Dynamic_Screening_Model - Atlas of Economic Models
 

Dynamic Screening Model

Introduction

The static model presented in Monopolist_Screening_Model assumes that agents interact with each other only once, or else that there is no option to renegotiate the contract after the first period. However, this is not a very realistic assumption since most economic relationships are based on repeated interactions over several time periods. Dynamic models analyse the how the optimal contract evolves over the duration of the relationship, which depends on the degree of commitment in the initial contract and how and when information is revealed to the uninformed party.

Full commitment leads to the most efficient contracts and means that Agents will reveal their type in the first period, but it is not realistic to assume that parties will agree ex-ante not to renegotiate allocations that are ex-post inefficient. Long-term commitment gives a more gradual revelation of information and ex-ante inefficiencies, but allocations are ex-post efficient. No commitment means Agents will be much more cautious in revealing their type and will expect higher compensation for doing so, thus allocations are less efficient than under other levels of commitment.

Model

At the start of the 'game' both the informed party $$ A $$ and the uninformed party $$ P $$ commit to the contract until some specified time. However, there may be the opportunity to renegotiate the terms of the contract by mutual consent. A contract that covers the duration of the relationship without the possibility of renegotiation is known as full commitment, and under these rules the game reduces to the static model. If there is less than full commitment there is now a dynamic element to the game whereby at the end of each period one or more parties may wish to renegotiate the contract, depending on what new information has been revealed during that time.

A contract signed at the beginning of an economic relationship (under less than full commitment so that renegotiation is possible if desired) is renegotiation proof if there are no incentives for players to multilaterally renegotiate in some future time period. The optimal contract under less than full commitment will be renegotiation proof, since circumstances under which renegotiation would be expected have been anticipated and built in to the provisions of the contract.

If the contract is complete (i.e. all variables which may have an impact at some point in the relationship have been taken into account) then it is beneficial to have full commitment. Any outcome which is feasible without full commitment can also be achieved with it, so agents cannot be worse off and may be better off under full commitment.

Example

Consider the simplest dynamic game, that with only two periods ($$ T=2 $$). As in the static model (see Monopolist_Screening_Model), there are two types of agents: $$ H $$ and $$ L $$ . Now utility functions are intertemporal, with $$ U_P(q,t) = \sum^T \delta^{t-1} (t - C(q)) $$ and $$ U_{i}(q,t) = \sum^T \delta^{t-1} (\theta_{i} q - t) $$ where $$ i=(H,L) $$ and $$ \delta $$ is the discount factor.

Agents interact with the monopolist (Principal) twice, athough their choice of contract $$ (q,t) $$ in the first period may reveal their type so one or both parties may wish to renegotiate the contract for the second period given this new information.

To summarise, the optimal contracts under asymmetric information in the static (one-period) model are $$ (q_H^{**},t_H^{**}) $$ and $$ (q_L^{**},t_L^{**}) $$ such that:

  • $$ q_H^{**} = q_H^{*} $$ and $$ q_L^{**} < q_L^{*} $$

  • $$ t_H^{**} < t_H^{*} $$ and $$ t_L^{**} = t_L^{*} $$

  • $$ U_H^{**} > \bar{U} $$ and $$ U_L^{**} = \bar{U} $$

  • $$ U_P^{**} < U_P^{*} $$

Solving

Full Commitment

Under full commitment there is no option to renegotiate at any stage of the relationship, so the optimal contract is to offer the static equilibrium as described above.

Long-term Commitment

Long-term commitment refers to the situation where the contract is expected to cover the duration of the economic relationship but there is the option to multilaterally renegotiate, and under these circumstances the optimal static contract is not renegotiation proof. If these contracts have been implemented at $$ t=0 $$ then by the end of the first period the monopolist would wish to renegotiate both contracts, since consumers have revealed their type through their choice of contract and the Principal makes higher profits under the optimal contracts for symmetric information $$ (q_H^*,t_H^*) $$ and $$ (q_L^*,t_L^*) $$ . L-type consumers would also be willing to renegotiate their contract as they are being 'rationed' with an inefficiently-low quality of the good. In other words, parties commit to the optimal ex-ante contract (in expectation over type) which is inefficient ex-post once agents' types have been realised.

However, if H-type consumers anticipate that the monopolist will renegotiate with L-type consumers at the end of the first period, they now prefer to pretend to be L-type in the first period and then accept the renegotiated (first-best for L-types) contract for future periods since $$ U_H^{**}(q_L^{**},t_L^{**}) + \sum_{t=2}^T U_H(q_L^{*},t_L^{*}) > \sum_{t=1}^T U_H(q_H^{**},t_H^{**}) $$ . In order for the optimal dynamic contract to be renegotiation proof, information about agents' types must be revealed gradually.

Instead of all H-type agents choosing the H-type contract in the first period and so revealing themselves as a true H-type, now only a proportion $$ x $$ do so. Thus at the beginning of the second period the Principal revises their beliefs about the proportion of true L-type consumers in the population $$ \gamma $$ to $$ \gamma_2 = \frac{\gamma}{\gamma+(1-\gamma)(1-x)} $$ .

In order to separate the different types, the principal offers a choice of either $$ q_H^{*} $$ or $$ q_t $$ where $$ q_t < q_H^{*} $$ . L-type agents always choose $$ q_t $$ , and H-type agents choose $$ q_H^{*} $$ if they did so in the first period and either $$ q_t $$ or $$ q_H^{*} $$ if they chose $$ q_t $$ in the first period. As this is only a 2-period game, the last period reverts to a static game where the quality given to L-type agents $$ q_2 $$ is characterised by $$ c'(q_L) = \theta_{L} - \frac{1 - \gamma_2}{\gamma_2} (\theta_{H} - \theta_{L}) $$ . As in the static model, this gives an inefficiently-low quality for L-types. This contract is renegotiation proof because H-types are now indifferent between $$ q_H^* $$ and $$ q_t $$ at the end of each period (their incentive compatibility constraint is binding) and provisions have been made in the terms of the contract to deal with the effect of some H-types pretending to be L-types.

No Commitment

If there is no commitment then either party can break the contract at the end of each period. The Principal would therefore offer the static second-best optimal contracts in the first period to induce agents to reveal their type, then in subsequent periods the first-best contracts. H-types lose the informational rent they were given under the second-best contracts. However, Agents realise that information they reveal by their choices in the first period will be used by the Principal to maximise their profits (i.e. leave Agents at reservation utility) so it becomes very costly for the Principal to induce Agents to reveal their type, known as the Ratchet Effect.

The Principal must bribe H-types to reveal themselves by giving them the present discounted value of the informational rent they could have expected to receive if they were given the second-best contract for the duration of the relationship. However, now L-types may pretend to be H-types in order to benefit from the bribe. This adds another incentive compatibility constraint to the problem.

The solution is a PBE such that a non-linear tariff $$ t_1(q_1) $$ is offered in the first period, and given Agents' choices the Principal revises their prior $$ \gamma $$ to $$ \gamma_2(q_1) $$ . In the second period a new non-linear tariff $$ t_2(q_1,q_2) $$ is offered, and the game repeats for the duration of the relationship. If $$ \delta $$ and $$ T $$ are both small (low value on future rents and short relationship) then the Ratchet Effect will not be of much importance since H-types do not have much to lose by revealing their type early, so although revelation will still happen in stages it will be fairly fast. However, if $$ \delta $$ and $$ T $$ are large then the Ratchet Effect will be significant as Agents are very reluctant to reveal their type.

Applications and Extensions