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8
Mar
2015

## DISPLACED CAPITAL: Theoretical Framework 4

Despite the higher costs of selling within the sector, the firm may choose to sell to other firms within the sector if the expected returns are large enough. Once the match is achieved, we make the common simplifying assumption that the selling price is equal to the valuation of the buyer. The valuations of the buyers, RA for insiders and RO for outsiders, are given as follows:

The f ’s are marginal revenue product shifters for each of the industries. For example, a decline in the demand for aerospace goods would be represented as a decline in fA. The S functions denote the goodness of the match of the capital’s characteristics, and are intended to capture the specificity of capital discussed above.8 Even within the same industry, capital from one firm may not be a perfect match for another firm. We assume that the value of a match with another industry insider rises with the selection currently available for sale, n. As shown above in equation (3), we also assume that it is always the case that the characteristics of the capital will be better suited to industry insiders than industry outsiders. If, however, fA falls far enough relative to fO then Ra can fall below RO.

We begin the analysis by deriving value functions using backward induction from the last unit sold. When the firm has only one unit left for sale, it can decide to draw from the insider pool or to sell to outsiders. The value function evaluated at the last unit left to sell (n=1) is given by:

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The first argument in the brackets is the expected value of taking a draw from inside the industry. The firm receives RA(1) with probability 9, and must pay a cost to search of C. There is a probability 1 – 9 of not making a match, and receiving the discounted value of having one unit left for sale the following period. If RO is greater than the first argument in the expression, the firm will decide to sell to outsiders. Let that value be given by:

If Ra(1) is sufficiently small that VA(1) < VO(1), the firm decides to sell its last unit to outsiders. Iterating backward from the last unit, we obtain the value of each choice when a firm is left with n units to sell, and finds it optimal to sell the last n-1 units to outsiders. These values are given by:

The following proposition establishes conditions under which the firm will sell some equipment to each group of buyers:

and the optimal policy is to sell at least one unit to sector A before selling all remaining units to outsiders.

Proof: See the technical appendix.

Despite the higher costs of selling within the sector, the firm may choose to sell to other firms within the sector if the expected returns are large enough. Once the match is achieved, we make the common simplifying assumption that the selling price is equal to the valuation of the buyer. The valuations of the buyers, RA for insiders and RO for outsiders, are given as follows: The f ’s are marginal revenue product shifters for each of the industries. For example, a decline in the demand for aerospace goods would be represented as a decline in fA. The S functions denote the goodness of the match of the capital’s characteristics, and are intended to capture the specificity of capital