Decentralization and Private Information with Mutual Organizations

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1 Decentralization and Private Information with Mutual Organizations Edward C. Prescott and Adam Blandin Arizona State University 09 April

2 Motivation Invisible hand works in standard environments But, many Private Info environments can have bad outcomes Multiple equilibria Inefficient equilibria Some argue these results necessitate government regulation We show: competition among MO s solves private info problem The invisible hand works in these environments as well! 2

3 Historical Review Spence in his signaling paper operated in the Theory of Value tradition not surprisingly given he was a student of Arrow Rothschild-Stiglitz in their adverse selection insurance paper operated in the Bertrand tradition Here we operate in something closer to the Edgeworth/core tradition There is competition between mutual organizations for members 3

4 Definition of Mutual Organization The residual claimants are the members and as a consequence there is no possibility of default Mutual insurance companies pay dividends to those insured if claims are less than premiums and pay less then fully on claims if claims exceed premiums 4

5 Thesis For a class of economies that include the Spence s signaling, the Rothschild-Stiglitz adverse selection insurance, and the Diamond-Dybvig private preference shock environments, with mutual organizations permitted: Equilibrium exists, is optimal, and is generically unique. 5

6 Class of Environments Finite number of types Types are private information i I Type measures λ i Expected utility maximization Single resource constraint Consider type-identical allocations only 6

7 Incentive-compatibility constraints are linear There is a single resource constraint, which is a linear inequality With lotteries, given expected utility maximization, the Rothschild-Stiglitz (R-S) environment has these properties 7

8 Underlying consumption space is a closed bounded set n S R+ Lotteries are on the Borel sigma algebra of S Preferences over lotteries, x, are ordered by v i u () x = v () s x( ds) i i The functions are continuous, strictly concave 8

9 x Incentive Constraints (IC s) is type i lottery for allocation x= {x } i i i I Feasibility requires IC's be satisfied u ( x ) u ( x ) all i, j i i i j and resource constraint (RC) be satisfied i λ t ( x ) 0 i i i The t i are transfers made They are affine functions. 9

10 Utility Possibility Set Utilities cone u = { u} i satisfying the IC s are a convex x(u) is the allocation which minimizes resources used to achieve utility u vector Rather than txu ( ( )) = { ti( xu i( ))} we abuse notation and write t(u) 10

11 Blocking Feasible allocation x is blocked by y if for some coalition B 1. All i types of B are weakly better off and at least one type better off 2. Under y total transfer by types in B is positive Proposition: Any non Pareto optimal allocation is blocked. Therefore the problem is to determine the set of Pareto allocations which are not blocked 11

12 Proposition: Any unblocked allocation minimizes transfers within the Pareto Optimum set of allocations. Proposition: The better off under y can not transfer more under y than under x Proof outline: If they could, there would be an allocation that is Pareto superior to Pareto optimal x 12

13 Existence Outline 1. Begin with Pareto optimal u (0) (1) 2. If blocked, can find Pareto optimum which blocks and has smaller tarnsfers in 1 u norm 3. Construct sequence with { u } with ( n) blocking. u u ( n) ( n+ 1) 4. There will be a convergent subsequence with limit u*, which is in the Pareto set and is unblocked given compactness 13

14 The Simplest Example: Two-Type Spence Environment u = c θ s c 0 is consumption and s 0 signal i i i i i i t = π c where i i i i π > 0 are productivities π > π and θ < θ Type 2 are the more productive ones and have lower disutility of signaling i i i 14 i Resource constraint: λ ( π c ) 0

15 45 IC

16 F F 45

17 Pareto Set F 45

18 F 45 π 1

19 Rothschild-Stiglitz: Adverse Selection Insurance Endowments of individuals: 0 < e 1 < e 2 Utility functions: u ( x ) = π v() c x ( dc) i i ie ie e Transfers: t ( x ) = π ( e c) x ( dc) i i ie ie e 19

20 A Diamond-Dybvig Environment: Private Preference Shocks Three dates, t = 0,1,2 υ : + One type ex ante, two types ex post with probability π i > 0: π + π = Types i are private information Endowment e > 0 in period t =0

21 A contract is a lottery pair {x i (dc 1 dc 2 )} i=1,2 The type utility functions are u ( x ) = v(c ) dx u ( x ) = v(c + c ) dx

22 Incentive Constraints (IC s) u ( x ) u ( x ) u ( x ) u ( x )

23 Resource Constraints (RC) i c π c 2 dx e i + 1 i A where A > 1

24 Pareto Optimum Pareto Optimum solves: max u( x ) dx x, x 1 2 π i i i i s.t. IC s and RC Given strict concavity of v, Pareto Optimum is unique. Note: RC is linear; IC s are linear

25 Implications Empirically we know centralized schemes to allocate resources plagued by inadequate information and unproductive rent-seeking. For a broad class of private information economies, the task of allocating resources efficiently can be left to small, decentralized units. Avoids problems associated with centralized decision-making.

26 Concluding Comments Analysis suggests with mutual arrangements permitted, the invisible hand works for signaling, adverse selection, and preference shock environments The single resource constraint is likely crucial Much remains to be done 26

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