Information Economics The Continuous-type Screening Model - PowerPoint PPT Presentation

Introduction Preliminaries Optimal contracts Implications Information Economics The Continuous-type Screening Model Ling-Chieh Kung Department of Information Management National Taiwan University CThe Continuous-type Screening Model 1 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Road map ◮ Introduction . ◮ Preliminaries. ◮ Optimal contracts. ◮ Implications. CThe Continuous-type Screening Model 2 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Screening ◮ Recall our monopoly pricing screening problem: ◮ There are two kinds of consumers: ◮ θ ∈ { θ L , θ H } where θ L < θ H , is the consumer’s private information. ◮ The seller believes that Pr( θ = θ L ) = β = 1 − Pr( θ = θ H ). ◮ When obtaining q units by paying t , a type- θ consumer’s utility is u ( q, t, θ ) = θv ( q ) − t. ◮ v ( q ) is strictly increasing and strictly concave. v (0) = 0. ◮ The unit production cost of the seller is c < θ L . ◮ By selling q units and receiving t , the seller earns t − cq . ◮ How would you price your product to maximize your expected profit? ◮ Because we assume that there are two kinds of consumers, this is a two-type screening model. CThe Continuous-type Screening Model 3 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Two-type screening ◮ The two-type screening problem can be formulated: � � � � max β t L − cq L + (1 − β ) t H − cq H q H ,t H ,q L ,t L s.t. θ H v ( q H ) − t H ≥ θ H v ( q L ) − t L θ L v ( q L ) − t L ≥ θ L v ( q H ) − t H θ H v ( q H ) − t H ≥ 0 θ L v ( q L ) − t L ≥ 0 . ◮ The first two are the incentive-compatible (truth-telling) constraints. ◮ The last two are the individual-rationality (participation) constraints. ◮ If θ H − θ L < β , the optimal menu { ( q ∗ L , t ∗ L ) , ( q ∗ H , t ∗ H ) } satisfies θ H � � 1 θ H v ′ ( q ∗ θ L v ′ ( q ∗ H ) = c and L ) = c . 1 − ( 1 − β θ H − θ L ) β θ L ◮ May we generalize this problem to n types ? CThe Continuous-type Screening Model 4 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications n -type screening ◮ Let θ ∈ { θ 1 , θ 2 , ..., θ n } , where θ 1 < θ 2 < · · · < θ n and Pr( θ = θ i ) = β i . ◮ Of course we have β i > 0 and � n i =1 β i = 1. ◮ The n -type screening problem can be formulated: n � max β i ( t i − cq i ) { q i ,t i } i =1 s.t. θ i v ( q i ) − t i ≥ θ i v ( q j ) − t j ∀ i = 1 , ..., n, j = 1 , ..., n θ i v ( q i ) − t i ≥ 0 ∀ i = 1 , ..., n. ◮ The first set is the set of IC constraints. ◮ The second set is the set of IR constraints. ◮ How to find the optimal menu? CThe Continuous-type Screening Model 5 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications n -type screening ◮ The n -type screening problem can be reduced to: n � max β i ( t i − cq i ) { q i ,t i } i =1 s.t. θ i v ( q i ) − t i ≥ θ i v ( q i − 1 ) − t i − 1 ∀ i = 2 , ..., n θ 1 v ( q 1 ) − t 1 ≥ 0 . ◮ Only local downward IC constraints (LDIC) are necessary. ◮ Only the IR constraint for the lowest type is necessary. ◮ Monotonicity, efficiency at top, and no rent at bottom still hold. ◮ May we generalize this problem to infinitely many types on a continuum ? CThe Continuous-type Screening Model 6 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Continuous-type screening ◮ Let θ ∈ S = [ θ 0 , θ 1 ], where θ 0 < θ 1 , with f and F as the pdf and cdf. ◮ The continuous-type screening problem can be formulated: � θ 1 � � max t ( θ ) − cq ( θ ) f ( θ ) dθ { q ( θ ) ,t ( θ ) } θ 0 θv ( q ( θ )) − t ( θ ) ≥ θv ( q (ˆ θ )) − t (ˆ ∀ θ ∈ S, ˆ s.t. θ ) θ ∈ S θv ( q ( θ )) − t ( θ ) ≥ 0 ∀ θ ∈ S. ◮ The first set is the set of IC constraints. ◮ The second set is the set of IR constraints. ◮ How to find the optimal menu? CThe Continuous-type Screening Model 7 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Road map ◮ Introduction. ◮ Preliminaries . ◮ Optimal contracts. ◮ Implications. CThe Continuous-type Screening Model 8 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Preliminaries ◮ Before we try to solve for the optimal menu, we need to get some mathematical tools. ◮ Hazard (failure) rates. ◮ Integration by parts. ◮ Envelope theorem. CThe Continuous-type Screening Model 9 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Failure (hazard) rates ◮ Consider a bulb whose life is X ≥ 0. Let X ∼ f, F . ◮ F ( t ) = Pr( X ≤ t ) is the probability for the bulb to fail by time t . ◮ F ( t + ǫ ) − F ( t ) is the probability for the bulb to fail within [ t, t + ǫ ]. ◮ f ( t ) = d dt F ( t ) = lim ǫ → 0 [ F ( t + ǫ ) − F ( t )] is the probability density for the bulb to fail at time t . ◮ The failure (hazard) rate of the bulb h ( t ) is the likelihood for the bulb to fail at time t , given that the bulb has not failed by time t : � � � Pr( X ∈ [ t, t + ǫ ] , X ≥ t ) � h ( t ) = lim ǫ → 0 Pr X ∈ [ t, t + ǫ ] � X ≥ t = lim Pr( X ≥ t ) ǫ → 0 Pr( X ∈ [ t, t + ǫ ]) f ( t ) = lim = 1 − F ( t ) . 1 − F ( t ) ǫ → 0 CThe Continuous-type Screening Model 10 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Failure (hazard) rates ◮ Some examples: ◮ If X ∼ Uni(0 , 1), we have f ( x ) = 1, F ( x ) = x , and thus h ( x ) = 1 1 − x . The hazard rate is increasing. ◮ If X ∼ Exp( λ ), we have f ( x ) = λe − λx , F ( x ) = 1 − e − λx , and thus h ( x ) = λ . The hazard rate is constant. ◮ In general, for a random variable with pdf f ( · ) and cdf F ( · ), its failure f ( · ) rate is h ( · ) = 1 − F ( · ) . ◮ For our private type θ , we impose the following assumption: Assumption 1 (Increasing failure rate (IFR)) The failure rate of θ is (weakly) increasing: Let H ( θ ) = 1 − F ( θ ) f ( θ ) , then H ( θ ) is (weakly) decreasing in θ . ◮ This is true for most of the well-known distributions (uniform, exponential, normal, gamma, beta, etc.). CThe Continuous-type Screening Model 11 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Integration by parts ◮ Let u ( x ) and v ( x ) be two functions of x defined over [ a, b ]. We have d � � � � ′ = u ( x ) v ′ ( x ) + v ( x ) u ′ ( x ) . u ( x ) v ( x ) = u ( x ) v ( x ) dx ◮ Integrating both sides with respect to x : � b � b � b � � d u ( x ) v ′ ( x ) dx + v ( x ) u ′ ( x ) dx u ( x ) v ( x ) dx = dx a a a � b � b �� b � � u ( x ) v ′ ( x ) dx = v ( x ) u ′ ( x ) dx. ⇔ u ( x ) v ( x ) − � � a a a ◮ The (abbreviated) formula of integration by parts : � � udv = uv − vdu. CThe Continuous-type Screening Model 12 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Integration by parts: examples � 1 ◮ Find xe x dx : 0 � 1 � 1 � 1 � e x dx e x e x x = x − dx = e − ( e − 1) = 1 . � ���� ���� ���� ���� � ���� ���� 0 0 0 u dv u v v du � 1 x 2 e x dx : ◮ Find 0 � 1 � 1 � 1 � x 2 e x dx = x 2 e x e x − 2 xdx = e − 2 × 1 = e − 2 . � � ���� ���� ���� ���� ���� � �� � 0 0 0 u u v v dv du CThe Continuous-type Screening Model 13 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications A parameter’s impact on the objective value ◮ Consider a function f ( x, θ ) and an optimization problem z ∗ ( θ ) = max f ( x, θ ) . x We will interpret x as the decision variable and θ as the parameter. z ∗ ( θ ) is the maximum attainable objective value given θ . ◮ Let x ∗ ( θ ) ∈ argmax x f ( x, θ ) be an optimal solution. Then we have z ∗ ( θ ) = f ( x ∗ ( θ ) , θ ) . ◮ Question: What is dθ z ∗ ( θ ), the impact of θ on the objective value? d ◮ One application: the impact of a parameter on the equilibrium utility. CThe Continuous-type Screening Model 14 / 35 Ling-Chieh Kung (NTU IM)

Introduction Preliminaries Optimal contracts Implications Envelope theorem ◮ An example: Let f ( x, θ ) = θ − ( x − θ ) 2 . Given θ fixed, we have x ∗ ( θ ) = θ and z ∗ ( θ ) = θ − ( θ − θ ) 2 = θ . Therefore, dθ z ∗ ( θ ) = 1. d ◮ To find dθ z ∗ ( θ ) in general: d ◮ Find x ∗ ( θ ), plug in x ∗ ( θ ), and then take the derivative. ◮ May we “reverse the order?” ◮ With the envelope theorem , we can: ◮ Find x ∗ ( θ ), take the derivative (typically easier), and then plug in x ∗ ( θ ). Proposition 1 (Envelope theorem) Given f ( x, θ ) , let x ∗ ( θ ) ∈ argmax x f ( x, θ ) and z ∗ ( θ ) = f ( x ∗ ( θ ) , θ ) . Then we have � dθz ∗ ( θ ) = ∂f ( x, θ ) d � . � ∂θ � x = x ∗ ( θ ) CThe Continuous-type Screening Model 15 / 35 Ling-Chieh Kung (NTU IM)