Dynamic inefficiency, endogenous growth, and Ponzi games

Journal of Monetary Economics 32 (1993) 79-104. NorthbHolland

Dynamic inefficiency, endogenous growth, and Ponzi games*

Ian King and Don Ferguson Uniwrsitj~ of’ Victoria. Victoria. B.C. VX W 3P5, Cmadu

Received February 1992. final version received May 1993

We show that in competitive endogenous growth models without externalities, balanced growth equilibria are dynamically efficient. With learning-by-doing externalities, dynamic inefficiency may exist, but due to the mix of capital rather than its scale. These results are obtained independent of whether or not lifetimes are finite or generations overlap. In OLG models with externalities we show that Ponzi games may be feasible even though equilibria are characterized by undersaving.

Key wrds: Growth; Technological change; Public finance; Debt

1. Introduction

The problem of dynamic inefficiency is well known, but not well understood. The balanced growth equilibrium of Diamond’s (1965) overlapping generations model exhibits capital overaccumulation if and only if the steady state return to capital (p) is smaller than the exogenously given rate of growth (9). Moreover, bubbles or ‘rational Ponzi games’ are possible in Diamond’s model whenever there is capital overaccumulation in the balanced growth equilibrium [Tirole (1985), O’Connell and Zeldes (1988)-J. Since bubbles or Ponzi games reduce

Corre.spondencr ro: Ian King, Department of Economics, University of Victoria. Victoria, B.C. VXW 3P5. Canada.

*Ian King’s research was funded by grants from the Social Sciences and Humanities Research Council and the University of Victoria. Earlier versions of this paper were presented at the Department of Economics at the University of Victoria, 1991, the Canadian Economics Association annual meetings, University of PEI, 1992, the meetings of the Society for Economic Dynamics and Control, University of Quebec in Montreal, 1992, the annual meetings of the European Economic Association, Trinity College, 1992. the Department of Economics at the University ofCiuelph, 1992, and the Department of Economics at the University of British Columbia, 1992. Thanks to Andrew Abel, Olivier Blanchard, Mick Devereux, Merwan Engineer, Peter Howitt, Peter Kennedy, Huiwen Lai. Serge Nadeau, Paul Romer, Bill Schworm, and Linda Welling for comments, corrections, and suggestions. Thanks also to Denby Wong for research assistance.

0304-3932;93/$06.00 0 1993--Elsevier Science Publishers B.V. All rights reserved

80 I. King and D. Ferguson. Dynamic inefficiency

capital holdings, then such schemes can remove the problem of capital overaccumulation. However, in representative agent models with exogenous growth, dynamic inefficiency, bubbles, and Ponzi games are all ruled out.’ The existence of overlapping generations is widely thought, therefore, to be crucial for the existence of dynamic inefficiency, bubbles, and Ponzi games in neoclassical

balanced growth environments. In this paper we argue that the above reasoning depends fundamentally upon

the assumption that the growth process is exogenous. We demonstrate that, in competitive endogenous balanced growth models without externalities2 p > y, the equilibrium is dynamically efficient, and bubbles or Ponzi games are not feasible, independent of whether or not lifetimes are finite or generations overlap. When learning-by-doing externalities are introduced in these settings [as, for example, in a balanced-growth version of Romer’s (1986) model], then dynamic inefficiency may exist. However, the source of dynamic inefficiency in these settings is not the scale of capital, but rather the mix of capital goods that is chosen in the equilibrium. Under these circumstances, simply scaling back capital will not improve welfare. In a model of this type with only one type of capital (as is commonly assumed) the equilibrium will be dynamically efficient. With externalities, there is a wedge between the social rate of return on capital (p”) and the private return (p), where p < p”. The equilibrium growth rate LJ will always be bound from above by ps, but not necessarily by p. Moreover, in the presence of these externalities, whether or not the equilibrium is dynamically inefficient, if agents live for two or more periods, undersaving will prevent the equilibrium from being Pareto optimal. Again, these results apply in both representative agent and overlapping generations environments.

In overlapping generations models of this sort (with externalities), bubbles and Ponzi games may be feasible even though there is undersaving in the equilibrium. In marked contrast to Tirole (1985) and O’Connell and Zeldes (1988) overaccumulation is nor required for the existence of feasible Ponzi games. This result arises because of the wedge between the social and private returns to capital. Capital overaccumulation would require 9 > pS (which is ruled out in these environments), but Ponzi games require only that g > p. As men- tioned above, rates of growth that lie between p and pS are possible. Under these circumstances Ponzi games are feasible, but they are never Pareto improving.

At the heart of these results is a distinction between growth models in which all factors are accumulated from current production and those in which one or more factors grow exogenously. This distinction is one that was also recognized

‘See Blanchard and Fischer (1989) for a textbook treatment of this topic.

‘As we discuss below, there are at least two endogenous growth models which fall into this class. The first is presented in Rebel0 (1991). the second is discussed in S&-i-Martin (1990) as a special case of Lucas’ (1988) model without externalities.

in traditional growth theory, von Neumann type models being an example of the former and Solow type models being an example of the latter. As it happens, much of the older literature on efficient balanced growth paths was presented in terms of general models that can be interpreted either way. Accordingly, we begin in section 2 with a discussion of some relevant results from ‘traditional’ multisectoral growth models, in the absence of externalities. Section 3 then recasts a number of endogenous growth models in terms of general multisectoral production models and shows how the results for traditional models can be applied to endogenous growth models with and without externalities. Section 4 discusses dynamic efficiency, Pareto optimality, Ponzi games, and bubbles in overlapping generations models with learning-by-doing externalities. Section 5 then introduces an example with closed-form solutions. The paper closes with some concluding comments in section 6.

Although our intention has been to present our results in the most general context possible, the wide diversity of endogenous growth models has forced us to limit attention to competitive models with convex technologies (we exclude increasing returns to scale) and to cases in which there exist balanced growth paths along which all inputs, outputs, and consumption grow at the same rate (which is consistent with Kaldor’s stylized facts). However, as we note along the way, it is sometimes possible to consider other types of balanced growth (in which growth rates differ) by mapping them into an alternate form which admits equal growth rates in transformed variab1es.j

2. Traditional multisectoral growth models

2.1. Production

In this section we are concerned with presenting a general framework that can be applied whether or not there is a factor that grows exogenously. The basic distinction is between ‘closed’ production models in which all factors are accumulated over time from each period’s production and ‘open’ production models in which one or more factors are determined exogenously in each period. As we shall see, this is an important distinction for the existence of dynamic inefficiency, not only in traditional growth models (which are dealt with later in this section), but also for a wide class of growth models of more recent vintage (which are the subject of the next section).

The technology for the economy will be represented by a set that describes the aggregate production activities. Specifically, the technology set at time t will be represented by s(t) = @(t)t, where r is a closed convex set with elements

‘In particular, see the discussion of Rebelo’s model in section 3.1.1

82

Y

t

0

Fig. I

(x, y) (x E R; and y E R;). The respective interpretations for closed and open models will be as follows.

Interpretation I: Closed model. 4(t) = 1 for all t and r is a cone, as illustrated in fig. 1 for the case of a single input and single output (r is the region between OB and the horizontal axis). In this case the only inputs are the capital stocks x(t) (accumulated from past production) and they yield outputs y(t) at the end of the period. To be feasible, (x(t), y(t)) must belong to 7.

Interpretation 2: Open model. 4(t) = I(t), where l(t) denotes the supply of efficiency units of labour services which is growing exogenously at the rate y (either as a result of growth in population or as a result of exogenous labour augmenting technological change). In this case (x(t), y(t)) E 7 denote the capital

I. King and D. Ferguson. Dynamics inefficienq 83

Y

X

Fig. 2

stocks and outputs per unit of labour services, with l(t).x(t) and I(t)_)(t) denoting the total capita1 stocks and outputs. 4 In general, as illustrated in fig. 2

which is patterned after the one-sector neoclassical model, the technology set for inputs and outputs per unit of labour services will not be a cone. (Again T is the set bounded by OB and the horizontal axis.)

It will be assumed that, in principle, each good can be used either as a consumption good or capita1 good or both. Letting c(t) E R'!+ denote the vector of consumption goods, then the feasible end of period allocations will be

c(t)+x(t+ l)=y(t)+x(t), (1)

41f T is derived from a constant returns to scale technology that is represented by a closed convex cone T that contains the feasible (I, Ix. /JI), then I E ((.x, ,v) (( I. .x, y) E Ti,

84 I. King and D. Ferpson, Dynamic inqjicienc~

if the model is closed, and

c(t) + (1 + g)x(t + 1) = y(t) + x(t),

if the model is open (all variables divided by the amount of labour services and g being the exogenous rate of growth in those services). To preserve as much generality as possible, at this stage we will not specify what governs consumption and investment choices; however, it is to be understood that such decisions are being made in some manner.

Under either interpretation, if growth occurs at the rate g, then a particular balanced growth path can be identified by (c, x, y, g), where (1 + g)i(c, x, y) will be the path of total consumption, capital, and output. Both (1) and (2) then take the common form

c + gx = y.

Assuming that production is statically efficient (as required by profit maximization), then, for the closed model in fig. 1, (x, y) will lie on the boundary of the technology set and (x(t), y(t)) will grow outward along the ray OB. Associated with this, gx(t) will be the level of investment and the vertical distance y(t ) - gx(t ) will be consumption, both of which will also be growing at this rate. Alternately, for the open model illustrated in fig. 2, all magnitudes will be stationary, with gx denoting investment per unit of labour services and y - gx denoting consumption per unit of labour services. In this case a balanced growth path is represented by a single point, with growth occurring as labour services grow.S

2.2. Prices

It will be assumed that factors are paid at the end of the period with p(t) denoting the price vector for that period’s products and r(t) the vector of rentals on the capital goods. Under either of the interpretations above, profit maximization implies that if (x(t), y(t)) is observed, then6

p(t).y(t) - r(t).x(t) 2 p(t)‘j(t) - r(t).I(t), V(zZ(t),j(t))~~. (4)

sAlthough diminishing returns to the nonlabour inputs implies that the set of feasible inputs and outputs per unit of labour services will not be a cone, it is possible that it may contain a cone as a subset. As discussed in footnote 8 below, this is the case considered by Jones and Manuelli (1990). In that case balanced growth could occur along a ray within the technology set; however, since the ray lies in the interior of the technology set, production would be statically inefficient. Consequently, growth paths of this type are precluded by profit maximization.

‘If the model is closed, then the expressions on either side of (4) are the profit associated with the respective production activities. If the model is open, then the expressions differ from per-capita profit only by the amount of the wage rate for labour services which will be the same on both sides.

I. King and D. Ferguson. Dynamic in<ficienc:r 85

In addition, if the model is closed, then the technology exhibits constant returns to scale in the accumulated factors. In that case, a competitive equilibrium will also be characterized by

p(t).y(t) - r(t).x(t) = 0. (5)

In terms of figs. 1 and 2 the maximal profit is represented by iso-profit lines supporting the technology set at the chosen activity.

Under either interpretation, equilibrium in the asset markets requires

Pi(l) - PiCr - l) Pitt 1

ri(t ) _ 0, I

PiCr)

Vi= 1,. . . ,n,

where, since there is no monetary asset, it is assumed that the money rate of interest on the right-hand side is zero. Along a balanced growth path relative prices remain constant, and consequently the rate of change in prices will be the same across all goods. From (6) it then follows that the own-rate of return ri(t)/pi(t) will also be the same. Letting p denote the common own-rate of return, along a balanced growth path we then have

p(t) = (1 + p))‘p for some p 2 0. (7)

Formally, in an open model we will say that (x, y, c, g) is a competitive balanced growth path if (x, y) E r, (3) holds, and if there exist (p, p) L 0 such that

P’Y-PP~x2p~~-pp~~:, v (2,j) E 5. (8)

In closed models only we will add the further requirement that

p.y-pp.x=o. (9)

2.3. Dynamic t@ciency

A feasible growth program starting from an endowment x(0) is said to be dynamically efficient if there is no other feasible growth path starting from the same endowment which provides at least as much of each consumption good in every period and more in some periods. The following result due to Gale and Rockwell (1975, p. 356) provides a clear characterization of dynamically efficient balanced growth programs for both open and closed models.

Proposition 1. Suppose that z is a closed convex set and that (x, y, c, g) is a competitive balanced growth path for some p > 0 and some p. Then (x, y, c, g) is

86 I. King and D. Ferguson. Dwamic ineficienc)

dynamically ejicient if

P ‘.4. (10)

Variations on this conclusion are well known, in particular the results on overaccumulation and underaccumulation that exist for the more familiar one-sector neoclassical growth model.

Less well appreciated is the fact that the ordering between the rate of growth and the rate of return is questionable only in the open version of the model. Under the interpretation as a closed model the zero profit condition (9) also characterizes a competitive equilibrium and we then have

Proposition 2. Consider a.feasible balanced growth path (x, y, c, g). If there e.xist

a price vector p > 0 and a rate of return p such that (9) holds and lf

p.c>O and p.x>O,

then

P ‘.4.

Proof From (3) and the nonnegativity of the price vector, we have

p.c + qp’x - p’y IO.

Since p. y = pp. x from (9) this is the same as

p’c + gp’x - pp’x IO,

from which it follows immediately that

p-,>E>O. QED

(11)

(12)

(13)

(14)

(15)

Since the method used by Gale and Rockwell to prove Proposition 1 does not preclude the zero profit restriction (9) we then have:

Proposition 3. Suppose that t is a closed convex cone. If (x, y, c, y) is a feasible balanced growth path and if there exist prices p > 0 and an interest rate p such that (8) (9) and (I 1) hold, then (x, y, c, g) is dynamically efficient.

The significance of this is to point out that in traditional growth models dynamic inefficiency due to the overaccumulation of capital (characterized by

I. King and D. Ferguson. Dynamic, inc$ic,icvc> 87

a rate of return on capital that is less than the growth rate) is really a problem of accumulating too much capital relative to the amount of labour (or any other factor that matters for production and is growing exogenously).’ If the model is closed, then only endogenous capital matters and, so long as the weak conditions of Proposition 2 are satisfied, capital cannot be overaccumulated.

The point at issue is clearly drawn in the simple cases illustrated in figs. I and 2. In either case competitive pricing yields maximal py - ppx that are representable by lines tangent to the boundary of the technology set. Accordingly, the rate of return on capital is given by the slope of the boundary at the point that is supported by those prices. As is apparent in fig. 1, if the model is closed, then so long as there is positive consumption (gx lies below OB) the rate of return on capital must exceed the rate of growth. However, in the case of the open model illustrated in fig. 2, the rate of return on capital may equal the rate of growth (as at the point shown), be less (to the right), or be greater (to the left).

3. Dynamic efficiency in endogenous growth models

3.1. Investment in human capital

The essence of most endogenous growth models is the introduction of a mechanism for the accumulation of labour services that converts what would other- wise be an open model into a closed model.* The most straightforward mechanism simply involves the inclusion of a technology for the production of labour services so that human capital becomes an accumulated factor. Labour services (in efficiency units of labour) can then be treated in the same way as any other form of capital and its growth is not constrained by the exogenous rate of growth in population. Such models can be mapped directly into the form of a traditional closed growth model and, so long as the conditions of Proposition 3 are met, balanced endogenous growth paths in such models will be dynamically efficient.

3.1.1. Rebelo's model

There are several examples of this type of growth process that also conform to our self-imposed limitation to convex economies. In particular, Rebel0 (1991)

‘This is also the point of the discussion surrounding Theorem 5 in Starrett (1970).

‘There is one exception Jones and Manuelli (I 990) present an open model in which r$(f ) does not change and in which there is no mechanism for the accumulation of human capital. Instead. their CONDITION G implies that the technology set (with inputs and outputs expressed per unit of labour services) contains a cone and consequently growth is not constrained by the exogenous supply of labour services, However, their model only approaches balanced growth asymptotically and can never actually attain balanced growth. For this reason. their model lies outside the domain of this paper. In addition, see footnote 5 above.

88 I. King and D. Ferguson, Dynamic ineficienc:,

has a two-sector model with a single capital good which he interprets as a composite of physical and human capital. The output of the first sector is a pure investment good that has no consumption use and the output of the second sector is a pure consumption good. In his notation, Z, is the stock of composite capital at time t, (5, is the proportion of capital allocated to production of the consumption good, C, is consumption, and T is a fixed endowment of land that enters only into the production of the consumption good. The model is then described by

Z 1+1 = A-&(1 - 4,) + z,, (16)

C, = B(&Z,)“T1 -‘, (17)

where 1 > TV > 0, A, and B are technological parameters. Despite the presence of the fixed factor in the consumption goods sector, if both sides of that equation are raised to the power l/z and if we redefine consumption as C:‘“, then the technology set will be a convex cone. Moreover, since transformed consumption varies monotonically with actual consumption, either can be used to assess the dynamic efficiency of growth.

To see more clearly how the results of the previous section can be applied, let y, and y, denote outputs of the investment good and consumption good respectively, x1 the stock of capital, and c2 the amount of (transformed) consumption. Then the technology set can be written as

T = I(xl~o~Yl~Y2)lYl = Ax*(l

where B = (BT’ ma)1ia. The associated tions will be

Wl = Yl, c2 = Y,,

- 4), y, = Bx,& 1 2 4 2 0, Xl 2 O),

(18)

balanced growth end-of-period alloca-

(19)

which is a particular instance of (3).9 It is not difficult to show that the other requirements of Proposition 3 are met and hence that growth is dynamically efficient. Moreover, since this is a special case, it is also clear that the same conclusion would apply if Rebelo’s model were to be extended in a number of directions.

As an aside, it can also be pointed out that more general log-linear trans- formations could be used to apply the arguments of the previous section to

“We have introduced a minor departure from Rebelo’s model in that there is no depreciation of capital. This is not a substantive matter either for the arguments in the previous section or for their application to Rebelo’s model.

1. King and D. Ferguson. Dynamic inqfic,ienq 89

a wider class of models in which returns to scale are not constant and growth rates differ across variables. (Note that in Rebelo’s model there are decreasing returns to the composite capital and consumption does not grow at the same rate as capital.)

3.1.2. Lucas’ model

A second example is provided by the presentation of Lucas’ model without externalities that is given in Sala-i-Martin (1990). Once again there are two sectors, however in this case the outputs are goods (used both for consumption and investment) and human capital. In the notation of Sala-i-Martin, h(r) is the stock of human capital, K(t) is the stock of physical capital, u(t) is the proportion of human capital allocated to the production of goods, and L is the given number of workers. The model is then described by

K(t + 1) + C(t) = AK(t)qu(t)h(t)L)‘-~ + K(t),

WC + 1) = $W)(l - u(t)) + h(t), (21)

where A, /I, and 4 are technological parameters. If we now let yl denote the output of goods, y, the output of human capital, and x1 and x2 the respective stocks, then the technology set is

7 = CCXlT x23 Yl, Y,)lY, = Axf(ux2L)‘-p, y, = 4x2(1 - u),

1 2 u 2 0, x1 2 0, x2 2 O}, (22)

and the corresponding balanced growth end-of-period allocation equations will be

Cl +sx1 =y13 sx2 = Y2. (23)

Further, if we examine the form of the market solution obtained by Sala-i- Martin for the case in which there is no externality (i.e., the case just presented), then both assets earn the same rate of return, profits are maximized, and the maximal profit is zero.” Consequently, Proposition 3 can be applied. More- over, it is again evident that this model could be extended in a number of different directions without affecting this conclusion.

“In the notation of Sala-i-Martin, the absence of externalities implies that $ = 0. If that is the case, then his (7.18) implies that ya + p = 4 (all in his notation). Consequently, the right-hand sides of his (7.16) and (7.17) are equal. With his $I denoting the rate of return on capital, it follows that (6) is satisfied with v denoting the price of goods and i the price of human capital. With this interpretation in hand, (7.6) can then be interpreted to imply profit maximization. Together with the adding-up property of constant returns to scale production functions, this also implies that profits are zero in both sectors.

90 I. King and D. Fcquson. Dynamic ine#ic’irnq

3.2. Learning by doing

In learning-by-doing models labour services are augmented as an unan- ticipated consequence of other activities and not as the result of deliberate production activity. For this reason, Proposition 3 cannot be applied directly and the analysis of such models is necessarily more involved. Nevertheless, as we shall see, the results for traditional growth models serve as a useful benchmark.

Formally, we suppose that the technology of the economy is represented by a closed convex cone T which contains elements (I(t), x(t )y(t)), where I(t) E R is the amount of labour services expressed in efficiency units. The distinctive feature of the model is the existence of a Romer-like learning-by-doing externality in the provision of labour services. The amount of services obtained from each worker is assumed to be a function of (vector) capital per worker. Letting h(x(t)/n(t)) denote that function [with total labour services I(r) = h(x(t)/n(t))n(t)], it will be assumed:

Assumption. h( .) is concave, homogeneous of degree one, and nondecreasing in

all of its arguments.

Here n(t ) = n(O)( 1 + gn)’ is the population at time t.’ ’ [The homogeneity of h( .) ensures that the social technology set T” defined below is a cone and, hence, allows for balanced growth. The concavity assumption ensures that T” is convex and, since concavity implies continuity, it also means that T” is closed. Together they allow us to invoke the arguments of the previous section.]

To individual agents h appears as a constant and the private technology is described by

T(h) = ((n(t), x(t), y(t))lMt), x(t), y(t)) E TJ, (24)

where hn(t) is the perceived flow of labour services. However, for society as a whole,

l(t) = Wt Mt )M) (25)

= Wt IX (26)

“This formulation for the supply of labour services was adopted to allow for balanced growth in the presence of exogenous population growth. Alternatively, we could follow Romer (1986) Hercowitz and Samson (1991), and Devereux and Smith (1991). and assume that population is stationary [n(f) = n(O), Vf] and that per-capita labour services depend only on the total stocks of capital. In that case I(I) = h(x(r))n(O) and, so long as h( ‘) satisfies the Assumption, all of the results below will also hold for this formulation. In particular, note the equivalence to (26) for n(0) = I.

I. King and D. Ferguson. Dynamic itwfic’irnc:~~ 91

where (26) follows from (25) by virtue of the homogeneity assumption on h( .). Consequently, the social technology is

T” = {(x(r), ~(t))l(W+)), x(t), y(t)) E T). (27)

Since T is a closed convex cone, it follows immediately that T(h) has the same properties. Moreover, the assumptions on h( .) imply that they also apply to T".

In terms of the treatment of traditional models, T" is the closed technology to which we might apply Proposition 3. However, before doing so it is necessary to consider the implications of the learning-by-doing externality for the profitabil- ity conditions. Specifically, (8) and (9) cannot be applied directly to characterize competitive growth paths. Instead, since choices are made out of the private technology set, we can only say that the observed production choices (n(t), -u(t), y(t)) at each point in time must satisfy

p(f).y(t) - r(t).x(t) - w(t)n(t) 2 p(t).jj(t) - r(t)..f(t) - w(t)fi(t),

(28)

V (fi(r ),Z(t 1, j?r 1) E T(h(f 11,

where

h(t 1 = Nx(t)ln(t )).

From the definition of the private technology set in (24) it follows that this is equivalent to stating that (I(r), x(r), y(t)) will be chosen to maximize

p(t).y(t) - r(t).x(t) - m(t)I(t) 2 p(t).j(t) - r(t).?(f) - tu(t)l(t),

(29)

V(k),-W),W)E T,

where

to(t) _= w(t )/h(t)

is the wage for an efficiency unit of labour. In a competitive equilibrium no profits will be earned and, consequently,

p(f).y(t) - r(t).x(t) - co(t)I(t) = 0 (30)

at each point in time.

92 I. King and D. Ferguson, Dynamic ineficienc)

This means that (1, x, y, c, g) is a competitive balanced (I, x, y) E T, (3) holds, and if there exist (0, p, p) 2 0 such that

p.y-pp.x-CIA=0

and

p.j-pp.2-d50, v (i, 2, j) E T.

growth path if

(31)

(32)

The central issue now is the relation that exists between these conditions and the competitive pricing conditions (8) and (9) that underly the efficiency of growth in closed production models. As a first step in establishing the relation, let us define a total rate of return on capital ps that appropriates the payments to labour as part of a ‘social return’ on capital. In particular, since 1 = h(x) from (26) if we define

p” E ( pp. x + oh(x)

> p.x ’ (33)

then, from (31)

p.y - psp.x = 0.

Using this rate of return, (9) is satisfied and we also know that:

(34)

Proposition 4. If p. x > 0 and p. c > 0, then

PS ’ 9. (35)

This suggests that overaccumulation will not occur; however, before Proposi- tion 3 can be applied it must be shown that there is no other (2, j) E T” that can earn a positive profit at the prices that support balanced growth paths in the presence of the externality.

In this context notice that from the definition of p’,

(36)

From (32) we know that the first term in (36) is nonpositive, however there is no basis on which to attach a definite sign to the second term. Consequently, even though (35) holds, dynamic inefficiency cannot be excluded.

I. King and D. Ferguson, Dynamic int$icienq 93

To sort matters out, a distinction can be made between ‘scale effects’ and ‘mix effects’ in the accumulation of capital goods. If we limit attention to vectors of capital goods that differ only in scale,

then, since h(x) is homogeneous of degree one, it follows that

N-f?) w -=- p.2 p.x’

and (36) is nonpositive. This provides a way of formalizing intuition based on (35) that the scale of capital utilization is not at issue. It also makes clear that if dynamic inefficiency does occur, it is due instead to an improper mix of capital goods. If there is only one capital good, as in all other Romer-like endogenous growth models that we are aware of, then capital stocks can differ only in scale and dynamic inefficiency of the type we have been considering cannot arise.

4. Overlapping generations, dynamic efficiency, Pareto optimality, Ponzi games, and bubbles

The arguments in the previous section have general application for any specification of agent preferences and longevity. In particular, there is nothing to preclude overlapping generations. As a result, in the Rebel0 and Lucas human capital models and in the Romer-like learning-by-doing model with one capital good, we can conclude that endogenous growth is dynamically efficient even if there are overlapping generations. This stands in marked contrast to the conclusions in Diamond (1965) for the exogenous growth version.

Nevertheless, in the learning-by-doing model there must be some sense in which the failure of agents to recognize the effect of capital accumulation on the productivity of labour leads to inefficiency in the scale of capital usage. How- ever, in this case the problem is not one of potential overaccumulation, but is instead one of underaccumulation. The question now is how this can be reconciled with the conclusion that the level of accumulation is not a source of dynamic inefficiency.

The heart of the matter is the longevity of agents and the distinction between efficiency in period consumption and Pareto efficiency in the lifetime utility of agents who live for more than one period. Since the productivity of labour today depends on capital stocks acquired in the previous period, it is not possible to realize the benefits of correcting for the externality within a single period. However, if agents live for two or more periods, then correcting for the externality can enhance their lifetime utility. By saving more today, their current

J.Mon-D

94 I. King and D. Ferguson. Dynamic inqjicienc~

welfare is reduced, but because of the external productivity benefit associated with a larger capital stock in the next period, the increase in their future welfare will more than compensate for the current loss.

The simplest framework for addressing these issues is an overlapping generations model in which agents live for two periods, but work only in the first. In this context, n(t) = n(O)(l + g,,)’ is the number of those who are born and work in period t. Agents’ preferences are assumed to be representable by

(39)

where cr and c2 are consumption vectors in the two periods, fl < 1 discounts future utility, and V(ci) is a subutility function that can also be thought of as defining a consumption aggregate. The subutility functions are assumed to be homogeneous of degree one, increasing, and concave. Along a balanced growth path the first-order conditions for consumption choices include

v(c1)-Tv(cl) - pp = 0, (40)

Bv(c2)-“wc2) - /@I(1 + P) = 0, (41)

where p is the marginal utility of first period income and Vv(ci) is the gradient of the consumption aggregate.’ 2

The Pareto inefficiency of competitive growth paths will be shown by considering a balanced growth path which is supported by prices which satisfy (31) or, equivalently, (34), and on which each generation chooses its privately optimal consumption path. We will then suppose that a single cohort deviates from the path. The capital stocks that are inherited by that cohort and subsequently left when it dies, as well as the consumption streams of all other cohorts, will be the same as on the balanced growth path. This ensures that the welfare of all other cohorts is unchanged. In this context, we will show that if the deviating cohort consumes less when young, then it is possible to increase its welfare.

Specifically, let (Z(t ), j(t), c(t )) denote the reference growth path, with the generational decomposition of F(t) being determined by the first-order conditions for the consumption choices and by the definitional relation

c(t) = n(t )c1(t) + n(t - l)cz(t), (42)

where ci(t ) is per capita consumption at time t of the members of the generation that is in period i of their lives. From the first-order conditions and the

‘*The first-order conditions also include the agent’s wealth constraint

I. King and D. Ferguson, D.wamic ineficienq 9s

homogeneity assumption on v( .), it follows that (Cr (t ), CZ(t)) = (1 + g,)‘(Fl, C2) for some (Cr , F2) that satisfy the first-order conditions and (42).

Suppose now that the cohort born at time t deviates from the balanced growth path by changing its first-period consumption. To keep the focus on the scale of capital utilization, it will be assumed that changes in capital accumulation take the form of equi-proportionate changes in all capital goods with 13 denoting the scaling factor. We then have

c(t) I j(r) + X(l) - iqr + l), (43)

where x(t) is inherited and j(t) is the output based on it. Since T" is a cone, Aj(t + 1) is feasible and, consequently, so is any consumption vector

c(t + 1) I ny(t + 1) + Z(t + 1) - x(t + 2) (44)

where X(t + 2) returns the model to the reference growth path for all subsequent generations. From (43), (44), and the form of the zero profit condition in (34) it follows that there are feasible changes in consumption for which

p*dc(t) + (1 + p”)p.dc(t + 1) = (p.j(t + 1) - p”p.x(t + 1))dl = 0.

(45)

Since the consumption paths of other cohorts are unchanged, (42) and (45) imply that there are feasible de,(t) and dcz(t + 1) such that

,n.dc,(t) + (1 + p”)p.dc,(t + 1) = 0. (46)

To relate this to the welfare of the cohort, note that the first-order conditions (40) and (41) imply

dCJ = /A p.dc,(t) +

From (46) we then have

dU = & (P - p”)p.dc,(t).

(47)

(48)

By construction [see (33)],

ps > p> (49)

96 1. King and D. Ferguson. Dynamic inqficienq

from which we have:

Proposition 5. Competitive balanced growth paths in the learning-by-doing model with overlapping generations are Pareto ineficient in lifetime utility. Pareto improvements can be made if young agents save more than they would choose to save in a competitive equilibrium.

With respect to Ponzi games, the fact that the social rate of return exceeds the rate of growth does not preclude the possibility that the private rate of return on capital is less than the rate of growth. If the latter does occur, then, so long as the amount of capital that is crowded out initially is not too great, it would be feasible for the government to issue debt to benefit one generation, and then pay the interest by issuing still more debt. However, in contrast to exogenous growth overlapping generations models, the fact that the private rate of return on capital is less than the rate of growth does not mean that capital has been overaccumulated. As we have seen, if anything, capital is underaccumulated. This distinction is fundamental for considering the welfare implications of Ponzi games.

In the exogenous growth OLG model, if a Ponzi game is feasible (p < g), then the crowding out of capital that occurs as the result of the debt issue is welfare-improving because it reduces the overaccumulation of capital. However, in the learning-by-doing endogenous growth model, because the social rate of return is greater than the private rate, (48) implies that the associated increase in consumption decreases welfare. Although feasible, Ponzi games are welfare- reducing.

Finally, since prices are governed by (7), along a balanced growth path we

have

l+q ’ P(t).x(t) = jyj ( > P’X. (50)

Consequently, if p < g, then

lim p(t).x(t) = co, t-r*.

(51)

which means that speculative bubbles are consistent with competitive equilibria.’ 3

‘jThis last point is a generalization of a result obtained by Yanagawa and Grossman (1992) in an independent and contemporaneous study.

I. King and D. Ferguson. Dynamic im$icirnc.t* 97

5. A closed form example

In this section we present an example of an overlapping generations multi- capital model with learning-by-doing externalities, which illustrates most of the major points being made about such models and which has a closed form solution. Consider a model in which there are two different types of physical capital: xl(t) and x,(t). These are perfect substitutes in production, but only x, (t ) contributes to human capital accumulation:r4

h(t ) = Xl (t )ln(t 1. (52)

For simplicity, we assume that there is only one type of output. The private production technology is given by”

y(t) = A@,(t) + x2(~))‘-“w)4~))“. (53)

Using (52) in (53) the social technology is

y(t) = A(x,(t) + X*(f))l-aX*(t)a. (54)

Let

X(t) =x,(t) + x2(t). (55)

Since firms take h(t) as given, they are indifferent about the mix of x,(t) and x2(f) in X(t). They choose n(t) and X(t) to maximize

z(r) = AX(t)‘-“(h(t)n(t))” - r(t)X(t) - w(t)n(t), (56)

which has first-order conditions

&4X(t)’ -ah(t )“n(t)- ’ = w(t), (57)

(1 - cc)AX(t)-“(h(t)n(t))” = r(t). (58)

“‘Alternatively, we could assume that both types of capital contribute to human capital accumulation, but with different coefficients. The same general results would be obtained. We use this simple example because it makes the point most clearly.

“The measure of firms is normalized to unity.

Suppose firms arbitrarily choose a fraction < of their capital to be of type x, . We can think of < as a mix parameter. Thus,

x,(t) = 5X(t) and x2(t) = (1 - 5)X(t). (59)

Using (52) and (59) in (57) and (58) allows us to write the first-order conditions of the firm as

cL4~“X(t )/n(t) = w(t ), (60)

(1 - x)A(” = r(t) = p. (6’)

Notice that the private rate of return on capita1 (p) is constant. We assume that preferences take the form

(62)

Worker-consumers in a representative generation individually supply one unit of labour inelastically when young, and choose c,(t) and cz(t + 1) [and hence savings s(t)] to maximize (62) subject to

c,(r) = w(t) - s(t), (63)

C2(l + 1) = (1 + p)s(t ). (64)

This implies a savings function of the form

s(t) = R ~ w(t 1, l+R

(65)

where

In equilibrium, aggregate savings must equal the total capita1 stock of the following period, so

n(t)s(t) = X(t + 1). (67)

Using (65) and (60) in (67) yields the endogenous balanced growth rate for levels of aggregate variables:

X(t + 1) R -=-tY,4y= 1 +g.

X(t ) l+R (68)

1. King and D. Ferguson, Dynamic int$cicvq~ 99

Since gn is defined as the growth rate of the population, then the equilibrium

growth in per-capita variables is given byr6

(1 +g) RzA<” -=

’ +“=(l +g.) (1 + R)(l +gn)’ (69)

where

R = fi’““‘[l + (1 _ x)A(“](l-“)!‘-‘. (70)

Notice that the equilibrium balanced growth rate of this economy is an increasing function of 5, the proportion of capital that is of type 1. The social rate of return on X(t) can be found by rewriting the social technology (54) using (59):

y(t) = Ar’“X(t). (71)

Hence,

dy(t ) ” = dX(r)

~ = At”. (72)

Clearly, the social return to xl(t) is A and the social return to x2(t) is zero. Notice that, from (61) and (72):

p = (1 - cc)@,

so that

p < ps.

(73)

(74)

That is, the private return to capital is smaller using (68) and (72) it is simple to show that

than the social return. Moreover.

(75)

Both of these orderings are independent of the value of 5.

5. I. Dynamic inqficienq

For any 5 < I, the equilibrium of this model is dynamically inefficient. That is, it is possible to increase per-capita consumption in every period. This

“Note from (69) that there will be positive balanced growth in per-capita variables if A is large enough to ensure that the right-hand side is greater than one.

loo I. King and D. Ferguson, Dynamic inc$icienc:1

inefficiency arises because the mix of the two types of capital is not optimal. The optimal mix sets 5 = 1. However, whether or not 5 = 1, simply scaling down X(t) in any period to increase current consumption without changing the mix will not allow for any increase in future consumption.

To see this, define c(t) as total consumption in period t. So

c(t) = n(t)c*(t) + n(t - l)cz(t). (76)

End-of-period allocations are constrained by

X(t) + AyX(t) = X(t + 1) + c(t). (77)

Clearly, if 5 < 1, any increase in 5 will increase current output, which will increase the left-hand side of this equation. Hence, c(t) can feasibly be increased without sacrificing any X(t + 1). However, given 5 and x(t), it is not possible to increase c(t) without decreasing X(t + 1) and, as we will see, c(t + 1). Suppose c(t) is increased marginally so that X(t + 1) decreases; this implies

de(t) = - dX(t + 1). (78)

Updating (77) and using (68) yields

X(t + 1) + AyaX(r + 1) = (1 + g)X(t + 1) + c(t + 1). (79)

Hence,

dc(t + 1) = (At” - g)dX(t + 1). (80)

Now using (72) and (78) yields

dc(t + 1) = (g - p”)dc(t). (81)

Since g < ps [by (75)], then this exercise will never allow dc(t + 1) to be positive, regardless of the value of 5.

5.2. Pareto ejiciency

If 5 = 1 in this model, there is no mix problem and the equilibrium is dynamically efficient. However, the equilibrium is not Pareto optimal even if 5 = 1. Marginally increasing the savings of the current young and investing it at the social return p” will increase the utility of the current generation. This is proved in section 4 above. The intuition behind this result is very simple, and can be illustrated using a standard two-period consumption diagram. In fig. 3,

1. King and D. Ferguson, Dynamic inefirienq 101

w+P7

w+P)

w

Fig. 3

c,(t) and c2(t + 1) are deflated on the horizontal and vertical axes respectively by X(t)/n(t). This allows for a stationary representation of the consumption choice in the growing economy. Let W = w(t)n(t)/X(t) = ct(‘A. Individuals face the budget constraint with intercepts Wand W(l + p). Their consumption choice occurs at E, with utility level 0. The budget constraint facing society, however, has intercepts W and W(l + p”), which lies everywhere above the private constraint except at the endowment point. Any reallocation which lies above U but not above the social budget constraint (that is, in the area defined by EFH) will raise the utility of this representative generation and be feasible. However, any allocation to the right of E will make future generations strictly worse off, since the capital stock will be reduced. Allocations in the shaded area EFG are all Pareto superior to the equilibrium allocation. The above argument considers a local perturbation around E: suppose the government moves this representative generation marginally away from E by reducing cl(t)n(t)/X(t) along a constraint with slope - (1 + p”). This is represented on the diagram by the dashed line through E. Since the slope of this constraint is steeper than that of the private constraint, the proposed allocation will necessarily lie in the shaded area of Pareto superior allocations. Notice that this argument works for any value of the mix parameter 5.

102 I. King and D. Ferguson. Dynamic itwficienc:,~

5.3. Government Ponzi games

In Diamond’s (1965) model, government Ponzi games (using debt) are feasible if and only if the equilibrium is dynamically inefficient [Tirole (1985), O’Connell and Zeldes (1988)]; that is, if and only if p < 9. Since debt crowds out capital in these models, Ponzi games are Pareto improving when feasible. That is, if governments can play Ponzi games then they should. In this section, we demonstrate that such Ponzi games may be feasible in this model, but they are never Pareto improving.

To see this, suppose that at time zero the government finances a lump-sum gift to the existing old generation by issuing one-period debt b(O), which must earn the same rate of return as capital, if both are to be held by consumers in equilibrium. In the subsequent periods, rather than using taxes to pay off the debt, the government simply uses a further debt issue, so that

h(t + 1) = (1 + r(t + l))h(t), vt = 0, I, 2, . .

The equilibrium condition (67) is now modified to

n(t)s(t) = X(t + 1) + b(r).

We now ask: under what conditions are such Ponzi games feasible?

Proposition 6. Ponzi games are ,feasible in this economy if and only !f

(9 9 > P,

(ii) b(WX(O) < g - P.

Proof Using (61) in (82) yields

b(t + 1) = (1 + p)b(t).

(82)

(83)

(84)

(85)

(86)

Using (65) and (60) in (83) yields

X(t + 1) = R(l + R)~‘c~4~‘=X(t) - b(c).

This implies

(87)

X(t + l)/b(t + 1) = R(l + R)- ‘olAS”(X(t)/b(t))(b(t)/b(t + 1))

- b(t )/b(t + 1). (88)

1. King and D. Fqwson, Dynamk inrffkienc:v 103

Using (86) and (68) this implies

X(r + l)/h(t + 1) = (1 + .4)(1 + #0-‘(X(f)/h(t)) - (1 + p)-‘. (89)

Feasibility requires that jX(t)/h(t)} be bound strictly above zero. Conditions (84) and (85) are necessary and sufficient for this. QED

Notice that the growth rate g in conditions (84) and (85) is the rate that would exist in the economy in the absence of debt. In this model LJ > p is a possibility, although by (75) y < ps. In general, the condition for the feasibility of Ponzi games is that the rate of growth of capital must be greater than the private return to capital (p), whereas the condition for dynamic inefficiency is that the rate of growth of capital must be greater than the social return to capital (p”). In Diamond’s model, these two rates of return are the same. However, when an externality of the Romer sort is introduced, as in this model, then a wedge is introduced so that pS > p, and the possibility arises that g may fall between these two values. Once again, this argument is independent of the value of 5.

6. Conclusion

This paper makes the following main points. In competitive endogenous balanced growth models without externalities, the balanced growth equilibrium will be dynamically efficient. If learning-by-doing externalities are present, then dynamic inefficiency may exist, but due to the mix of capital rather than its scale. Under these circumstances, simply scaling back capital (as Ponzi games would do) will not improve welfare. Moreover, in the presence of learning-by-doing externalities, whether or not the equilibrium is dynamically efficient, undersaving will prevent the equilibrium from being Pareto optimal. All of these results are obtained independent of whether or not lifetimes are finite or generations overlap.

In the context of exogenous growth models, Ponzi games are feasible only if there is capital overaccumulation and, if played, they are welfare-improving. For these reasons they have been viewed more as a curiosity than as a matter of substantive policy concern. However, it is now clear that if growth is endogenous, then Ponzi games are consistent with capital underaccumulation, and if current generations choose to take advantage of them, then future generations will suffer.

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Documents

Dynamic inefficiency, endogenous growth, and Ponzi games