Advanced microeconomics - Appunti completi (ENG)

Advanced microeconomics – lecture notes

We will consider a specific issue of microeconomic which is economics of information. Information is an

important factor in the decision-making process of consumers and firms. In reality, most of the time,

economic agents make choices under imperfect (or asymmetric) information conditions. This may happen

in several situations, like a job interview where the HR has incomplete information about us so we try to

display the positive information and to hide the negative information. Akerlof is an important reference.

The starting point: Akerlof argues that when there is a used-car market, the buyer has different information

compared to the seller, and thus the seller has an incentive to sell “lemons” at the same price so good cars

are going to exit the market and only bad cars will be sold (adverse selection, the worst).

What is contract theory about?

Contract because all examples are based on two people that agree on something. We have cases of

asymmetric information: two parties have to trade but one of them knows something that the other does

not know. Think of a dentist, unless you are a dentist you can’t evaluate anything. Other words are:

• Strategic interaction: each party, when deciding his behaviour, takes into account the response of

the other party to his own action. This is extremely important in contracts;

• Risk: something bad or good can happen in the world;

• Uncertainty: we may ignore which type of counterpart we are interacting with;

• Rationality: each party plays the optimal strategy, i.e. individual (expected) utility maximisation;

• Optimal contract: the contract resulting from the interaction between two optimising parties.

Example: insurance

I buy insurance, covering the damages I might cause to other people while driving my car ("RCA").

Insurance company is the principal (P) and insured people are the agent (A). A pays a premium to P. A can

choose between different behaviours:

• Drive slowly;

• Drive fast.

He does not bear the economic consequence of driving fast because damages are paid by P. The economic

cost of fast goes to P. The probability of an accident increases together with the expected liability of P.

Therefore, A can be induced to drive fast, if he likes to do so, at the expense of P: moral hazard. The

problem is that P does not observe the behaviour of A. Thus the premium cannot depend on the action of

A: the action is not contractible. The premium can depend on the outcome of the action, which is observed

by P: e.g., the number of accidents in one period determines the premium in the next period ("bonus

malus"). This mechanism can induce A to drive slowly: the contract is incentive compatible (IC). However,

this mechanism (IC) introduces some risk for A: he might suffer accidents and pay a penalty even if driving

slowly. Such risk decreases his utility, creating an agency cost (AC) due to AI.

Another insurance example

I buy insurance, covering the risk of somebody stealing my car. The agent can choose between different

behaviours: high or low effort to avoid theft. A’s action cannot be observed by the insurance company (P),

so it is not contractible. A can be induced to exert low effort at the expense of P: moral hazard. P designs a

contract with partial insurance (a downpayment, sharing risk) to induce A to exert high effort (IC). Agency

cost: A bears some risk. This results in sub-optimal risk sharing between a risk neutral insurer and risk

averse insured individuals. Hidden action problem: essential elements

P and A must sign a contract. After signing, A has to choose among different actions that affect the

utility/profit of P. A may be tempted to choose the action he prefers, harming P: moral hazard. AI: P does

not observe the action taken by A. Thus the contract (price/penalties, etc.) cannot depend on A’s action,

which is not contractible. P can design a contract depending on the outcome of A’s action to induce him to

behave well: an incentive compatible contract. Incentive compatibility (IC) is costly, generating an agency

cost (AC) (e.g., sub-optimal risk sharing). Public health and lending are other examples of moral hazard.

Hidden information and credit market

Bank (P) and borrowing firms (A). In the population of firms each firm can be either high risk or low risk:

exogenous types. AI: the bank does not observe the type of each firm. The bank must offer the same price

(interest rate) to all firms in a risk class. This price might be too high for low-risk firms, so they do not

borrow and the bank ends up lending only to high-risk firms: a case of adverse selection. To avoid AS, the

bank may limit the quantity of credit, leading to credit rationing (an equilibrium excess demand). Agency

cost: some trades are not made. Firms do not receive enough credit and must forgo some investment

opportunities. This also happens in health insurance (only unhealthy people sign at that premium).

Hidden information problem: essential elements

There is a population of A with different types affecting the utility/profit of P. Each A knows his own

exogenous type (the firm doesn’t know). AI: P does not observe the type of each A, but just has info about

the distribution of types. P must offer the same contract to all A. AS: best agents may not accept the

contract and thus P just trades with worst agents. Agency cost:

• Some trades may not happen (credit rationing);

• Some distortions in contracts are introduced (partial insurance).

Usually, two contracts are offered: one with full insurance (but with a high premium) and one with partial

insurance (with a low premium). Partial insurance functions as a screening to distinguish between safer and

riskier individuals. This implies an AC: sub-optimal risk sharing. In conclusion, we have two main issues:

1. Hidden action: P does not observe the action taken by A, which produces the moral hazard (MH,

after the contract is signed) problem;

2. Hidden information: there are different types of A in the population and P does not observe the

type of each A, which produces the adverse selection (AS, before the contract is signed) problem.

We will see what happens when everyone knows everything, when there is MH and when we have AS.

Moral hazard

The presence of opportunistic post-contractual behaviour, due to informational asymmetry, manifests as

hidden information or hidden action. The latter gives rise to the phenomenon of moral hazard. Typical

solutions: direct controls, collateral requirements, or incentive contracts that tie compensation to an

observable variable. Adverse selection

The presence of opportunistic pre-contractual behaviour: the classic lemons market. Two contractual

solutions can be employed:

1. Screening: the uninformed party offers a menu of contracts so that only the appropriate type

accepts;

2. Signalling: the informed party takes an observable action that reveals its private characteristics.

Elements of the problem

The main elements are:

• Principal;

• Agent;

• Basic assumption: the contract stipulates that the agent acts in the interest of the principal.

Traditionally: the principal offers the contract terms, the agent evaluates the offer, the agent signs. There is

no bargain power. A stylized situation

The principal formulates the contract to be offered to the agent. The agent accepts if and only if its

expected utility exceeds its reservation utility (utility that we should receive at least in order to accept the

contract, like a salary). The agent acts on behalf of the principal’s interests. The agent’s interests do not

coincide with those of the principal. We denote the agent by A and the principal by P.

Conflicting interests between P and A

The cost for one party is the gain for the other. The agent’s wage () represents the cost incurred by the

principal. The agent’s effort () entails a cost for the agent and a benefit for the principal. Information: the

set of conditions that are contractually verifiable.

The benchmark model

This is the model where everyone knows everything. Given P and A (the players) and defining N (nature,

something we can’t control like the geopolitical environment):

1. P defines the contract;

2. A accepts or reject the contract;

3. A exerts the effort: in this case it is verifiable;

4. N determines the state of the world: profit of the firm depends both on nature (random variable)

and effort of A (example of the barman and the rain);

5. Outcome is determined and payoffs delivered.

We know everything about the agent but there is an uncertainty about what happens in the world (N). We

can’t avoid this risk also at level 0. How to solve it? We use a strategic interaction and backward induction

(P wants A to sign so he must put in the shoes of A internalizing his behaviour).

Moral hazard (structure)

The action of A cannot be verified or A receives private information after signing the contract:

1. P defines the contract;

2. A accepts or reject the contract;

3. A exerts a non-verifiable effort: like driving slow or fast;

4. N determines the state of the world: if it rains it’s more probable an accident driving fast;

5. Outcome is determined and payoffs delivered.

We must take into account the behaviour of the agent that changes with N. The problem is very complex.

Adverse selection (structure)

There exists a private information before signing the contract:

1. N determines the type of player A (only A can see it);

2. P defines the contract;

3. A accepts or reject the contract;

4. A exerts the effort;

5. N determines the state of the world;

6. Outcome is determined and payoffs delivered.

Signalling: agent’s private information

Similar to adverse selection but before the contract is signed, A sends a signal to P to influence her beliefs

about A’s type:

1. N determines the type of player A (only A can see it);

2. A sends a signal to P;

3. P defines the contract;

4. A accepts or reject the contract;

5. A exerts the effort;

6. N determines the state of the world;

7. Outcome is determined and payoffs delivered.

Signalling: principal’s private information

Similar to adverse selection but before the contract is signed, P sends a signal to A to influence her beliefs

about P’s type:

1. N determines the type of player P (only P can see it);

2. P defines the contract that uses a signal;

3. A accepts or reject the contract;

4. A exerts the effort;

5. N determines the state of the world;

6. Outcome is determined and payoffs delivered.

Screening: agent’s private information

Similar to adverse selection but P uses contract to screen agents:

1. N determines the type of player A (only A can see it);

2. P defines the contract to screen A’s type;

3. A accepts or reject the contract;

4. A exerts the effort;

5. N determines the state of the world;

6. Outcome is determined and payoffs delivered.

Some basic assumptions

The choices of P and A are made under conditions of uncertainty. The choices of P and A depend on the

shape of their objective functions. That is, the curvature of the functions. For simplicity, we adopt the von

Neumann-Morgenstern expected utility model.

Risk and uncertainty vs ambiguity (in a very literal sense)

Risk/uncertainty: when the probabilities of events are known (e.g., coin toss, lottery). Ambiguity instead is

when the probabilities are not known (e.g., climate change, multiple scenarios). We consider risk and

uncertainty. Expected utility

Expected utility means evaluating every possible utility and computing the average. The evaluation

depends on: risk preferences, preferences, risk perception, and, in general, on how we compare the

time

alternatives. Let’s look at these two alternatives:

Which do you prefer, A or B? Same expected value:

A

Possible event Outcome Probability 0.5 × 100 + 0.5 × 0 = 50

1 100 50%

2 0 50% But A involves risk. If you prefer B: you are risk averse. If you

prefer A: you are risk-loving. If you are indifferent: you are

B risk neutral. Think it as a lottery where you can get 100 with

Possible event Outcome Probability 50% probability or 50 with 100% probability. Many of us will

1 50 100% choose the second alternative.

In expected terms, you get a very similar amount of money.

Risk aversion

We don’t like risky environment. Many agents are risk averse and others are risk lovers (think at betting).

How a utility function represent risk aversion is extremely important.

Assume that you have some utility for receiving some money. On the

axis we have money and on axis we have utility. Utility functions are

concave, they increase at a lower rate:

• If you are very poor: the very first euro delivers you high utility;

• If you have saved a lot of money: the additional money doesn’t

make you happy as the first one. The utility that additional money give

to you is lower and lower.

Imagine this graph as a lottery: you can get 0 or 90 with equal probability, this utility function gives the

utility for these values (2 and 12). The alternative is to receive for sure 45. The utility of receiving these

money for sure is the pink point at the top. Any point on the straight line represents a weighted average of

utility of receiving 0 or 90 (at the far left we can imagine that the probability of getting 0 is 90%).

Participating into to this lottery I receive half of the time 2 and half of the time 12 so we place into to

middle point (pink). The upper point is the expected utility of getting 45$ for sure. One is above the other

so we prefer receiving 45$ for sure than participating into that lottery because This type of

($45) > .

utility function represents risk averse agents. A risk averse agent doesn’t like participating in a lottery if he

can have the same amount of money in expected terms. The utility function is concave (required). Basically

and The probability of winning 0 is while is the probability of winning 90.

(0) = 2 (90) = 12. 1 −

(1

The expected utility is If is very high I get

() = ∙ (0) + − ) ∙ (90) = ∙ 2 + (1 − ) ∙ 90.

close to the left side of the straight line (which is the expected utility for participating into the lottery). If

is very low I get close to the right side. The first order derivative of a concave utility function (with respect

to outcomes/money) is positive but the second order derivative is negative.

Risk neutral

A single agent is usually risk averse. Firms are usually risk neutral. The insurance company doesn’t care

about the single accident, they care at the aggregate so they average risk. In that case, participating in a

lottery or having the amount for sure is exactly the same. The utility function is linear. ($45) = = 7.

Risk lover

Risk lover agents prefer the lottery to the sure amount (they enjoy the game). The utility of having 45$ for

sure is lower than the utility of participating into the lotter. The utility function is convex.

Economics of information: the basic model

We know the type of agent, we can observe his actions but we cannot know what happens in the world.

Consider a consumer with bundle How does utility change when we increase slightly? The rate

( , ).

! " #

of change is the marginal utility of good : ( , ) ( , )

! " ! "

= , =

! "

! "

The marginal rate of substitution (MRS) is:

!

= −

"

This explain how much I would like to substitute good 1 and good 2.

The basic model (2 agents, conflicting interests and attitudes toward risk)

Let’s move to the basic model with symmetric information. We have 2 agents (P and A) and conflicting

interests (otherwise we immediately agree on the contract). The information is symmetric and the agents

have different attitudes toward risk (the final agreement depends on the risky preferences of both).

The object of the contract () and the outcomes

We are in the job market. Principal proposes an employment contract (P has all information about you so

the question is about the effort which is inserted into the contract and can be controlled). The agent exerts

effort The possible outcomes are different: Outcome is different depending on the states

. ∈ { , , … }.

! "

of the world that impact the profit of the firm, remember the beer-weather case.

The effort

depends on and a random component. The probability of a state of the world depends on the effort:

# ()

= [ = |]

# #

Of course the probabilities sum up to one: $ () ()

I = 1 , > 0

# #

#%!

We cannot rule out any result for any given effort level. We are in an uncertain setting.

Symmetric information (sharing the same prior)

P and A share the same prior about the random component. Both look the same weather forecasts for

example and they also know how the effort affects the probability of low and high profit.

Principal’s utility function (goals)

are the profits that the principal gets, it depends both on effort and weather. However, the principal also

has to pay wages. The net profit is The principal has a utility of this money (risk-averse/neutral) and

− .

the utility function of the principal of receiving an amount of money is It is concave and

− ( − ).

increasing. Agent’s utility function

The utility function of the agent is impacted by 2 elements: and

:

(, ) = () − ()

It’s not just about the money. We must also think at the money compared to the effort (10h/day or

1h/day). The utility of the wage is and the disutility from effort is Then:

() ().

1. this utility function is concave. The marginal utility is positive while the second order

():

derivative is negative. The agent is risk averse;

2. costs are convex. The more you exert effort, the higher this additional hour is costly. Both

():

derivatives are positive (any additional unit is more costly than the previous on