Topics in Contract Theory Lecture 5. Property Rights Theory. The key question we are staring from is: What are ownership/property rights?

Transcription

1 Leonardo Felli 15 January, 2002 Topics in Contract Theory Lecture 5 Property Rights Theory The key question we are staring from is: What are ownership/property rights? For an answer we need to distinguish between: specific rights over a physical asset, that can be disposed of by means of a contract (they are contractible); residual rights over a physical asset, these are all the rights that cannot be disposed of, at an ex-ant stage, by means of a contract. 1

2 Topics in Contract Theory 2 Definition: Ownership of an asset corresponds to the entitlement to the residual rights of control over the asset. In the event that a contingency occurs that is not disciplined by the contract the owner has the right to dispose at will of the asset. Example: the rental contract of an apartment. In the event an event occurs that is not disposed of in the rental contract the owner of the apartment has the right to decide what happens to the property. Notice that since all residual rights are allocated at the same time a contract can dispose of them in a residual manner. Ownership (the allocation of the residual rights) is itself contractible.

3 Topics in Contract Theory 3 Consider a simple incomplete contract model with physical assets. Let a 1 and a 2 denote two physical assets. Consider two individuals m 1 and m 2. We can interpret m 1 and m 2 as two managers operating the physical assets. The transaction: m 2, with the use of a 2, supplies a single unit of input (widget) to m 1 ; m 1, with the use of a 1, uses this widget to produce output that is sold on the output market.

4 Topics in Contract Theory 4 Timing: Both m 1 and m 2 make a relationship specific investment i and e, respectively; A contract to trade the widget is agreed between m 1 and m 2 ; Trade occurs and output is realized. The incomplete contract assumption is that before relationship specific investments are made friction prevent the parties from writing a contract on (i, e) and the nature of the widget that is traded. Underlying assumption: there exists uncertainty about the nature of the widget to trade that is resolved once investments are made.

5 Topics in Contract Theory 5 The parties are assumed to be risk neutral with unlimited wealth so that they can purchase any asset they desire. The rest of the incomplete contract assumption is that no specific right (the use) of the assets a 1 and a 2 can be disciplined by an ex-ante contract (written before investments are made). The only ex-ante contract that can be written on the assets is an ownership contract that disposes of the entire residual rights on the assets. This essentially means that the owner of one of the assets a i can use it in ny way he/she pleases.

6 Topics in Contract Theory 6 The ex-ante ownership contracts we analyze are: non-integration: m 1 owns a 1 and m 2 owns a 2 ; type 1 integration: m 1 owns a 1 and a 2 ; type 2 integration: m 2 owns a 1 and a 2 ; Other ownership structures include random allocation of ownership, joint allocation of ownership etc. The ex0ante investments i and e cost i and e, respectively. Assume that the ex-ante investment i affects m 1 s revenue both if m 1 trades with m 2 and he does not.

7 Topics in Contract Theory 7 If trade occurs between m 1 and m 2 at the price p then m 1 s revenue is R(i) while m 1 s ex-post payoff and ex-ante payoffs are: R(i) p, R(i) p i If trade does not occur m 1 buys a widget from an outside supplier at a price p. Then m 1 s revenue is r(i, A) while m 1 s ex-post and ex-ante payoffs are: r(i, A) p, r(i, A) p i Let r(, ) denote the revenue in the absence and R( ) revenue in the presence of m 2 s specific investment.

8 Topics in Contract Theory 8 Let A denote the set of assets available to m 1 when trade does not occur: A {a 1, a 2 } In case of non-integration A = {a 1 }, In case of type 1 integration A = {a 1, a 2 }, In case of type 2 integration A =,

9 Topics in Contract Theory 9 Similarly, the ex-ante investment e affects m 2 s revenue both if m 2 trades with m 1 and he does not.

10 Topics in Contract Theory 10 If trade occurs at the price p then m 2 s cost is C(e) while m 2 s ex-post payoff and ex-ante payoffs are: p C(e), p C(e) e If trade does not occur m 2 sells the widget to an outside buyer a price p. Then m 2 s cost is c(e, B) while m 2 s ex-post and ex-ante payoffs are: p c(e, B), p c(e, B) e Then c(, ) denote the cost in the absence and C( ) the cost in the presence of m 1 s specific investment.

11 Topics in Contract Theory 11 Moreover, B denote the set of assets available to m 2 when trade does not occur: B {a 1, a 2 } In case of non-integration B = {a 2 }, In case of type 1 integration B =, In case of type 2 integration B = {a 1, a 2 }, The total ex-post surplus if trade occurs is: R(i) C(e) while the total surplus if trade does not occur is: r(i, A) c(e, B)

12 Topics in Contract Theory 12 We assume that the investments are indeed specific (there are always ex-post gains-from-trade): R(i) C(e) > r(i, A) c(e, B) 0, i, e, A, B We also assume specificity in a marginal sense: R (i) > r (i, a 1, a 2 ) r (i, a 1 ) r (i, ) i C (e) > c (e, a 1, a 2 ) c (e, a 2 ) c (e, ) e Where x is the absolute value of x (cost are decreasing in investment) and r (i, A) = r(i, A), c (e, B) = i c(e, B) e We also assume that: R > 0, R < 0, C < 0, C > 0 r 0, r 0, c 0, c 0

13 Topics in Contract Theory 13 Notice that: R (i) > r (i, a 1, a 2 ) implies that i is at least partly specific to m 2 s human capital e, r (i, a 1, a 2 ) r (i, a 1 ) r (i, ) implies that i may or may not be specific to the assets a 1 and a 2, r (i, a 1, a 2 ) = r (i, a 1 ) > r (i, ) implies that i is specific to a 1 but not to a 2. We are assuming that i and e are investments in human capital and not physical capital this is implied by R(i) > r(i, a 1, a 2 )

14 Topics in Contract Theory 14 The underlying assumption is that human capital is inalienable (slavery contracts are illegal). We now solve the SPE of the model proceeding backward. Consider the negotiation of the ex-post contract for given investment choice (i, e) and given asset ownership structure (A, B). At the contracting stage Coase Theorem holds and hence the contracting parties m 1 and m 2 will exploit all gains from trade and achieve an agreement that leads to trade. We assume that the parties use a generalized Nash bargaining solution with outside option to split the surplus: each party is guaranteed his outside option and a fix share of the gains from trade.

15 Topics in Contract Theory 15 Assume that the parties choose p so as to split equally the gains from trade [R(i) C(e)] [r(i, A) c(e, B)] The parties payoffs are then: π 1 = R(i) p = [r(i, A) p] [R(i) C(e) r(i, A) + c(e, B)] π 2 = p C(e) = [ p c(e, B)] [R(i) C(e) r(i, A) + c(e, B)] Alternatively: π 1 = p [R(i) + r(i, A) C(e) + c(e, B)] π 2 = p + 1 [ C(e) c(e, B) + R(i) r(i, A)] 2

16 Topics in Contract Theory 16 The price of the widget is then: p = p [R(i) + C(e) r(i, A) c(e, B)] Notice that this price shows the possibility that a free rider problem arises: if R is increased by 1 unit then the price p is only increased only by 1/2 unit, if C is decreased by 1 unit then the price p decreases only by 1/2 unit. The ex-ante efficient level of investments (i, e ) solve: max i,e R(i) C(e) i e The first order conditions imply: R (i ) = 1, C (e ) = 1

17 Topics in Contract Theory 17 Consider now the equilibrium investments (î, ê). The investment î is the solution to the problem: max i π 1 i = = p [R(i) + r(i, A) C(e) + c(e, B)] i The investment ê is the solution to the problem: max e π 2 e = = p [ C(e) c(e, B) + R(i) r(i, A)] e The first order conditions of these two problems are: 1 2 R (î) r (î, A) = C (ê) c (ê, B) = 1

18 Topics in Contract Theory 18 Result 1. Grossman and Hart 1986) Under any ownership structure, there is underinvestment in relationship-specific investments: î < i, ê < e Proof: It follows from the strict concavity of R( ) and strict convexity of C( ) and R (i) > r (i, A), C (e) > c (e, B) that imply: R (î) > 1 2 R (î) r (î, A) = 1 C (ê) > 1 2 C (ê) c (ê, B) = 1 The intuition is the standard free-rider problem.

19 Topics in Contract Theory 19 If i is increased of di the social surplus increases of R (i) di while m 1 s payoff increases of 1 2 [R (i) + r (i, A)] di < R (i)di the rest of the increase is expropriated by m 2. Notice also that the returns of the returns to m 1 depend on r (i, A). In other words the m 1 s equilibrium incentives to invest in i depend on the allocation of ownership rights on the physical assets a 1 and a 2 : the set A. Therefore ownership of physical assets affects efficiency through the effect that A has on m 1 s outside option r(i, A) (the returns in the case of no trade).

20 Topics in Contract Theory 20 Consider now the three ownership allocations mentioned above the equilibrium investments are: non-integration (i 0, e 0 ): 1 2 R (i 0 ) r (i 0, a 1 ) = C (e 0 ) c (e 0, a 2 ) = 1 type 1 integration (i 1, e 1 ): 1 2 R (i 1 ) r (i 1, a 1, a 2 ) = C (e 1 ) c (e 1, ) = 1 type 2 integration (i 2, e 2 ): 1 2 R (i 2 ) r (i 2, ) = C (e 2 ) c (e 2, a 1, a 2 ) = 1

21 Topics in Contract Theory 21 Result 2. Grossman and Hart 1986) The three ownership structures non-integration, type 1 integration and type 2 integration are such that the parties relationship-specific investments satisfy: i > i 1 i 0 i 2, e > e 2 e 0 e 1 We can now move to the analysis of the optimal ownership structure. This is easily obtained comparing S 0 = R(i 0 ) C(e 0 ) i 0 e 0 S 1 = R(i 1 ) C(e 1 ) i 1 e 1 S 2 = R(i 2 ) C(e 2 ) i 2 e 2 Notice that the ownership structure only indirectly affects the social surplus through its effect on the equilibrium incentives of m 1 and m 2 to invest.

22 Topics in Contract Theory 22 Definition 1. The investment choice of m 1 is inelastic in the range 1/2 ρ 1 if the solution to the problem max i ρ R(i) i is independent of ρ in this range. The investment choice of m 2 is inelastic in the range 1/2 σ 1 if the solution to the problem min e σ C(e) + e is independent of σ in this range.

23 Topics in Contract Theory 23 Result 3. If m 2 s investment decision is inelastic then type 1 integration is optimal. Alternatively, if m 1 s investment decision is inelastic then type 2 integration is optimal. Proof: Since e is inelastic then m 2 sets e = ē, where C (ē) + 1 = σ C (ē) + 1 σ [1/2, 1] for every ownership structure. Hence C(ē) ē is invariant to the ownership allocation. Since R(i) i does depend on the ownership allocation from result 2 above it is optimal to maximize the incentives of m 1. It is optimal to allocate all assets to m 1 : i = i 1.

24 Topics in Contract Theory 24 Definition 2. The investment choice of m 1 is relatively unproductive if R(i) can be replaced by θ R(i) + (1 θ) i and r(i, A) can be replaced by θ r(i, A) + (1 θ) i for every A {(a 1 ), (a 1, a 2 ), } where θ > 0 and is small. The investment choice of m 2 is relatively unproductive if C(e) can be replaced by θ C(e) (1 θ) e and c(e, B) can be replaced by θ c(e, A) (1 θ) e for every B {(a 2 ), (a 1, a 2 ), } where θ > 0 and is small.

25 Topics in Contract Theory 25 Result 4. If m 2 s investment becomes relatively unproductive, and for all i r (i, a 1, a 2 ) > r (i, a 1 ) Then for θ small enough type 1 integration is optimal. Alternatively, if m 1 s investment decision is relatively unproductive, and for all e c (e, a 1, a 2 ) > c (e, a 2 ) Then for θ small enough type 2 integration is optimal. Proof: Assume that e is relatively unproductive then m 2 s first order conditions under any ownership structure become: 1 2 θ C (ê) (1 θ) θ c (ê, B) + 1 (1 θ) = 1 2

26 Topics in Contract Theory 26 or 1 2 C (ê) c (ê, B) = 1 independently of θ. In other words e is independent of θ. However the net surplus is not independent of θ: S = R(i) i θ C(e)+(1 θ)e e = = R(i) i θ (C(e) e) Clearly, for θ small S R(i) i Since R(i) i does depend on the ownership allocation from result 2 above it is optimal to maximize the incentives of m 1. It is optimal to allocate all assets to m 1 : i = i 1.

27 Topics in Contract Theory 27 Definition 3. Assets a 1 and a 2 are independent if: r (i, a 1, a 2 ) = r (i, a 1 ), c (e, a 1, a 2 ) = c (e, a 2 ), i e Result 5. If assets a 1 and a 2 are independent then non-integration is optimal. Proof: Since assets are independent then from result 2 we get: i 1 = i 0, e 2 = e 0 Since from result 2 we also have: i 0 i 2, e 0 e 1 We conclude that S 0 is the highest possible surplus.

28 Topics in Contract Theory 28 Definition 4. Assets a 1 and a 2 are strictly complementary if either: or r (i, a 1 ) = r (i, ), c (e, a 2 ) = c (e, ), i e Result 6. If assets a 1 and a 2 are strictly complementary then some form of integration is optimal. Proof: Assume that r (i, a 1 ) = r (i, ) for all i. Then from result 2 above we get: i 0 = i 2 and e 2 e 0 Which implies that type 2 integration dominates nonintegration: S 2 S 0. The same argument shows that when for every e we have c (e, a 2 ) = c (e, ) then type 1 integration dominates non-integration: S 1 S 0.

29 Topics in Contract Theory 29 Definition 5. The human capital investment of m 1 is essential if: c (e, a 1, a 2 ) = c (e, ), e The human capital investment of m 2 is essential if: r (i, a 1, a 2 ) = r (i, ), i Result 7. If m 1 s human capital investment is essential then type 1 integration is optimal. If m 2 s human capital investment is essential then type 2 integration is optimal. If both m 1 s and m 2 s human capital investment are essential then all ownership structures are equally optimal.

30 Topics in Contract Theory 30 Proof: If m 1 s human capital is essential then from result 2 above we have: e 2 = e 0 = e 1 and i 1 i 0 i 2 Then type 1 integration is optimal: S 1 S 0 S 2 If both i and e are essential then from result 2 we have: e 2 = e 0 = e 1 and i 1 = i 0 = i 2 Therefore we have: S 2 = S 0 = S 1. Definition 6. The joint ownership allocation is such that neither m 1 nor m 2 can dispose of any asset without the permission of the other party. This implies that under joint ownership: A =, and B =

31 Topics in Contract Theory 31 Result 8. Joint ownership is never optimal unless: r (i, A) = r (i, ), i, A and c (e, B) = c (e, ), e, B Proof: If the condition above is not satisfied then the first order conditions of the joint ownership case imply: i 1 i j, e 2 e j with at least one inequality strict. Therefore either S 1 > S j or S 2 > S j.