, respectively, and (k*)P and (k*)R are the corresponding steady states. If the poor country is initially at kP (0) and the rich country at kR(0), the former will grow slower than the latter (the vertical distance CD is larger than AB).

A refined test of the Solow-Swan model makes use of the conditional convergence hypothesis (CCH) according to which similar countries should converge. Barro and

Figure 14.5. Conditional growth convergence

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The Foundation of Modern Macroeconomics

Sala-i-Martin (1995, pp. 27-28) show that convergence does appear to take place for the twenty original OECD countries and a fortiori for the different states in the US. This suggests that the CCH is not grossly at odds with the data, which is good news for the Solow-Swan model (and bad news for some of the endogenous growth models discussed below).

14.3.3 The speed of adjustment

The convergence property is not the only testable implication of the Solow-Swan model. Apart from testing whether economies converge, another issue concerns how fast they converge. In order to study this issue further we follow Burmeister and Dobell (1970, pp. 53-56) and Barro and Sala-i-Martin (1995, pp. 37-39, 53) by focusing on the Cobb-Douglas case for which f = k(t)", and the fundamental differential equation (14.15) becomes:

An exact solution to this differential equation can be obtained by using a transformation of variables, i.e. by rewriting (14.27) in terms of the capital-output ratio,

x(t) k(t)/y(t) = k(t)l -a:

Hence, the half-life of the divergence = D equals t i/ 2 = log 2/fi = 0.693//3.5 Some back-of-the-envelope computations based on representative values of n1. = 0.01 (per annum), n A = 0.02, 8 = 0.05, and a = 1/3 yield the value of /3 = 0.0533 (5.33% per annum) and an estimated half-life of t112 = 13 years. Transition is thus relatively fast, at least from a growth perspective.6 As Barro and Sala-i-Martin (1995, p. 38) indicate, however, this estimate is far too high to accord with empirical evidence.

5 See also Chapter 7 where we compute the convergence speed of the unemployment rate in a discrete-time setting.

6 Note that Sato (1963) actually complains about the startlingly low transition speed implied by the Solow-Swan model. His object of study is fiscal policy and business cycle phenomena. In this context convergence of 5% per annum is slow. Hence the different conclusion.

They suggest that f i 5.33%). So here is a ro to generate a realistic share must be unr( = 0.02)! One way t broad measure of cap the approach taken

14.3.4 Human cal

Mankiw, Romer, al, using real world data the model appears entirely satisfactory. than the actual car nology assumption i capital input. They ai drum of the Solow-5 to include human ca

Y(t) = K(t)"KH,.

where H(t) is the stoc of the two types of L model, productivity and t(t) I L(t) = ) al be written in effects

k(t) = sKy(t) + h(t) = 5Hy(t)

where h(t) H(t)/[.a accumulate physic well as the deprecia there are decreasing I the model possesses By using (14.31)-(14

1-aycah

k* =( sK 'ti + n

By substituting k' we obtain an estimal

:oes appear to take place the different states in the h the data, which is good )f the endogenous growth

Lion of the Solow-Swan nother issue concerns how

. - e follow Burmeister and 1995, pp. 37-39, 53) by , and the fundamental

(14.27)

-;ned by using a transfor- ' the capital-output ratio,

(14.28)

(14.29)

do to which the economy 7) measures the speed of 100% of the divergence )f tc :

(14.30)

og 21p= 0.693/13.5 Some re values of nL = 0.01 (per of /3 = 0.0533 (5.33% per cition is thus relatively -i-Martin (1995, p. 38) with empirical evidence.

the unemployment rate in a

•lsition speed implied by the le phenomena. In this context

Chapter 14: Theories of Economic Growth

They suggest that p is more likely to be in the range of 2% per annum (instead of 5.33%). So here is a real problem confronting the Solow-Swan model. In order for it to generate a realistic convergence rate of 2%, for given values of 6 and n, the capital share must be unrealistically high (a value of a = 4 actually yields an estimate of p= 0.02)! One way to get the Solow-Swan model in line with reality is to assume a broad measure of capital to include human as well as physical capital. This is indeed the approach taken by Mankiw, Romer, and Weil (1992).

14.3.4 Human capital to the rescue

Mankiw, Romer, and Weil (1992, p. 415) start their highly influential analysis by using real world data to estimate the textbook Solow model. They show that, though the model appears to fit the data quite well, some of the parameter estimates are not entirely satisfactory. For example, the estimated capital coefficient is much larger than the actual capital share of about one third. So either their Cobb-Douglas technology assumption is inappropriate or there is a serious mis-measurement of the capital input. They adopt the latter stance and suggest that the convergence conundrum of the Solow-Swan model disappears if the production function is modified to include human capital:

where H(t) is the stock of human capital and aK and aH are the efficiency parameters of the two types of capital (0 < 1). In close accordance with the Solow-Swan model, productivity and population growth are both exponential (A(t)/A(t) nA and L(t)/L(t) = nL ) and the accumulation equations for the two types of capital can be written in effective labour units as:

where h(t) H(t)/0(t)L(t)], n E-- nA -1- nL, and sK and sH represent the propensities to accumulate physical and human capital, respectively. The production functions as well as the depreciation rate of the two types of capital are assumed to be equal. Since there are decreasing returns to the two types of capital in combination (aK + aH < 1) the model possesses a steady state for which k(t) = h(t) = 0, k(t) = k*, and h(t) = h*. By using (14.31)-(14.33) we obtain:

By substituting k* and h* into the (logarithm of the) production function (14.31) we obtain an estimable expression for per capita output along the balanced growth

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The Foundation of Modern Macroeconomics path:

Mankiw et al. (1992, p. 417) suggest approximate guesses for aK = 3 and aH between and 9. The latter guess is based on the observation that in the US manufacturing sector the minimum wage is between a third and a half of the average wage. By interpreting the minimum wage as the return to labour without any human capital (so-called "raw" labour), this means that between half and two thirds of the total payment to labour represents the return to human capital. Since an income share

of (1 - aK) is left after payments to owners of physical capital are taken care of, this implies 1. (1 - aK) < aH < 4(1 - aK) or 3 < an- <

As a result of the inclusion of human capital, the model is much better equipped to explain large cross-country income differences for relatively small differences between savings rates (sK and sH ) and population growth rates (n). This is apparent from equation (14.35). An increase in sic, for example, induces higher income in efficiency units just as in the standard Solow-Swan model (see (14.32)) but also raised the stock of human wealth in efficiency units. By adding human capital to the model, the elasticity of sK in (14.35) is of the order of unity rather than one half which is predicted by the standard Solow-Swan model. A similar conclusion holds for a change in n. An increase in n reduces income because both physical and human capital are spread out over more souls and the elasticity of (n + 8) is not -4, as in the Solow-Swan model, but a staggering -2! See Romer (1996, pp. 134-135) for a further numerical example.

Not surprisingly, the inclusion of a human capital variable works pretty well empirically; the estimated coefficient for aH is highly significant and lies between 0.28 and 0.37 (Mankiw et al., 1992, p. 420). The convergence property of the augmented Solow-Swan model is also much better. The convergence speed is now defined as p (1 aK — aH)(n + 8) which can be made in accordance with the observed empirical estimate of = 0.02 without too much trouble. Hence, by this very simple and intuitively plausible adjustment the Solow-Swan model can be salvaged from the dustbin of history. The speed of convergence it implies can be made to fit the real world.

7 Ingenious as it is, this approach to estimating the income share of human capital is not without dangers, especially in Europe where the minimum wage is policy manipulated rather than market determined.

14.4 Macroec

4

The Solow-Swan n such as the effect Ricardian equivale Solow-Swan mode

14.4.1 Fiscal pol

Suppose that the demand in the goo

Y(t) = C(t) +

Aggregate saving is

S(t) = s [Y(t)

-

where T(t) is the government defit investment, i.e. GO by:

B(t) = r(t)B(t)

where B(t) is govt competitive condi productivity of cal

r(t) = f'(k(t)) -

By writing all ca: condensed to the f

k(t) = f(k(t)) - = sf (k(t)) b(t) = [f"(k(t))

where r(t)--_E-T(t)i. Under pure tax t b(t) = 0), the govt

8 This result is de,-

'tx = 3 and aH between i the US manufacturing ,f the average wage. By lout any human capital two thirds of the total Since an income share

are taken care of, this

; much better equipped tively small differences

.'s (n). This is apparent luces higher income in 1 (see (14.32)) but also 'cling human capital to unity rather than one

I.A similar conclusion

ise both physical and :itv of (n + 6) is not -1, (1996, pp. 134-135)

able works pretty well ant and lies between ce property of the augrgence speed is now accordance with the )uble. Hence, by this an model can be salit implies can be made

an capital is not without ►ulated rather than market

Chapter 14: Theories of Economic Growth

14.4 Macroeconomic Applications

The Solow-Swan model can also be used to study traditional macroeconomic issues such as the effect of fiscal policy and the issue of debt versus tax financing and Ricardian equivalence. In order to keep things simple, we return to the standard Solow-Swan model in which there is only physical capital.

14.4.1 Fiscal policy in the Solow model

Suppose that the government consumes G(t) units of output so that aggregate demand in the goods market is:

Aggregate saving is proportional to after-tax income, so that (14.2) is modified to:

government deficit must be compensated for by an excess of private saving over investment, i.e. G(t) - T(t) = S(t) - I(t). The government budget identity is given by:

where B(t) is government debt and r(t) is the real interest rate which, under the competitive conditions assumed in the Solow-Swan model, equals the net marginal productivity of capital: 8

By writing all variables in terms of effective labour units, the model can be condensed to the following two equations:

where t (t) T(t)/N(t), g(t) G(t)/N(t), and b(t) B(t)/N(t).

Under pure tax financing and in the absence of initial government debt (b(t) =

-

The Foundation of Modern Macroeconomics

(6 + n) k (t)

sf (k (t))

s[ f (k (t))— g (t)]

— sg (t)

Figure 14.6. Fiscal policy in the Solow–Swan model

this expression into (14.40) we obtain:

The economy can be analysed with the aid of (14.40) alone—see Figure 14.6. In the absence of government consumption, the unique (and stable) steady-state equilibrium is at point E0. An increase in government consumption shifts the net investment line down which results in multiple equilibria (or even no equilibria). Of these equilibria, the one at point A is unstable and that at E1 is stable. Fiscal policy crowds out the physical capital stock. At impact, private consumption and net investment (in efficiency units of labour) both fall (dc(0) < 0 and dk(0) < 0) but output is unchanged (dy(0) = 0). Over time, as the capital stock dwindles, output and private consumption per effective labour unit fall:

Next we consider the issue of bond financing. If the government increases its consumption without at the same time raising r (t) by the same amount, a primary deficit will be opened up which, according to (14.41), will lead to an ever-increasing explosive process for government debt (since r > n by assumption in (14.41)). In order to avoid this economically rather uninteresting result, we postulate a debt

b(t)

b '

Figur-4

SOIo

stabilization rule, a I

r(t) = To +

1

By substituting (14.1

b(t) = [f'(k(t))

The dynamic prop, diagram in (k, b) sp,t, the following express

k(t) = sf (k(t)) –

The slope of the k =1

db(t)\

dk(t) kco=0 = -

The k = 0 line is uid., with positive (negatii stock, an increase in sumption, and renue equilibrium features horizontal arrows u. ,

9 Equation (14.46) is stal which is negative.

n) k (t)

sf (k (t))

s[f (k (t))— g (t)]

k (t)

el

(14.42)

le—see Figure 14.6. In the stable) steady-state equiumption shifts the net is (or even no equilibria). at at E1 is stable. Fiscal private consumption and

(0) < 0 and dk(0) < 0) but al stock dwindles, output

(14.43)

(14.44)

rnment increases its con- , me amount, a primary lead to an ever-increasing :cumption in (14.41)). In

.1t, we postulate a debt

Chapter 14: Theories of Economic Growth

b (t)

k (t)

Figure 14.7. Ricardian non-equivalence in the

Solow–Swan model

stabilization rule, a variation of which was suggested by Buiter (1988, p. 288):

The dynamic properties of the economy can be illustrated with the aid of a phase diagram in (k, b) space—see Figure 14.7. By combining (14.40) and (14.45) we obtain the following expression:

The slope of the k = 0 line is obtained from (14.47) in the usual fashion:

dk(t))0-)=0 8 + n – sf'

The k = 0 line is upward sloping and points above (below) this line are associated with positive (negative) net investment, i.e. k > 0 (< 0). Ceteris paribus the capital stock, an increase in the level of debt raises tax receipts (by (14.45)), reduces consumption, and renders net investment positive. As a result, the new capital stock equilibrium features a higher capital stock. The dynamic forces are indicated by horizontal arrows in Figure 14.7.

9 Equation (14.46) is stable because the coefficient for b(t) on the right-hand side is equal to (r—n)— f , which is negative.

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The Foundation of Modern Macroeconomics

The b 0 line is obtained from (14.46). It is horizontal if debt is zero initially but with a positive initial debt level, it is downward sloping because of the diminishing marginal productivity of capital:

For points above (below) the b = 0 line there is a government surplus (deficit) so that debt falls (rises). This is indicated with vertical arrows in Figure 14.7. The Buiter rule thus ensures that the economy follows a stable (and possibly cyclical) adjustment pattern, as can be verified by graphical means.

Now consider the typical Ricardian equivalence experiment, consisting of a postponement of taxation. In the model this amounts to a reduction in To. This creates a primary deficit at impact. (g (t) > ro) so that government debt starts to rise. In terms of Figure 14.7, both the k = 0 line and the b = 0 line shift up, the former by more than the latter. In the long run, government debt, the capital stock, and output (all measured in efficiency units of labour) rise as a result of the tax cut.

Jacobian matrix of the two-by-two system of differential equations, and Xi and A2 are the two characteristic roots. As tr(A) Al +A2 = (r - n)] - [n + 3 — Sr] < 0, both roots are negative, i.e. as we already verified by graphical means, the system is stable. Clearly, Ricardian equivalence does not hold in the Solow-Swan model. A temporary tax cut boosts consumption, depresses investment, and thus has real effects.

14.5 The Ramsey Model

Up to now we have assumed an ad hoc savings function according to which aggregate saving is a constant fraction of income (see (14.2)). Whilst the underlying consumption function works rather well empirically, there are serious theoretical objections that can be raised against it. In Chapter 6, for example, it was shown that a forward-looking "representative" agent would condition consumption not on some measure of disposable income but rather on lifetime wealth, comprising the sum of financial and human wealth. In this section we investigate the implications for growth of the intertemporal consumption theory.

14.5.1 The represe

Assume that the repre fect foresight. The con depends on the con , tive but diminishing n In addition the follow

lim U' [c(t)] = 00-4)

The consumer derives] inelastically supply I labour supply grows , The consumer's utility and future felicity. N, •

00

A(0) f U[c(t)Je

where A(0) is lifetime u consumer holds financ The budget identity is

C(t) + A(t)

where W(t) is the real (14.54) says that the si hand side) is equal to rewriting (14.54) in per

without further restrit. borrow all it likes at the initely and thus be abl, economically nonsensic

lim a(t) exp

o

Intuitively, (14.56) s,. assets and is not allow,

1 ° Alternatively, one mi ..-' • across time via operative bet] interpretation.

11 Under the extended- .

12 Compare the discussion in equality form is an out