8 October 1996;

The Royal Swedish Academy of Sciences has decided to
award **the Bank of Sweden Prize in Economic Sciences in Memory
of Alfred Nobel, 1996**, to

**Professor James A. Mirrlees**,
University of Cambridge, U.K. and

**Professor William Vickrey**, Columbia University, New York,
USA

(deceased October 10, 1996)

*for their fundamental contributions to
the economic theory of incentives under asymmetric
information.*

**Information and Incentives**

One of the most important and liveliest areas of economic
research in recent years addresses situations where
decision-makers have different information. Such *informational
asymmetries* occur in a great many contexts. For example, a
bank does not have complete information about lenders' future
income; the owners of a firm may not have the same detailed
information about costs and competitive conditions as the
managing director; an insurance company cannot fully observe
policyholders' responsibility for insured property and external
events which affect the risk of damage; an auctioneer does not
have complete information about the willingness to pay of
potential buyers; the government has to devise an income tax
system without much knowledge about the productivity of
individual citizens; etc.

Incomplete and asymmetrically distributed information has fundamental consequences, particularly in the sense that an informational advantage can often be exploited strategically. Research on the economics of information has therefore focused on the question of how contracts and institutions can be designed to handle different incentive and control problems. This has generated a better understanding of insurance markets, credit markets, auctions, the internal organization of firms, wage forms, tax systems, social insurance, competitive conditions, political institutions, etc.

This year's laureates have laid the
foundation for examining these seemingly quite disparate areas
through their analytical work on issues where informational
asymmetries are a key component. An essential part of **William
Vickrey**'s research has concerned the properties of different
types of **auctions**, and how they can best be designed so as
to generate economic efficiency. His endeavors have provided the
basis for a lively field of research which, more recently, has
also been extended to practical applications such as auctions of
treasury bonds and band spectrum licenses. In the mid 1940s,
Vickrey also formulated a model indicating how income taxation
can be designed to attain a balance between efficiency and
equity. A quarter of a century later, interest in this model was
renewed when **James Mirrlees** found a more complete solution
to the problems associated with *optimal income taxes*.
Mirrlees soon realized that his method could also be applied to
many other similar problems. It has become a principal
constituent of the modern analysis of complex information and
incentive problems.

**1. William Vickrey**

William Vickrey was born in 1914 in Victoria, British Columbia.
He received his Bachelor of Science degree from Yale University
in 1935. That same year, he began his postgraduate studies at
Columbia University, New York, where he received his Master's
degree in 1937 and his Ph.D. in 1947. He has been affiliated with
the faculty of Columbia University since 1946, and served as a
tax advisor for various organizations between 1937 and 1947. He
was Professor Emeritus at Columbia University.

Throughout his professional life, Vickrey was engaged primarily in research and teaching; he was also appointed to fact-finding commissions by a number of public authorities. For example, he designed a fare system for the New York subway (1951) and worked on city planning and transportation in India, Argentina and Venezuela (1962-63). Among Vickrey's better-known activities as a consultant are also his studies of the transportation and traffic system in Washington DC in the late 1950s and his service as an advisor to the UN on taxation.

William Vickrey has above all contributed to enhancing our knowledge about the efficient use of resources in the public sector. Notwithstanding his numerous and far-reaching practical applications, it is the depth of his theoretical endeavors which gives lasting value to his scientific lifework. Vickrey has a characteristic style of writing; almost in passing, he interposes deep insights in seemingly routine economic arguments. This has sometimes inferred that his theoretical contributions did not become apparent until long after they were written. Vickrey's creativity is particularly remarkable in his studies of social choice and resource allocation mechanisms. He has also carried out important research on public-sector pricing as well as transportation and urban economics.

**1.1. Auctions and Resource
Allocation**

One of Vickrey's foremost contributions is his article
"Counterspeculation, Auctions, and Competitive Sealed Tenders"
(1961), where he analyzed different kinds of auctions and
introduced the so-called second-price auction or, as it came to
be known, the Vickrey auction. His study was inspired by Abba P.
Lerner's suggestion that, in markets with imperfect competition,
efficiency-enhancing institutions should be developed so as to
generate a resource allocation similar to that under perfect
competition. Vickrey's analysis proved important not only to the
theory of auctions, but also in providing general insights into
the design of allocation mechanisms aimed at creating socially
desirable incentives when information is incomplete. It is
related, for example, to the problem of inducing people to state
their true willingness to pay for public services.

In a second-price auction, an object (a resource, a right) is auctioned off by means of sealed bidding, where the highest bidder gets to buy the item, but at the second highest bid. This mechanism elicits the individual's true willingness to pay. By bidding above his own willingness to pay, an individual runs the risk that someone else will bid likewise, and he could be forced to buy the object at a loss. If he instead bids below his own willingness to pay, he runs the risk of someone else buying the item at a lower price than the amount he himself is willing to pay. In this type of auction, a truthful bid is thus the dominant strategy. The auction also distributes the good in a socially, efficient way; the object goes to the bidder with the highest willingness to pay, and the bidder in question pays the social opportunity cost which is the second highest bid.

An analogous idea underlies the so-called Clarke-Groves mechanism for eliciting truthful tenders for public projects, which was worked out in detail by Clarke (1971) and Groves and Loeb (1975). Vickrey anticipated this important result by a substantial time margin.

The efficiency aspect was Vickrey's main interest when the second-price auction was introduced in his 1961 article. Another issue that interested him is which type of auction would yield the highest expected price. Apart from the Vickrey auction, he studied three other auctions. An English auction begins with the auctioneer requesting a first bid, and ends when he can no longer elicit a higher bid. The object goes to the highest bidder, who pays his tender. A Dutch auction starts at a price which far surpasses the amount any of the participants are willing to pay. The price is gradually reduced until someone shouts "mine" and then gets to buy the object at the proposed price. A first-price auction relies on sealed bidding, where the highest bidder buys the good at his tendered price.

Vickrey noted that the English auction and
the second-price auction are strategically equivalent and,
likewise, that the Dutch auction and the first-price auction are
strategically equivalent. He also showed that if bidders'
valuations of the object are statistically independent and
uniformly distributed over an interval, then - in Nash
equilibrium - all four auctions result in the same expected price
paid to the seller. This important finding has later been
extended and has become known as the *revenue equivalence
theorem*.

A distinguishing feature of the second-price (and hence also the English) auction is that it only requires individual rationality. That is, a participant's optimal bid in such an auction is independent of his expectations about others' bids. This contrasts with the first-price (and hence also the Dutch) auction, in which optimality depends on (equilibrium) expectations, making the outcome less robust.

In his article "Auctions and Bidding Games" (1962), Vickrey elaborated on the game-theoretic aspects of auctions, and generalized his earlier analysis to auctions over multiple (identical) objects as well as to a broader class of distributions for the willingness to pay.

In his work on auctions, Vickrey studied
games under first-order incomplete information, thereby
anticipating the 1994 economics laureate John Harsanyi's more
thorough research on games with incomplete
information.^{1} In recent years,
analysis of auctions as allocation mechanisms has become a very
active field of research.

1. Incomplete information of the first order arises in an auction game if i) the bidders do not know one another's willingness to pay, but instead assign probability distributions to them, and ii) the bidders know these probability distributions. Incomplete information of higher orders implies, roughly speaking, that the bidders have expectations about each other's expectations, etc.

**1.2. Optimal Income Taxation**

Vickrey's most creative endeavor regarding taxation is his
article "Measuring Marginal Utility by Reactions to Risk" (1945),
which addresses the balance between efficiency and equity in
designing an income tax system. It is explicitly formulated in
the form of - what economists now call - an optimal income tax
problem.

Edgeworth's classical essay (1897) on the structure of income taxation is based on a utilitarian welfare perspective. His intention was to examine how a system of redistributive taxes should be designed in order to maximize social welfare, defined as the sum of all individuals' utility (welfare). Assuming diminishing marginal utility of consumption, he found support for a progressive tax schedule. If all individuals derive the same utility from consumption, a welfare optimum requires the marginal utility of consumption to be the same for all individuals. If their gross incomes differ, then net income has to be equalized.

Vickrey (1945) emphasized that such
taxation does not give individuals incentives to work. His
analysis takes the same utilitarian approach as Edgeworth did;
the government wants to maximize the sum of all individuals'
utility. Vickrey assumes that all individuals have the same
preferences over productive effort *e* and consumption
*y*, but different abilities to produce goods and services.
He also assumes that the productive effort *e* required of
an individual with ability *t* to produce quantity *x*
is given by a function *w*, or *e = w(x,t)*.

The government's aim is to maximize the sum
of all individuals' utility, given that total tax revenue reaches
a predetermined level. Vickrey presupposes that redistribution
takes place by means of a tax function *f* which determines
each individual's consumption *y* based on his production
*x*, that is, by the equation *y = f(x)*. The tax paid
is thus *x-y*. Given the tax function *f*, each
individual *t* chooses his production *x* so that his
utility is maximized. Let *x*(t)* be individual *t*'s
optimal choice of productive effort, and let *v(t)* be the
resulting (maximum) utility level for the individual.

The objective of the government's choice is
the function *f*, where *f* is chosen so that the
"sum"^{2}

(1) *v(t)dt*

of all individual's utility is maximized, under the constraint that the total tax revenue

(2) *(x*(t) -f(x*(t)))dt*

is equal to a given level. Vickrey derived the so-called Euler equation for this calculus of variation problem, a necessary condition for an optimal tax function. But the equation became implicit and rather complicated; Vickrey concluded the formal part of the discussion by declaring: "Expanding this expression...produces a completely unwieldy expression. Thus even in this simplified form the problem resists any facile solution." As it turned out, this conclusion was too pessimistic.

In his paper "The Problem of Progression" (1968), Vickrey returned to the problem of creating the right balance between incentives and distributional objectives in the income tax system, although he was not able to specify the degree of progressivity in the optimal tax structure. He also considered the technical and political difficulties in constructing a tax system which is progressive in any real sense.

It proved to be James Mirrlees who, independently of Vickrey's article from 1945, gave the optimal income tax problem a new and precise formulation and developed a methodology for analyzing the properties of the solution (see Section 2.1).

2. For technical reasons, Vickrey chose to treat the population as a continuum, whereby sums can be replaced by more manageable integrals.

**1.3. Other Contributions**

The above-mentioned article from 1945 contains another pioneering
theoretical contribution. While considering methods of measuring
utility through individuals' decisions under uncertainty, Vickrey
introduced the approach of letting the individual himself
evaluate income distributions by assuming that he can, with equal
probability, end up in any other individual's position in the
distribution. This approach, where societies are evaluated under
the *veil of ignorance*, was later rediscovered and
developed by John Harsanyi in two studies in the early 1950s
(1953, 1955), where he introduced the concept of the *original
position*. The same idea is taken up in John Rawls's book,
*A Theory of Justice* (1971).^{3}

In his paper "Utility, Strategy, and Social Decision Rules" (1960), Vickrey discusses Kenneth Arrow's impossibility theorem and corrects an error in Arrow's analysis. He also asserted that the condition that the preference ordering should be independent of irrelevant alternatives is sufficient for ensuring that a voting procedure will not be manipulable, in the sense that no one should be able to sway the outcome of the vote to his advantage by stating untruthful preferences. Since Arrow showed that all of his conditions are impossible to fulfill simultaneously, Vickrey guessed, perceptively, that a voting procedure which fulfills the remaining requirements is manipulable. Almost 15 years later, Allan Gibbard (1973) and Mark Satterthwaite (1975) succeeded in showing that Vickrey had been right. According to Gibbard: "Indeed the proof in this paper proceeds roughly by confirming Vickrey's conjecture." More specifically, they demonstrated that every voting procedure, which is not determined by a dictator and has at least three outcomes, is manipulable. In the literature, this is known as the Gibbard-Satterthwaite impossibility theorem.

Vickrey's doctoral thesis, "An Agenda for
Progressive Taxation", was published in 1947. The agenda consists
of 21 proposals for reforming the U.S. income tax system. His
objective was to design a stable tax base, which he regarded as a
prerequisite for a progressive tax system. His dissertation
contains an innovation which is quite likely to be part of
economists' intellectual baggage for a long time. In fact, the
concept of *cumulative averaging* was first presented in
Vickrey's article "Averaging of Income for Tax Purposes" (1939).
The idea is that taxation should be neutral with respect to the
point in time when income is realized, which eliminates the
incentive for taxpayers to vary transaction times for tax
purposes. Once the principle was formulated, consistent logic and
practical considerations remained. Vickrey found the sought-after
neutrality by regarding all tax payments as deposits in an
interest-bearing account in the taxpayer's name. Under these
conditions, the tax owed at a certain point in time would be
equal to the tax on the total income accumulated, including the
interest income on the tax account, minus the accumulated tax
already deposited in the account. The idea was never put into
practice, however.^{4}

*Marginal cost pricing of public
services* permeates Vickrey's scientific production. About 40
of his studies address efficient pricing of public services. He
has not only made significant theoretical contributions but,
unlike most excellent theorists, he has also followed up on his
proposals all the way to their implementation.

A problem with marginal cost pricing in public enterprises is that full cost coverage cannot be attained if the marginal cost decreases with the volume of production. This problem had already been dealt with in 1926 by Frank Ramsey, who solved the pricing problem under a given budget restriction. The solution is called Ramsey pricing and implies, roughly speaking, that prices are inversely proportional to the price elasticity of the demand for the various goods.

In his classical inquiry into pricing of subway transportation in New York, "A Proposal for Revising New York's Subway Fare System" (1955), Vickrey realized that marginal cost pricing would burden the subway system with a substantial budget deficit, which would have to be covered through taxation. Since taxes have a social cost in that they distort incentives, he tried to estimate the marginal cost of the additional taxes required to cover the deficit. The fare should then be set so as to achieve a balance between the social losses from raising the price above the marginal cost and the subsequent gains from lessening the need for tax revenue. The analysis is an early example of how Ramsey pricing can be implemented in practice. It also improved on Ramsey's original representation of the problem; Vickrey evaluated pricing policy against the marginal cost of tax revenue and not against a given budget restriction.

Characteristically, Vickrey's New York study also contains some interesting practical details. For instance, he suggested that each passenger should pay a maximum fare (25 cents) when boarding the subway, using a ticket magnetically coded with the time and place. When exiting, the passenger would insert the ticket in a turnstile and be refunded the difference between the maximum fare and the real cost.

Vickrey has also done other work in urban economics, the most important of which is perhaps his article entitled "Congestion Theory and Transport Investment" (1969). The model is relatively simple. A group of commuters who start work at the same time have to pass a "bottleneck" on their way to the office. If the number of arrivals at the bottleneck exceeds its capacity, a line develops. When the commuters decide what time to leave home, they weigh the gains in travel time against the costs of getting up too early or arriving late for work. Vickrey realized that travel costs in equilibrium are independent of when the journey begins. In other words, the cost of getting up early for the commuter who leaves home first is equal to the cost of standing in line at the bottleneck for the commuter who gets to work on time. Under different road tariff systems, applications of this equilibrium concept allow a solution for determining the length of the queue over time and an equilibrium pattern for departure times. Vickrey showed that the equilibrium tariff can be used for profitable "tax switching". Commuters are by definition indifferent between paying a road toll and not having to stand in line, or not paying a toll and standing in line. What is more, there is no change in arrival times at work. Hence, the tax has neither excess burden nor burden, and the tax revenue can be used to reduce other tolls for motorists, thereby improving their situation. As usual, Vickrey was ahead of his time. His model was rediscovered by road engineers and economists in the early 1980s.

3. Vickrey returned to the veil of ignorance in other respects before Rawls had finished his book; see Vickrey (1968).

4. An incomplete predecessor was the income tax system in Wisconsin during the period 1928-1932.

**2. James A. Mirrlees**

James A. Mirrlees was born in 1936 in Minnigaff, Scotland. He
received his Master's degree in Mathematics in Edinburgh in 1957,
and his Ph.D. from the University of Cambridge in 1963. He was
Edgeworth Professor of Economics at Oxford University between
1969 and 1995, and currently holds a professorship in Economics
at the University of Cambridge.

Mirrlees has made several far-reaching contributions in public economics, particularly in optimal taxation and development. One of his most important contributions, carried out jointly with Peter Diamond, consists of a very thorough and almost definitive treatment of optimal commodity taxes (see Section 2.3). Mirrlees subsequently applied this production efficiency result in analyses, coauthored with Ian Little, of cost-benefit rules in developing economies. These rules have since become common property in economics. Mirrlees's foremost achievement, however, is related to information and incentive theory. His study of optimal income taxes was the first complete analysis of the optimal design of economic policy under information asymmetries. In solving this problem, he developed a methodology which has become a paradigm in the economics of asymmetric information.

**2.1. Optimal Income Taxation**

Mirrlees's model for optimal income taxation is very similar to
Vickrey's. It is assumed that the government's objective is to
maximize the sum of all individuals' utility, and all individuals
have the same preferences for leisure and consumption, but
different abilities to produce goods and services. Individuals
know their productivity, but this information is not shared by
the government.

As in the description of Vickrey's model,
let *x* denote an individual's pretax income, or
*production,* and *y* his aftertax income, or
*consumption*. Individual differences in productivity are
captured by an index *t* distributed over some interval. An
individual with a higher *t* can produce a given quantity
*x* of goods and services with less effort or in less
working hours. Let the scale for *t* be such that an
individual of type *t* with productivity *e*, for
example measured in the number of working hours per week,
produces *x = et*; this corresponds to *e = w(x,t) =
x/t* in Vickrey's analysis. The utility of an individual who
consumes *c* and works *e* is *u(c,e)*, where the
utility function *u* is assumed to be increasing in its
first argument and decreasing in its second. By normalizing the
length of the workweek to 1, the residual *z = 1-e*
represents the individual's leisure.

The government's objective is to maximize
the sum of all individuals' utility under the constraint that
their combined consumption does not exceed total
production.^{5} If the government
could observe each individual's type *t*, then it would only
have to require each individual to produce and consume in
accordance with the solution to this optimization problem. If
leisure is a normal good (i.e., the individual chooses to consume
more of it with rising income), the solution would result in
lower utility for individuals with a higher productivity index.
This follows from the fact that the government on the one hand
wants to equate individual differences in the marginal utility of
consumption (so as to maximize the sum of their utility), while,
on the other hand, it wants to require more productive
individuals to work more (so as to facilitate meeting the
resource restriction). At the optimum, the balance between these
two objectives would imply a transfer of utility from more to
less productive individuals.

The fact that the resulting individual
utility would be decreasing in the productivity index illustrates
the underlying incentive problem: if the government cannot
observe an individual's type t and would attempt to implement the
above solution, it would be in the self-interest of all
individuals to claim that their productivity is lower than is
actually the case. The first-best solution outlined here is
therefore not feasible when individuals' productivity is their
private information. Instead, Mirrlees tried to find a
second-best solution presupposing that individuals' productivity
is in fact their own private information, and that redistribution
has to take place through a tax schedule common to all
individuals. Just as in Vickrey's model, the tax schedule
determines aftertax income *y* as a function *f* of
pretax income *x*. Given such a tax function *f*, each
type of individual *t* chooses his pretax income
(production) *x* in order to achieve maximum utility
*u(f(x),x/t)*. For individual *t*, let *x*(t)* and
*y*(t)* be the optimal choice of own production and
consumption, and let *v(t)* be his resulting (maximum)
utility.

The individual's utility maximization
problem is shown in Figure 1. The horizontal axis represents the
individual's production, *x = et*, and the vertical axis his
consumption *y*. The dashed curves designate levels of the
utility function *u*, and the solid irregular line is the
tax function *f*. The upper bound on working hours *e*
is normalized to 1, so that the upper bound on *t*'s
production is *t*. The shaded area under the graph of the
tax function is the individual's choice set. The optimal utility
level *v(t)* is achieved at the point (indicated in the
figure) where the highest level curve which intersects the choice
set is tangent to the tax function (the boundary of the choice
set).

The tax schedule *f* is thus the
object of the government's choice, i.e., its policy instrument.
Each property of the tax schedule, such as its progressivity, has
to be derived from the government's optimization problem; none of
these properties can be assumed at the outset. This implies, for
instance, that the usual convexity assumptions which are crucial
in economic models cannot be made.^{6} Owing to this circumstance, the government's
optimization problem, containing all individuals' optimization
problems as constraints, becomes unwieldy.

Mirrlees's breakthrough was to impose an
empirically reasonable restriction on the utility function. Under
this restriction, the government's optimization problem can be
reformulated so that the constraint incorporating all
individuals' optimization problems can be replaced by a
differential equation for the utility derivative *dv(t)/dt*.
This equation indicates how the resulting individual utility
level changes when there is a marginal change in the type of
individual *t*. Mirrlees made the astute observation that
this optimization problem may be regarded as a dynamic control
problem of a type that is well known from applied mathematics. He
interpreted *t* as "time", the utility level *v(t)* as
the "state" at time *t*, the differential equation as the
"law of motion" for the state variable *v(t)*, and
individual *t*'s production *x(t)* as the control
variable. He then proceeded to solve this dynamic control problem
using Pontryagin's maximum principle.

The solution to the optimization problem,
the quantities and for each type of
individual *t*, can be respecified as a tax function, *y =
f(x)*, by eliminating the index *t* from the set of
equations *y =* and *x =*.
The government's optimal policy instrument *f* under
asymmetric information can then be identified.

Mirrlees showed that the differential
equation for *v(t)* guarantees an outcome whereby each type
of individual *t* will choose the production
which is optimal from the government's point of view, that is,
*x*(t) =* .

This approach has subsequently been applied in several basic models with asymmetric information. In general, however, there is no analytical solution; instead, the optimal tax function has to be obtained by numerical methods. Here, of course, the degree of progressivity will be lower than it is according to Edgeworth's result. In some income intervals, the result can even be a regressive tax schedule.

The key assumption that enables this
drastic simplification and reformulation of the optimization
problem is the so-called *single-crossing* condition. The
essential requirement of this condition with respect to the
utility function u is that if two level curves in the
*(x,y)* plane, one for individual *t* and the other for
individual *t'*, intersect, then the level curve for the
more productive individual has a smaller slope; see Figure 1. In
other words, two level curves cannot intersect more than once -
thereof the name of the condition. In relation to an (increasing)
tax function *f*, this implies that more productive
individuals will find it optimal to produce more; cf. Figure 1.
In more economic terms, the essential requirement of this
condition is that if increased effort is imposed on an
individual, then in order for the individual to remain
indifferent, he needs a greater increase in consumption as
compensation, the higher his initial effort. The single-crossing
condition has become central to several microeconomic and game
theoretic models of incentives under asymmetric information.
Mirrlees was the first to identify the condition and explain its
analytical significance.

The optimization problem respecified by
Mirrlees yields the same result as if the government had asked
all individuals to report their productivity index *t*, and
had then imposed corresponding production and consumption
objectives, and . These individual
objectives, stated in advance for each productivity index
*t*, would constitute a binding commitment on the part of
the government as to how the private information reported by
individuals would be used. Given this commitment from the
government, every individual would fulfill his objective
voluntarily.

Mirrlees's analysis thus proved to contain
the germ of a very profound and general insight. An economic
allocation mechanism which asks all individuals to provide their
private information is called *direct*. If truthful
reporting is optimal for all individuals, the mechanism is called
*incentive compatible*. The above argument indicates that
the allocations which can be achieved through a tax function
*f* under the single-crossing condition can be attained by a
direct and incentive-compatible allocation mechanism. The
so-called *revelation principle* signifies that for every
desired allocation, there is a direct and incentive-compatible
mechanism which implements it.^{7}
This principle allows the analyst to limit the search for an
optimal mechanism to the relatively small subclass of direct and
incentive-compatible mechanisms. It usually represents a
considerable simplification of the analytical problem. Other
researchers later developed a general form of the revelation
principle, which has had a far-reaching influence on
microeconomic theory; cf. e.g. Dasgupta, Hammond and Maskin
(1979) and Myerson (1982). Nowadays, it is the standard method
for designing optimal allocation mechanisms under asymmetric
information.

Early applications of the methodology in Mirrlees (1971) may be found in Spence and Zeckhauser (1971) on health insurance, Sheshinski (1972) on linear income taxation, Spence (1977) on nonlinear pricing, and Mussa and Rosen (1978) on product quality, etc.

5. Since it is assumed that there is a
continuum of types of individuals, the sums take the analytical
form of integrals over the type index *t*.

6. In Figure 1, note that since there is more than one tangential point between the level curves and the graph of the tax function, multiple optima can be obtained for individuals. This would not be possible under conventional convexity assumptions.

7. The equilibrium condition for implementing the mechanism is important in this context. In Mirrlees's model, the desired allocation can be effectuated under the weakest equilibrium condition used, i.e., "solution in undominated strategies". This is also the case in the Vickrey auction; see Section 1.1.

**2.2. Moral Hazard**

The informational asymmetry in optimal taxation concerns both the
type *t* of the individual and his or her work effort
*e*. In Mirrlees's model, the government is only assumed to
know the product *x = te*. Another type of incentive problem
is dealt with in the above-mentioned paper by Spence and
Zeckhauser regarding the optimal design of health insurance. This
is usually referred to as *moral hazard*. In a series of
studies during the 1970s, Mirrlees analyzed this kind of problem,
with applications to insurance and to incentives in organizations
and firms.

A moral-hazard problem between two parties
is comprised of: a variable *x*, which is observable to both
parties; the "outcome", which depends on one party's - the
*agent*'s - action *a*, which the other party - the
principal - cannot observe; and a stochastic variable .
The outcome *x* is thus a function of *a* and
*; x = g(a, )*.^{8} In an insurance context, where the term moral
hazard originated, can be some potentially harmful
effect on the object (such as weather or attempted theft),
*a* the care taken by the policyholder, and *x* the
resulting damages to the object. In the relation between the
owners and the management of a firm, *a* can be the
executive's work effort, *x* profits, and
production and market conditions. In both cases,
represents external factors which may be difficult or impossible
for the principal to monitor.

Following the 1972 economics laureate Kenneth Arrow (1963), moral-hazard problems have been modelled in exactly this way; see e.g. Spence and Zeckhauser (1971). However, severe technical problems have been encountered, similar to those noted by Vickrey in connection with optimal income taxation. The approach was too cumbersome. Mirrlees realized that each action carried out by the agent gives rise to a probability distribution over the outcomes. The agent's choice of action can thus be regarded as a choice of probability distribution. This seemingly simple reformulation turned out to permit a more powerful analysis. Mirrlees proceeded to develop this reasoning in a series of papers (1974,1975, 1976, 1979).

According to the terminology currently used
in this area, a *contract* between an economic agent and a
principal is a rule, or function *h*, which for every
outcome *x* indicates (positive or negative) compensation
*y* to the agent: *y = h(x)*. Here, the compensation
rule *h* is the object of the principal's choice - in the
same way that the tax schedule *f* is the government's
choice in income taxation. The principal (the insurance company
or the owner of a firm) is assumed to choose the compensation
rule with the objective of maximizing his own expected utility,
under two constraints. First, since the contract relation is
assumed to be voluntary, the principal has to ensure that the
agent is willing to accept the contract, i.e., choose the
function *h* so that the agent has at least as high expected
utility from accepting the contract as from rejecting it. This is
known as the *incentive compatibility* constraint or the
*participation constraint*. The second constraint is that
once the agent has signed the contract, his action *a* is
optimal for him under the contract.

As in the case of the income tax schedule
*f*, analytically desirable properties of the function
*h* cannot be assumed at the outset; they have to be derived
from the optimality conditions.^{9} Again, there is an obstacle to further
analysis, and once again it was Mirrlees who was able to overcome
it.

He began by replacing the agent's
optimality constraint with an equation expressing a first-order
condition for the agent's action to be optimal. When this
equation is inserted into the principal's first-order condition,
the resulting equation contains the ratio between both parties'
marginal utility. This ratio had already appeared in the earlier
Arrow-Debreu model of optimal risk sharing under symmetric
information, i.e., without the informational asymmetry outlined
here. Mirrlees's equation also contains a new term which can be
interpreted as the derivative of a *likelihood ratio*. Let
*(x|a)* be the probability density of the outcome
*x*, given the agent's action *a*. This can be written
as the derivative of the logarithm of *(x|a)*
with respect to *a*. The term expresses the information
about the agent's action contained in the outcome *x*, and
thus the extent to which risk sharing has to be restricted
because information is asymmetric to the disadvantage of the
principal.

This is where Mirrlees's reinterpretation
of the agent's choice - from a choice of action *a* to a
choice of distribution function for the outcome *x* -
becomes crucial. This interpretation can be used to determine the
cost of compensating the agent so that his effort will come close
to that which is optimal for the principal. Mirrlees showed that
it is cheap for the principal to implement the desired outcome if
the agent is sensitive to "punishment" (low or negative
compensation) and if, in addition, the derivative above becomes
very large in bad outcomes. The optimal contract will then
approximate the solution to the Arrow-Debreu *first-best*
solution under symmetric information. This occurs simply by
letting the contract stipulate severe punishment in bad outcomes.
The result is that the policyholder takes care of the insured
object almost as if it were uninsured, and the executive manages
the firm almost as if it were his own. Under the above
assumptions, bad outcomes are in fact highly informative; the
principal knows almost for certain that the agent has not exerted
himself to any large extent. Mirrlees (1974) provided similar
examples, and Mirrlees (1975, 1976) established general
results.

Prior to Mirrlees, other researchers had
replaced the agent's optimality conditions with an equation for
first-order conditions; see e.g. Ross (1973). But it was Mirrlees
(1975) who showed that this method - which in general only yields
a necessary but not sufficient condition for optimality, and can
be applied only if the function *h* is "well behaved" - is
problematic. As in the case of nonlinear optimal taxation,
nonlinear contracts easily lead to nonconvexities. Mirrlees
identified two conditions which allow the first-order aspect to
be applied: the *monotone likelihood ratio* and the
*convex distribution function property*.

8. This function *g* and the
probability distribution of are assumed to be
*common knowledge*- to both parties. In the standard model,
it is assumed that the agent does not know the value of
when he chooses action *a*.

9. In general, the optimal reward is not even monotonically related to the outcome.

**2.3. Other Contributions**

Mirrlees (1976) applied the method sketched above to the internal
organization of firms, with an emphasis on incentive pay schemes
and hierarchical structures. This work contributed substantially
to formalizing the 1991 economics laureate Ronald Coase's insight
that the size of a firm is determined by a balance between the
costs of internal control and the costs of market transactions. A
noteworthy result of Mirrlees's analysis is that compensation at
lower levels in a firm should take the form of fixed wages,
whereas profit-related pay is preferable at top management
levels.

In collaboration with Peter Diamond, Mirrlees has also carried out a comprehensive analysis of commodity taxes in a second-best world. Due to the fact that commodity taxes create a "tax wedge" between consumer and producer prices, taxpayers are prepared to pay more in order to avoid the taxes than the amount which accrues to the government in tax revenue. Given that this social cost cannot be fully prevented, taxes have to be chosen in the second-best way.

The general theory of optimal taxation in a second-best economy encompasses few clear-cut recommendations. If one condition for social efficiency is violated, as a rule there is reason to violate others as well. However, Diamond and Mirrlees (1971) obtained a highly universal result. Under relatively general conditions, it is desirable to maintain production efficiency. In concrete terms, this means that taxes should not be levied on factors of production.

The underlying intuition is relatively simple. Assume that final consumption varies continuously with tax rates. Diamond and Mirrlees showed that, in general, there exists a change in the tax rates which is advantageous to all consumers. If production is not efficient at the outset, i.e., it is in the interior of the production set, there will be a sufficiently small change in tax rates which is favorable to all consumers and still keeps the economy within the production set (due to continuity). Hence, the initial situation could not have been optimal.

As already indicated, this result has had
widespread consequences for project appraisal and economic policy
in developing countries. Jointly with Little, Mirrlees has set up
criteria for evaluating projects based on the desirability of
efficiency in production; see Little and Mirrlees (1974). On a
theoretical level, the analysis has been generalized in various
directions in Mirrlees (1972) and Diamond and Mirrlees
(1976).

**3. Summary Statement**

James A. Mirrlees and William Vickrey have been awarded the 1996
Bank of Sweden Prize in Economic Sciences in Memory of Alfred
Nobel for their fundamental contributions to the economic theory
of incentives under asymmetric information.

**4. References**

Arrow, K.J., (1963) Uncertainty and the Welfare Economics of
Medical Care, *American Economic Review 53, 941-969*.

Clarke, E., (1971) Multipart Pricing of Public Goods, *Public
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Dasgupta, P.S., Hammond, P.J., and Maskin, E., (1979) The
Implementation of Social Choice Rules: Some General Results on
Incentive Compatibility, *Review of Economic Studies 46,
185-216*.

Diamond, P., and Mirrlees, J.A., (1971) Optimal Taxation and
Public Production I: Production Efficiency, II: Tax Rules,
*American Economic Review 61, 8-27, and 261-278*.

Diamond, P., and Mirrlees, J.A.,(1976) Private Constant Returns
and Public Shadow Prices, *Review of Economic Studies 43,
41-78*.

Edgeworth, F.Y., (1897) The Pure Theory of Taxation, *Economic
Journal 7, 46-70, 226-238 och 550-571*.

Gibbard, A., (1972) Manipulation of Voting Schemes: A General
Result, *Econometrica 41, 587-601*.

Groves, T., and Loeb, M., (1975) Incentives and Public Inputs,
*Journal of Public Economics 4, 311-326*.

Harsanyi, J., (1953) Cardinal Utility in Welfare Economics and in
the Theory of Risk Taking, *Journal of Political Economy 61,
434-435*.

Harsanyi, J., (1955) Cardinal Welfare, Individalistic Ethics, and
Interpersonal Comparisons of Utility, *Journal of Political
Economy 63, 309-321*

Little, I.M.D., and Mirrlees, J.A., (1974) *Project Appraisal
and Planning for Developing Countries*, New York: Basic
Books.

Mirrlees, J.A., (1971) An Exploration in the Theory of Optimal
Income Taxation, *
Review of Economic Studies 38, 175-208.* Mirrlees, J.A.,
(1972) On Producer Taxation,

Mirrlees, J.A., (1974) Notes on Welfare Economics, Information and Uncertainty, in Balch, M., Mc Fadden, D., and Wu, S., (eds.)

Mirrlees, J.A., (1975) The Theory of Moral Hazard and Unobservable Behavior: Part I, Nuffield College, Oxford, mimeographed.

Mirrlees, J.A., (1976) The Optimal Structure of Incentives and Authority within an Organisation,

Mirrlees, J.A., (1979) The Implications of Moral Hazard for Optimal Insurance, paper presented at a symposium in honor of Karl Borch,

Mussa, M., and Rosen, S., (1978) Monopoly and Product Quality,

Myerson, R., (1982) Optimal Coordination Mechanisms in Generalized Principal-Agent Problems,

Journal of Mathematical Economics 10, 67-81.

Ross, S., (1973) The Economic Theory of Agency: The Principals Problem,

Satterthwaite, M., (1975) Strategy Proofness and Arrows Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions,

Sheshinski, E., (1972) The Optimal Linear Income Tax,

Spence, M.,(1977) Non-Linear Prices and Economic Welfare,

Spence, M., and Zeckhauser, R., (1971) Insurance, Information, and Individual Action,

Vickrey, W., (1939) Averaging of Income for Income Tax Purposes,

Vickrey, W., (1945) Measuring Marginal Utility by Reactions to Risk,

Vickrey, W., (1947)

Vickrey, W., (1955) A Proposal for Revising New Yorks Subway Fare System,

Vickrey, W., (1960) Utility, Strategy, and Social Decision Rules,

Vickrey, W., (1961) Counterspeculation, Auctions, and Competitive Sealed Tenders,

Vickrey, W., (1962) Auction and Bidding Games, in

Vickrey, W., (1968) The Problem of Progression,

Vickrey, W., (1969) Congestion Theory and Transport Investment,

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