Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

I

' if debt is zero initially but because of the diminishing

(14.49)

ent surplus (deficit) so that Figure 14.7. The Buiter rule ^ csibly cyclical) adjustment I

riment, consisting of a postxluction in To. This creates a t debt starts to rise. In terms ft up, the former by more apital stock, and output (all

f the tax cut.

(14.50)

(14.51)

0 is the determinant of the equations, and A1 and A2

- (r - n)] - [n + 3 - sf'] < 0, raphical means, the system in the Solow-Swan model.

, tment, and thus has real

according to which aggre- ' ). Whilst the underlying re are serious theoretical for example, it was shown "tion consumption not on me wealth, comprising the "vestigate the implications

Chapter 14: Theories of Economic Growth

14.5.1 The representative consumer

Assume that the representative consumer is infinitely lived l° and blessed with perfect foresight. The consumer experiences instantaneous utility (or "felicity") which depends on the consumption flow c(t). The felicity function, U(c(t)), exhibits positive but diminishing marginal utility and thus satisfies U'(c(t)) > 0 and U"(c(t)) < 0. In addition the following Inada-style conditions are imposed:

lim U'[c(t)] = +oo, lim Ujc(t)] = 0. (14.52)

c(t)-,o c(t)-,00

The consumer derives no felicity from the consumption of leisure and is assumed to inelastically supply L(t) units of labour to a competitive labour market. As before,

The consumer's utility functional is defined as the discounted integral of present and future felicity. Normalizing the present by t = 0 ("today") we obtain:

A(0) f U [c(t)]e- Pt dt , p > 0, (14.53)

where A(0) is lifetime utility and p is the pure rate of time preference. At time t, the consumer holds financial assets totalling A(t) and yielding a rate of return of The budget identity is thus given by:

C(t) + A(t) r(t)A(t) + W (t)L(t), (14.54)

where W(t) is the real wage and C(t) c(t)L(t) is aggregate consumption. Equation (14.54) says that the sum of income from financial assets and labour (the righthand side) is equal to the sum of consumption and saving (the left-hand side). By rewriting (14.54) in per capita form we obtain:

a(t) [r(t) - n] a(t) + W (t) - c(t), (14.55)

where a(t) A(t)/ L(t). As it stands, (14.55) is still no more than an identity, i.e. without further restrictions it is rather meaningless. Indeed, if the household can borrow all it likes at the going interest rate r(t) it will simply accumulate debt indefinitely and thus be able to finance any arbitrary consumption path. To avoid this economically nonsensical outcome, we need to impose a solvency condition:

Intuitively, (14.56) says that the consumer does not plan to "expire" with positive assets and is not allowed by the capital market to die hopelessly indebted. 12

10 Alternatively, one might assume a representative family dynasty, the members of which are linked across time via operative bequests. See Barro and Sala-i-Martin (1995, p. 60) and Chapter 6 for this interpretation.

11 Under the extended-family interpretation the family grows exponentially at rate 0L .

12 Compare the discussion in Barro and Sala-i-Martin (1995, pp. 62-66). Strictly speaking (14.56) in equality form is an outcome of household maximizing behaviour rather than an a priori restriction.

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The Foundation of Modern Macroeconomics

By integrating (14.55) over the (infinite) lifetime of the agent and taking into account the solvency condition (14.56), we obtain the household lifetime budget constraint:

CO

where a(0) is the initial level of financial assets, h(0) is human wealth, and R(t) is a discounting factor:

Equation (14.59) shows that human wealth is the present value of the real wage, i.e. the market value of the agent's time endowment. From the viewpoint of the consumer, the right-hand side of (14.57) is given and acts as a restriction on the time paths for consumption that are feasible.

The consumer chooses a time path for c(t) in order to attain a maximum lifetime utility level A(0) (given in (14.53)), subject to the lifetime budget restriction (14.57). The first-order conditions are (14.57) and:

where A is the marginal utility of wealth, i.e. the Lagrange multiplier associated with the lifetime budget restriction (14.57). The left-hand side of (14.60) represents the marginal contribution to lifetime utility (evaluated from the perspective of "today", i.e. t = 0) of consumption in period t. The right-hand side of (14.60) is the lifetime marginal utility cost of consuming c(t) rather than saving it. The marginal unit of c(t) costs exp ( - [R(t) - nt]) from the perspective of today. This cost is translated into utility terms by multiplying it by the marginal utility of wealth. 13

Since the marginal utility of wealth is constant (i.e. it does not depend on t), differentiation of (14.60) yields an expression for the optimal time profile of consumption:

where we have used the fact that dR(t) I dt = r(t) (see (14.58)) and where 0[.] is the elasticity of marginal utility which is positive for all positive consumption levels

By using (14.56) we avoid getting bogged down in technical issues. See also Chapter 6 for an intuitive discussion of the solvency condition in macroeconomics.

13 See Dixit (1990, ch.10) for intuitive discussions of apparently intractable first-order conditions.

because of the strict co

0 [c(t)] = U" [c( t

The intertemporal sui. , tionship, the expressioi equation:

1 dc(t) = a [c(t c(t) dt

Intuitively, if a [.] is 10 the household to ad( case the willingness to marginal utility is high, opposite holds if a [.1 is I that a small interest gar As it stands, (14.63 ) rendering (14.63) difficL for consumption impo! There are two useful 1L4.I

U [c(t)]

and the iso-elastic

c(t) rl

U (t) ] = log" c( t

It is not difficult to very two functional forms respective Euler equatio

dc(t)

dt =[r(t)a

1 dc(t) = a [r(t) - c(t) dt

So both these utility fur,, But what about the clos

14 The second line in (14.' is to use L'HOpital's rule for a

agent and taking into isehold lifetime budget

(14.57)

n wealth, andR(t) is a

(14.58)

(14.59)

value of the real wage, m the viewpoint of the as a restriction on the

n a maximum lifetime fidget restriction (14.57).

(14.60)

. !tiplier associated with If (14.60) represents the ! perspective of "today", (14.60) is the lifetime it. The marginal unit of This cost is translated

wealth. 13

t does not depend on optimal time profile of

(14.61)

and where 0[.] is the

-e consumption levels

<- Chapter 6 for an intuitive

'le first-order conditions.

The intertemporal substitution elasticity, a [1, is the inverse of 0 [1. By using this relationship, the expression in (14.61) can be rewritten to yield the consumption Euler equation:

Intuitively, if GT] is low, a large interest gap (r(t) - n - p) is needed to induce the household to adopt an upward-sloping time profile for consumption. In that case the willingness to substitute consumption across time is low, the elasticity of marginal utility is high, and the marginal utility function has a lot of curvature. The opposite holds if a [.] is high. Then, the marginal utility function is almost linear so that a small interest gap can explain a large slope of the consumption profile.

As it stands, (14.63) is of little use to us because still depends on consumption, rendering (14.63) difficult to work with and the derivation of a closed-form solution for consumption impossible. For this reason an explicit form for U[.] is chosen. There are two useful functional forms, i.e. the exponential utility function:

It is not difficult to verify that the substitution elasticities corresponding with these two functional forms are, respectively, a [.] = alc(t) and a [.] = a, so that the respective Euler equations are:

So both these utility functions lead to very simple expressions for the Euler equation. But what about the closed-form solution for consumption itself?

14 The second line in (14.65) is obtained from the first line by letting 1/a approach unity. The trick is to use L'FlOpital's rule for calculating limits of the 0 0 type:

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The Foundation of Modern Macroeconomics

We focus on the iso-elastic case, leaving the exponential case as an exercise for the reader. First we note that (14.67) can be integrated to yield future consumption c(t) in terms of current consumption c(0):

By substituting this expression into the household budget constraint (14.57) we obtain in a few steps:

00

c(0)e[R(0-nt- pt] e-[R(t)-nt] dt = a(0) + h(0)

where A(0) -1 is the propensity to consume out of total wealth:

According to (14.69), consumption in the planning period is proportional to total wealth. Some special cases merit attention. If a = 1 (so that U[.] in (14.65) is logarithmic), A(0) -1 = p and the household consumes a constant fraction of total wealth in the current period. Income and substitution effects of a change in the interest rate exactly cancel in this case (see also Chapter 6). Another special case is often used in the international context. If a country is small in world financial markets and thus faces a constant world interest rate r* it follows from (14.58) that

and from (14.70) that A(0) -1 = a p (1— o- )(r* — n). (Of course restrictions on the parameters must ensure that A(0) remains positive.)

14.5.2 The representative firm

Perfectly competitive firms produce a homogeneous good by using capital and labour. Since there are constant returns to scale to the production factors taken together (see (P1)) there is no need to distinguish individual firms and we can make use of the notion of a representative firm, which makes use of technology as summarized by the production function in (14.6). (We abstract from technical progress to keep things simple.)

The stockmarket value of the firm is given by the discounted value of its cash flows:

where R(t) is the discounting factor given in (14.58), I (t) is gross investment by the firm (see equation (14.4)), and sI is an investment subsidy to be used below (in

this section we ass, subject to the capit,.. of the firm's choice S4 i.e. there are no ac, discussion of such co! we find that the obje

V(0) = K(0) + .f

where K(0) is the in about factor inputs is

and K(t) yields the fail I

FL [K(t), L(t)] = 11

By substituting the 1.. the linear homogene K(0). In the absence c the (replacement) v By writing the pre rewrite the margin,:

FK [K(t), L(t)] = f

We now have all the sake of convenience associated with an b. combines equations ( is obtained by comma

14.5.3 The phase

The model in Table 1 phase portrait which rants some additions space for which the p

15 In deriving (14.72, I.

Jo1') [K(t) — r(t)K(t)ic

where we have used the

16 We use F = FKK f expression.

ntial case as an exercise for I to yield future consumption

(14.68)

idget constraint (14.57) we

(14.69)

ii wealth:

(14.70)

eriod is proportional to total c , that U[.] in (14.65) is log- a constant fraction of total In effects of a change in the er 6). Another special case y is small in world financial

•it follows from (14.58) that

•- ri). (Of course restrictions t ive. )

good by using capital and production factors taken klual firms and we can make use of technology as sum- : ct from technical progress

•scounted value of its cash

, (14.71)

p

1(t) is gross investment by

!isidy to be used below (in

I

Chapter 14: Theories of Economic Growth

this section we assume sI = 0). The firm maximizes its stockmarket value (14.71) subject to the capital accumulation constraint (14.4). Implicit in the formulation of the firm's choice set is the notion that it can vary its desired capital stock at will, i.e. there are no adjustment costs on investment (see Chapter 4 and below for a discussion of such costs). Indeed, by substituting (14.4) into (14.71) and integrating we find that the objective function for the firm can be written as:ls

where K(0) is the initial capital stock. Equation (14.72) shows that the firm's decision about factor inputs is essentially a static one. Maximization of V(0) by choice of L(t) and K(t) yields the familiar marginal productivity conditions for labour and capital:

By substituting the marginal productivity conditions (14.73) into (14.72) and noting the linear homogeneity property of the production function we find that V(0) = K(0). In the absence of adjustment costs on investment the value of the firm equals the (replacement) value of its capital stock and Tobin's q is unity.

By writing the production function in the intensive form (see (14.8)) we can rewrite the marginal products of capital and labour as follows: 16

We now have all the ingredients of the model and we summarize them for the sake of convenience in Table 14.1. Equation (T1.1) is the rewritten Euler equation associated with an iso-elastic felicity function (see the expression in (14.67)). (T1.2) combines equations (14.3)-(14.5) and is written in the intensive form. Finally, (T1.3) is obtained by combining the relevant conditions in (14.73) and (14.74).

14.5.3 The phase diagram

The model in Table 14.1 can be analysed to a large extent by means of its associated phase portrait which is given in Figure 14.8. The construction of this diagram warrants some additional comment. The k(t) = 0 line represents points in (c(t), k(t)) space for which the per capita capital stock is in equilibrium . The Inada conditions

15 In deriving (14.72) the key thing to note is:

[K(t) - r(t)K(t)je-R(t) dt = f d[K(t)e-R(1= —K(0),

where we have used the fact that limK(0_,,, K(t)e-R(t) = 0 in the final step.

16 We use F = FKK + FLL, which follows from Euler's theorem, and FK = f' to derive the second expression.

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The Foundation of Modern Macroeconomics

Table 14.1. The Ramsey growth model

Notes: c(t) is per capita consumption, k(t) is the capital-labour ratio, and r(t) is the interest rate. Capital depreciates at a constant rate 8 and the population grows exponentially with rate n.

k (0=0

kKR kGR kM k (t)

Figure 14.8. Phase diagram of the Ramsey model

ensure that it passes through the origin and is vertical there (see point A 1 ). Golden rule consumption occurs at point A2 where the k(t) = 0 line reaches its maximum:

dk(t) k(t)=O

The maximum attainable capital-labour ratio, kmAx , occurs at point

capita consumption is zero and total output is needed for replacement investment:

f (kmAx)

(14.76)

kmAx = 8 +11.

Finally, the capital dynamics depends on whether there is more or less capital than the golden rule prescribes:

This has been indicated by horizontal arrows in Figure 14.8.

The -(t) = 0 line r, is flat. In view of (T1.1 the rate of time prefer the superscript "KR" re result. The Keynes—Rai ratio (see (T1.3)). Her

I

f,(kKR) = 8 n

The comparison of (14 lies to the left of k". Ramsey capital—labour the modified golden ru..

14.5.4 Efficiency pr,

Perhaps the most impc possibility of dynamic in the Solow—Swan 1: because there are no mi so there is no reason t( economics. 1

The efficiency prope the equivalence of the solution chosen by a bl imize lifetime utility the production functi,

The Hamiltonian ass,

7-i(t) -=- U [c(t)Je- '

where ,u(t) is the co-sta ing the social optimui..

aR(t)ac(t) =

it(t) = 87-c(t) = ak(t)

where the superscript ' interest rate can be dc:,

17 As well as an initial cos and capital, and a transversal (1971, pp. 405-416).

(T1.1)

(T1.2)

(T1.3)

r(t) is the interest rate. Capital •i rate n.

A3

AMAX k (t)

model

ere (see point A1). Golden ^ e reaches its maximum:

(14.75)

irs at point A3, where per replacement investment:

(14.76)

more or less capital than

(14.77)

8.

Chapter 14: Theories of Economic Growth

The e(t) = 0 line represents points for which the per capita consumption profile is flat. In view of (T1.1) this occurs at the point for which the interest rate equals the rate of time preference plus the rate of population growth, r" p + n, where the superscript "KR" refers to "Keynes—Ramsey", who were the first to discover this result. The Keynes—Ramsey interest rate is associated with a unique capital-labour

The comparison of (14.75) and (14.78) reveals that r(kKR) exceeds r(kGR), i.e. kla lies to the left of kGR. Finally, we note that the expression determining the Keynes— Ramsey capital—labour ratio (namely (14.78)) is often referred to in the literature as

the modified golden rule.

14.5.4 Efficiency properties of the Ramsey model

Perhaps the most important property of the Ramsey model is that it precludes the possibility of dynamic inefficiency and oversaving, phenomena which are possible in the Solow—Swan model. Intuitively, this result is perhaps not that surprising because there are no missing markets, distortions, and external effects in the model so there is no reason to suspect violation of the fundamental theorems of welfare economics.

The efficiency property of the Ramsey model can be demonstrated by proving the equivalence of the market outcome (discussed in the previous section) and the solution chosen by a benevolent social planner. Such a social planner would maximize lifetime utility of the representative agent (A(0) given in (14.53)) subject to the production function (14.6) and the capital accumulation constraint (14.4). 17

The Hamiltonian associated with the command optimum is given by:

7-1(t) U [c(t)] e- Pt + au(t) (k(t)) — c(t) — (n + 8)k(t)] , (14.79)

where ,u(t) is the co-state variable. The first-order necessary conditions characterizing the social optimum are:

where the superscript "SO" denotes socially optimal values. The socially optimal interest rate can be defined as rs° (t) r k[ sorr)]— 8, so that (14.79)—(14.80) can be

17 As well as an initial condition for the capital stock, non-negativity constraints for consumption and capital, and a transversality condition. See Blanchard and Fischer (1989, pp. 38-43) and Intriligator (1971, pp. 405-416).

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The Foundation of Modern Macroeconomics

combined to yield an easily interpretable expression for the optimal time profile of consumption:

Equation (14.82) has exactly the same form as (14.63) so that the planning solution and market outcome coincide. 18 Hence, by removing the ad hoc saving function from the Solow-Swan model there is no possibility of oversaving any more.

14.5.5 Transitional dynamics and convergence in the Ramsey model

As was demonstrated graphically with the aid of Figure 14.8, the Ramsey model is saddle-point stable. An exact solution for the saddle path can in general not be obtained, however, rendering the study of the convergence properties of the model slightly more complicated than was the case for the Solow-Swan model. By linearizing the model around the initial steady state, E 0 , however, the approximate transitional dynamics can be studied in a relatively straightforward manner.

After linearizing the model in Table 14.1 we obtain the following system of firstorder differential equation:

where the superscript "k" denotes initial steady-state values. The Jacobian matrix on the right-hand side of (14.83) is denoted by A. Since tr(A)-a7 A l + A2 = p > 0 and IA I ---- A1X2 = a c*f"(k*) < 0, where A l and A2 are the characteristic roots of A, equation (14.83) confirms saddle-point stability, i.e. ),. 1 and A2 have opposite signs. The absolute value of the stable (negative) characteristic root determines the approximate convergence speed of the economic system. After some manipulation we obtain the following expression:

18 We have also used the fact that the initial condition and the capital accumulation constraint are the same for the market and planning solutions. This implies that the levels of the interest rate, capital, and consumption also coincide for the two solutions.

where ala =_-- (1 - labour in the produ. income (both evalual model predicts a co:. mate of about 2% pe not immediately apps model also predicts to ters. This has been dei 14.2. We calibrate the of pure time preferenc at 2% (n = 0.02), and state implies r* = p -- By varying the capital and the production vergence speed /3. As even faster conver,_ function and the pros that 010-KL = 1) then fi is a staggering 10.9791 and the felicity functi4 come anywhere near

14.5.6 An open-ecc

Up to this point we hi resentation of the Ra: clears the domestic It tionship with the ca; which is small in wt.):

-r the optimal time profile of

n)]

so that the planning solution the ad hoc saving function oversaving any more.

e in the Ramsey model

pre 14.8, the Ramsey model 1,11e path can in general not 3nvergence properties of the )r the Solow—Swan model. By r. ), however, the approximate raightforward manner.

the following system of first-

(14.83)

values. The Jacobian matrix

", ce + A2 = p > 0 re the characteristic roots of i.e. Al and A2 have opposite teristic root determines the mi. After some manipulation

(14.84)

1]

vital accumulation constraint are levels of the interest rate, capital,

Chapter 14: Theories of Economic Growth

Table 14.2. Convergence speed in the

Ramsey model

where an (1 — WK) f'/( — kf") is the substitution elasticity between capital and labour in the production function and WK kr If is the capital share in national income (both evaluated in the initial steady state). Recall that the Solow—Swan model predicts a convergence speed which exceeds the empirically relevant estimate of about 2% per annum by quite a margin (see section 3.3). Although it is not immediately apparent from the formula in (14.84) it turns out that the Ramsey model also predicts too high a rate of convergence for realistic values of the parameters. This has been demonstrated by means of some numerical simulations in Table 14.2. We calibrate the steady state of a fictional economy as follows. We set the rate of pure time preference at 3% per annum (p = 0.03), the rate of population growth at 2% (n = 0.02), and the depreciation rate of capital at 5% (8 = 0.05). The steady state implies r* = p + n, (k/y)* = WK/(r* + 8), and (c/y)* = 1 — (8 + n)WK/(r* + 6).

By varying the capital share (WK) and the ratio of elasticities of the felicity function and the production function (a/an) we obtain a number of estimates for the convergence speed /3. As is clear from the results in Table 14.2, the Ramsey predicts even faster convergence than the Solow model! For example, if both the felicity function and the production function feature a unitary substitution elasticity (so that a = 1) then for the realistic capital share of WK = the convergence speed is a staggering 10.97% per annum. Only if the capital share is unrealistically high and the felicity function is relatively inelastic (so that 0- /an is low) does the model come anywhere near to matching the empirically observed speed of convergence.

145.6 An open-economy Ramsey model

Up to this point we have focused attention on the traditional closed-economy representation of the Ramsey model. In a closed economy, the domestic interest rate clears the domestic rental market for physical capital and thus bears a close relationship with the capital-labour ratio; see equation (T1.3). In an open economy, which is small in world financial markets, on the other hand, the interest rate is

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The Foundation of Modern Macroeconomics

fully determined abroad and is thus exogenous. It is clear that the marginal productivity condition for capital (equation (T1.3)) can only hold for a small open economy if the physical capital stock is perfectly mobile across countries! Indeed, a small increase in the world interest rate must be accompanied by an immediate and instantaneous outflow of physical capital in order to restore equality between the domestic marginal product of capital and the world interest rate.

Apart from the fact that perfect mobility of physical capital is extremely unrealistic, it also has a very unfortunate implication in that it renders the convergence speed of the economy infinitely large! In technical terms, capital is changed from a slow-moving (predetermined) variable to a jumping variable. The traditional solution to this problem is to assume that physical capital is firm specific and thus cannot move costlessly and instantaneously. Financial capital, such as bonds and ownership claims of domestic assets, is of course perfectly mobile in this context so that yields on domestic and foreign assets are equalized. In technical terms imperfect mobility of physical capital is modelled by assuming that the firm must incur installation costs associated with the investment process.

The small open economy assumption also causes a complication on the consumption side of the Ramsey model. Indeed, as was shown above, the representative household chooses its optimal consumption profile according to the Euler equation (T1.1). But if the rate of interest is exogenous (i.e. r(t) = r*, where r* is the world interest rate) then consumption can only ever attain a steady state (4t) = 0) if the world interest rate happens to be equal to the exogenous population growth plus the rate of time preference, i.e. r* = p n must be satisfied. In any other case, the country either follows an ever-decreasing path of per capita consumption if its citizens are impatient (p +n > r*) or the country saves so much that it eventually ceases being small in world financial markets (with very patient citizens, p n < r*). In order to avoid these difficulties we assume that the following "knife-edge" condition holds:

p n = r*• (14.85)

An immediate consequence of (14.85) in combination with (T1.1) is that per capita consumption of the representative household is completely smoothed over time, i.e. c(t)/c(t) = 0 for all time periods.

We now consider the behaviour of the representative (domestic) firm facing adjustment costs for investment. The stockmarket value of the firm is still given by (14.71) but net and gross investment are now related according to a concave installation function:

where C.) rep We assume n._

> 0, and

The firm c in order to n„ initial conditi necessary col.,

W (t) = FL q(t)(1)' I(

q(t) =[r(1

where q(t) is 1 value of instal As was demi can be written

Y(t)

where X(t) is installation cc abstract from stock of net fo equal to gross The current a( earnings on fc is thus:

AF(t) = r*

which can be

ilF(t) =

where we hay freely borrow

19 See Chapter costs.

20 See the Inte order conditioi