Contract_theory_Notes_1.pdf - Markets with asymmetric...

Markets with asymmetric information and
contract theory
Ivan Susin, [email protected]
Higher School of Economics, St-Petersburg, Russia
1 Formal issues
The grading formula:
1 , 6 + 2 , 8 + 2 , 8 + 2 , 8 + [1]
First homework will be due November 24th 23:59
Second homework will be due December 8th 23:59
Late submissions will receive 50% penalty.
Homeworks are to be typeset in L
A
T
E
X(I recommend using overleaf.com ,
it works really smoothly) and submitted to e-mail. Handwritten papers are
incredibly hard to read and formulas typeset in Word are not much better.
I’ll provide a template along with the problem set itself.
Tests will be given on the lectures. First one will be given on 20th
November. Second one will be given on 4th of December.
2 Dictionary
2.1 Letters and symbols.
θ “theta” (usually pronounced ’тэта’ in Russian) usually denotes a type.
μ (’мю’), ρ (rho, ’ро’), λ (lambda, ’лямбда’) or rarely η (eta ’эта’) and
ν (nu, ’ню’) denote probabilities and beliefs.
As usual, p refers to price, U to utility (when principal is the firm, π is
the profit), C - to costs.
1

Indexes: i - is our usual generic type. j - one of other types, ( - i ) - all other
types. Over-line, underline or number (
¯
θ , θ , θ
1
) denotes particular type.
2.2 Terms
Binding constraint - the one that is equality, we cannot maximize further
the terms in it without violating it. Non-binding constraint - there is strict
inequality, this constraint currently does not affect equilibrium.
Principal-agent framework. Both are a gents, only Agent is A gent.
Principal moves first and proposes the contract (or menu of contracts ).
She maximizes her utility over all possible contracts, so it is usually our job
to understand what contracts are possible given the problem. Then we have
to select the contract that is best for her and is possible.
Agent chooses to sign or not to sign a contract or, if there is a menu of
contracts, chooses particular one. There are different types of agents. Type
is a game-theoretic idea that captures all the information that agent has. We
might have from just two types to uncountable continuum of types (if private
information is a number in [0 , 1] or R , for example).
Those types may be publicly known - situation that is usually known as
“First Best” or FB, because usually it is the simplest and Pareto-better
than otherwise. If only the Agent (not the principal) knows his type for sure,
we are in the “Second Best” world (and we usually are). In SB principal
still can propose something to induce the Agent to reveal the information.
There is an important idea, called “revelation principle” , that roughly says
that if the principal can commit to the contract (to make it irrevocable once
the Agent reveals information), any contract can be written in a way where
Agent is truthful. Here a little trick happens - space of possible contracts is
weird and may be hard to describe. This problem will be discussed in more
detail in mechanism design course, in particular “implementation theory” is
devoted to this issue. For now we’ll explicitly fix set of contracts to be linear