Dynamic Mechanisms without Money

Transcription

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2 Dynamic Mechanisms without Money Yingni Guo, 1 Johannes Hörner 2 1 Northwestern 2 Yale July 11, 2015

3 Features - Agent sole able to evaluate the (changing) state of the world. 2 / 1

4 Features - Agent sole able to evaluate the (changing) state of the world. - Principal with commitment has the decision rights. 2 / 1

5 Features - Agent sole able to evaluate the (changing) state of the world. - Principal with commitment has the decision rights. - No transfers to facilitate truth-telling. 2 / 1

6 Features - Agent sole able to evaluate the (changing) state of the world. - Principal with commitment has the decision rights. - No transfers to facilitate truth-telling. - No hard (possibly statistical) evidence either. 2 / 1

7 Some Examples - Patients want the nurses attention. 3 / 1

8 Some Examples - Patients want the nurses attention. - Managers want the go-ahead for their projects. 3 / 1

9 Some Examples - Patients want the nurses attention. - Managers want the go-ahead for their projects. - State departments want to expand. 3 / 1

10 Some Examples - Patients want the nurses attention. - Managers want the go-ahead for their projects. - State departments want to expand. - Cities/States want more resources. 3 / 1

11 Objective - Is there a simple optimal policy? 4 / 1

12 Objective - Is there a simple optimal policy? - How does utility/inefficiency evolve over time? 4 / 1

13 Objective - Is there a simple optimal policy? - How does utility/inefficiency evolve over time? - How does this depend on the lack of transfers? 4 / 1

14 Related Literature 1. Linking incentives: - Fang and Norman (2006), Jackson & Sonnenschein (2007), Cohn (2010), Hortala-Vallve (2010), Eilat & Pauzner (2011), Miralles (2012), Frankel (2014). - Radner (1981), Rubinstein (1979), Rubinstein & Yaari (1983). 2. Dynamic mechanism design: - Battaglini (2005), Zhang (2012), Pavan, Segal & Toikka (2014), Battaglini & Lamba (2014). - Fernandes & Phelan (2000), Doepke & Townsend (2006). 3. Virtual budgets/ Chip Mechanisms: - Möbius (2001), Hauser & Hopenhayn (2008), Abdulkadiroğlu & Bagwell (2012), Johnson (2014). 4. Dynamic contracting ( Immizeration ): - Townsend (1982), Spear & Srivastava (1987), Thomas & Worrall (1990). 5 / 1

15 Related Literature 1. Linking incentives: - Fang and Norman (2006), Jackson & Sonnenschein (2007), Cohn (2010), Hortala-Vallve (2010), Eilat & Pauzner (2011), Miralles (2012), Frankel (2014). - Radner (1981), Rubinstein (1979), Rubinstein & Yaari (1983). 2. Dynamic mechanism design: - Battaglini (2005), Zhang (2012), Pavan, Segal & Toikka (2014), Battaglini & Lamba (2014). - Fernandes & Phelan (2000), Doepke & Townsend (2006). 3. Virtual budgets/ Chip Mechanisms: - Möbius (2001), Hauser & Hopenhayn (2008), Abdulkadiroğlu & Bagwell (2012), Johnson (2014). 4. Dynamic contracting ( Immizeration ): - Townsend (1982), Spear & Srivastava (1987), Thomas & Worrall (1990). 6 / 1

16 The Model

17 Discrete time n = 0, 1, / 1

18 Discrete time n = 0, 1,.... Agent s type=value is v n {l, h}. 8 / 1

19 Discrete time n = 0, 1,.... Agent s type=value is v n {l, h}. Cost to supply unit is c > 0, with h > c > l > 0. 8 / 1

20 Discrete time n = 0, 1,.... Agent s type=value is v n {l, h}. Cost to supply unit is c > 0, with h > c > l > 0. Values follow a Markov chain, with: P[v n+1 = h v n = h] = 1 ρ h, P[v n+1 = l v n = l] = 1 ρ l. Assume 1 ρ h ρ l. (h is more likely to be followed by h than l is.) 8 / 1

21 Discrete time n = 0, 1,.... Agent s type=value is v n {l, h}. Cost to supply unit is c > 0, with h > c > l > 0. Values follow a Markov chain, with: P[v n+1 = h v n = h] = 1 ρ h, P[v n+1 = l v n = l] = 1 ρ l. Assume 1 ρ h ρ l. (h is more likely to be followed by h than l is.) The (invariant) probability of h and the (unconditional) expected value of the unit are q := ρ l /(ρ h + ρ l ), µ := E[v] = qh + (1 q)l. 8 / 1

22 Discount factor δ (0, 1). 9 / 1

23 Discount factor δ (0, 1). Agent s Utility: U = (1 δ) δ n x n v n, n=0 Principal s Payoff (Welfare): W = (1 δ) δ n x n (v n c), n=0 where x n {0, 1} is the principal s decision to supply in period n. 9 / 1

24 Discount factor δ (0, 1). Agent s Utility: U = (1 δ) δ n x n v n, n=0 Principal s Payoff (Welfare): W = (1 δ) δ n x n (v n c), n=0 where x n {0, 1} is the principal s decision to supply in period n. In each period, the agent makes a report m, and the principal acts. 9 / 1

25 Discount factor δ (0, 1). Agent s Utility: U = (1 δ) δ n x n v n, n=0 Principal s Payoff (Welfare): W = (1 δ) δ n x n (v n c), n=0 where x n {0, 1} is the principal s decision to supply in period n. In each period, the agent makes a report m, and the principal acts. Revelation Principle m M := {l, h}, and Agent tells the truth. 9 / 1

26 Discount factor δ (0, 1). Agent s Utility: U = (1 δ) δ n x n v n, n=0 Principal s Payoff (Welfare): W = (1 δ) δ n x n (v n c), n=0 where x n {0, 1} is the principal s decision to supply in period n. In each period, the agent makes a report m, and the principal acts. Revelation Principle m M := {l, h}, and Agent tells the truth. A policy is a map x = (x n ) n 0, x n : M n ({0, 1}). 9 / 1

27 Roadmap 1. i.i.d., Binary Values. 10 / 1

28 Roadmap 1. i.i.d., Binary Values. 2. Markov, Binary Values. 10 / 1

29 i.i.d. Values

30 Wlog, the agent s ex ante utility is a valid state variable. 12 / 1

31 Wlog, the agent s ex ante utility is a valid state variable. A policy is represented as a map U (p m, U m ), m = l, h, with p m [0, 1], U m [0, µ]. Note that all utilities are in [0, µ]. 12 / 1

32 The Optimization Program The principal s problem is a Markov decision problem. 13 / 1

33 The Optimization Program The principal s problem is a Markov decision problem. The optimality equation is, for any U [0, µ], W (U) = sup {(1 δ) (qp h (h c) + (1 q)p l (l c)) (p m,u m) + δ (qw (U h ) + (1 q)w (U l ))}, 13 / 1

34 The Optimization Program The principal s problem is a Markov decision problem. The optimality equation is, for any U [0, µ], W (U) = s.t. ( Promise Keeping ) sup {(1 δ) (qp h (h c) + (1 q)p l (l c)) (p m,u m) + δ (qw (U h ) + (1 q)w (U l ))}, U = (1 δ) (qp h h + (1 q)p l l) + δ (qu h + (1 q)u l ), 13 / 1

35 The Optimization Program The principal s problem is a Markov decision problem. The optimality equation is, for any U [0, µ], W (U) = s.t. ( Promise Keeping ) sup {(1 δ) (qp h (h c) + (1 q)p l (l c)) (p m,u m) + δ (qw (U h ) + (1 q)w (U l ))}, U = (1 δ) (qp h h + (1 q)p l l) + δ (qu h + (1 q)u l ), and, for m = l, h, m m ( Incentive Constraint-m ) (1 δ)p m m + δu m (1 δ)p m m + δu m. 13 / 1

36 Complete Information Same program, without incentive constraints. 14 / 1

37 Complete Information Same program, without incentive constraints. If initial utility is freely chosen, (p h, p l ) = (1, 0) and U = qh. 14 / 1

38 Complete Information Same program, without incentive constraints. If initial utility is freely chosen, (p h, p l ) = (1, 0) and U = qh. Instead, taking U as given, stationary policy. p h = U qh, p l = 0 if U [0, qh], p h = 1, p l = U qh (1 q)l if U [qh, µ]. ( 1 c W (U) = h) U if U [0, qh], ( ) ( 1 c U + cq h l 1) if U (qh, µ]. l 14 / 1

39 Complete Information W (U) 0.06 W W = h c h W = l c l qh µ U Figure: Complete information payoff, (δ, l, h, q, c) = (.95,.4,.6,.6,.5)

40 Second-Best: Two Observations 1. Efficient allocation as long as possible. Caveat: efficient allocation infeasible if U, µ U too small. 16 / 1

41 Second-Best: Two Observations 1. Efficient allocation as long as possible. Caveat: efficient allocation infeasible if U, µ U too small. 2. One IC always binds: l pretending h. 16 / 1

42 Second-Best: Two Observations 1. Efficient allocation as long as possible. Caveat: efficient allocation infeasible if U, µ U too small. 2. One IC always binds: l pretending h. Differs from standard adverse selection model where h mimicks l. 16 / 1

43 Dynamics, I Two equations, IC-l and PK, and two unknowns, U h, U l : 17 / 1

44 Dynamics, I Two equations, IC-l and PK, and two unknowns, U h, U l : Both types are willing to send m = h, so: U = (1 δ)µ + δu h. 17 / 1

45 Dynamics, I Two equations, IC-l and PK, and two unknowns, U h, U l : Both types are willing to send m = h, so: U = (1 δ)µ + δu h. Hence, for all U, U h < U. 17 / 1

46 Dynamics, I Two equations, IC-l and PK, and two unknowns, U h, U l : Both types are willing to send m = h, so: U = (1 δ)µ + δu h. Hence, for all U, U h < U. The high type has (1 δ)(h l) excess utility over sending m = l: U = (1 δ)q(h l) + δu l. 17 / 1

47 Dynamics, I Two equations, IC-l and PK, and two unknowns, U h, U l : Both types are willing to send m = h, so: U = (1 δ)µ + δu h. Hence, for all U, U h < U. The high type has (1 δ)(h l) excess utility over sending m = l: U = (1 δ)q(h l) + δu l. Hence: U l < U iff U < q(h l) =: U: Utility is trapped below U. 17 / 1

48 Dynamics, II Region [0, U): transient, leading to {0}. 18 / 1

49 Dynamics, II Region [0, U): transient, leading to {0}. Region [U, 1): transient, leading to either [0, U) or {µ}. 18 / 1

50 Dynamics, II Region [0, U): transient, leading to {0}. Region [U, 1): transient, leading to either [0, U) or {µ}. Drift? Given by PK! U = (1 δ)qh + δ (qu h + (1 q)u l ). }{{} E[U ] 18 / 1

51 Dynamics, II Region [0, U): transient, leading to {0}. Region [U, 1): transient, leading to either [0, U) or {µ}. Drift? Given by PK! U = (1 δ)qh + δ (qu h + (1 q)u l ). }{{} E[U ] U drifts up/down according to U qh. 18 / 1

52 Formally An optimal policy is: { } { U p h (U) = min 1,, p l (U) = max 0, 1 µ U }. (1 δ)µ (1 δ)l 19 / 1

53 Formally An optimal policy is: { } { U p h (U) = min 1,, p l (U) = max 0, 1 µ U }. (1 δ)µ (1 δ)l Payoff W is: - linear and equal to W for U U; - strictly concave and below W for U (U, µ); 19 / 1

54 Formally An optimal policy is: { } { U p h (U) = min 1,, p l (U) = max 0, 1 µ U }. (1 δ)µ (1 δ)l Payoff W is: - linear and equal to W for U U; - strictly concave and below W for U (U, µ); - C 1 over (0, µ), with lim U µ W (U) = lim U µ W (U). 19 / 1

55 Formally An optimal policy is: { } { U p h (U) = min 1,, p l (U) = max 0, 1 µ U }. (1 δ)µ (1 δ)l Payoff W is: - linear and equal to W for U U; - strictly concave and below W for U (U, µ); - C 1 over (0, µ), with lim U µ W (U) = lim U µ W (U). Optimal choice of U 0 = U solves W (U) = / 1

56 Second-Best W (U) 0.06 W 0.05 W U U qh µ U Figure: Payoff as a function of utility, (δ, l, h, q, c) = (.95,.4,.6,.6,.5)

57 A Martingale W (U) =(1 δ)q(h c) ( ( ) U (1 δ)µ + δ qw + (1 q)w δ and so by differentiation, W (U n ) = E[W (U n+1 )]. Hence, probability α of absorption at U = 0 solves ( U (1 δ)u δ )), or W (0) {}}{ h c h α + W (µ) {}}{ l c (1 α) = 0, l α 1 α = h/l (h c)/(c l). 21 / 1

58 Implementation Let f := (1 δ)u, and g := (1 δ)µ f. 22 / 1

59 Implementation Let f := (1 δ)u, and g := (1 δ)µ f. Give the agent a budget of U. 22 / 1

60 Implementation Let f := (1 δ)u, and g := (1 δ)µ f. Give the agent a budget of U. In each period: 1. Charge him a fixed fee of f ; 22 / 1

61 Implementation Let f := (1 δ)u, and g := (1 δ)µ f. Give the agent a budget of U. In each period: 1. Charge him a fixed fee of f ; 2. If he asks for the item, charge g in addition; 22 / 1

62 Implementation Let f := (1 δ)u, and g := (1 δ)µ f. Give the agent a budget of U. In each period: 1. Charge him a fixed fee of f ; 2. If he asks for the item, charge g in addition; 3. Give him a yield at rate r = 1 δ δ. 22 / 1

63 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) 1. Is the agent a prophet or a forecaster? (Is the problem static or dynamic?) 23 / 1

64 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) 1. Is the agent a prophet or a forecaster? (Is the problem static or dynamic?) 2. Are token mechanisms optimal? 23 / 1

65 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) A prophetic agent (static) A forecasting agent (dynamic) 24 / 1

66 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) A prophetic agent (static) A forecasting agent (dynamic) Token mechanisms 24 / 1

67 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) A prophetic agent (static) A forecasting agent (dynamic) Token mechanisms Asymptotically optimal as T (or δ 1). 24 / 1

68 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) A prophetic agent (static) A forecasting agent (dynamic) Token mechanisms Asymptotically optimal as T (or δ 1). The difference in information plays no role. 24 / 1

69 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) A prophetic agent (static) A forecasting agent (dynamic) Token mechanisms Asymptotically optimal as T (or δ 1). The difference in information plays no role. The loss is of the order O(1/ T). 24 / 1

70 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) The question of how the two mechanisms compare is ambiguous a priori. 25 / 1

71 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) The question of how the two mechanisms compare is ambiguous a priori. Lemma It holds that W (U ) q(h c) = O(1 δ). In the case of a prophetic agent, the average loss converges to zero at rate O( 1 δ). 25 / 1

72 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) A prophetic agent (static) A forecasting agent (dynamic) Token mechanisms Asymptotically optimal as T (or δ 1). The difference in information plays no role. The loss is of the order O(1/ T). 26 / 1

73 A Comparison with Token Mechanisms as in Jackson and Sonnenschein (2007) A prophetic agent (static) A forecasting agent (dynamic) Token mechanisms Asymptotically optimal as T (or δ 1). The difference in information plays no role. The loss is of the order O(1/ T). Optimal mechanisms O( 1 δ) O(1 δ) 26 / 1

74 Markovian Values

75 Set-Up Values follow a Markov chain, with: P[v n+1 = h v n = h] = 1 ρ h, P[v n+1 = l v n = l] = 1 ρ l. Assume 1 ρ h > ρ l. 28 / 1

76 Set-Up Values follow a Markov chain, with: P[v n+1 = h v n = h] = 1 ρ h, P[v n+1 = l v n = l] = 1 ρ l. Assume 1 ρ h > ρ l. All else: the same. 28 / 1

77 Reduction to Dynamic Programming, I Revelation Principle still applies. 29 / 1

78 Reduction to Dynamic Programming, I Revelation Principle still applies. But promised utility is no longer a valid state variable. What is? 29 / 1

79 Reduction to Dynamic Programming, I Revelation Principle still applies. But promised utility is no longer a valid state variable. What is? State variables: 1. A pair of promised interim utilities: U h, U l. 2. The belief of the principal, φ = P[v = h] [0, 1]. 29 / 1

80 Reduction to Dynamic Programming, I Revelation Principle still applies. But promised utility is no longer a valid state variable. What is? State variables: 1. A pair of promised interim utilities: U h, U l. 2. The belief of the principal, φ = P[v = h] [0, 1]. Choice variables, given (φ, U h, U l ): 1. Supply decision: p = (p h, p l ) [0, 1] / 1

81 Reduction to Dynamic Programming, I Revelation Principle still applies. But promised utility is no longer a valid state variable. What is? State variables: 1. A pair of promised interim utilities: U h, U l. 2. The belief of the principal, φ = P[v = h] [0, 1]. Choice variables, given (φ, U h, U l ): 1. Supply decision: p = (p h, p l ) [0, 1] Promised pair tomorrow: for m = h, l: (U h (m), U l (m)) R / 1

82 Reduction to Dynamic Programming, II The optimality equation becomes W (U h, U l, φ) = sup {φ ((1 δ)p h (h c) + δw (U h (h), U l (h), 1 ρ h )) + (1 φ) ((1 δ)p l (l c) + δw (U h (l), U l (l), ρ l ))}, over (p h, p l, U h (h), U l (h), U h (l), U l (l)) s.t. U h = (1 δ)p h h + δ(1 ρ h )U h (h) + δρ h U l (h) (1 δ)p l h + δ(1 ρ h )U h (l) + δρ h U l (l), and similarly for U l. 30 / 1

83 Incentive Feasibility Let us ignore optimality and focus on incentives. 31 / 1

84 Incentive Feasibility Let us ignore optimality and focus on incentives. In the i.i.d. case, all U [0, µ] are achievable by some IC-policy. 31 / 1

85 Incentive Feasibility Let us ignore optimality and focus on incentives. In the i.i.d. case, all U [0, µ] are achievable by some IC-policy. What are the IC-feasible pairs (U h, U l )? (e.g., (0, µ) is not.) 31 / 1

86 Incentive Feasibility Let us ignore optimality and focus on incentives. In the i.i.d. case, all U [0, µ] are achievable by some IC-policy. What are the IC-feasible pairs (U h, U l )? (e.g., (0, µ) is not.) 1. Ignore reports; produce for first N periods ( frontloading ); 31 / 1

87 Incentive Feasibility Let us ignore optimality and focus on incentives. In the i.i.d. case, all U [0, µ] are achievable by some IC-policy. What are the IC-feasible pairs (U h, U l )? (e.g., (0, µ) is not.) 1. Ignore reports; produce for first N periods ( frontloading ); 2. Ignore reports; start producing after period N ( backloading ). 31 / 1

88 Incentive Feasibility Let us ignore optimality and focus on incentives. In the i.i.d. case, all U [0, µ] are achievable by some IC-policy. What are the IC-feasible pairs (U h, U l )? (e.g., (0, µ) is not.) 1. Ignore reports; produce for first N periods ( frontloading ); 2. Ignore reports; start producing after period N ( backloading ). Given the same U h, the values of U l are not the same under frontand backloading. U l is higher under backloading. 31 / 1

89 Incentive Feasibility Let us ignore optimality and focus on incentives. In the i.i.d. case, all U [0, µ] are achievable by some IC-policy. What are the IC-feasible pairs (U h, U l )? (e.g., (0, µ) is not.) 1. Ignore reports; produce for first N periods ( frontloading ); 2. Ignore reports; start producing after period N ( backloading ). Given the same U h, the values of U l are not the same under frontand backloading. U l is higher under backloading. Front- and backloading define the boundaries of the IC-feasible set. 31 / 1

90 U l Backloading V Frontloading (µ h, µ l ) U h Figure: The set V for parameters (δ, ρ h, ρ l, l, h) = (9/10, 1/3, 1/4, 1/4, 1)

91 U l Backloading V Frontloading U h Figure: The set V for parameters (δ, ρ h, ρ l, l, h) = (9/10, 1/3, 1/4, 1/4, 1) v (µ h, µ l )

92 Optimal Policy Recall the policy is defined for all (U h, U l ) V, φ {ρ l, 1 ρ h }. 33 / 1

93 Optimal Policy Recall the policy is defined for all (U h, U l ) V, φ {ρ l, 1 ρ h }. The optimal policy is simple and independent of beliefs. 33 / 1

94 Optimal Policy Recall the policy is defined for all (U h, U l ) V, φ {ρ l, 1 ρ h }. The optimal policy is simple and independent of beliefs. I will focus on the reachable subset of states given the optimal U. 33 / 1

95 The Lower Boundary U lies on the lower boundary (frontloading policies). 34 / 1

96 The Lower Boundary U lies on the lower boundary (frontloading policies). This does not imply that frontloading occurs: Any policy s.t. IC-l binds in every period, and s.t. p h = 1 ( whenever possible ) yields utilities on this boundary. 34 / 1

97 The Lower Boundary U lies on the lower boundary (frontloading policies). This does not imply that frontloading occurs: Any policy s.t. IC-l binds in every period, and s.t. p h = 1 ( whenever possible ) yields utilities on this boundary. They differ in terms of the principal s payoff, and utility dynamics. 34 / 1

98 Optimal Policy (on Lower Boundary) Choose efficient allocation { p h = min 1, U h (1 δ)h } {, p l = max 0, 1 µ } l U l, (1 δ)l and continuation utilities on the lower boundary s.t. IC-l binds. Jump to Global Policy 35 / 1

99 U l (µ h, µ l ) U h

100 U l h (µ h, µ l ) U h

101 U l h l (µ h, µ l ) U h

102 U l h (µ h, µ l ) U h

103 U l (µ h, µ l ) 0.1 h l U h

104 Dynamics It holds that U(h) U. 37 / 1

105 Dynamics It holds that U(h) U. U(l) U if and only if U is low enough. 37 / 1

106 Dynamics It holds that U(h) U. U(l) U if and only if U is low enough. U is drifting up iff ρ h ρ h + ρ l U l + ρ l ρ h + ρ l U h ρ l ρ h + ρ l h. ( Long-run efficient utility below current long-run promise. ) 37 / 1

107 Dynamics It holds that U(h) U. U(l) U if and only if U is low enough. U is drifting up iff ρ h ρ h + ρ l U l + ρ l ρ h + ρ l U h ρ l ρ h + ρ l h. ( Long-run efficient utility below current long-run promise. ) Utility is eventually absorbed at U {0, µ}. We know of no formula for absorption probability. 37 / 1

108 Implementation The obvious budget unit is: # of consecutive periods the agent can claim the unit, no questions asked: (B n, γ n ) N [0, 1]. 38 / 1

109 Implementation The obvious budget unit is: # of consecutive periods the agent can claim the unit, no questions asked: (B n, γ n ) N [0, 1]. Not asking for the unit leads to the revised promise where U l (B n, γ n ) δ = E l [ Uvn+1 (B n+1, γ n+1 ) ]. E l [ Uvn+1 (B n+1, γ n+1 ) ] = (1 ρ l )U l (B n+1, γ n+1 )+ρ l U h (B n+1, γ n+1 ) is the expected utility from tomorrow s (B n+1, γ n+1 ) given l today. 38 / 1

110 Implementation The obvious budget unit is: # of consecutive periods the agent can claim the unit, no questions asked: (B n, γ n ) N [0, 1]. Not asking for the unit leads to the revised promise where U l (B n, γ n ) δ = E l [ Uvn+1 (B n+1, γ n+1 ) ]. E l [ Uvn+1 (B n+1, γ n+1 ) ] = (1 ρ l )U l (B n+1, γ n+1 )+ρ l U h (B n+1, γ n+1 ) is the expected utility from tomorrow s (B n+1, γ n+1 ) given l today. Asking for it leads to a payment (1 δ)l, as before: U l (B n, γ n ) (1 δ)l δ = E l [ Uvn+1 (B n+1, γ n+1 ) ]. 38 / 1

111 Continuous-Time Limit Lack of smoothness prevents easy comparative statics. 39 / 1

112 Continuous-Time Limit Lack of smoothness prevents easy comparative statics. Let ρ h λ h, ρ l λ l, r (1 δ) and take / 1

113 Continuous-Time Limit Lack of smoothness prevents easy comparative statics. Let ρ h λ h, ρ l λ l, r (1 δ) and take 0. Flow values evolve according to a two-state Markov chain with parameters λ l, λ h. 39 / 1

114 Continuous-Time Limit Lack of smoothness prevents easy comparative statics. Let ρ h λ h, ρ l λ l, r (1 δ) and take 0. Flow values evolve according to a two-state Markov chain with parameters λ l, λ h. Two simplifications: 1. No kinks: lower boundary parameterized by τ R + ; 39 / 1

115 Continuous-Time Limit Lack of smoothness prevents easy comparative statics. Let ρ h λ h, ρ l λ l, r (1 δ) and take 0. Flow values evolve according to a two-state Markov chain with parameters λ l, λ h. Two simplifications: 1. No kinks: lower boundary parameterized by τ R + ; 2. Degenerate beliefs: φ {0, 1}. 39 / 1

116 U l 0.5 (µ h, µ l ) V U h Figure: Incentive-feasible set for (r, λ h, λ l, l, h) = (1, 10/4, 1/4, 1/4, 1)

117 Payoff Dynamics Payoffs (conditional on the point belief) satisfy the coupled ODE: and (r + λ h )W h (τ) = r(h c) + λ h W l (τ) W h(τ), (r + λ l )W l (τ) = λ l W h (τ) + g(τ) µ q(h l)e (λ h+λ l )τ W l (τ), where g(τ) := q(h l)e (λ h+λ l )τ + le rτ µ, and W (0) = / 1

118 Proposition The value function of the principal is given by W 1 (τ) W 1 (τ) w 0 (τ) h l hl crµ W 1 (τ) + w 0 (τ) h l hl c ˆτ 1 + rµ t f (s)ds e τ 0 τˆτ w 0 2(t) t λ h +λ l e 2rt g(t) ˆτ τ dt if τ [0, ˆτ), if τ [ˆτ, τ 0 ), f (s)ds τ 0 dt e t f (s)ds τ 0 w 0 2 dt (t) t λ h +λ l e 2rt f (s)ds τ 0 g(t) dt if τ τ 0, where W 1 (τ) := (1 e rτ )(1 c/h)µ, w 0 (τ) := µe rτ (1 q)l, f (τ) := r (λ h + λ l ) w 0(τ) g(τ) erτ, and τ 0 is the positive root of w 0, and ˆτ of g. 42 / 1

119 W, W W p = p = 1/4 τ 0.06 Figure: Payoff; (λ l, λ h, r, l, h, c) = (p/4, 10p/4, 1, 1/4, 1, 2/5), p = 1, 1/4

120 Persistence, Convergence Lemma The value W (τ) decreases pointwise in persistence 1/p, where λ h = p λ h, λ l = p λ l, for some fixed λ h, λ l. 44 / 1

121 Persistence, Convergence Lemma The value W (τ) decreases pointwise in persistence 1/p, where λ h = p λ h, λ l = p λ l, for some fixed λ h, λ l. Lemma It holds that max W (τ) q(h c) = O(r). τ 44 / 1

122 A Comparison with the Transfer Case With transfers, efficiency is trivial (the agent pays c for the unit). 45 / 1

123 A Comparison with the Transfer Case With transfers, efficiency is trivial (the agent pays c for the unit). Revenue maximization (Battaglini, 2005). 1. Inefficiency decreases over time. 2. Efficiency achieved along all histories, asymptotically. 45 / 1

124 A Comparison with the Transfer Case With transfers, efficiency is trivial (the agent pays c for the unit). Revenue maximization (Battaglini, 2005). 1. Inefficiency decreases over time. 2. Efficiency achieved along all histories, asymptotically. Logic is different: 1. With transfers, IC-h becomes the problematic constraint (because transfers are used to extract surplus). 45 / 1

125 A Comparison with the Transfer Case With transfers, efficiency is trivial (the agent pays c for the unit). Revenue maximization (Battaglini, 2005). 1. Inefficiency decreases over time. 2. Efficiency achieved along all histories, asymptotically. Logic is different: 1. With transfers, IC-h becomes the problematic constraint (because transfers are used to extract surplus). 2. Information rents in period n can be extracted in period 0 via higher prices (as the expected rent is insensitive to the initial value when n is large). 45 / 1

126 A Comparison with the Transfer Case Transfers have two benefits: 46 / 1

127 A Comparison with the Transfer Case Transfers have two benefits: 1. Promises made in period n can be cleared then. 46 / 1

128 A Comparison with the Transfer Case Transfers have two benefits: 1. Promises made in period n can be cleared then. 2. Clearing earlier reduces cost of future information rents. 46 / 1

129 All That is Missing Statistical signals.

130 All That is Missing Statistical signals. 2. More than one agent (allocation problems).

131 All That is Missing Statistical signals. 2. More than one agent (allocation problems). 3. More than one good (matching problems).

132 Thank You!

133 U h U l V v V v v 0 V h Figure: The set V, V, (δ, ρ h, ρ l, l, h) = (9/10, 1/3, 1/4, 1/4, 1)

134 U h U l V V P V h h l h l h l h l Figure: Optimal policy for (δ, ρ h, ρ l, l, h) = (9/10, 1/3, 1/4, 1/4, 1)

135 Optimal Policy Let P t, P b denote the top and bottom boundary of V. Let V t, V b denote the regions of V above (below) some polygonal chain P (omitted here).

136 Optimal Policy Let P t, P b denote the top and bottom boundary of V. Let V t, V b denote the regions of V above (below) some polygonal chain P (omitted here). For all U = (U h, U l ) V, set { p l = max 0, 1 µ } { } l U l U h, p h = min 1,, (1 δ)l (1 δ)h and U(h) P b, U(l) { Pb if U V b, P t if U P t. Furthermore, if U V t, U(l) is chosen so that IC-h binds. Jump Back

137 General i.i.d. Distributions What remains the same: - Convergence to either 0 or µ (so inefficiency is pushed back). - Martingale property of W. - Slopes match W at the end points. What changes: - Strict concavity on all [0, µ], W < W on (0, µ). - Policy no longer efficient as long as possible; not even cut-off. - (F(v) = v α, α 1): - U U : 0 < v 1 < v 2 < 1 such that { 0 if v v 1, p(v) = 1 if v v 2, and continuously increasing on [v 1, v 2 ]. - U < U : same, excepts v 2 = 1.

138 Complete Information Efficient policy yields v m: { v h = (1 δ)h + δ(1 ρ h )v h + δρ hv v l = δ(1 ρ l )v l + δρ l v h. Policy and payoff are piecewise linear. Increasing in each U m iff U m v m. l,

139 U l Figure: Impact of persistence U h

140 U h Independent U l Figure: Impact of persistence

141 U h Independent Very persistent U l Figure: Impact of persistence