url
9
Mar
2015

## DISPLACED CAPITAL: Theoretical Framework 5

Proposition 1 establishes conditions under which there will be two stages to the selling process. Consider Figure 1, which shows graphs of VO(n) and VA(n) against n. Note that time moves backwards as n increases, since higher n means that the firm has more units left to sell. The two conditions together guarantee that VO(n) and VA(n) cross, and that there is at least one crossing where VA(n) crosses VO(n) from below. If the conditions hold, the first stage of the selling process involves some search for sector A buyers, which we call the “private liquidation sale.” The second stage involves selling all remaining units at once to sector O. We call this stage the “public auction.” If it is the case that VA(n) is less than VO(n) for all n<N, then the firm will skip the first stage and only hold the public auction.

In general, the first stage need not involve only sales to sector A buyers. During the first stage, the firm is faced with the following tradeoff: Selling some units to outsiders now has the advantage of speeding up the time when the firm can realize revenues from the public auction, but has the disadvantage of decreasing the selection available to sector A buyers, hence reducing the expected revenue from those sales. When will the firm not sell units to outsiders during the second stage? The following lemma specifies a sufficient condition to ensure that the gains to speeding up the process by selling to outsiders early are outweighed by the lost revenue from selling to insiders.
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Lemma. If, in addition to the conditions in Proposition 1, the following condition holds:

then the sale will take place in two stages, with a first stage involving only search for sector A buyers and the second stage involving the sale of all remaining units at once to sector O.

Proof: See the technical appendix

While this corollary is not necessary for explaining many of the empirical results, it is useful for clarifying the effects of certain parameters on the nature of the sales process, as shown in the following proposition.

Then, the firm will sell more units to sector A,

(a) the higher is b

(b) the higher is 9

(c) the lower is RO

(d) the higher is fA

Proof: See the technical appendix.

Proposition 2 shows that as a firm becomes more patient, it will search longer for sector A buyers and delay the public auction. This result can explain how a firm’s discount factor determines how much capital it sells to industry insiders. The proposition also shows that the firm will search longer for sector A buyers if the efficiency of searching for sector A buyers is higher (higher 9) or if the marginal revenue product of capital in sector A is higher.

Proposition 1 establishes conditions under which there will be two stages to the selling process. Consider Figure 1, which shows graphs of VO(n) and VA(n) against n. Note that time moves backwards as n increases, since higher n means that the firm has more units left to sell. The two conditions together guarantee that VO(n) and VA(n) cross, and that there is at least one crossing where VA(n) crosses VO(n) from below. If the conditions hold, the first stage of the selling process involves some search for sector A buyers, which we call the “private liquidation sale.” The second stage involves selling all remaining units at once to sector O. We call this stage the “public auction.” If it is