The following proposition describes the feasibility of … first-best allocations under an optimally designed monetary mechanism. We show that for sufficiently high ... the first best assignment can be supported if. Definition 9 The ME e-money mechanism implements ... first-best with limited participation if.
ME is self-…financed with limited participation according to the allocation of …rst-best (q ; zb; zs; nb; ns). The following proposition defines the condition under which an optimal limited-participation electronic money mechanism can be best achieved. Proposition 4 There is an ME electronic money mechanism that implements the best … with bounded participation if and only if .
Therefore, this example shows that when is small and U(q ) [1. 1 )]C(q) is small (but remains positive, as assumed), the ...first-best allocation cannot be implemented by any limited-participation e-money mechanism. When the bargaining power of buyers is high ( ), an optimal monetary mechanism can implement the ... first best. When buyers have moderate bargaining power and money growth is high, an e-money mechanism with limited participation is essential to implement the …first best.
ML is self-financed with limited transferability according to the allocation …rst-best (q ; zb; zs; nb; ns). Proposition 10 (Essentiality of limited transferability electronic money) (a) If = 1 and < , then the distribution … best money can be implemented by a limited transferability electronic money mechanism when. Assume that this is not the case, and denote (q ; zb; zs) as the ...rst-best allocation implemented.
The following proposition characterizes the set of an Andolfatto's mechanism that implements the ... first best. On the other hand, if ( ), we can construct an e-money mechanism ME that implements the ... first best with limited participation. Thus (q ; zb;0; nb;0) is also incentive compatible for buyers under ME, and ME implements the ... first best with limited participation.
The following proposition characterizes the set of membership-reward-deposit mechanisms that implement the ... first best. Thus (q ; zb; zs; nb; ns) is incentive compatible for buyers under ML, and therefore implements the ... first best. So, from the proof of Proposition 8, there exists an e-money mechanismML that implements the …first best for the given.
The following proposition characterizes the set of exchange-reward-deposit mechanisms that implement the best ….
Optimal Money Mechanism
In this section, we show that all results under proportional bargaining extend to a competitive pricing environment. In order to induce buyers to participate in the mechanism (ie, eb = 1), a satisfactory incentive-compatible allocation (q; d; zb; zs) is required. Similarly, using the linearity of Ws(z) and ignoring the constant terms, we can reformulate the seller's problem into a CM angle.
A seller who chooses to participate and intends to have a post-transfer balance of z must fetch z+Ts(z)b from CM and declare b. Thus, in order to induce sellers to participate in the mechanism, a satisfactory incentive-compatible allocation (q; d; zb; zs) is required. Here the LHS captures the payo¤ for participating in the mechanism and the RHS captures the payo¤ for skipping it.
By abuse of notation, define the share of buyer surplus in the …best case as U(q ) C0(q )q. The following proposition describes the feasibility of ... first best allocations under an optimally designed monetary mechanism. Proposition 16 In competitive pricing, there exists a monetary mechanism M that performs …first best if and only if.
On the other hand, if , we can construct a monetary mechanismM that implements the …first best. Proposition 17 Under competitive pricing, if a monetary mechanismM fTb(z); Ts(z); g implements …first best, then >1.
Electronic Money with Limited Participation
To get sellers to participate in the mechanism, it is necessary to have an incentive compatible allocation(q; d; nb; ns) satisfactory. The following proposition establishes the condition that the first best can be achieved by an optimal e-money mechanism with limited participation. Proposition 18 Under competitive pricing, there exists an e-money mechanismME that implements the …first best with limited participation if and only if.
First, we want to show that if < , then there is no mechanism for ME electronic money that implements the first best with limited participation. Suppose this is not the case, then there is an ME electronic money mechanism that implements a. The fact that (q; d; nb; ns) is incentive compatible for sellers and buyers implies that 0 and d =nb.
On the other hand, if , we can construct an e-money mechanism ME that implements the ... first best with limited participation. Then it is straightforward to verify that (q ; d ; nb;0) is incentive compatible for sellers under ME and that ME is self-financed with limited participation. Thus (q ; d ; nb; 0) is also incentive compatible for buyers under ME, and ME implements the ... first best with limited participation.
Proposition 19 Under competitive pricing, if there exists a money mechanism M that implements the … first best with, then there also exists an e-money mechanism ME that implements the … first best with limited participation under the same. So we have , and so from the previous proposition there exists an e-money mechanism ME that implements the ... first best. Proposition 20 If 2 ; , then under competitive pricing, … first-best allocation (i) cannot be implemented by any monetary mechanism;. ii) can be implemented through an e-money mechanism with limited participation among some.
Proposition 21 Given under competitive prices if an e-money mechanism with limited participation exists ME =fTb(n); Ts(n)g, but not any monetary mechanism M=fTb(z); Ts(z); g that implements the …first best, then Ts(n)>0 and Tb(n)<0.
Electronic Money with Limited Transferability
Proposition 22 Under competitive pricing, there exists some and an e-money mechanismML that implements the …first best with limited tradability if and only if either (a) or (b) ( + 1)U(q )> C(q. If (a) does not hold, then there exists ML implementing the …first best if and only if For the sake of brevity, we only show the last part that if <, then ML implements the …first best if and only if .
So (q; dz; dn; zb; zs; nb; ns) is an incentive compatible for buyers under ML, and therefore implements the ...first best. To state the "only if" part, let's assume there is ML out there that implements the ...first best for some. Under the < premise, the ...first term on the last line is negative, so the second term must be positive in order not to be smaller than the non-negative ...first line.
If there exists an e-money mechanism ME that implements the ... first best with limited participation, then there also exists an e-money mechanism ML that implements the ... first best with limited portability under the same. If there exists an e-money mechanismML that implements the ... first best with limited portability, then there also exists an e-money mechanism ME that implements the ... first best with limited participation under the same. Then, from the proof of the previous statement, there exists an e-money mechanism ML that implements the ... first best for the given.
Then by the previous proposition, there exists ME that also implements the … first best for the given. Proposition 25 Given, under competitive prices, . a) if there exists an e-money mechanism with limited portabilityML =f b; s; Tb(n); Ts(n)g, but not any e-money mechanism with limited participation ME, which implements the Coalition-Fixed Trade and the Friedman Government in the Lagos-Wright Model.” Journal of Political Economy.
Some Results on the Optimality and Implementation of the Friedman Rule in the Search Theory of Money." Journal of Economic Theory.
