Theoretical Economics Letters, 2012, 2, 470-473
http://dx.doi.org/10.4236/tel.2012.25088 Published Online December 2012 (http://www.SciRP.org/journal/tel)
Theft and Welfare in General Equilibrium: A Theoretical
Note
Thomas Randolph Beard1, George S. Ford2, Liliana V. Stern1, Michael L. Stern1
1
Department of Economics, Auburn University, Auburn, USA
Phoenix Center for Advanced Legal and Economic Public Policy Studies, Washington DC, USA
Email: beardtr@auburn.edu, ford@phoenix-center.org, sternli@auburn.edu, sternml@auburn.edu
2
Received September 20, 2012; revised October 21, 2012; accepted November 23, 2012
ABSTRACT
We show that in a dynamic general equilibrium model theft lowers social welfare even if it is costless to steal, there is
no theft prevention cost, and all stolen goods are immediately returned to society. Theft lowers social welfare because it
distorts the investment decision, resulting in undercapitalization and a lower steady-state level of capital. This sheds a
new light on the literature originated by Tullock [1].
Keywords: General Equilibrium; Theft; Efficiency
1. Introduction
Although theft of property receives nearly universal
condemnation by non-economists, the status of theft in
the neoclassical economic framework is far more nuanced. Perhaps the clearest example of this circumstance
can be found in the famous exchange between George
Stigler and Paul Samuelson concerning the status of what
was then termed the “New Welfare Economics”, of
which Samuelson [2,3] was an architect and Stigler [4]
an early critic. The difficulty, which still haunts such
discussions, is really a closely related set of complications which, however, are logically independent. In the
first place, when agent A steals something from agent B,
the “first-order” welfare consequences are ambiguous:
Perhaps B values it more than A did, and theft merely
represents another way in which goods are distributed
among agents. If money is stolen, then theft might
represent a “pure transfer”, i.e. an activity which is
assumed to have no welfare consequences. Second, as
emphasized by, for example, Tullock [1, p. 230], the
“theft industry” attracts resources both as “investments”
by thieves (e.g., burglars’ tools), and precautions by
potential victims (e.g., burglar alarms and security guards):
“This equilibrium, however, would be extremely costly
to the society in spite of the fact that the activity of theft
only involves transfers. The cost to society would be the
investments of capital and labor in the activity of theft
and in protection against theft. This lesson has been
learned by almost all societies that have adopted a
collective method of reducing this sort of income
transfer.”
Copyright © 2012 SciRes.
Tullock’s assertion that theft “only involves transfers”
echoes the point made much earlier by Stigler [4, p. 356]:
“If these theorems are applied to the problem of international trade, for example, they show that income (of
all countries together) is maximized by free trade”; and
criticized later by Samuelson [2, p. 606]: “This lack of
emphasis explains the occurrence of what can only be a
momentary lapse, which leads the author (Stigler) to state
that ‘income (of all countries together) is maximized by
free trade’. Aside from the meaningless statement in the
parentheses, the statement is wrong from almost any
point of view...” More recently, one can see a reprise of
many of these same issues in the reception of Kaplow
and Shavell’s “Fairness versus Welfare” [5], which
proposes that legal analysis be based solely on consequentialist welfare arguments, rather than deontological
principle, as in Dolinko [6] and Dorff [7].
What, then, are economists of a primarily neoclassical
bent supposed to conclude about the welfare consequences of theft? Arguments which one might view as
relatively persuasive in the partial-equilibrium framework adopted by Tullock [1] are less convincing in the
general equilibrium case. If, as claimed, thefts involve
“transfers” so that they have no undesirable welfare
consequences per se, should the social attitude towards
stealing spring primarily (if only partially) from a disapproval of the resources used up to affect it? Is
stealing socially undesirable because it is privately
costly?
We address this question by including theft within a
classic dynamic general equilibrium model. Our analysis
is designed to answer a very specific question: SimultaTEL
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T. R. BEARD ET AL.
neously supposing that some output is stolen, that theft is
costless, that no resources are used up in the act of stealing, and that those who steal the goods value them to
precisely the same degree as those who are robbed, does
the existence of theft impose any cost on society? The
answer, we will argue, is “yes”, and this conclusion
arises because theft alters the marginal incentives of
producers to produce the good which is stolen, leading to
a welfare-dominated sequence of capital investments and
lower (Pareto inferior) social outcomes. We conclude
that theft is undesirable in a very general sense, and this
conclusion is derived within a strictly neoclassical GE
framework. Moreover, this conclusion does not depend
on the existence of costs of either theft, or theft prevention.
2. A Simple General Equilibrium Model
That Incorporates Theft
We adapt the dynamic GE model in Becker [8] to the
problem at hand. In this model, an infinitely-lived representative household has perfect foresight, and selects a
sequence of consumption and labor supply decisions in
order to maximize a discounted sum of utility. The
household’s choices are constrained by a typical intertemporal budget constraint, and the capital stock evolves
in the conventional manner. The single consumption good
is supplied by a competitive industry under constant
returns to scale. We are interested in the properties of the
consumption path under steady-state conditions, and the
relationship of this path to the level of theft in the economy.
Let st denote the amount of the output good stolen in
period t that the household receives. The household takes
as given the initial capital stock k0, and the sequence of
prices of capital services, labor services, and the levels of
theft rt , wt , st for t 0,1, 2, , . The household chooses
the consumption and labor sequences ct , lt t 0 to maximize:
t u ct
t 0
(1)
subject to the constraints:
ct kt 1 rt kt wt lt st
0 lt 1
ct , kt 0
(2)
(3)
(4)
where β is the household’s discount factor and u is the
household’s instantaneous utility function. The utility
function is assumed to be strictly increasing, concave,
differentiable, and to have a sufficiently large margin at
the origin for interior solutions. The maximal labor supply is normalized to one unit and for simplicity, there is
no disutility from work. Hence, lt 1 in every period.
Copyright © 2012 SciRes.
The household’s problem yields the following Euler
equation as a necessary condition:
u ct rt 1u ct 1
(5)
In order to illustrate as simply as possible the welfare
consequences of theft, we suppose that some fraction θ
of the output of the economy is “stolen”, in the sense that
the seller is not compensated for it. However, in general
equilibrium nothing “disappears”, and the form of the
budget constraint given in (2) implies that, in equilibrium,
our representative household receives the benefit of the
theft in the same manner as it receives other income
st f .
Thus, the representative competitive firm is assumed
to maximize profits given the market prices of labor and
capital inputs, and the extent of theft θ in the economy.
The firm’s profit (t) in period t is:
πt 1 f kt , lt rt kt wt lt ,
(6)
where f is the production function which is assumed to be
concave, strictly increasing, differentiable, and exhibit
constant returns to scale (a common assumption that implies there are no profits in equilibrium, and thus no need
to specify the ownership of the firm). Note that theft will
decrease the returns to capital and labor, as profit maximization will require the equilibrium conditions
1 f k rt and 1 fl wt where the subscripts
on f indicate marginal products of capital and labor, respectively. If we combine (5) with the marginal condition
for profit maximizing with respect to capital, we immediately observe the effect of the distortion caused by theft.
We have:
u ct 1 f k u ct 1 .
(7)
Equation (7) illustrates the distortionary effect of theft
even when the stolen goods are returned to the household
without deterioration, theft is costless, and no resources
are used up to prevent it. Theft, in effect, is equivalent to a
decrease in the discount factor, resulting in greater
household impatience and lower levels of capital accumulation, as we will see below.
We consider next the steady state of the economy, in
which ct ct 1 c and kt kt 1 k . Equation (7)
evaluated at the steady state implies:
1
f k k , 1
β 1 θ
(8)
Equation (8) has several immediate implications. First,
the steady state marginal product of capital is higher the
higher is the rate of theft, θ. This in turn suggests that the
steady state capital stock is lower in an economy with
higher levels of theft, even when theft is costless and fully
redistributed to the household. Lower capital also implies
lower wages.
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How, though, does the prevalence of theft affect the
welfare of the household? Using the equilibrium budget
constraint we can examine the steady state level of consumption:
c f k ,1 k
(9)
Differentiating this expression with respect to the
steady state capital level and utilizing the expression for
the marginal product of capital in Equation (8) yields:
c
1
1 0
k 1
(10)
The decrease in the steady state capital stock resulting
from the theft causes a decrease in the steady state level of
consumption. The theft operates like an increase in
consumer impatience, generating an increase in initial
consumption, decrease in capital, and an ultimate decline
in the long-run level of consumption. The remaining
question is whether the change in the consumption stream
causes a decrease in overall lifetime utility.
To resolve this question, we need to find an optimal
growth model formulation which is equivalent to the
general equilibrium problem above. Becker [8] establishes that dynamic equilibria of the type described above
are mathematically equivalent to optimal growth models
which utilize the modified discount factor
ˆ 1 . The corresponding optimal growth problem is given by:
max ˆ t u ct
t 0
ct kt 1 f kt , lt
(11)
subject to the constraints:
0 lt 1 and ct , kt 0
(12)
(13)
The optimal growth problem described in (11)-(13)
has exactly the same feasible set as the problem with the
true discount factor and this fact has immediate implications. Suppose that some sequence of consumption cˆt
is optimal for the planning problem with the discount
factor ˆ . The cˆt sequence is also feasible for the
“true” discount factor β. Yet, since cˆt is clearly distinct from the optimal path with β as the discount factor,
we conclude that the presence of theft, even costless theft
that is returned fully to society, leads to a reduction in the
lifetime utility of the representative household.
reasons why “theft is bad”, at least in some credible,
neoclassical welfare calculation. This problem is, indeed,
a bit harder than it seems because there are many ways in
which theft might impact welfare. Many commentators
have noted that costly activities intended to facilitate
theft, or to prevent it, will reduce welfare at least
compared to cases where theft does not occur, so long as
that theft does not somehow reallocate resources to those
who value them more. Certainly there is a tradition, made
explicit in Tullock [1] that theft is “a transfer”. Be this as
it may, it is fair to say that everybody except economists
(and presumably thieves) is reasonably convinced that
theft is a bad thing in some aggregative, global sense. For
example, Siwek [9] estimates that piracy in the copyright
industry alone costs the US economy $58 billion in aggregate output each year.
We show in this note that everybody is correct. This
conclusion is more useful than might be apparent,
however, because we explicitly ignore certain costs that
might be associated with theft, which, if included, might
“tilt the board” in favor of finding that theft is indeed a
bad thing. In particular, we assume throughout that the
level of theft is exogenous (so one need not worry about
affecting it), that theft is costless to undertake, and no
expenditures occur to prevent it, and that the goods thus
stolen are immediately and costlessly returned to “society” (our representative household), who benefits fully
from it. Even with all of these assumptions, each of
which might be viewed as favoring a finding that theft is
socially costless, we find a persistent distortion caused by
theft which leads to undercapitalization and lower social
welfare within an analytic framework that is widely used
in economic theory. Utilizing an idea from Becker [8],
we demonstrate that the existence of theft acts to distort
the investment decision in a manner similar to that arising from more aggressive discounting of the future.
Our analysis supports the general social taboo associated with theft, and it does this on purely neoclassical
grounds. Theft can be viewed as a form of societal impatience, with all of the associated dynamic implications. In
this way we hope to retire one of the perennially misunderstood aspects of the “New Welfare Economics”, with
all due respect to the late George Stigler.
REFERENCES
[1]
G. Tullock, “The Welfare Costs of Tariffs, Monopolies
and Theft,” Western Economic Journal, Vol. 5, No. 3,
1967, pp. 224-232.
[2]
P. Samuelson, “Further Commentary on Welfare Economics,” American Economic Review, Vol. 33, No. 3, 1943,
pp. 604-607.
[3]
P. Samuelson, “The Gains from International Trade,”
Canadian Journal of Economics and Political Science,
Vol. 5, No. 2, 1939, pp. 195-205. doi:10.2307/137133
3. Concluding Remarks
Welfare economics has presented a number of important
(and not so important) puzzles over the last sixty years.
One puzzle, which has persisted for a long time in a
variety of literatures influenced by Tullock’s famous
paper on rent-seeking and welfare, concerns the precise
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T. R. BEARD ET AL.
[4]
G. Stigler, “The New Welfare Economics,” American
Economic Review, Vol. 33, No. 2, 1943, pp. 335-359.
[5]
L. Kaplow and S. Shavell, “Fairness versus Welfare,”
Harvard University Press, Cambridge, 2002.
[6]
D. Dolinko, “Review Essay: The Perils of Welfare Economics,” Northwestern Law Review, Vol. 97, No. 1, 2002,
pp. 351-393.
[7]
M. B. Dorff, “Why Welfare Depends on Fairness: A Re-
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473
ply to Kaplow and Shavell,” Southern California Law
Review, Vol. 75, 2002, p. 847.
[8]
R. Becker, “Capital Income Taxation and Perfect Foresight,” Journal of Public Economics, Vol. 26, No. 2, 1985,
pp. 147-167. doi:10.1016/0047-2727(85)90002-7
[9]
S. Siwek, “The True Cost of Copyright Industry Piracy to
the US Economy,” Policy Report #189, Institute for Policy Innovation, 2007.
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