Welfare analysis

One key contribution of this paper is to compare the welfare under different levels of information sharing. To achieve this, we firstly calculate welfares under a given level of information sharing. Define social welfare 𝑉𝑉 as the sum of the net value functions of all projects’ holders selling the projects and all investors financing the projects. Then, we can get the welfare of the four equilibriums as following:

181 1. Pooling equilibrium: 𝑉𝑉𝑝𝑝 ₌𝜆𝜆𝑅𝑅𝑔𝑔 + (1 − 𝜆𝜆)𝑅𝑅𝑏𝑏 𝑅𝑅 − 1 2. No-Trade equilibrium: 𝑉𝑉𝑛𝑛 _{= 0}

3. Pure valuation equilibrium:

𝑉𝑉𝑣𝑣 _{= 𝐻𝐻�(𝜆𝜆}∗ ₍𝑅𝑅𝑔𝑔 − 𝑅𝑅

𝑅𝑅 ) − 𝑐𝑐)

4. Mixed equilibrium:

𝑉𝑉𝑓𝑓 ₌ 𝜆𝜆𝑅𝑅𝑔𝑔 + (1 − 𝜆𝜆)𝑅𝑅𝑏𝑏

𝑅𝑅 − 1 − 𝑐𝑐𝐻𝐻�

The welfare of pooling equilibrium is the return when all projects are originated minus the origination cost 1. It is straightforward to see the welfare of no-trade equilibrium is 0 since no projects are financed. For pure valuation equilibrium, only sophisticated investors get positive profit which is the multiple of unit of valuation capacity used 𝐻𝐻� and the net profit per unit of valuation can gain is (𝜆𝜆∗ (𝑅𝑅𝑔𝑔−𝑅𝑅

𝑅𝑅 ) − 𝑐𝑐). For the mixed equilibrium, all projects are sold and

sophisticated investors used 𝐻𝐻� units of valuation capacity at a cost of 𝑐𝑐 . Therefore, the aggregate welfare is the return from all projects minus the origination cost 1 and valuation costs 𝑐𝑐𝐻𝐻�. By comparing the welfare for different regions, we can find consistent results with Fishman & Parker (2015). In the regions of multiple equilibriums, pooling equilibrium gives higher welfare. Even in region of pure equilibrium, the strategy of buying all projects still can generate higher welfare. Mixed equilibrium is always less efficient than buying all projects,

182 since in addition to buying all projects, valuation costs are used by sophisticated investors as a tool to drive down the alternative cost. The more valuations are done, the less good projects in pooling and thus less alternative cost offered by unsophisticated investors. Valuation is a social waste and both good projects and bad projects holders bear the cost. Pure valuation equilibrium is always less efficient than pooling equilibrium in multiplicity since the benefit of including unvalued good projects and saved valuation costs outweigh the cost of originate bad projects in this region. In region that pure valuation equilibrium dominates, the strategy of buying all

projects is more efficient as long as 𝑐𝑐 ≥ 𝑐𝑐𝑆𝑆𝑓𝑓𝑓𝑓 = (1−𝜆𝜆∗𝐻𝐻�)𝑅𝑅−(1−𝜆𝜆)𝑅𝑅𝑏𝑏−(𝜆𝜆−𝜆𝜆∗𝐻𝐻�)𝑅𝑅𝑔𝑔

𝐻𝐻�𝑅𝑅 , i.e. 𝑉𝑉𝑣𝑣 ≤ 𝑉𝑉𝑝𝑝. This

is illustrated as the region above the dotted line in figure 1.

The next step is comparing the welfares under different levels of information sharing. Without losing generality, we assume in the regions of multiple equilibria, equilibrium with valuation always dominates. Our result depends on whether there is an equilibrium shift. The cost constraints and the market quality constraints of equilibriums will change correspondingly with regard to information sharing and therefore shift the regions of all equilibriums. It is important to look back to table 2 and explore how those constraints change with regard to the information sharing. When 𝜎𝜎 increases, more information is shared. Sophisticated investors’ valuation accuracy improves given less wastes occur. i.e. 𝜆𝜆∗ increases. Given such improvement, the marginal benefit of using valuation would increase. Therefore, for equilibrium with valuation, the upper boundaries of valuation cost raise since they can afford higher valuation costs. No-trading equilibrium could shift to pure valuation equilibrium (region 4). Pooling equilibrium could shift to pure valuation equilibrium (region 1 and 2) or mixed equilibrium9 (region 3). With regard to market quality constraints, the boundary between

9 _{No trading equilibrium will never shift to a mixed equilibrium since lower boundary of mixed equilibrium’s}

market quality constraint is strictly higher than the upper boundary of no trading equilibrium’s market quality constraint.

183 pooling equilibrium and no trading equilibrium is independent of the information sharing. However, it will shift the boundary of market quality constraints between pure valuation equilibrium and mixed equilibrium towards left. The implication is that with more information sharing, the accuracy of valuation improves. Therefore, this leads to an overall increase in the size of good projects get valued and bought. Subsequently, the quality in unsophisticated market pool decreases. Unsophisticated investors may find the pooling projects are no longer profitable to originate and cease investments. This indicates a shift from mixed equilibrium to a pure equilibrium (region 7).

Notice that, only the welfare of pure valuation equilibrium depends on the information sharing. If there is no change of equilibrium type, increasing information sharing will only increase the welfare of pure equilibrium (region 5) since duplicated information acquisition is avoided. If there are changes in equilibrium type, there are four possibilities.

1. No-trade equilibrium to pure valuation equilibrium(region 4) 2. Pooling equilibrium to pure valuation equilibrium(region 1 and 2) 3. Mixed equilibrium to pure valuation equilibrium(region 7)

4. Pooling equilibrium to mixed equilibrium(region 3)

[Insert Figure 2]

Unlike the no-trade equilibrium where no surplus is created, in pure valuation equilibrium sophisticated investors gain positive surplus. Therefore, a shift from no-trade equilibrium to pure valuation equilibrium will increase social welfare. The comparison of welfare in pooling equilibrium and welfare in pure valuation equilibrium is identical to previous single

information sharing level case. For any valuation cost 𝑐𝑐 ≥ 𝑐𝑐𝑆𝑆𝑓𝑓𝑓𝑓 =(1−𝜆𝜆∗𝐻𝐻�)𝑅𝑅−(1−𝜆𝜆)𝑅𝑅𝑏𝑏−(𝜆𝜆−𝜆𝜆∗𝐻𝐻�)𝑅𝑅𝑔𝑔

184 the shift from pooling equilibrium to pure valuation equilibrium will decrease social welfare and vice versa. A shift from mixed equilibrium to pure valuation equilibrium will always increase the social welfare. And a shift from pooling equilibrium to mixed equilibrium will always decrease the social welfare since additional valuation costs occur. Figures 2 illustrates the overview for welfare changes comparing a partial information sharing regime and a complete information sharing regime. The green color indicates the regions have an increase in social welfare after information sharing shift from partial sharing to complete sharing. The red color indicates the regions have a decrease in social welfare after information sharing shift from partial sharing to complete sharing. The white regions are those social welfares stay unchanged. In summary, we will get Proposition 1 as following10_.

Proposition 1

Assume in the region of multiple equilibriums, equilibriums with valuation always dominate. When 𝑐𝑐 ≥ 𝑐𝑐𝑆𝑆𝑓𝑓𝑓𝑓, perfect information sharing will decrease the social welfare for regions switching from pooling equilibrium to equilibrium with valuation. The rest of valuation region will have an increase or keep unchanged in social welfare.

One noticeable result is that for those regions shift from pooling equilibrium to valuation regimes and 𝑐𝑐 ≥ 𝑐𝑐𝑆𝑆𝑓𝑓𝑓𝑓, the social welfare actually decreases. Unlike previous literature social inefficiency is caused by competition among sophisticated investors, here the social inefficiency is caused when sophisticated investors become interested in acquiring information at the cost of other market participants. This is because when information sharing improves, it creates the incentive for all sophisticated investors to do valuation. When valuation leads to the positive surplus for them, it reduces the overall social welfare since the cost of valuation and cost of passing away good projects due to limited valuation capacity exceeds the profit of

185 avoiding bad projects at aggregate level. However, the sophisticated investors only care if the profits from good projects are sufficient to cover the cost of valuation. They don't care passing away unvalued goods projects since they know the valuation capacity is limited at the first place.