Enclosure.

Taub, Bart ; Zhao, Rui

I. INTRODUCTION

It seems that you cannot construct macroeconomic reforms on sand. Capitalism requires the bedrock of property and other legal institutions, yet most people outside the West have no solid property rights--De Soto (2002).

De Soto (1989) has highlighted how tenuous property rights are in poor countries. To buy a house or to start a business in Peru is a major undertaking in no small part because the ownership of a piece of land might be ill defined or because the myriad licensing requirements of a business effectively block its legitimate formation. In rich countries, the notion of the transfer of property is an elementary process because the delineation of property rights is at the very core of the state's purpose. If the consequences of poor property rights are so grim, countries wishing to emerge from poverty would support them without outside prodding and with far greater alacrity than they do in practice. Why don't they? We provide an answer.

In our view, property rights internalize the portion of the return to capital that is otherwise treated as common property. The internalization is realized via a multilateral agreement that can be implemented and sustained only when agents are sufficiently patient. In a standard growth model, the degree of patience is endogenous: patience increases as an economy get richer. Property rights are therefore the consequence of growth as well as its cause. The development of property rights is an endogenous process, which cannot be separated from the process of economic development itself.

The development of property rights in our model resembles the enclosure of public fields in England prior to the Industrial Revolution (McCloskey 1972). We therefore term the transition to well-defined property rights "enclosure."

We do not differentiate capital into physical capital and human capital. For example, we view intellectual output such as inventions as a kind of commons and the institution of patent protection as the equivalent of enclosure. The same reasoning can apply to physical capital: roads are potentially usable by nonowners, and property rights enforcement can enable user fees to be collected. We thus model ownership of capital as diffuse: capital nominally owned by a firm not only is used directly by that firm but is also an element of the production functions of other firms. This production technology is used to approximately represent the fact that without clearly defined property rights, a part of the benefits of investment is shared as common property. Embedded in this representation are technology spillovers as documented by Griliches (1958) and Bernstein (1988) and externalities of human capital as advocated by Lucas (1990). In order to emphasize the fungibility of the extent of these spillovers, we refer to the spillover component of firms' production functions as complementary capital.

When firms agree to pay each other the marginal value of the complementary capital that they use, they are encouraged to accumulate additional capital; we equate the extent of these payments with the extent of property rights and interpret the unwillingness of firms to make the payments as the absence of property rights. We examine whether a multilateral agreement to make such payments can be supported in a non-cooperative repeated game setting. Our main finding is that property rights cannot be supported in poor economies but may eventually become supportable as capital accumulates.

The driving force behind the result is the folk theorem. The folk theorem establishes that cooperation can be an equilibrium outcome in a repeated game if the players have sufficient patience. Patience plays a role because the gains from defection are short-lived and the subsequent losses long-lived. In the model, a period of free riding, during which a defecting firm is paid by other firms but does not pay others, is followed by permanent exclusion from the property rights agreement, in which access to its complementary capital is limited. When agents are patient, the long-run punishment is discounted less severely so that it can dominate the short-term gains from defection and therefore deter it.

In standard repeated games, the discount rate or the degree of patience is fixed. But during economic growth, the discount rate, which is identical to the marginal product of capital, is not fixed: it declines as capital accumulates. In our model, poor economies with low capital have high marginal products of capital and therefore high discount rates. (1) They therefore cannot sustain the cooperative outcome. As they accumulate capital, their discount rate shrinks, and when they cross a threshold level of capital, the discount rate reaches the point at which cooperation is sustainable. Property rights then form--that is, enclosure takes place.

Clearly, capital accumulation would be accelerated if the overall marginal return on capital increases due to enclosure. Conversely, capital accumulation supports enclosure. Because of this positive feedback between property rights and capital accumulation, our model predicts that rich countries, which can sustain property rights, can sometimes grow faster than poor countries.

Tornell and Velasco (1992) and Benhabib and Rustichini (1996) also study growth consequences when the proceeds of private investment can be appropriated by others. In both models, the resources available for investment are common property. The extent to which the proceeds of investment are subject to appropriation is more extreme than in our own model, where it is limited by the degree of spillover. Because the outcomes of noncooperative actions are much more dire in the Tornell-Valasco and Benhabib-Rustichini models, it is easier to sustain cooperation.

The results in the Tornell-Valasco and Benhabib-Rustichini models are driven by the choice of parameters. For households to refrain from consuming and to invest, both preferences and technology matter. In the Tornell-Velasco model, parameters capture the willingness to invest via the subjective rate of time discount and intertemporal elasticity of substitution, and the parameters also affect the incentive to invest via the marginal product of capital, and they assume these parameters to be constant. The best supportable cooperative solution therefore does not change as capital accumulates. Benhabib and Rustichini, on the other hand, study various combinations of parameters using simplified examples. Depending on the combination of parameters, cooperation can deteriorate or improve as capital accumulates. In our examples, we use the prevailing functional form from the growth literature; we find that cooperation improves over a country's development path.

Another literature studies the formation of property rights-promoting institutions as the outcome of voting in an economy with heterogeneous agents who can increase their incomes by either investing or appropriating resources from a common pool. The voting mechanism, expropriation technology, and the nature of heterogeneity vary. Chong and Gradstein (2004) analyze the incentives of a median voter and index heterogeneity by how much income can be engaged in expropriative activity. In a similar setting, Hoffand Stiglitz (2005) use majority voting, with agents differing in their ability to expropriate. In both papers, the development of property rights depends on the extent of heterogeneity. Their models are structured so that there is feedback between poor institutions and inequality, which can then reinforce each other. This literature thus finds that good or bad growth outcomes depend on initial conditions of inequality. In our own model, we assume homogeneous agents, and our results do not depend on distributional states. The transformation from bad to good institutions is destined to occur as an economy grows over time: it is a matter of when, not if.

Our study also supplements the existing literature on intellectual property rights, such as Grossman and Helpman (1991a, 1991b, 1994), Helpman (1993), and Grossman and Lai (2004). In these models, the ability to garner monopoly profits is the only incentive for a firm to invest in knowledge. The externality of knowledge takes the form of costless imitation, which, if not discouraged by property rights, drives the profits of the initial investor to zero. In this setup, there is an optimal level of property rights protection that a benevolent government could establish at any time by balancing the tradeoff between the loss of consumer surplus due to monopoly power and the loss of investment incentives due to the absence of property rights protections. We go beyond this literature in two senses. First, in our model, the extent of property rights is endogenous and cannot be consciously imposed by a government. Second, our model is dynamic, and so we can say when property rights form and characterize their extent.

The theory of property rights is also central in the new institutional economics literature. This literature springs in part from a thesis of Demsetz (1967) that property rights develop when a resource becomes economically valuable. The increase in the value of a resource could be due to discoveries, such as the discovery of the Comstock lode (Libecap 1978); government deregulation in the case of airport landing slots (Riker and Sened 1991); or technological change, such as improving salmon fishing productivity and precipitating the institutionalization of fishing rights (Higgs 1982). Our mechanism similarly triggers enclosure when a resource becomes sufficiently valuable: lower discounting increases the firm values with property rights. But more central to our model is that the increased value simultaneously initiates the commitment that is the foundation of property rights.

In this paper, we set out a simple demonstration of our concept. In the next section, we set out the basic structure of our model. We then describe the efficient growth path as a benchmark. Using shooting methods to numerically simulate growth paths, we then demonstrate our central result that enclosure occurs after a period of autarky. Aside from establishing that enclosure does occur, we also establish that investment can accelerate in preenclosure economies and continues at a high rate once enclosure occurs. In our concluding sections, we provide some limited empirical support for our model.

II. THE MODEL

Our model is an extension of the standard, deterministic, neoclassical, growth model. We model households and firms separately. Households behave in an entirely conventional way, holding equity in the firms. The households do not make physical investment decisions and perceive growth only through changes in the value of their equity holdings. The endogenous discounting that drives the model occurs on the firm side. Firms make investment decisions, and the discounting effects of the model appear through the interest rate, which is driven by the marginal product of capital.

Time is continuous. There is a continuum of identical households with unit measure. The instantaneous discount rate of households is 9. Households obtain utility from consuming the single good with instantaneous felicity function u(c).

There is a continuum of firms, also with unit measure. Production requires capital but no labor. We do not differentiate capital into physical capital and human capital.

We posit a production function with two types of capital: firm specific and complementary. (2) The firm-specific capital corresponds to the usual expression of capital in a production function. Complementary capital is a function of the average of all other firms' capital and enters the production function separately. (3) Because there is a continuum of identical firms, each individual firm has no direct impact on the complementary capital of other firms.

For intuition about complementary capital, it is useful to think of the capital as cattle that are used for milk production. Cattle are typically individually owned, but their productive value includes not only the direct harvest of milk but also their value in breeding. An animal that is a good milk producer can be bred with an animal from a line with disease resistance, resulting in calves that combine both attributes: both parent animals thus provide each other with complementary capital.

In developed economies such breeding is highly formalized, with pedigrees carefully kept and which help establish the market value of breeding rights for an animal, and significant payments are made for the use of a breeding animal. There is a penumbra of rights surrounding cattle breeding in the advanced economies: a farmer cannot make use of his neighbor's bulls without committing trespass on well-defined private property; there are formal courts for redress, and the ownership of the genetic material of a cow or a bull is itself covered by property law akin to patent and trademark law, not just the animal per se.

By contrast, in pastoralist cultures such as the Maasai of Africa, there is no formal record keeping, and breeding is usually uncontrolled, with cattle mating within herds or with neighboring herds at communal watering areas and pastures; no payments are made for the use of breeding animals. (4)

A. Notation

We denote the capital stock for firm i as [k.sup.i] and firm i's complementary capital as [k.sup.-i] The production function for firm i is [f.sup.i]([k.sup.i],[k.sup.-i]), net of depreciation. We require that the production function display decreasing returns to scale in all capital. Each firm's investment [x.sup.i] is subject to convex adjustment costs, [phi]([x.sup.i]), where [phi](0) = 0, [x.sup.i][phi]' ([x.sup.i]) > 0, and [phi]"([x..sup.i]) > 0. (5)

The price of output is normalized to 1. The market value of firm i is [q.sup.i.sub.t]. The instantaneous interest rate is [r.sub.t]. Both firms and households can borrow and lend at this rate. To simplify notation, use [R.sup.t.sub.s] = [[integral].sup.t.sub.s][r.sub.[tau]]d[tau] to denote the discounting between time s and t [less than or equal to] s.

In the remainder of this section, we first set out and solve the household's and firm's problems, conditional on an interest rate path. We then impose market clearing and characterize equilibrium when market clearing and a pre-specified property rights agreement is imposed. In Section III, we then analyze the central planner's problem. The central planner takes account of the marginal product of complementary capital and therefore generates faster growth. Finally, in Section IV, we explore how a property rights agreement in which payments for complementary capital are linked to the marginal product of complementary capital affects growth. In Section V, we turn to calibrated numerical experiments in order to characterize the transition from autarky to a property rights agreement.

B. The Household's Problem

Because all households are identical, we can aggregate them into a single representative household that holds shares in all the firms and consumes the aggregated output of all firms.

The share of firm i's stock held by the representative household is [[alpha].sup.i]. The representative household maximizes discounted utility in continuous time for one good:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The budget constraint states that the flow of consumption and portfolio adjustments must equal income from profit and capital gains.

Because the model is deterministic, the household's portfolio choice decision problem can be simplified. Let [r.sup.i.sub.t] denote the instantaneous holding return of stock for firm i, that is,

[r.sup.i.sub.t] = ([[pi].sup.i.sub.t] + [[??].sup.i.sub.t])/[q.sup.i.t].

The household's optimal portfolio holding is the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The household's problem is well defined if and only if the holding returns are equalized and are equal to the interest rate. That is,

[r.sup.i.sub.t] = [r.sub.t] for all i.

Therefore, from the household's perspective, the portfolio can be aggregated into a single asset [a.sub.t], with rate of return [r.sub.t]. The representative household's problem can then be restated as follows:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(2) [[??].sub.t] = [r.sub.t][a.sub.t] - [c.sub.t].

The solution to the household problem is then characterized by the law of motion (2) and the Euler equation (3):

(3) [[??].sub.t]/[c.sub.t] = - u'([c.sub.t])/(u"([c.sub.t])[c.sub.t])([r.sub.t] - [rho]),

which we express as the growth rate of consumption as the product of the intertemporal elasticity of substitution and the interest rate net of time preference.

C. Property Rights and the Firm's Problem

Since the capital of firm i affects the productivity of other firms and vice versa, firms can implement a multilateral agreement to ensure compensation for this productivity--property rights. We model this agreement as specifying a per-unit fee [[omega].sup.i.sub.t] that is paid to firm i by other firms for the use of firm i's capital as complementary capital, and a per-unit fee [[omega].sup.-i.sub.t] that is paid to other firms by firm i at time t. If [[omega].sup.i.sub.t] = [[omega].sup.-i.sub.t] = 0, then no such agreement exists at time t. We define characteristics of agreements in the following definition:

DEFINITION 1. A property rights agreement is regular if [[omega].sup.i.sub.t] and [[omega].sup.-i.sub.t] are piecewise continuous in t and both converge to a constant over time. A property rights agreement is symmetric if [omega]it = [[omega].sup.i.sub.t] and [[omega].sup.-i.sub.t] = [[omega].sup.-j.sub.t] for all i, j and [[omega].sup.i.sub.t] = [[omega].sup.-i.sub.t.

In our analysis, we consider only regular and symmetric agreements.

If no property rights agreement exists at time t, firms may engage in activities to deter free riding by other firms. We abstractly represent the effect of such activity as a reduction in the available complementary capital (6): [k.sup.-i.sub.t] = [theta] [[integral].sub.j[not equal to]i][k.sup.j.sub.t]dj, where 0 < [theta] < 1. Once mutual property rights are established, [k.sup.-i] = [[integral].sub.j[not equal to]i][k.sup.j.sub.t]dj.

In the cattle husbandry setting, defection could consist of using an animal for breeding and failing to pay for it. In advanced economies, cattle breeding associations record and certify pedigrees, limiting this kind of cheating. Exclusion can be effected in a concrete way: illicit progeny cannot be certified for breeding, and if a suspicion were established that the farmer involved knowingly provided a substandard or illicitly bred animal, he would be excluded from membership and breeding privileges with members' breeding stock. (7)

Let [K.sub.s] denote the capital of all firms at time s, that is, [K.sub.s] [equivalent to] {[k.sup.i.sub.s]}i. Let [[OMEGA].sub.s] [equivalent to][{[[omega].sup.i.sub.t], [[omega].sup.-i.sub.t]}.sup.[infinity].sub.t=s]. denote a property rights agreement from time s on. Given the property rights agreement [[OMEGA].sub.s], the objective of the firm i is to choose an investment plan [x.sup.i.sub.t] to maximize its market value.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is, the representative firm maximizes the discounted value of output net of investment and adjustment costs of investment, along with net payments from complementary capital.

Let [f.sup.i.sub.l] denote the marginal product of firm i's own capital and [f.sup.i.sub.2] the marginal contribution of its complementary capital. Then, the solution of the firm's problem satisfies the following differential equation.

(4) [phi]"([x.sup.i.sub.t])[[??].sup.i.subt] = (1 + [phi]'([x.sup.i.sub.t]))[r.sub.t] - [f.sup.i.sub.l]([k.sup.i.sub.t], [k.sup.-i.sub.t]) - [[omega].sup.i.sub.t.]

If adjustment costs are zero, the equation reduces to the standard equation of the marginal product of the firm's own capital and in addition its receipts from other firms with the equilibrium interest rate. The higher agreed payment [[omega].sub.i] induces firms to attain a higher level of capital in the long run, an incentive that is absent without property rights.

D. Symmetric Equilibrium

We next construct the equilibrium under the assumption that a property rights [OMEGA] is in force, that is, [[omega].sup.i.sub.t] and [[omega].sup.-1.sub.t] are positive and predetermined, and under the assumption that firms act symmetrically.

In equilibrium, the goods market and the capital market must clear and prices must be consistent with the optimization of firms and consumers.

The goods market-clearing condition is that consumption, investment, and the adjustment cost of investment must be financed by output:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The assets held by the representative household must equal the value of the firms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Recalling the Euler conditions of consumers in Equation (3) and firms in Equation (4), we have the requirement that the interest rate satisfy the condition:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume that all firms have the same production function and initial capital. Under a symmetric property rights agreement, all firms then choose the same investment plan, that is,

(7) [c.sub.t], + [x.sup.i.sub.t] + [phi]([x.sup.i.sub.t]) = [f.sup.i]([k.sup.i.sub.t], [k.sup.-i.sub.t]).

The profit, [[pi].sup.i.sub.t], and market value, [q.sup.i.sub.t], of a typical firm are:

[[pi].sup.i.sub.t] = [f.sup.i]([k.sup.i.sub.t], [k.sup.-i.sub.t]) - [phi]([x.sup.i.sub.t]) = [c.sub.t]

[q.sup.i.sub.t] = [a.sub.t].

Combining these equations, we obtain the equilibrium condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which implies that the interest rate facing firm i, [r.sup.i.sub.t], is identical to [r.sub.t] for all i, ratifying the simplification of the household's problem mentioned in the previous section.

Totally differentiating the goods market-clearing condition, Equation (5), yields:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solving this equation for b and substituting into Equation (6) yields:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The symmetric equilibrium is thus characterized by Equation (9) and by:

(10) [[??].sup.i.sub.t] = [x.sup.i.sub.t],

where [c.sub.t] is determined by the market-clearing condition, Equation (5).

If [[omega].sup.i.sub.t] converges to a constant [omega] over time, the steady state of the economy is given by the following equations:

(11) [f.sup.i.sub.1]([k.sup.i], [k.sup.-i]) + [omega] = [rho]

(12) c = [f.sup.i]([k.sup.i], [k.sup.-i]),

where [k.sup.-i] = [k.sup.i] if [omega] > 0, otherwise [k.sup.-i] = [theta][k.sup.i].

Imposing our symmetry assumption so that [[omega].sup.i.sub.t] = [[omega].sup.-i.sub.t] = [omega] for all t, we can use phase diagram methods to study the qualitative features of the model. The global phase diagram of this problem is complicated due to the presence of the adjustment cost. One can show that the [[??].sup.i] = 0 locus intersects with the [[??].sup.i] = 0 locus only once at the steady state, but the locus is not necessarily monotonically decreasing. (8) As long as the complementarity between the firm's own capital and the complementary capital is not too high, however, the slope of [[??].sup.i] = 0 locus is locally negative around the intersection of the two loci, as shown in Figure 1. Therefore, the growth path has standard behavior in the neighborhood of the steady state; in our computed examples, this holds globally as well.

III. THE SOCIAL OPTIMUM AND AUTARKY

We next analyze the social planner's problem and autarky as a benchmark. A social planner would force firms to internalize the externalities associated with complementary capital and attain the efficient neoclassical growth path.

A social planner solves the following problem: maximize the discounted utility of the representative consumer through the choice of consumption and investment:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The symmetric solution satisfies the differential equation:

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as well as the market-clearing condition (Equation 7) and the law of motion (Equation 10).

[FIGURE 1 OMITTED]

The socially optimal capital at the steady state, [k.sup.op], satisfies [f.sup.i.sub.1]([k.sup.op], [k.sup.op]) + [f.sup.i.sub.2]([k.sup.op], [k.sup.op]) = [rho], that is, the marginal product of complementary capital is added to the marginal product of firm-specific capital. Under autarky, in which firms free ride on each other so that [[omega].sup.i.sub.t] = [[omega].sup.-i.sub.t] = 0, the steady-state capital under autarky, [k.sup.au], satisfies [f.sup.i.sub.1]([k.sup.au], [theta][k.sup.au]) = [rho]. The socially optimal capital level at the steady state is therefore higher than the competitive equilibrium under autarky (as shown in Figure 2).

The social optimum can be achieved with a property rights agreement that pays the marginal contribution of complementary capital at every instant, that is, [[omega].sup.i.sub.t] = [[omega].sup.-i.sub.t] = [f.sup.i.sub.2]([k.sup.i], [k.sup.i]). As we later establish, it might not be possible to attain the social optimum, but it might yet be possible to attain an intermediate state in which there are positive but suboptimal payments for complementary capital. We make the following definition.

[FIGURE 2 OMITTED]

DEFINITION 2. A complete property rights agreement pays the marginal contribution of complementary capital at every instant, that is, [[omega].sup.i.sub.t] = [[omega].sup.-i.sub.t] = [f.sup.i.sub.2]([k.sup.i], [k.sup.i]). A partial property rights agreement is such that:

[[omega].sup.i.t] = v[f.sup.i.sub.2]([k.sup.i.sub.t], [k.sup.-i.sub.t]),

with v [member of] (0, 1).

We treat the completeness index v of property rights agreements as a fixed constant that is determined by the maximum sustainable v at the steady state. However, v is potentially endogenous in the sense that lower values of v, but not a zero value, might be feasible along the transition path to the steady state, and this would in turn affect the locus of the transition path. Our model, in which the transition from a zero value of v to the steady-state value occurs at a discrete moment, must be seen as an approximation to this model.

IV. THE TRANSITION TO PROPERTY RIGHTS

Not all property rights agreements can be enforced. Firms have incentives to compromise the property rights of other firms by reneging on their payments to them for complementary capital. This reneging constitutes defection in the game sense.

We assume that the detection of defection takes time. (9) Until a defecting firm is detected, it receives payments from other firms but does not pay other firms for the complementary capital that they provide. After a lag of [tau], the defection is detected and the defecting firm no longer receives payments from other firms, and at that time, the firm is permanently excluded from the agreement. Thenceforth, the value of its complementary capital is reduced as well because other firms take measures to prevent its use--only the fraction [theta] of complementary capital can then be used.

We can cast the delay in the detection of defection in terms of the cattle husbandry example. In that context, a farmer who purchases breeding rights from the owner of a bull for a single cow could secretly breed the bull with extra cows, without paying the fees for the breeding with the extra cows. Detection would occur if the progeny reflected the unique traits of the bull or if the owner attempted to register the calves for a pedigree; the essential point is that detection would occur only after a lag of at least the gestation period for a calf. (10)

The gain from defection is twofold: the first is the immediate but short-run retention of payments that were made to other firms under the contract and the second is the possible increase of profit obtained by either low investment or sale of the excess capital previously accumulated. This latter gain accrues because the firm's autarkic steady-state capital is lower than the property rights contract steady-state capital. The cost of defection is the loss of full utilization of complementary capital as [theta] shrinks when defection is detected, and lower long-run profits entailed by moving toward a lower steady-state capital.

Because the set of firms is a continuum, the effect of one firm's defection is negligible. The consequence of this assumption is that the interest rate used by the defecting firm to calculate its discounted payoffs from defection is still the equilibrium rate under full cooperation, and the defecting firm therefore does not affect the investment decision of nondefecting firms.

With the game structured in this way, we can analyze the off-equilibrium-path payoff of a defecting firm. For each level of capital, we consider an equilibrium path in which a property rights agreement is in force and examine the payoff of a firm that defects. If a representative firm prefers such a deviation, then because of symmetry so will all firms, and the proposed equilibrium path is not sustainable at that level of capital. However, if there is a threshold level of capital such that no firm wishes to defect once that level is attained, the equilibrium path is sustainable.

A. The Problem of a Defecting Firm

Given a regular and symmetric property rights agreement, where [[omega].sub.t] > 0 for all t, a firm defecting at date s solves the following problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the notation [??] and [??] denotes the defection path, distinct from the nondefection path followed by other firms.

The optimal investment plan of the defecting firm is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When we calculate the value of defection, the investment path during the initial stage of defection (t < [tau]) is pasted to the postdetection path (t [greater than or equal to] [tau]).

B. The Transition Path

We now combine the analysis of defection with our earlier construction of a property rights agreement to determine when property rights agreements can be sustained.

DEFINITION 3. A property rights agreement [[OMEGA].sub.s] is enforceable at time s with capital [K.sub.s] if [q.sup.i.sub.s]([K.sub.s], [[OMEGA].sub.s]) [greater than or equal to] [[??].sup.i.sub.s]([K.sub.s], [[OMEGA].sub.s]). A property, rights agreement is enforceable starting from time s if it is enforceable for all t [greater than or equal to] s.

Thus, an agreement is enforceable if there is no gain from defection.

The enforceability of the contract depends on the harshness of the punishment, which is parameterized by time lag of detection, [tau], and ability to block free riding, [theta]. The smaller are [tau] and [theta], the less incentive there is for a firm to defect.

The enforceability of the contract also depends on the average current capital stock: the lower the level of capital, the higher the equilibrium interest rate. Future losses accruing from the loss of complementarity during the punishment phase are therefore discounted at a higher rate when the capital stock is low, reducing the severity of the anticipated punishment. The contractual arrangement is therefore not supportable at low capital levels but becomes supportable as capital reaches a threshold level.

The resulting equilibrium path is illustrated in Figure 3.

The figure shows two growth paths. The path terminating at A corresponds to a pure autarky equilibrium in which enclosure never occurs. The second path, terminating at B, illustrates the enclosure process. The path has two legs. The second leg, terminating at B, corresponds to the equilibrium path under a property rights agreement. At low levels of capital, this path is not enforceable; the economy starts on an autarky leg, with a property rights agreement commencing only when capital attains [k.sup.*], the enclosure threshold. Observe that even though there is a period of autarky, the rate of investment on this preenclosure leg is higher than it would be under a permanent autarky regime because enclosure is anticipated.

[FIGURE 3 OMITTED]

V. NUMERICAL EXPERIMENTS

In this section, we use numerical examples to demonstrate the enclosure process. We consider a specification of the model with power utility, quadratic adjustment cost, and Cobb-Douglas production function.

u(c) = [c.sup.1-[sigma]]/(1 - [sigma]) [phi](x) = [delta][chi square] [f.sup.i]([k.sup.i, [k.sup.-i]) = [([k.sup.i]).sup.[alpha]][([k.sup.-i]).sup.[gamma]].

Notice that there is no explicit total factor productivity (TFP) term in the production function; the existence of complementary capital in the production function implies a TFP term of A [equivalent to] [([k.sup.-i]).sup.[gamma]].

We then examine partial property rights equilibria, in which we restrict v to be a constant. We discuss the consequences of relaxing this restriction later in this section.

The parameter values for our experiments are listed in Table 1. The coefficient of relative risk aversion, [alpha], is .5. (11) The subjective discount factor [rho] = .04 implies a 4% interest rate at the steady state. The share of own capital [alpha] is .3, which is roughly the capital share for the U.S. economy. In the baseline example, the share of complementary capital [gamma] is low. Later on we discuss robustness of our results with respect to this parameter. The parameters [tau] and [theta] are chosen so that the transition path is nontrivial, that is, so that enclosure actually occurs. (12)

A. Algorithm

Both the competitive equilibrium under a property rights agreement and the optimal control problem of a defecting firm display the saddle path property. The transversality conditions force the equilibrium path and the defection path to follow saddle trajectories and converge to their corresponding steady states.

The shooting algorithm we use finds these saddle paths. Given initial capital [k.sub.0], the shooting algorithm begins with a conjecture of an initial value for [x.sub.0], then follows the path generated by the differential equation system for ([??], [??]) (9-10), that is, under the assumption that the property rights agreement holds and there is no defection. If capital exceeds the steady-state level, which is known analytically, along this conjectured path for large values of t, then we conclude that the conjecture was incorrect, the initial guess of [x.sub.0] is adjusted, and the process repeated. The initial guess of [x.sub.0] is iteratively adjusted in this fashion to get the terminal capital as close to the steady state as possible.

Once the saddle path is reasonably approximated, we can compute the paths of the interest rate [r.sub.t] and profit [[pi].sup.i.sub.t] of a firm that abides by the property rights agreement, as well as the profit [[??].sup.i.sub.t] of a defecting firm. When computing the saddle path of a defecting firm, the interest rate and capital of all the other firms remain the equilibrium values under cooperation because each firm is negligible due to the continuum assumption. This interest rate is then used to calculate the profit of the defecting firm.

The shooting process is challenging because the value of a defecting firm must be calculated at every point along the saddle path. The saddle path of a defecting firm differs from the main saddle path and must take account of the nonconstant future path of the interest rate; that rate is itself endogenous to the stock of capital.

We mark the enclosure threshold by the level of capital that leaves firms indifferent between defecting and not defecting. Below that level, we assume autarky and above it we assume that the partial property rights agreement stays in force, with the constant v.

B. An Upper Bound on v

We determine v numerically by calculating the highest value of v that is sustainable against defection at the steady state. The steady-state capital [k.sup.ss] is such that under a property rights agreement with completeness v,

[f.sup.i.sub.1]([k.sup.ss], [k.sup.ss]) + v[f.sup.i.sub.2]([k.sup.ss], [k.sup.ss]) = [rho].

A cooperating firm has value [f.sup.i]([k.sup.ss], [k.sup.ss])/[rho] at the steady state. The value of a defecting firm is then computed numerically using the method described in the previous subsection.

A property rights agreement with degree of completeness v is sustainable at the steady state only if the value of a cooperative firm is higher than the value of a defecting firm. Figure 4 plots the value difference between a cooperating firm and a defecting firm for different values of v at the steady state. The figure illustrates that the marginal value of cooperation decreases as the degree of completeness increases. The value of defecting increases as v increases due to the short-run gain from avoiding the payment [[omega].sup.-i]. For larger v, property rights agreements are therefore not sustainable, even at the steady state. In our baseline experiments, the maximum sustainable v is about 60%. If we interpret the contractual agreement as patent protection, when the interest rate is 4%, v = 60% roughly corresponds to a patent with lifespan of 23 yr--that is, close to its actual value. (13)

[FIGURE 4 OMITTED]

C. Sustainability of Partial Property Rights along the Saddle Path

For a fixed degree of completeness v < [v.sup.max], we can numerically determine when enclosure will occur by comparing the value of a cooperative firm against the value of a defecting firm. Figure 5 plots the value differences for various levels of capital with v = .55. Because of the patience effect, the incentive for cooperation is monotonically increasing in capital.

In the early stages of economic development, capital is low, with the interest rate consequently high. Since the gain from cooperation is in the future while the gain from defection is immediate, the initial high interest rate leads firms to prefer defection. Only when capital reaches a threshold point [k.sup.*](v), and the associated interest rate is low enough, will defection be dominated by cooperation. From that point on, the partial property rights agreement becomes enforceable.

[FIGURE 5 OMITTED]

D. The Transition Path and Enclosure

Because an economy with low initial capital anticipates enclosure, firms will not stay on the pure autarky-equilibrium saddle path even in the early preenclosure stage of development. As we pointed out in Figure 3, the equilibrium path still satisfies the system (9-10) under autarky but smoothly pastes to the saddle path of an equilibrium under a partial property rights agreement at the point where capital reaches the threshold value [k.sup.*](v).

Figure 6 plots this transition path for three different values of v. One can see that the critical value [k.sup.*] increases when the degree of completeness, v, increases. (14) In other words, as an economy accumulates more capital, it could also sustain a more complete property rights agreement. The more complete property rights agreement leads to higher steady-state capital. This gradual enclosure of the marginal contribution of complementary capital counters the force of the diminishing marginal return on capital, rendering the growth rate and investment rate of a more developed economy high.

[FIGURE 6 OMITTED]

In order to illustrate the effect of enclosure on growth and investment, we compare growth rates and investment rates in an economy that has a partial property rights agreement to one without such an agreement (Figure 7). The dashed lines correspond to autarkic paths and the solid lines correspond to paths in which autarky is followed by enclosure and the property rights path. It is worth emphasizing that if autarky prevailed permanently, a standard growth path would emerge: investment would steadily decrease as the autarkic steady state was approached. In the economy with a property rights agreement, the pasting of the pre-property rights path to the property rights path results in an increase of the growth rate and investment. There is a preenclosure investment "frenzy"; investment takes off in Rostow's (1956) sense before enclosure and continues at a high rate thereafter.

[FIGURE 7 OMITTED]

A more complete model would entail continuous and gradual enclosure, with ever-increasing degrees of internalization of complementarities. This more complete model would have the degree of internalization determined by firms having a zero marginal incentive to defect at all times. Thus, v will in general also be a function of the state, increasing as the steady state is approached. Our simulations, which show a sharp transition between the absence of a property rights agreement and their institution, approximate a model in which the completeness of property rights, characterized by v, increase steadily over time. In a model with a continuously increasing v, the sharp transition would disappear: the interest rate in particular would follow a smoother path and would decline in the long run at a much slower rate than it would under pure autarky or under a pure neoclassical regime; this slower decline would far better match the empirical long-run path of interest rates.

E. Sensitivity Analysis of [gamma]

Prescott (1998) stated that "(t)he neoclassical growth model accounts for differences across countries only if total factor productivity differs across countries." Our model implies that cross-country differences in TFP are partially driven by the quantity of capital: TFP comprises the contribution of complementary capital, [k.sup.[gamma]]. We use this observation to figure out a reasonable range of values for [gamma].

Let [A.sub.it] and [k.sub.it] be the measured Solow residual and capital per worker for country i at time t, respectively. We fitted a power curve in the form of [A.sub.it] = [A.sub.t][k.sup.[gamma].sub.it], allowing a common component [A.sub.t], not contained in our model, as well, with [[gamma].sub.t] as the parameter of interest.

Using data from the Penn World Table (Mark 5.6) for the years 1965-1990 for 64 countries that have data on the physical capital stock, assuming that the parameters are fixed over time, and pooling all the data together, we estimated value of [gamma], to be .30. Allowing for evolution of the parameters over time and estimating [gamma] year by year, the value of [gamma] falls between .24 and .34.

The value of .1 for [gamma] in our baseline numerical model is smaller than the estimated value. However, all the results we presented in the previous subsections continue to hold qualitatively for larger values of [gamma]. (15) One regularity is that as [gamma] increases, the upper bound of the degree of completeness of property rights v first decreases slightly and then increases. Holding all other parameters fixed, the increase of the power of the externality through the increased share of complementary capital increases the unit payment under the

property rights agreement, which increases the value of defection and makes it harder to enforce property rights agreements with higher completeness v. On the other hand, the increased share of complementary capital dramatically increases the level of the steady-state capital under the more complete property rights agreement, which works to sustain cooperation. Initially, the first effect is strong, but eventually, the second effect dominates (Table 2).

VI. EMPIRICAL DISCUSSION

Our model predicts that rich countries are able to sustain more complete property rights agreements, and more complete property rights increase subsequent growth rates. Empirically, it is extremely difficult to compare the strength of property rights on a consistent basis across countries. There are few attempts to develop qualitative rankings based on legal documents, especially those about intellectual property rights. Rapp and Rozek (1990) and Ginarte and Park (1997) are among them. Both studies focus exclusively on patents. Ginarte and Park extended the methodology of Rapp and Rozek significantly and developed a panel of indexes by rating the patent laws of most countries every 5 yr from 1960 to 1990. (16) They also showed that a significant positive correlation between patent rights and gross domestic product (GDP) per capita exists.

Ginarte and Park (1997) demonstrated a positive correlation between patent rights and income per capita. The more recent empirical study by Djankov et al. (2004), which examines the efficiency of legal systems, also provides some indirect evidence for our thesis. Their paper compares the efficacy of legal systems in providing clear and practical property rights to parties in ordinary commercial activities: eviction of tenants for nonpayment and collection of non-sufficient-funds checks. Djankov et al. carefully surveyed law firms in more than 100 countries in order to measure costs and durations in carrying out these court transactions. Although their focus is on the effects of English and French legal systems on the outcomes, their regressions also establish that per-capita GDP is negatively related to property rights in the sense of their definition.

Baier, Dwyer, and Tamura (2005) extend the growth accounting literature by adding measures of human capital and developing data over a very long term (~150 yr) in order to extract the low-frequency component. They find that growth in TFP is smaller than in previous estimates but is still far from zero. They note that in some regions, TFP has actually shrunk and draw the conclusion that institutions, not technology, drive TFP growth. That institutions drive growth is a conclusion we also draw--although we do not go so far as to predict negative TFP growth--and moreover, we explain why institutions are so tightly connected with wealth.

Similarly, Alfaro, Kalemli-Ozcan, and Volosovych (2005) examine cross-country capital flows. They find that equity inflows per capita are much lower in poor countries than in rich countries, when elementary theory predicts the opposite. Our model explains the pattern as stemming from impaired property rights in poor countries. (17) They then measure institutional quality, following the literature exemplified by La Porta et al. (1998), finding that institutional quality explains growth better than GDP. However, institutional quality is highly correlated with GDP, which is again predicted by the enclosure model. They also present time series graphs documenting the secular upward trend in institutional quality--again, in agreement with our theory.

Godoy and Stiglitz (2006) provide further evidence of the linkage between property rights and growth. They carry out an empirical analysis of the outcomes of privatization in post-Soviet Eastern Europe. They disentangle the effects of the speed of privatization from the effects of the quality of property rights institutions on the outcomes. They find support for the proposition that good institutions are the primary drivers of successful privatization, rather than the speed of privatization.

Our results rest on the impatience of economic agents in poor countries. In the model, impatience is expressed in higher interest rates. Is it true that poor countries have higher interest rates than rich countries? Cross-country comparisons do show that real interest rates and GDP per capita are negatively correlated (Figure 8), while time series evidence is mixed. (18)

McCloskey and Nash (1984) showed that the interest rate, measured by seasonal dynamics of the price of grain in medieval England, was high relative to post industrial revolution interest rates. Siegel (1992) carefully constructed the real risk-free rate series for both the United States and the United Kingdom between 1800 and 1990. He showed that the short rate went through wide swings historically and is highly correlated across the two countries. Though it is true that the interest rate mostly declined in the nineteenth century, the same cannot be said since then. This is not surprising as interest rates are also affected by technological progress and other factors besides capital accumulation. Cross-country comparisons, on the other hand, could bypass that problem by holding these extraneous effects fixed but are also subject to error: the cross-sectional interest rates shown in Figure 8 are not perfectly comparable across countries, the maturities of the underlining financial instruments range from 1 to 6 mo, and the spreads between borrowing rates and lending rates differ tremendously across countries. Nevertheless, the negative correlation is robust for different time periods and is present whether we use borrowing rates or lending rates.

[FIGURE 8 OMITTED]

VIII CONCLUSIONS

We view our model as meeting a challenge posed by North (1996):

 There is no mystery why the field of development
 has failed to develop during the five decades
 since the end of the Second World War.
 Neoclassical theory is simply an inappropriate
 tool to analyze and prescribe policies that will
 induce development. It is concerned with the
 operation of markets, not with how markets
 develop. How can one prescribe policies when
 one doesn't understand how economies
 develop? The very methods employed by neoclassical
 economists have dictated the subject
 matter and militated against such a development.
 That theory, in the pristine form that gave
 it mathematical precision and elegance, modeled
 a frictionless and static world. When
 applied to economic history and development,
 it focused on technological development and
 more recently human capital investment but
 ignored the incentive structure embodied in
 institutions that determined the extent of societal
 investment in those factors. In the analysis of
 economic performance through time it contained
 two erroneous assumptions: first, that
 institutions do not matter and, second, that time
 does not matter.

Our model is indeed a neoclassical model that does in fact endogenize the formation of institutions over time.

Our model is highly specialized: it is symmetric, but actual capital is distributed highly asymmetrically, and the institution of property rights is highly asynchronous. It is also deterministic. An extended model with random technological changes would allow the marginal product of capital to increase at random times; by decreasing the marginal product, and perforce patience, this could have the effect of sundering rather than strengthening property rights, with the consequence that growth might be slowed, or at least that it would fail to accelerate, to the degree predicted by elementary theory.

An extended model would entail continuous and gradual enclosure, with ever-increasing degrees of internalization of complementarities, and would also encompass heterogeneous initial capital across firms or countries. This more complete model would have the degree of internalization determined by firms having a zero marginal incentive to defect at all times and would result in a smoother, more slowly decreasing path of interest rates. The results in our present model suggest that there might be extended periods in which countries of intermediate or high wealth experience acceleration of growth, rather than the deceleration of standard theory.

ABBREVIATIONS

GDP: Gross Domestic Product

TFP: Total Factor Productivity

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(1.) Our theory rests on the fact that interest rates and development are negatively correlated. Some evidence of this is provided in Section VI.

(2.) The empirical evidence for the existence of positive externalities in technology is compelling: calculations of the rate of return to investment in research and development show it to be extraordinarily high, far higher than market rates (see, e.g., Griliches [1958] and Bernstein [1988]). Technology spillovers documented by Bernstcin and Yan (1997) and Coe, Helpman, and Hoffmaister (1997)--the ability of poorer countries to incorporate technology from developed economies through imitation--are apparently an engine of growth. By contrast, our model views these spillovers as symptoms of incompletely developed property rights.

(3.) Separately incorporating firm-specific and aggregate capital harks back as far as the papers of Arrow (1962), with later applications such as Romer (1986). Barro and Sala-i-Martin (1997) and Eeckhout and Jovanovic (2002) also incorporate capital externalities in the production function in order to study various growth anomalies.

(4.) See Janssen-Tapken et al. (2006).

(5.) Without adjustment costs, because of the continuum of firms assumption, each firm would engage in "bang-bang" investment, either zero or infinite. Adjustment costs guarantee a smooth growth path.

(6.) Activity to limit the uncompensated use of complementary capital is in reality costly, and we are suppressing the explicit representation of this cost, and the means by which free riding is impeded, in this version of the model. We would argue, however, that this cost is potentially small: in the context of cattle husbandry, breeding associations make it cheap to exclude defectors. A farmer who illicitly bred cattle would be unable to certify the pedigrees of the extra progeny. The record keeping for this certification, and the provision of pedigrees, is relatively costless.

(7.) A broader example of property rights agreements is the use of patent cross-licensing agreements. Firms operating in the same industry often have patents that are complementary with the patents of competitors. Defection would then entail a firm's deliberate infringement of a patent, knowing that it will eventually be detected, which in practice takes time. Exclusion then takes the concrete form of exclusion from cross-licensing, or ultimately a lawsuit. In poor countries, the lack of patent protection limits the recourse of firms; exclusion from cross-licensing agreements is then not an option.

(8.) The slope of [[??].sup.i] = 0 locus is A/B, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(9.) As with our abstraction of the effort to limit the uncompensated use of complementary capital, we are abstracting away from explicitly modeling the effort needed to monitor and respond to defection.

(10.) And as with our discussion of patents in Footnote 7, detection of patent infringement takes time.

(11.) Increasing the value of the coefficient of relative risk aversion would not change the results much. Because the coefficient also indexes the intertemporal elasticity of substitution, a higher value would lead to slower convergence toward the steady state but would not affect the ultimate behavior of the transition path very much, nor would the steady state itself be affected.

(12.) As with the model of Benhabib and Rustichini, for some parameters, a growth trap is possible, that is, enclosure never occurs because defection is always optimal.

(13.) That is, the present value of 23 yr of a flow at an interest rate of 4% is about 60% of the value of an infinite term for that same flow.

(14.) This increase is not very significant in the baseline model, but it is magnified when the value of [gamma] is increased.

(15.) The computed example becomes numerically unstable for [gamma] values that are higher than [alpha], the share of own capital in production.

(16.) The Ginarte-Park index is based on five components: duration of protection, extent of coverage, membership in international patent agreements, provisions for loss of protection, and enforcement measure. Some components are further broken down into characteristics that they think closely describe their effective strength. All scores are aggregated with equal weights. The final index has a minimum possible score of 0.0 and a maximum of 5.0.

(17.) Our extension of the enclosure model to a multi-country setting elaborates on this point further.

(18.) There is no uncertainty in our model, and hence, discounting and the interest rate are identical. The addition of the hazard rate for theft and expropriation, a significant risk in low property rights settings such as the pastoralist culture of the Maasai, might increase discounting even more.

Taub: Professor, Department of Economics, University of Illinois at Urbana-Champaign, 470E Wohlers Hall, Champaign, IL. Phone 217-333-4828, Fax 217-244-6571, E-mail [email protected]

Zhao: Assistant Professor, Department of Economics, University of Illinois at Urbana-Champaign, 343I Wohlers Hall, Champaign, IL. Phone 217-333-4508, Fax 217-244-6571, E-mail [email protected]

TABLE 1
Parameter Values

Parameter Value

Risk aversion coefficient, [sigma] .5
Subjective discount rate, [rho] .04
Adjustment cost, [delta] .04
Own capital, [alpha] .3
Complementary capital, [gamma] .1
Detection time lag, [tau] 2
Ability to block free riding, [theta] .9

TABLE 2
Sensitivity Analysis of [gamma]

[gamma] .1 .2 .3 .4
v .59 .54 .60 1.00