Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

:Jew that the marginal proly hold for a small open across countries! Indeed, ompanied by an immediate to restore equality between

interest rate.

capital is extremely unreal- : it renders the convergence capital is changed from a able. The traditional solu- al is firm specific and thus capital, such as bonds and v mobile in this context so

J.In technical terms imperthat the firm must incur

ss.

i complication on the convn above, the representative Drding to the Euler equation = r*, where r* is the world steady state (c(t) = 0) if the

s population growth plus ed. In any other case, the 'pita consumption if its citiuch that it eventually ceases ent citizens, p + n < r*). In ng "knife-edge" condition

(14.85)

Kith (T1.1) is that per capita ' 'tely smoothed over time,

I

tive (domestic) firm facing ue of the firm is still given ted according to a concave

(14.86)

Chapter 14: Theories of Economic Growth

where (I)(.) represents the presence of installation costs associated with investment. We assume that the installation cost function satisfies the usual properties: 43(0) = 0, VC) > 0, and (t."(.) < 0. 19

The firm chooses time paths for investment, labour demand, and the capital stock in order to maximize V(0) subject to the capital accumulation identity (14.86), an initial condition for the capital stock, and a transversality condition. The first-order necessary conditions are the constraint (14.86) and: 2°

where q(t) is Tobin's q (its current value, q(0), measures the marginal (and average) value of installed capital, K(0), i.e. V(0) = q(0)K(0)).

As was demonstrated in Chapter 11, gross domestic product in an open economy can be written as follows:

where X(t) is net exports (i.e. the trade balance) and gross investment (inclusive of installation costs) appears in the national income identity. Note furthermore that we abstract from government consumption for convenience. Designating AF (t) as the stock of net foreign assets in the hands of domestic agents, gross national product is equal to gross domestic product plus interest earnings on net foreign assets, r*AF(t). The current account of the balance of payments is equal to net exports plus interest earnings on foreign assets. The dynamic equation for the stock of net foreign assets is thus:

which can be written in per capita form as:

where we have used the fact that p = r* — n (see (14.85)). Although the country can freely borrow from (or lend to) the rest of the world, it must obey an intertemporal

19 See Chapters 2 and 4 for an extensive discussion of the theory of investment based on adjustment costs.

20 See the Intermezzo on Tobin's q-theory of investment in Chapter 4 for a derivation of these firstorder conditions.

433

The Foundation of Modern Macroeconomics

solvency condition of the form:

Equations (14.92) and (14.93) in combination imply that there is a relationship between the initial level of net foreign assets per capita, aF,0, and the present value of future trade balances:

To the extent that the country possesses positive net foreign assets (aF,o > 0), it can afford to run present and future trade balance deficits. All that nation-wide solvency requires is that the present value of these trade balance deficits (the right-hand side of (14.94)) add up to the initial level of net foreign assets (left-hand side of (14.94)).

We now possess all the ingredients of the open-economy Ramsey model and we restate its key equations for the sake of convenience in Table 14.3.

Equation (T3.1) shows that per capita consumption is completely smoothed over time. As was pointed out above, this result is a direct consequence of the assumption expressed in (14.85). Equation (T3.2) implicitly determines the optimal investmentcapital ratio as a function of (subsidy-adjusted) Tobin's q. Equation (T3.3) gives the dynamic evolution of Tobin's q and (T3.4) does the same for the capital stock per worker. Finally, (T3.5) is the current account equation which is obtained by substituting the production function, f (k(t)), into (14.92).

Model solution and convergence speed

The model is quite unlike the growth models that were studied up to this point because it contains a zero root and thus displays hysteretic properties in the sense that the steady state depends on the initial conditions. 21 Technically, the model

Table 14.3. The Ramsey model for the open economy

Notes: c(t) is per capita consumption, k(t) is the capital-labour ratio, q(t) is Tobin's q, i(t) is gross investment per worker, .51 is an investment subsidy, and AF(t) is net foreign assets per worker. See also Table 14.1.

21 See Turnovsky (1995, ch. 12), Sen and Turnovsky (1990), and Giavazzi and Wyplosz (1985) for a further discussion. See also Chapter 2 above for an example of a hysteretic model in discrete time.

solution proceeds as autonomous subs\ once the solutions lc into the nation-wide capita consumption. Since the model is

it around the steady s (T3.4). To keep the ITII function:

i(t) 1

•k(t) - 1 -

with 0 < cri < 1. _ tion. The lower is Cr, the international ma investment demanu

i(t)

k(t) = g(q(t), .51)

By inserting (14.96) expression for the i:.

[ k(t) =

4(0 [

The Jacobian matrix ( elements by 8y . Thk. . tive trace (equal to p)

This implies that the sign.22 Denoting the it follows from (14.9;

i.e. the unstable R. tion speed in the ecl of the investment investment and the a

If the initial capita steady state provith.

22 Recall that the trace product of the charactedi

studied up to this point tic properties in the sense 21 Technically, the model

(T3.1)

(T3.2)

(T3.3)

(T3.4)

(T3.5)

robin's q, i(t) is gross investment

-ker. See also Table 14.1.

lli and Wyplosz (1985) for a c model in discrete time.

The Jacobian matrix on the right-hand side of (14.97) is denoted by A/ and its typical elements by 84 . The investment system is saddle-point stable because Al has a posi-

tive trace (equal to p) and a negative determinant (equal to (1 "(V)/[((r)201]). This implies that the characteristic roots of Al are real, distinct, and opposite in

sign.22 Denoting the stable and unstable roots by, respectively -Xi < 0 and > 0 it follows from (14.97) that:

A2 — = tr(An = X2 = p Xi > p, (14.98)

i.e. the unstable root equals the pure rate of time preference (p) plus the transition speed in the economy (represented by X1). Note that the adjustment speed of the investment system (X1) is finite due to the existence of installation costs of investment and the associated short-run immobility of capital.

If the initial capital stock is denoted by k0, then the system converges to the steady state provided it is on the saddle path. Deferring the technical details of the

22 Recall that the trace and determinant of the Jacobian matrix equal, respectively, the sum and the product of the characteristic roots.

The Foundation of Modern Macroeconomics

Figure 14.9. Investment in the open economy

derivation to the appendix of this chapter, we find that the solution to (14.97) is:

The solution path is illustrated in Figure 14.9. For the initial capital stock, k0, Tobin's q is above its equilibrium level and the economy moves gradually towards the steady-state equilibrium E0.

Now that we know the dynamic paths for the capital stock and Tobin's q (and thus, by (14.96), the implied path for investment) we can work out the restriction implied by national solvency. First, we linearize the production function, and the investment function (14.96) around the steady state:

el)

where g* g(q*,si) and g; gq (q* , si). By using (14.99)—(14.101) we find the (approximate) path for i(t) – y(t):

consumption stays constant during the transition, i.e. c(t) = c*. By using this result

as well as equat., following express

aF ,o = —6.* + P

c* +

c* + i*

I

where we have u,

from the steady-st (14.103) can be re

C2

aF ,o + (T) kc

2

As Sen and Turnoff resents the initial I interpreted as natit aF,0 + k0 , plus the starting from the i

The striking ft depends on the A. alluded to above. I the model cons's:

q* 4), i(±*

k*

f'(k * ) =

pa; + f (k* )

which jointly L, structure of the m( hysteresis and are

B In particular, (14 (14.106) determines h. (14.108) are and. hysteretic property ex)

IC(t) = 0

=0

k (t)

imy

the solution to (14.97) is:

(14.99)

(14.100)

1 capital stock, k0, Tobin's yes gradually towards the

k and Tobin's q (and thus, out the restriction implied Unction, y(t) = f (k(t), and

(14.101)

', 9)-(14.101) we find the

t) -

(14.102)

,.1) shows that per capita = c*. By using this result

Chapter 14: Theories of Economic Growth

as well as equation (14.102) in the nation-wide solvency condition we obtain the following expression:

C* cx)

aF,0 — f [i(t) — y(t)] e— Pt dt P o

where we have used the fact that p + A (see (14.98)) in the final step. It follows from the steady-state version of (14.92) that - y* (since er'F' = 0) so that (14.103) can be rewritten as follows:

As Sen and Turnovsky (1990, p. 287) point out, the left-hand side of (14.104) represents the initial value of total resources available to the economy and can thus be interpreted as national wealth. National wealth consists of initial non-human wealth, aF,0 + ko, plus the present value of resources generated by capital accumulation starting from the initial capital stock, ko.

The striking feature of the open-economy Ramsey model is that its steady state depends on the initial stock of assets, aF,o and ko. This is the hysteretic property alluded to above. In the steady state we have that c (t) = 4(t) = k(t) = ago = 0 and the model consists of equation (14.104) as well as:

which jointly determine the steady-state values q*, i* , k*, c*, and a'F' . Given the structure of the model, only consumption and the net stock of foreign assets display hysteresis and are thus a function of the initial conditions. 23

23 In particular, (14.107) determines i* I k* as a function of n + 3, (14.105) then determines q* and (14.106) determines k* (and thus i*). The only variables remaining to be determined by (14.104) and (14.108) are c* and (4. Sen and Turnovsky (1990) show that if labour supply is endogenous, the hysteretic property extends to investment and the capital stock also.

437

The Foundation of Modern Macroeconomics

Effects of an investment subsidy

We are now in the position to use the model to study the effects of an investment subsidy on the macroeconomy. To keep things simple we restrict attention to the case of an unanticipated and permanent increase in the investment subsidy. It is most convenient to determine the long-run effect first. Equation (14.107) shows that is constant, so that it follows from (14.105) that q* is proportional to (1 — sj). Hence, Tobin's q falls in the long run:

Equation (14.106) can be used to derive the long-run effect on the stock of capital per worker:

Hence, investment and the capital stock (both measured per worker) rise equiproportionally in the long run. The national wealth constraint (14.104) shows that the composition of wealth changes also, i.e. the increase in the domestic capital stock leads to a reduction in the long-run stock of net foreign assets:

The net effect on consumption is ambiguous.

The transitional effects of the policy shock can be studied with the aid of Figure 14.10. In that figure, ko is the initial capital stock per worker, the economy is at point A and is heading towards the steady state at E0 (where the steady-state capital stock per worker is 4). The long-run effect on the capital stock is positive (see (14.110)) and saddle-point stability requires that the economy must be on the stable arm of the saddle path. By using the expression for the saddle path (given in (14.100)) we obtain the impact effect for Tobin's q:24

24 We have used equations (14.109) and (14.110) in going from the first to the second line. In going from the second to the third line we have used some results for the characteristic roots, i.e. Ai X2 = -f" (k*)312 and A2 = p +

q (t)

q0*

q1 *

Figure '

The impact effect on 1 side is negative whi., fact that both the k(t) in the investment sul the (ilk) ratio is consti (see (14.96)), Tobin's

line shifts down. At tJ

4(t) = 0 line.

In Figure 14.10 drawn under the assui on Tobin's q (given i jumps from point A ti the new steady-state t Why is the impact

the ambiguity arises economy, A. 1 . If adju is relatively high, phl goods are close substil goods and thus also tl 1993, p. 13). The opp gram shows, howe \ - takes place (as B lies to El over time.

'he effects of an investment we restrict attention to the le investment subsidy. It is Equation (14.107) shows ) that q* is proportional to

(14.109)

ect on the stock of capital

(14.110)

- ed per worker) rise equiistraint (14.104) shows that se in the domestic capital oreign assets:

(14.111)

led with the aid of Figure ker, the economy is at point steady-state capital stock ck is positive (see (14.110)) st be on the stable arm of path (given in (14.100)) we

(14.112)

the first to the second line. In for the characteristic roots, i.e.

Chapter 14: Theories of Economic Growth

q (t)

Figure 14.10. An investment subsidy with high mobility of physical capital

The impact effect on Tobin's q is ambiguous because the first term on the right-hand side is negative whilst the second term is positive. The ambiguity arises from the fact that both the k(t) = 0 line and the /At) = 0 lines shift as a result of the increase in the investment subsidy. Recall that the k(t) = 0 line represents points for which the (ilk) ratio is constant. Since an increase in .s / leads to a higher desired (ilk) ratio (see (14.96)), Tobin's q must fall to restore capital stock equilibrium, i.e. the k(t) = 0 line shifts down. At the same time, the boost in si leads to an upward shift in the 4(t) = 0 line.

In Figure 14.10 the new steady-state equilibrium is at E1 and the saddle path is drawn under the assumption that the capital stock effect is dominated by the effect on Tobin's q (given in (14.112)), so that 0. At impact the economy jumps from point A to point B, after which gradual adjustment takes place towards the new steady-state equilibrium E 1 .

Why is the impact effect on Tobin's q ambiguous? Equation (14.112) shows that the ambiguity arises because dq(0)1 cis' depends on the adjustment speed in the economy, 1 . If adjustment costs on investment are relatively low (a/ ti 0), then Al is relatively high, physical capital is highly mobile, and installed and new capital goods are close substitutes. The investment subsidy reduces the price of new capital goods and thus also the value of the installed capital stock in that case (Bovenberg, 1993, p. 13). The opposite holds if adjustment costs are severe (cvi 1). As the diagram shows, however, regardless of the sign of net capital accumulation takes place (as B lies above the new k(t) = 0 line) and the economy moves from B to E1 over time.

439

The Foundation of Modern Macroeconomics

14.5.7 Fiscal policy in the Ramsey model

We now return to the closed-economy Ramsey model summarized in Table 14.1 and investigate the effects of government consumption at impact, during transition, and in the long run. This ultimately leads into a discussion of Ricardian equivalence. We assume that government consumption has no productivity-enhancing effects and, to the extent that it affects the welfare of the representative agent, does so in a weakly separable manner. 25 The only change that is made to the Ramsey model relates to equation (T1.2) which is replaced by:

k(t) = f (k(t)) - c(t) - g(t) - (3 + n)k(t), (14.113)

where g(t) G(t)/L(t) is per capita government consumption. Government consumption withdraws resources which are no longer available for private consumption or replacement of the capital stock. As a result, for a given level of per capita public consumption, g(t) = g, the k(t) = 0 line can be drawn as in Figure 14.11. Several conclusions can be drawn already. First, the existence of positive government consumption does not reinstate the possibility of dynamic inefficiency in the Ramsey model. The golden-rule capital stock per worker is not affected by g, although of course the golden-rule per capita consumption level is affected. Second, the issue of multiple equilibria also does not arise in the Ramsey model with government consumption. In contrast to the situation in the Solow model, provided an equilibrium exists in the Ramsey model it is unique and saddle-point stable.

An unanticipated and permanent increase in the level of government consumption per worker shifts the k(t) = 0 line down, say to (k(t) = 0)1. Since the shock comes as a complete surprise to the representative household, it reacts to the increased level of taxes (needed to finance the additional government consumption) by cutting back private consumption. The representative household feels poorer as a result of the shock and, as consumption is a normal good, reduces it one-for-one:

for all t E [0, oo). There is no transitional dynamics because the shock itself has no long-run effect on the capital stock and there are no anticipation effects. In terms of Figure 14.11 the economy jumps from E0 to E1 .

With a temporary increase in g there are non-trivial transition effects. The representative household anticipates the temporarily higher taxes but spreads the negative effect on human wealth out over the entire lifetime consumption path. As a result,

25 See Turnovsky and Fisher (1995) for the more general cases. With weak separability we mean that the marginal utility of private consumption does not depend on the level of government consumption.

tiic impact

a -1 < -

In terry o: shock Li rate rises, ar

:n A to b initial level w the c. temporary I engineer a •

1%. 11 an . ing transiti,

• -,.ce the acid a grads down wa rd-

A', after ‘+- at precisely

: d.

13" to El.

7arized in Table 14.1 and ct, during transition, and Ricardian equivalence. ivity-enhancing effects ntative agent, does so in ie to the Ramsey model

(14.113)

)tion. Government confor private consumpgiven level of per capita n as in Figure 14.11. Sevof positive government inefficiency in the Ramffected by g, although of ed. Second, the issue of with government con- - rovided an equilibrium

table.

wernment consumption the shock comes as cts to the increased level

nsumption) by cutting 'eels poorer as a result of it one-for-one:

(14.114)

11 the shock itself has no ipation effects. In terms

11)

on effects. The represen- )ut spreads the negative ption path. As a result,

—k separability we mean that DI government consumption.

Chapter 14: Theories of Economic Growth

Figure 14.11. Fiscal policy in the Ramsey model

the impact effect on private consumption is still negative but less than one-for-one:

In terms of Figure 14.11 the economy jumps from E0 to point A. Immediately after the shock the household starts to dissave so that the capital stock falls, the interest rate rises, and (by (T1.1)) the consumption path rises over time. The economy moves from A to B which is reached at the time government consumption is cut back to its initial level again. This cut in g (and the associated taxes) releases resources which allow the capital stock to return to its constant steady-state level. As a result of the temporary boost in government consumption, the policy maker has managed to engineer a temporary decline in output per worker.

With an anticipated and permanent increase in g, the opposite effect occurs during transition. Consumption falls by less than one-for-one (as in (14.115)), but since the government consumption has not risen yet it leads to additional saving and a gradual increase in the capital stock, a reduction in the interest rate, and a downward-sloping consumption profile. At impact the economy jumps from E 0 to A', after which it gradually moves from A' to B' during transition. Point B' is reached at precisely the time the policy is enacted. As g is increased, net saving turns into net dissaving and the capital stock starts to fall. The economy moves from point B' to E1 .

441

The Foundation of Modern Macroeconomics

Ricardian equivalence once again

Ricardian equivalence (see Chapter 6) clearly holds in the Ramsey model as can be demonstrated quite easily. The government budget identity (in per capita form) is given in (14.41). Like the representative household, the government must also remain solvent so that it faces an intertemporal solvency condition of the following form:26

By combining (14.41) and (14.116), we obtain the government budget restriction:

To the extent that there is a pre-existing government debt (b(0) > 0), solvency requires that this debt must be equal to the present value of future primary surpluses.

In principle, there are infinitely many paths for r (t) and g(t) (and hence for the primary deficit), for which (14.117) holds.

The budget identity of the representative agent is given in (14.55). It is modified to take into account that lump-sum taxes are levied on the agent:

By using (14.118) in combination with the household solvency condition (14.56), the household budget restriction is obtained as in (14.57), but with a tax-modified " definition of human wealth:

By substituting the government budget restriction (14.117) into (14.119), the expression for human wealth can be rewritten as:

The path of lump-sum taxes completely vanishes from the expression for human wealth. Since b(0) and the path for g(t) are given, the particular path for lump-sum taxes does not affect the total amount of resources available to the representative agent. As a result, the agent's real consumption plans are not affected either.

26 By substituting (14.39) into (14.41) and integrating the resulting expression we obtain:

lim b(t)c [RW- nt] — b(0) = f [g(t) — 4)] e-[R(t)-nt] dt,

where we have also used (14.58). The first term on the left-hand side is the government solvency condition. By imposing this condition the government budget restriction (14.117) is obtained.

By using (14.120 A

fo°° c(t)e-[R(t)-,

This expression .. breaking article. Ui as household we to reveal the houi

14.5.8 Overlapp

In the previous so vis-a-vis fiscal poll, tion which arises for these classical connectedness of individual generat

Up to now we 1 dynastic family. Si (as the dynastic fa the form of new have infinite lives of their birth. Thi suggested by Wc4. "extensive" rather

Since different birth rate (genera assume that a R consumption in

subject to the buc

à(v, , r)

and the intertemi

lim a(v,, )e

t-,00

where c(v, r) and timer of the rep ing in (14.122) ft