This note studies the exploitation of a common proerty resource when firms choose a technology before exploiting the resource. The sequential choice leads to overinvestment. Moreover, exploitation levels are higher than they would be if technology and labour input were chosen simultaneously. The tragedy of the commons is aggravated.common property resources

Rainald Borck

English

Research Papers in Economics

Investment and the tragedy of the commons 
Rainald Borck
Humboldt University Berlin
Abstract
This note studies the exploitation of a common proerty resource when firms choose a
technology before exploiting the resource. The sequential choice leads to overinvestment.
Moreover, exploitation levels are higher than they would be if technology and labour input
were chosen simultaneously. The tragedy of the commons is aggravated.
I would like to thank Pio Baake for helpful comments.
Citation: Borck, Rainald, (2001) "Investment and the tragedy of the commons." Economics Bulletin, Vol. 4, No. 2 pp. 1−6
Submitted: February 1, 2001.  Accepted: March 19, 2001.
URL: http://www.economicsbulletin.com/2001/volume4/EB−01D60001A.pdf1. Introduction
Common property resources are subject to the well known “tragedy of the com-
mons” (Hardin, 1968). If I ﬁsh on the open sea, I diminish your catch, and the
neglect of this externality leads to overﬁshing.
In this note, we show that the problem may get worse when ﬁrms using com-
mon pool resources ﬁrst choose an investment which determines the productivity
of their labour input. For instance, farmers have to make substantial investments
in tractors, milking machines, and other equipment before grazing their cows on
the commons; ﬁshermen must buy vessels before going out to ﬁsh. To a large
extent, these investments are irreversible.
The literature has implicitly assumed that the choice of technology is made
simultaneously with the choice of factor input. However, when ﬁrms commit to
a technology before choosing inputs, there is an additional externality: When my
labour force gets more productive, I will exploit the resource more and thereby
reduce the proﬁts of all other ﬁrms. Hence, the level of investment is too high
from a social point of view.
The idea that commitment changes the incentives of players is of course fa-
miliar in economics. Applications range from bargaining (Schelling, 1960) to
industrial organisation (e.g., Fershtman and Judd, 1987), strategic trade policy
(Brander and Spencer, 1985), and monetary policy (Rogoﬀ, 1985). Konrad (1994)
shows that individuals may strategically choose to be poor in order to free ride
on others’ contributions in a game of private provision of public goods. Similar
reasoning is applied to the common pool problem in this note.
2. The Model
Suppose for simplicity there are two ﬁrms which use a common property resource,
say, a ﬁshing ground. The total catch, X, is a function of aggregate labour
employed, L := L1 + L2, where Li is labour input of ﬁrm i: X = F(L);F 0 >
0;F 00 < 0. Firm i’s proﬁt depends on labour supplied and on its productivity
parameter, µi. Let c(µi) be the cost of productivity increases, with c(0) = 0;c0;c00 >
0. Firm i’s proﬁt function is:
¼i = µiLi
F(L)
L
¡ wLi ¡ c(µi); (1)where w is the constant wage rate. The assumption that ﬁrms’ catch rates are
proportional to their input levels is made for convenience and is common in the
literature.
Suppose ﬁrst that the choices of labour and technology are made simultane-
ously. The ﬁrst order conditions are:
d¼i
dLi
= µi
µ
Li
L
F
0 +
Lj
L
F
L
¶
¡ w = 0; (2)
d¼i
dµi
= Li
F
L
¡ c
0 = 0: (3)
These equations characterize the Nash equilibrium of the simultaneous move
game. The resulting equilibrium levels of L and µ will be denoted by (LN;µN),
where µ := µ1 + µ2 is the aggregate productivity parameter. We will concentrate
throughout the paper on symmetric equilibria.
3. Sequential Choice
Suppose now that ﬁrms commit to a technology before deciding on the level of
exploitation of the common pool resource. We now have a two stage game: At
the ﬁrst stage, the irreversible technology choice is made, and at the second stage,
each ﬁrm chooses how much labour to employ for a given technology. Assuming
complete information, we solve backwards for a subgame perfect equilibrium.
The ﬁrst order condition at the second stage, for given technology choices, is
completely analogous to (2). Total diﬀerentiation shows that the reaction func-
tions, Ri := Li(Lj;µi;w), are downward sloping. Let L1(µ1;µ2;w) and L2(µ1;µ2;w)
be the resulting equilibrium levels of labour for given technology choices. Substi-
tute into the proﬁt function to get the “indirect proﬁt function” Πi(µ1;µ2;w). We
assume that the equilibrium is locally stable. This implies dLi=dµi > 0;dLj=dµi <
0.
Firm i’s technology choice then satisﬁes the ﬁrst order condition
dΠi
dµi
= µi
Li
L
µ
F
0 ¡
F
L
¶
dLj
dµi
+ Li
F
L
¡ c
0 = 0: (4)
Denote the outcome of the sequential game by (LS;µS).
2Proposition 1. At the subgame perfect equilibrium of the sequential game, tech-
nology choices and labour inputs are larger than at the Nash equilibrium of the
simultaneous move game.
Proof. Since F 0 < F=L and dLj=dµi < 0, at the sequential equilibrium, we have
Li
F
L
¡ c
0 < 0: (5)
The LHS of this inequality corresponds to d¼i=dµi with simultaneous technology
and input choices. Since d2¼i=(dµidLi) > 0, the result follows.
The welfare consequences are readily deduced. Let ¼ := ¼1 + ¼2 be aggregate
proﬁt. Suppose L1 = L2; and µ1 = µ2. We have
d¼
dL
= µF
0 ¡ w; (6)
d¼
dµ
= F ¡ 2c
0: (7)
Let L¤ and µ¤ be the choice of labour and technology where (6) and (7) are
zero.
Note that, conditional on the suboptimal labour choice LN, µN is optimal from
the social point of view. This is because in the simultaneous move game, there is
no technological spillover between the ﬁrms. In the sequential game, evaluate (4)
at µ¤, using symmetry:
dΠi
dµi
¯
¯
¯
¯
µ¤
=
µi
2
µ
F
0 ¡
F
L
¶
dLj
dµi
> 0: (8)
Thus, ﬁrms have an incentive to overinvest in technology.
Proposition 2. Aggregate proﬁts are higher at the Nash equilibrium of the si-
multaneous move game than at the subgame perfect equilibrium of the sequential
game.
Proof. The assumptions on F and c imply that both (6) and (7) are negative iﬀ
L > L¤ and µ > µ¤. Since LS > LN > L¤;µS > µN > µ¤, the result follows.
34. Policy
An obvious implication is that the sequential move game requires diﬀerent policies
to internalize the externalities. The standard prescription is a unit tax on labour.
The ﬁrm’s optimisation problem is then
max¼i = µiLi
F(L)
L
¡ (w + ti)Li ¡ c(µi): (9)
Let us assume ﬁrms are symmetric, and the equilibrium is symmetric too, i.e.,
L1 = L2;µ1 = µ2. Then we have the following.
Proposition 3. In the simultaneous move game, the optimal tax rate just equals
(minus) the marginal externality (i.e., loss of proﬁts) imposed on the other ﬁrm:
t
¤
i = ¡
d¼j
dLi
= ¡µj
Lj
L
µ
F
0 ¡
F
L
¶
> 0: (10)
Proof. The ﬁrm’s ﬁrst order conditions are
d¼i
dLi
= µi
µ
Li
L
F
0 +
Lj
L
F
L
¶
¡ (w + t) = 0; (11)
d¼i
dµi
= Li
F
L
¡ c
0 = 0: (12)
Inserting (10) into (11), using symmetry, gives the result.
In the sequential move game, the formula remains valid, conditional on the
choice of technology. If there is no corrective policy to ensure optimal investment,
however, the taxation of labour according to this rule will not achieve the ﬁrst
best. In fact, it needs to be supplemented by a tax on investment. Let ¿ be the
unit tax rate on investment. The ﬁrm’s problem now is
max¼i = µiLi
F(L)
L
¡ (w + ti)Li ¡ c(µi) ¡ ¿iµi: (13)
Proposition 4. The optimal tax system in the sequential move game consists of
tax rates t¤
i;¿¤
i , where
¿
¤
i =
dΠj
dµi
dLj=dµi
dLi=dµi
= µi
Li
L
µ
F
0 ¡
F
L
¶
dLj
dµi
> 0: (14)
4Proof. The ﬁrst order condition for the ﬁrst stage is now
dΠi
dµi
= µi
Li
L
µ
F
0 ¡
F
L
¶
dLj
dµi
+ Li
F
L
¡ c
0 ¡ ¿i = 0: (15)
Substitute (14) into (15), which shows that with this tax rate, (7) is zero, and
hence, the individual technology choice is socially optimal. Diﬀerentiating Πj
gives
dΠj
dµi
= µj
Lj
L
µ
F
0 ¡
F
L
¶
dLi
dµi
< 0; (16)
which, using symmetry, gives the result.
Note that, since labour input is a function of µ, one really needs a tax on
labour and technology: While one might set a tax rate, say, ˆ ¿, which would result
in L(µ(ˆ ¿)) = L¤, the corresponding technology level would not equal µ¤. Likewise,
a tax on labour alone would not suﬃce.
5. Conclusion
This note has shown that there is an extended tragedy of the commons when
ﬁrms make irreversible investments before choosing their input levels. An im-
portant assumption was that technology is a private good. Many technological
improvements, however, have clear public good properties. For instance, the ben-
eﬁts of research and development activities are often available to all ﬁrms in an
industry (Kamien, Muller and Zang, 1992, study the incentives of ﬁrms to inter-
nalize these externalities in cartels or joint ventures). If this is the case, there
is a tendency to underinvest in technology, which would counteract the tendency
of overexploitation. On the other hand, there may also be negative externalities
in R & D patent races (see Reinganum, 1989, for an overview): If I am the ﬁrst
to get a patent on an invention, you can’t get it. This externality is akin to the
common pool problem. This would lead to overinvestment and therefore make
the tragedy even more tragic.
5References
Brander, J. A. and Spencer, B. J. (1985). “Export subsidies and international
market share rivalry”, Journal of International Economics 18, 83–100.
Fershtman, C. and Judd, K. L. (1987). “Equilibrium incentives in oligopoly”,
American Economic Review 77, 927–940.
Hardin, G. (1968). “The tragedy of the commons”, Science 162, 243–248.
Kamien, M. I., Muller, E. and Zang, I. (1992). “Research joint ventures and R &
D cartels”, American Economic Review 82, 1293–1306.
Konrad, K. A. (1994). The strategic advantage of being poor: Private and public
provision of public goods, Economica 61, 79–92.
Reinganum, J. F. (1989). “The timing of innovation: Research, development, and
diﬀusion”, in R. Schmalensee and R. D. Willig (eds), Handbook of Industrial
Organization, Vol. 1, North-Holland, Amsterdam, pp. 849–908.
Rogoﬀ, K. (1985). “The optimal degree of commitment to an intermediate mon-
etary target”, Quarterly Journal of Economics 100, 1169–1190.
Schelling, T. (1960). The Strategy of Conﬂict, Harvard University Press, Cam-
bridge, MA.
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Investment and the tragedy of the commons

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Investment and the tragedy of the commons

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