Monopolist_Screening_Model - Atlas of Economic Models
 

Monopolist Screening Model

Introduction

The Monopolist Screening Model illustrates the key ideas behind Principal Agent adverse selection problems. In the general case, the Agent A is of a particular type but this is imperfectly observed by the Principal P. The Principal's problem is then to offer Agents the optimal contract (price and quality) such that Agents choose the contract designed for them and the Principal maximises profits. In this model, Agents are consumers who vary by preference for quality (and willingness-to-pay for quality) and the Prinicpal is a monopolist supplier of a particular good who can vary both the price and quality of this good.

Model

There are two 'types' of Agents: those with a preference for high quality goods H and those with a preference for low quality (or lower willingness-to-pay for quality) goods L. The utility function of an Agent of type i is given by $$ U_{i}(q,t) = \theta_{i} q - t $$ such that $$ \theta_H > \theta_L $$, t is the price paid by the Agent to the Principal and q is the quality of the good provided. It is assumed that Agents (and the Principal) have a reservation utility of 0: $$ \bar{U_j} = 0 $$.

The Spence-Mirrlees Condition implies that the indifference curves of different Agent types (in the (q,t) space) can cross only once, hence it is also referred to as the single crossing property. For a given quality q, Agents with a higher preference for quality i.e. a higher value of $$ \theta $$ have a steeper indifference curve than those with a lower preference for quality, intuitively meaning that Agents with higher $$ \theta $$ are willing to pay more for a given increase in q than Agents with a lower value of $$ \theta $$. Formally, this can be written as $$ \frac{\partialU^2}{\partialq\partial\theta} > 0 $$. The Spence-Mirrlees Condition is fundamental to this model as it means that Agents can be separated by type by offering high quality high price contracts to those with high values of $$ \theta $$, as shown below.

The Principal can provide a certain quality of the good $$ q $$ at a cost of $$ C(q) $$ such that $$ C'(q)>0 $$ and $$ C"(q)>0 $$ i.e. strictly convex. The Principal's payoff for a given contract $$ (q,t) $$ is $$ U_P(q,t) = t - C(q) $$.

Mechanism Design

The Revelation Principle states that any allocation rule implementable by any mechanism can also be implemented by a Direct Truthful Mechanism, so when looking for the optimal contract only Direct Truthful Mechanisms need be considered. See Mechanism_Design for more.

Solving

First-Best Solution

It is useful to solve this problem for the first-best, i.e. under perfect information, in order to obtain a benchmark to evaluate the efficiency properties of solutions under asymmetic information. If each Agent's type ($$ \theta_i $$) can be perfectly observed by the Principal, the Principal's problem now becomes to select a pair of contracts (ti,qi) which maximise profits subject to the participation constraint (also knows as the individual rationality constraint) that Agents are at least at their reservation utility level: $$ U_{i}(q,t) = \theta_{i} q - t \ge 0 $$. Written using a Lagrangian expression this becomes:

  • $$ L = t_i - C(q_i) - \lambda_i(\theta_{i} q_i - t_i) $$

which gives the first order conditions:

  1. $$ \frac{\partialL}{\partialt_i} = 1 + \lambda_i = 0 \Rightarrow \lambda_i = -1 $$
  2. $$ \frac{\partialL}{\partialq_i} = -C'(q_i) - \lambda_i \theta_i = 0 \Rightarrow \theta_i = C'(q_i) $$ i.e. marginal benefit from another unit of quality equals marginal cost
  3. $$ \frac{\partialL}{\partial\lambda_i} = \theta_i q_i - t_i = 0 \Rightarrow t_i = \theta_i q_i \Rightarrow \theta_{i} q_i - t_i = 0 $$ i.e. the participation constraint is binding

These conditions imply that q*H > q*L (from (2) using $$ \theta_H > \theta_L $$) and t*H > t*L (from (3) using q*H > q*L).

Second-Best Solution: Asymmetric Information

Under asymmetric information the Principal cannot observe the type of individual Agents, but does know the proportion of each type in the consumer population where $$ \gamma $$ is the proportion of L consumers. This makes the contracts (q*H,t*H) and (q*L,t*L) sub-optimal for the Principal since H consumers would choose the L contract (as it would give them positive utility) and thus the Principal's profits would not be maximised:

  • $$ U_H(q*_L,t*_L) = \theta_H q*_L - t*_L > \theta_L q*_L - t*_L = 0 $$

NB: * denotes First-Best solutions, ** denotes Second-Best solutions

The problem now becomes to choose a set of contracts (q**H,t**H) and (q**L,t**L) such that the incentive feasibility constraints are satisfied:

  • $$ Max_{q_H, t_H, q_L, t_L} [(1 - \gamma) (t_H - C(q_H)) + (\gamma)(t_L - C(q_L))] $$ subject to (1) and (2) below:
  1. Participation constraints - the Agent must be at least at reservation utility

    1. $$ \theta_{H} q_H - t_H \ge 0 $$
    2. $$ \theta_{L} q_L - t_L \ge 0 $$
  2. Incentive compatibility constraints - the Agent must be as least as well off with the contract designed for their type as with the contract designed for the other type

    1. $$ \theta_{H} q_H - t_H \ge \theta_{H} q_L - t_L $$
    2. $$ \theta_{L} q_L - t_L \ge \theta_{L} q_H - t_H $$

At the solution constraints 1.ii. and 2.i are binding:

  • 1.ii: $$ \theta_{L} q_L - t_L = 0 $$ i.e the L consumer is at reservation utility
    • $$ \Rightarrow t_L = \theta_{L} q_L $$
  • 2.i: $$ \theta_{H} q_H - t_H = \theta_{H} q_L - t_L $$ i.e the H consumer is indifferent between the contract designed for H and the contract designed for L
    • $$ \Rightarrow t_H = \theta_{H} q_H - (\theta_{H} - \theta_{L}) q_L $$

These values are then substituted in to the Principal's profit maximisation problem, which reduces to two variables (qH and qL) and gives the first order conditions:

  1. $$ \theta_{H} = c'(q_H) \Rightarrow q**_H = q*_H$$
  2. $$ c'(q_L) = \theta_{L} - \frac{1 - \gamma}{\gamma} (\theta_{H} - \theta_{L}) \Rightarrow c'(q**_L) < \theta_{L} \Rightarrow q**_L < q*_L $$

    • H-type consumer are given informational rent since t**H < t*H i.e. the price paid by H to the Principal is less than their willingness-to-pay. This is to induce them to reveal their type and not to pretend to be L-type consumers.

    • L-type consumers face inefficiently bad quality of the good (q**L < q*L) as this again ensures that H-type consumers do not wish to pretend to be low-type consumers. The difference between q*L and q**L depends positively on $$ (\theta_{H} - \theta_{L}) $$ and negatively on the proportion of L-type consumers $$ \gamma $$.

      • It is assumed that $$ \gamma $$ is sufficiently high such that q**L > 0 , but if this is not the case then it becomes optimal for the Principal to offer only a single contract designed for H-type consumers i.e. the First-Best contract (q*H,t*H).

    • q**L < q**H and t**L < t**H.

The General Case

In the section above it was assumed that there were only two discrete types of consumer. However, the following properties are common to models with any number of discrete consumer types:

  • The highest type $$ \theta_{highest} $$ gets an efficient allocation: q**highest = q*highest.

  • All but the highest type get an inefficient allocation: $$ q**_i < q*_i $$ iff $$ i \neq highest $$.

  • The incentive compatibility constraint binds for all but the lowest type i.e. consumers of a particular type are indifferent between the contract designed for them and the contract designed for the next-lowest type.
  • All but the lowest type get an informational rent, which increases with type $$ \theta_i $$ : $$ t**_i < t*_i $$ iff $$ i \neq lowest $$.

  • The lowest type $$ \theta_{lowest} $$ gets 0 informational rent: t**lowest = t*lowest

Applications and Extensions

Monopoly Airline

  • Two types of consumers with utility functions as described above:
    • Business-class passengers $$ \theta_H $$ - high willingness-to-pay for quality
    • Economy-class passengers $$ \theta_ $$ - low willingness-to-pay for quality
  • Asymmetric information - the airline (which has a monopoly on the particular route) is unable to observe which type a particular passenger is, so offers two Second-Best contracts (q**H,t**H) and (q**L,t**L) which satisfy the following properties:

    • q**H = q*H and $$ t_H = \theta_{H} q_H - (\theta_{H} - \theta_{L}) q_L $$ - business-class passengers have the efficient quality of air travel and get informational rent in the form of lower prices than their willingness-to-pay for a particular quality.

    • q**L < q*L and $$ t_L = \theta_{L} q_L $$ - economy-class passengers have an inefficiently bad quality of air travel (as many of us know!) and get no informational rent as price paid equals their willingness-to-pay for a particular quality.

Extensions to this model add multiple time periods and the option for renegotiation of the contract: Dynamic_Screening_Model .