THE ECONOMIC RECORD, VOL. 70, NO. 21 I , DECEMBER 1994. 368-372

On Optimal Enclosure and Optimal Timing of Enclosure*

NGO VAN LONG McGill University, Montreal, Canada

This paper demonstrates that the enforcement of property rights by enclosing properties under common access is. in general, socially suboptimal: the timing of enclosure may be inappropriate. causing inefficiency. It is argued that the firstfundomental theorem of welfare economics applies for a given collection of production sets. while enclosure implies a choice over collections of produc- tion sets.

I Introduction There seems to be a widespread belief that

scarce resources should be properly priced, and that if owners of these resources find it profitable to enforce property rights then the outcome is socially efficient. Recently this belief has been challenged by David de Meza and J.R. Could (1992). They show that a competitive equilibrium with enforced property rights may well be socially inefficient. They argue that their result is not log- ically incompatible with the first fundamental theorem of welfare economics (which states that a competitive equilibrium is Pareto efficient), by appealing to the fact that in their model ‘the elim- ination of externalities by enclosure is costly’

The purpose of the present paper is twofold. First, we show that in an intertemporal framework (in contrast to the static framework of de Meza and Could), a Competitive equilibrium may exist in which all sites are immediately enclosed in the first period, while the socially desirable time of enclosure is in some later period. Second, we identify the source of inefficiency of the compet- itive equilibrium, arguing that passing from a non-

(p.579).

’ Thanks arc due to Richard Comes, Murray Kemp, Henklis Polemarchakis, Koji Shimomura and a referee for helpful comments.

enclosure regime to a full enclosure regime involves effectively a change in production sets for some producers. Thus a competitive equilib- rium with enclosure is Pareto optimal only with respect to the new production sets. but can be inefficient when we think of the economy as having a choice over production sets.

I! The Model There are M identical ‘sites’, each of which can

be used as a ‘common’, or privately owned and enclosed. At site i , the variable input (say, a com- posite input consisting of one unit of effective labour and one unit of freely available lamb) is applied to the fixed capital (land) to produce an output (say, wool). The production function is

Y , = f(qJ ( 1 )

where f is strictly concave and increasing, with f (0) = 0 and q, denoting the quantity of the vari- able input. The marginal product. f ( q , ) . is a decreasing function.

Labour is homogeneous in the grazing activity, but non-homogeneous in the other activity, which we call ‘service’. More precisely, we assume that there is an ability index 8 such that a unit of labour of type 8 can either contribute one unit of effective labour in grazing (independent of the value of 8), or produce W(8) units of service.

368

1994. The Economic Society of Austnlia. ISSN 0013-0249.

19w ON OPTIMAL ENCLOSURE 369

Higher 8 types produce more service per unit of labour: W ( 8 ) > 0. There is a continuum of workers, indexed by 8, and uniformly distributed over the interval [O,l]. Labour is inelastically sup- plied. Let N ( t ) denote the workforce at time t. We assume that the workforce grows exponentially.

N(r) = N(O)e", n b 0 (2) Consumers' preferences are such that a unit of service is a perfect substitute for a unit of wool. Their prices are normalized at unity.

( i ) A Private Ownership Equilibrium With Cost- lessly Enforced Property Rights

With property rights costlessly enforced, at time I there are e'(r)N(t)/M workers hired at each site, and [ 1 - W(t ) ]N(r ) workers are producers of serv- ices. The value of W ( t ) satisfies the condition

(3) f ( e - ( t ) N ( r ) / M ) = W(W( t ) ) .

Aggregate value of output is

A(t) = MAB'(r)N(f)/M) + W ) w ( 8 ) d e e * w

(4)

q'(1) = e - ( t )N( f ) /M (3 '1

I' We define:

( i i ) A Common Property Equilibrium Consider the opposite extreme, where property

rights are not enforceable. In this case, if qi units of labour exploit site i , then each unit earnsf(q,)/ q, (i.e. its avenge product). The common-property equilibrium is characterized by the equation

f (8"(f)N(t) /M) = B"(t)(N(t)/M w (8"(t)). ( 5 )

qO(t) = B"(r)N(t)/M (5 '1 We define

Remark 1 If n > 0, then B"(t) and W(r) are decreasing functions of f.

(iii) The Enforcement of Property Rights by Enclosure

If all sites arc enclosed, the owner of each site

B(r) = f(q'(t)) - q'(Ow(W).) (6) (It is easy to verify that B'(t) > 0 if n > 0). Assume that the once-and-for-all cost of enclosure is $X per

earns

site.l Then each site owner prefers to enclose his site at date 0 (rather than never enclosing) if he takes the time path W ( r ) and hence W(B'(r)) as given and if X is smaller than the cumulative gain from immediate enclosure, G(0). where

G(0) = B(t)e-"dt. (7) 16 Of course, each site owner may consider the option of delaying the enclosure even if G(0) > X . Such a delay would be profitable if the mar- ginal gain of delaying were to exceed the marginal cost of doing so, i.e. if rX .> B(0). In that case, the privately optimal enclosure time. tf', would be characterized by the condition

rX = B(tf'). (8) In what follows. we assume that X < G(0). and

rX 4 B(0) (9) so that immediate enclosure is optimal for each site owner, given that each expects all other site owners to enclose at time 0 also.

( iv) The Social Optimum For simplicity we assume utility is linear in

consumption and there is no disutility associated with work. The flow of social benefit (the sum of consumers and producers surplus) if all sites are common properties at time t is then simply

F(r) = M,vo(t)N(o/w + W ) W ( 8 ) d e I:,., (10)

I' e*(t)

&., ( 1 1 )

The gross social gain from enclosure is then A(?) - s"(t), and the gross social gain per site is

g(r) = f ( q W + ( W / M ) w(e)de

- f(qO(t)) - (N(t)/M) W ( 0 ) d e

It is easy to show that g(t) is smaller than the private gain B(t). given that W'(B) > 0.

I It may be argued that in a continuous time model. set-up costs cannot be incurred at a point of time, but over an interval of time. This interval can be taken as a given constant or can be treated as a choice variable. Alternatively. we can take the limit as the interval tends to zero. See Hawick. Kemp and Long (1986) for a discussion of set-up costs.

370 ECONOMIC RECORD DECEMBER

because W(8) is an increasing function, and 8" >

Q.E.D. It follows from lemma 1 that there exists a range of values for X such that it is privately profitable to enclose immediately, although from society's point of view, if the choice is between immediate enclosure and allowing free access for ever, the second alternative is preferable. A numerical example is given in Section 111 to illustrate this possibility. It is shown in that example that

x < G(0) (12)

rx < B(0) (13) so that there exists a competitive equilibrium with immediate enclosure, and that

8..

and

X > [ g(t)e-" dt (14) so that allowing permanent free access dominates immediate enclosure.

Condition (14) however does not imply that per- manent free access is the best alternative. It may be optimal to enclose all sites at some later time. In other words, there may exist some t* such that

x < JI' e-~I-'.) g(f)dr. (15)

Assume for simplicity that all sites must be enclosed simultaneously. (The problem of optimal sequential enclosure times is beyond the scope of the present note.) Then the socially optimal enclo- sure time is determined by the following maxi- mization problem. Choose f that maximizes

V = [* S"(f)e-"dt + A(t)e-"dt - e-"' XM I: (16) The first-order condition is

.!?(fa) - A( [ ' ) + rxM = 0 (17) i.e.

- g(r') + rx = 0 (17') and the second-order condition is

- g'( t ' ) s 0 (18) Lemma 2 The second-order condition (18) is sat- isfied if W ( 8 ) is sufficiently large and if n > 0.

Proof

g ' ( t ) = f(q')(dq'/dt) + (1/M) W(B)de(dN/dr) II - ( N / M ) W(8'(t)) (de'ldt) - f(q")(dqo/dt)

- (1/M) W(B)de(dN/dr) 1,1 + ( N / M ) W(e"(t))(d B"/dt), (19)

where

dq'ldt = (N/M)(de'/dt) + (O'/M)(dN/df)

and

dq'ldt = (N/M')(dO"/dt) + (B"/M)(dN/dt).

Using the fact that f ( q ' ) = W(8') we have

(MinN)gv) = v ( q w - f(qo)eo + w(e)de i ,

- eo[w(eo) - j ~ q o ) ~ * ~ [ w ( e o ) - m0) + w w ( e o ) ]

i: (20)

where the integral in (2) exceeds W(W)[€I" - 0.1 From lemmas 1 and 2. we have our main result:

Proposition 1 There exists a competitive equilib- rium with immediate enclosure of all sites, and a social optimum with enclosure at a later time, pro- vided W(8) is sufficiently large.

1994 ON OPTIMAL ENCLOSURE 37 1

where y = [ r ( l + a) - n(l - a)]/(l + a) (30) F(r)/M = ( 1 - a)-l( 1 - a)(a-lP(l+a)

(N/M)(I -aY(I +a1 +

- (In) (N/mll-aWl+a) ( 1 - a) -U(I+a l

( N W

(31)

g ( r ) = D(N/M)(I-aHl+a) (32)

D = [ l / ( l - a)] - [(l - a)-a(i+a) + I l l 2 (33)

Remark 2 The expression D in (33) is positive, because it has the same sign as

(34) and it is easily verified that for 0 < a < I , h( 1 + a) > [(a - 1)/( 1 + a)] ln( 1 - a).

A private ownership equilibrium with immedi- ate enclosure occurs if

where

( 1 + a) - ( 1 - a)(a-lfil+a)

yX -= [&(I - a)](N,/M)(l-aY(l+a) (35) and

It is socially inefficient to enclose all sites at t = 0 if

(37) Condition (37) is also sufficient for the dominance of permanent open access over immediate enclosure.

The socially optimal enclosure time (assuming simultaneous enclosure of all sites) is t'. given by

(38) If NJM = 1. r = . 1 1 , n = .03, a = 0.5, X = 9, then ( 3 3 , (36) and (37) are satisfied. and the optimal enclosure time is t' = 9.

YX =. D(N /M)c I -aY( I +ai

rX = D ( N /w(l-aMl+a) @I-aw'fll+a'

IV Competitive Ineficiency: An Explanation The key elements of the model are (i) increasing

opportunity cost of labour, and (ii) cost of fencing. There are two ways of explaining the inefficiency displayed in our model. The first way is to argue that the decision of the site owners does not take into account the loss of surpluses that would accrue to poachers (or their hired labour) under the free access scenario. Some might object to this explanation on the grounds that it seems to rely on 'pecuniary externalities'. We therefore turn to a second explanation. using the Arrow-Debreu model.

For simplicity, let's take the case of zero pop- ulation growth. Then our model says that if the cost of enclosure, X , falls within a certain range, there exists a competitive equilibrium that is not a Pareto optimum (we assume for simplicity that all individuals hold equal shares of all assets, including human capital). Does this constitute a contradiction of the fundamental theorem of welfare economics which says that if a competi- tive equilibrium exists, then it is a Pareto Optimum? The answer is 'no'. The fundamental theorem asserts optimality for a given technology (i.e. a given collection of production sets), but says nothing about the passage from one technol- ogy to another (i.e. from one collection of pro- duction sets to another).

In the private property equilibrium, the produc- tion sets can be specified as follows.

Firm i , that operates site i , can produce an output f ( q i ) if it uses two inputs: 'labour', in amount q,, and 'fence', in amount yi 3 X. If yi < X , then the firm's output is zero. There are a large number of potential poachers for each site. A poacherj at site i can hire at most one unit of labour, i.e. Lji < 1. His output is

312 ECONOMIC RECORD DECEMBER

q, = 6,Lf(4,)/qn1L,, (39)

q, = EL,,, where he takes q, as given, though in fact

and Si is either 0 or I: 6, = 0 if a fence is in place, and 6 = 1 otherwise. Clearly, the action of firm i has an 'external effect' in that it changes the production sets of the poachers. (This is to be dis- tinguished from the other source of externalities. namely the action of each poacher at site i influ- ences the output of all other poachers at the m e site.) When the problem is seen this way, it is clear that 'the tragedy of enclosure' is a possibility and it might be worse than the tragedy of the commons.

REFERENCES

de Meza. David, and Gould. J.R. (1992). 'The Social Efficiency of Rivatc Decisions to Enforre Property Rights', Journal of Political Economy 100, 561-80.

Hamwick. J, Kcmp, M.C. and Long. N.V. (1986). 'Set- up Costs and the Theory of Exhaustible Resources', Journal of Environmental Economics and Manage- ment 13, 212-24.

APPENDIX

Prwf o$ Remark 2

Let D = 1/(1 - a) - [ ( I - a)-211*-i + IJn Then D > 0 if and only if

U(I - a) - 1 > ( I - 1.e.

( I + a)/(l - a) > ( I - a)-U"+aB

i.e.

( I + a) > ( I - a)la-l~~l+a~

Finally. In( I + a) is an increasing function of a and [(a - I )/( 1 + a)]ln( 1 - a) is P decreasing function of a for 0 < a < I . and the two functions are equal at a = 0.