The level of education is highly related to the future possibilities to find a job.
It is obvious that the people with university degrees have better chances that the other ones.
A person with higher education degree has larger chances to go into the labour market and for a better
position his well being will growth too. As well as the well being of people depending on him.

## The Paper

The educational process is important not
only for the main purpose to create an intelligent labour market capable to offer
highly trained people to fulfil complex requests of modern world.

The signals offered by the labour force
are due to the education received and reveal skills, wishes and other types of information
that help the individual to evaluate himself for all his life. They also show information
to a potential employer, information that help him to compare the abilities of a
large number of individuals that wishes to be hired. The employer’s opinion about
an individual just form knowing a certain signal is not perfect but their recognition
is used instead of interviews, tests or training period.

If an individual passes a difficult exam
(mathematics) with a high grade, the fact may represent a signal strong enough so
an individual can work in IT where the information changes very fast.

The problem of educational signals is
not recent; it represents the subject of many debates, especially since Michael
Spence published his master thesis [5] in 1974. Previous papers belong to Arrow
[1], Fields [2] and Thurov [7].

A complex educational signals is made by Stiglitz [6].

In the Principal-Agent model, there is a bidirectional relation between an institution
(university, doctoral school, master, etc) and an individual (student, master student
or PhD candidate) which has a contract as a result. The contract shows the demands
and the rights of the two parties.

The individual (agent) makes an effort to obtain some results (passing exams,
finishing research projects) and is rewarded for it.

In the situation of incomplete information (results can not be known for certain),
*X *will be the set of possible results:

_{
}_{} is a
possible result (a possible value of the income obtained by a research institution
from papers and studies publishing).

We shall consider a state where individual decisions are not based on exactly knowing
the results of the individuals’ actions and not even on the utility of the results.

More possible results may be predicted together with different probabilities.

The probabilities can be objective or subjective.

The objective probabilities that don’t differ from one person to another represent
the relative frequency of an even appearing.

The subjective or Bayesian probability shows the relative frequency by which an
individual thinks that a certain event happens or differs from a person to another.

Frank Knight proposed the following classification scheme of the incomplete information
problems:

Both parties of the contract are risk averse or are indifferent to risk.

The attitude towards risk is characterized by a VNM utility function, both for the
Agent and the Principal.

We assume that the necessary time needed by the Agent to produce a production unit
is _{
} (before
the ending of the course) or the effort, production cost, etc.

If the Agent wishes to go to a certain school, he must pay a sum of money *S*
(at the beginning) and then a sum of *a* for each monetary unit earned as a
consequence of the degree held by the Agent. Let *t* be the necessary time
to produce a unit or to earn a monetary unit after the graduation. Obviously, we
have_{}

Next, we shall present a special type of contract adapted from [3, pg 149].

*Definition1. *A contract in symmetric information is given by the couple_{}

The market demand_{}_{} are expressed
as functions of average cost (or average time) denoted by *x*. If the price
of a unit produced is_{}

_{
}
(1)

*Proposition 1.* If _{
}_{}

*Proof*

The derivative of _{
} is
zero for _{
}

_{
}

or

_{
}
(2)

The revenue function from (1) derived
with respect to *x* becomes:

_{
}

_{
}

as in relation (2).

The Principal’s objective is to maximize
the revenues and it can be written as:

_{
}

s.t.

_{
}
(3)

_{
}

*Theorem 1.* The solution of the program (3) (the optimal contract under symmetric
information) is Pareto optimal and is given by the couple_{}

Proof

Using Kuhn-Tucker method, the multipliers _{
} and
_{
} are
attached to the constraints in (3).

The Lagrangean function is:

_{
}

Searching for an interior optimum, we set the partial derivative with respect to
*S* to zero:

_{
} or
_{
}

That is: _{
}

Then _{
} (the
first constraint is binding).

We can rewrite the program (3) in the
following form:

_{
}

This provides us:

_{
}

_{
} and
_{
}

The partial derivatives of the Lagrangean
function are:

_{
}

_{
}

or

_{}
(4)

(4) corresponds to the condition for Pareto
optimality satisfied by the optimal contract under symmetric information.

Next, we consider the same problem, but
in the case of asymmetric information, where the Agent has hidden information about
the contract. For instance, he knows how important is the production plan received
from the Decident. Further, we suppose that the type of the program is good (G)
- with probability _{
} - or
bad - with the probability _{
}

*Definition2.* A contract under asymmetric information is given by the couples:

_{
}

We have _{
} and
_{
}

We can formulate now the Principal’s program
(i.e., maximizing expected revenues):

_{
}

s.t.

_{
}
(5)

_{
}
(6)

_{
}
(7)

_{
}
(8)

*Theorem 2.* The optimal contract under asymmetric information is characterized
by:

_{
} and
_{
}

Proof

The Kuhn-Tucker multipliers _{
} and
_{
}_{}_{} correspond to _{} variables.

The restriction (7) is a consequence of
the restrictions given by (5) and (8) (if the problem has admissible solution).
To prove this, we have:

_{
}

_{
}

The Lagrangean function becomes:

_{
}

_{
}

_{
}
_{
}

_{
}

The first order conditions for an interior optimum are:

_{
}
(9)

or_{}

The first conclusion is that the restriction (5) is binding. Using this we find
that:

_{
}
(10)

_{
}
(11)

Adding the terms from (9) and (11), we
obtain:

_{
} or
_{
}

One result is that the restriction (8) is binding. Thus:

_{
}
(12)

We shall use partial derivatives for the
Lagrangean function respected to the variables _{
} and
_{
} and
we have:

_{
}

or

_{
}

But _{
} (from
(9)) and the precedent equation become:

_{
}
(12)

(12) shows that _{
} must
be stated.

For _{
}

i) _{
}_{}

ii) _{
}

_{
}

Because the variables’ coefficients _{
} and
_{
} are
strictly positive, we have, _{
} and
_{
}

_{
}

_{
}

By conveniently combining the terms and
knowing that _{
}

_{
}

so that _{
}

Obviously, _{
}_{}

_{
} Because_{}

If_{}_{}
implies _{
} or_{}

This is obtained from (13), because _{
}

_{
} or
_{
}

The first term is strictly positive, while
the second is negative. Thus, _{
}

Finally, we can characterize the optimal contract.

_{
}

_{
}

_{
}

Using (10), we obtain:

_{
}

or

_{
}

Thus, the optimal contract under asymmetric
information is no longer Pareto optimal.