Entry strategies of partnerships versus conventional firms.

Moretto, Michele ; Rossini, Gianpaolo

1. Introduction

According to the U.S. Census taxonomy, firms can be classified in two broad categories: nonemployer and employer.

"Nonemployers are businesses without paid employees. Most Nonemployers are self-employed individuals operating very small unincorporated businesses" (U.S. Census Bureau 2003).

The nonemployer category accounts for nearly three fourths of all businesses and contains enterprises of three distinct legal or organizational forms or both: individual proprietorship, partnership, and corporation, (1) all without employees. Among them, the most common are the first two.

Employers are enterprises that maximize profit and display separation between employees and owners. We may dub them conventional firms (as in Pencavel and Craig 1994) or, simply, firms.

Because nonemployers do not live, on average, longer than employers, (Taylor 1999, Parker 2004), we can proxy net entry between 1997 and 2001 in the United States using the number of establishments. Nonemployer net entry is more than twice that of employer. Between the U.S. Censuses (2) of 1997 and 2001, the number of nonemployers grew by 10% compared with 3% of employers. Moreover, nonemployer business is smaller (average receipt is $43,638 in 2002) than employer business ($3,872,141). (3) Last but not least, nonemployers are quite common in expanding sectors, such as services and advanced industries.

U.S. Census data show that the most dynamic and fastest growing group among the nonemployers is partnerships. (4) It is on this more popular and successful subcategory that we concentrate our study, the main aim of which is the comparison of entry strategies and sizes of conventional firms (Fs) and partnerships (PAs). Our interest in PAs is due to their diffusion in advanced industries and their apparent flexibility resulting from small dimension and swift entry policies.

We conduct our analysis by taking advantage of the similarities between the internal organization of a PA and that of a labor-managed enterprise, (5) in which owners and employees coincide and share the governance of the firm on an equal level, maximizing individual dividend.

Our contribution is cast within real option literature, which started with the seminal works of Brennan and Schwartz (1985) and McDonald and Siegel (1986). On the basis of the analogy between security options and the opportunities to invest in real assets, (6) these contributions underline the crucial role of investment timing when there are sunk costs and uncertainty over future rewards. Irreversibility and uncertainty induce entry only when the investment value exceeds that of the option to wait, once we apply the "bad news principle of irreversible investment" (Bernanke 1983).

In a dynamic setting, in which a new venture project is carried out at distinct times and at distinct entry-trigger market prices, most differences between the PA and the F are due to uncertainty and sunk costs. The PA enters at less favorable conditions than the F because the trigger price increases in peculiar fashions for the two enterprises as uncertainty unfolds. Higher risk makes the investment return more volatile, and the value of the entry option goes up, as well as the incentive to wait.

In a PA, each member shares the enterprise risk with colleagues and bears only a fraction of the corresponding cost. The consequence is a higher value of the investment option without any increase in the incentive to delay entry.

In Fs, the entire risk is borne by shareholders. Therefore, entry might occur later.

Our theoretical research on PAs and Fs provides fresh interpretations of two facts observed in U.S. data: (i) the smaller dimension of PAs, in terms of average receipts, and (ii) the recent growth of PAs during a period of intense financial volatility. (7)

The paper is organized as follows: In the next section, we set up the basic entry option model for the two types of enterprise. In the third and fourth sections, we find their stock market values. In section 5, we compare their different entry strategies. In section 6, we assess the effect of uncertainty on entry and optimal size. In section 7, we supplement the theoretical inquiry with a numerical example. Section 8 concludes.

2. A Start-Up Option

We first go through the entry strategies of the two firms that are supposed to own a startup option that allows them to begin producing a good and then sell in on the market. To this purpose, each firm has to bear a sunk cost, which is internally financed. Workers of F get the market unit wage w, which is the opportunity cost of joining the PA. Firms operate in an uncertain market environment. Decisions are taken on an infinite time horizon: in the PA by members, in the F by shareholders.

We begin by comparing entry policies and options. Each enterprise is supposed to own an infinitely lived investment project. We model entry with a set of common assumptions plus some specific hypotheses referring to each enterprise.

ASSUMPTION 1. The project, corresponding to the start-up decision, is of finite size and requires an exogenous investment I to be borne if the enterprise enters, by shareholders in F and by partners in PA.

ASSUMPTION 2. Once entered, the investment becomes irreversibly sunk. It can be neither changed, temporarily stopped, or shut down. (8)

ASSUMPTION 3. Once the project is implemented, the instantaneous short-run revenue of the project is

R([p.sub.t]; [L.sub.t]) [equivalent to] [p.sub.t]Q([L.sub.t]), (1)

where [p.sub.t] is the market output price, [L.sub.t] is labor, Q([L.sub.t]) is the short-run production function, with the usual properties: Q(0) = 0, Q'([L.sub.t]) > 0, Q" ([L.sub.t]) < 0, and L [member of] [[L.bar], [bar.L]].

ASSUMPTION 4. The uncertain market price evolves according to the following trendless stochastic differential equation:

[dp.sub.t] = [sigma][p.sub.t] [dB.sub.t] with [sigma] > 0 and [p.sub.0] = p, (2)

where [dB.sub.t] is the standard increment of a Wiener process (or Brownian motion), uncorrelated over time and satisfying the conditions that E([dB.sub.t]) = 0 and E([dB.sup.2.sub.t]) = dt (Dixit 1993). Therefore E([dp.sub.t]) = 0 and E([dp.sup.2.sub.t]) = [([sigma][[pi].sub.t).sup.2] dt, i.e., starting from the initial value[p.sub.0], the random position of the price [p.sub.t] at time t > 0 has a normal distribution with mean [p.sub.0] and variance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which increases as we look further and further into the future. The process "has no memory" (i.e., it is Markovian), and hence (i) at any time t, the observed [p.sub.t] is the best predictor of future prices and (ii) [p.sub.t] moves at any t + 1 upward or downward with equal probability. By the Markov property of the process [p.sub.t], the results do not change qualitatively assuming a positive (or negative) price trend.

ASSUMPTION 5. The market unitary wage w is constant.

ASSUMPTION 6. For the PA, the investment is set and financed by the founding members, as if there were a market for memberships operating according to standard financial canons (Sertel 1993, 1997). For the F, shareholders are involved.

ASSUMPTION 7. Members of the PA are homogeneous. They invest in the project and maximize the discounted value of expected individual net dividends. They receive an income that can be thought of as a kind of "supplemented wage," equal to dividends plus the opportunity cost of being a member (i.e., the market wage w).

ASSUMPTION 8. In Fs, the entrepreneur maximizes the discounted value of expected cash flows. In PAs, the objective is the individual discounted value of expected cash flows. This assumption is consistent with canonical modeling of profit-maximizing conventional firms and labor-managed enterprises (Bonin and Putterman 1987).

ASSUMPTION 9. Size (L), corresponding to the number of members for the PA and to labor force for the F, is set at entry and held fixed afterwards. As a matter of fact, new enterprises are usually small. It seems plausible to assume that, at their entrance, they choose the size of the labor force to hire and shun from adjusting it to variation of demand, preferring alternative ways that do not damage fresh internal organization.

3. The Value of a Partnership

If the market price of the product is high enough, PA enters setting the optimal size (L). The decision process requires a backward procedure. First, for any L, the value of the individual option to enter has to be computed. Subsequently, we have the choice of L that maximizes the individual option value at entry. The discounted value of expected net individual dividend is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where [E.sub.0](x) is the expectation operator, with the information available at time 0, [rho] is the riskless interest rate (9) and w/[rho] is the discounted flow of the market wage (i.e., the minimum that the PA grants its members). This salary corresponds to a participation constraint: Below it, members are better off supplying their labor in the market rather than founding a new PA.

Members of a PA of size L decide whether and when to start the new project by solving an optimal stopping time problem and choosing the investment timing, which maximizes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

By Assumption 7, PA associates are homogeneous. Each one holds an option to invest corresponding to Equation 4 and has an interest in exercising it cooperatively at the same time because members have just founded the firm of the optimal size, and they have no incentive to behave noncooperatively from the beginning.

PA associates wait up to time T, when [p.sub.t], starting from [p.sub.0], reaches an upper value, say [p.sub.PA]. Then, they invest. T is a random variable with distribution obtained from Equation 2. If we assume that [p.sub.PA] exists, taking expectation of Equation 4 and using the distribution of T, we are able to write the member's value function, before investing, as (Dixit and Pindyck 1994; Dixit, Pindyck, and Sodal 1999) (10)

[f.sub.PA] = (y([p.sub.PA]; l) - w / [rho]) (E.sub.0)[e.sup.[rho]T | [p.sub.0 = p]] = (y([p.sub.PA]; l) - w / [rho])[(p / [p.sub.PA]).sup.[beta]] for p < [p.sub.PA]. (5)

The member's option value, Equation 5, represents the expected net individual dividend of the project, that is, y([P.sub.PA]; L) - w/[rho], multiplied by the expected discount factor, [(p/[p.sub.PA]).sup.[beta]]. Then, the optimal investing rule implies that [f.sub.PA](p; L) > y(p; L) - w/[rho] for all p < [p.sub.PA]. By algebra, Equation 5 can be written as

[f.sub.PA](p; L) = Q(L) / L [[p.sub.PA] / [rho] - AC(L)] [(p / [p.sub.PA]).sup.[beta]]

for p < [p.sub.PA] and [p.sub.PA] [greater than or equal to] [rho]AC(L), (6)

where AC(L) [equivalent to] [(wL/[rho]) + I]/Q(L) is the long-run average total cost. AC(L) stands for the (deterministic) break-even rule implicit in the traditional accept/reject net present value (NPV) model; that is, entry occurs if the discounted cash flow generated by the project is weakly larger than the long-run average cost.

Therefore, the member option value has a simple economic interpretation: it is the NPV of the project evaluated at the time of entry divided by the number of associates; that is, [([p.sub.PA]/ [rho]) - AC(L)]Q(L)/L multiplied by the expected discount factor [(p/[p.sub.PA).sup.[beta]].

Furthermore, from Equation 6, the option value to invest of each partner goes to 0 in two extreme cases: (i) when the optimal price threshold [p.sub.PA] [equivalent to] [rho]AC(L) and (ii) when the optimal trigger price [p.sub.PA] [right arrow] [infinity]. In the latter case, the option vanishes because it is optimal to delay investment indefinitely (i.e., never enter). In the former case, the option value evaporates because of lack of flexibility: Each partner carries out the project if and only if p > [rho]AC(L).

4. The Value of a Conventional Firm

The procedure is the same as before. First, the entrepreneur's value of the option to invest for any given L is obtained. Subsequently, the labor force L is chosen at the optimal entry time. From Assumption 9, dictating that the conventional firm selects its project from a set of ventures with total cost (w/[rho])L + K, we know whether and when the project is ignited from the solutions of the following optimal stopping time problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where the market value of a project of dimension L is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Rather than solving Equation 7, simple direct inspection of Equations 7 and 4, plus the properties of the stopping time T, lead to

[F.sub.F](p; L) = [f.sub.PA](P; L)L, (8)

where [f.sub.PA](p; L) is the value of the project for the Lth associate of the PA, given by Equation 4. Then, by Equation 6, we get

[F.sub.F](p; L) = Q(L)[[p.sub.F] / [rho] - AC(L)][(p / [p.sub.F]).sup.[beta]] for p < [p.sub.F] and [p.sub.F] [greater than or equal to] [rho]AC(L), (9)

where [p.sub.F] is the optimal threshold that triggers the investment of F. The remarks made for Equation 6 extend to Equation 9: Also for F, the option value to invest goes to 0 when [p.sub.F] = [rho]AC(L) or when [p.sub.F] [right arrow] [infinity].

As before, the entrepreneur's value of the option to invest is the NP V of the project evaluated at entry, that is, [([p.sub.PA]/[rho]) - AC(L)]Q(L), multiplied by the expected discount factor [(p/[p.sub.PA).sup.[beta]].

5. Entry Strategies

Maximizing Equations 6 and 9 with respect to both [p.sub.PA] and [p.sub.F], we obtain the optimal entry policies. The optimal investment strategy for both firms requires investing as soon as the market price exceeds the break-even threshold:

[p.sub.i](L) [equivalent to] [beta] / [beta] - 1 [rho]AC(L) for I = PA, F. (10)

Because these break-even prices are given by [rho]AC(L) multiplied by [beta]/([beta] - 1) > 1, the entry rule (Eqn. 10) states that the enterprises should not invest before the value of the project has exceeded the long-run average cost by a definite magnitude. This is the fundamental result of irreversible investment under uncertainty: With new observations on market profitability obtained by waiting, enterprises reduce downside risk (Dixit and Pindyck 1994, p. 142).

Finally, substituting Equation 10 back into Equations 6 and 9 and maximizing with respect to L, we have:

LEMMA 1. The optimal entry size of the PA can be obtained from

[L.sub.PA]Q'([L.sub.PA]) / Q([L.sub.PA]) = 1 - ([beta] - 1) / [beta] I / [(w/[rho])[L.sub.PA] + I]; (11)

whereas, for the F, it comes from

[L.sub.F]Q'(L.sub.F) / Q(L.sub.F) = ([beta] - 1) / [beta] (1 - I / [(w/[rho])[L.sub.F] + I]). (12)

PROOF. See discussion and proof contained in the Appendix.

If an interior solution exists, we can compare the entry strategies of F and PA, setting first the optimal dimension at entry and then the entry trigger price. On the basis of Lemma 1, we can show the following:

PROPOSITION 1. (a) Over the range that the second-order condition holds, F is operating with a larger dimension than PA; that is,

[L.sub.PA] < [??] < [L.sub.F],

where [??] = arg min AC(L) is the minimum deterministic efficient scale, equal for both PA and F. (b) the entry trigger prices of PA and F react in different ways; that is,

[partial derivative][p.sub.F] / [partial derivative][L.sub.F] > 0 [partial derivative][p.sub.PA] / [partial derivative][L.sub.PA] < 0.

PROOF. See the Appendix.

To appreciate the intuition behind this result, we go back to Lemma 1, rewriting the first-order conditions for the optimal dimension at entry, Equations 11 and 12. By multiplying both sides of Equation 12 by [p.sub.F]([L.sub.F]) and simplifying, we get

[p.sub.F]([L.sub.F])Q'([L.sub.F]) = w. (13)

Then F, at entry, decides the optimal dimension equating the nominal marginal product to the market wage w. Similarly, we obtain

[p.sub.PA]([L.sub.PA])Q'([L.sub.PA]) = w + [f.sub.PA]([L.sub.PA]) > W, (14)

where [f.sub.PA]([L.sub.PA]) [equivalent to] [1/([beta] - 1)][w + [rho](I/[L.sub.PA])]. Unlike F, PA chooses the optimal size equating the nominal marginal product to the "supplemented wage," which exceeds the market wage w, even if PAs do not usually pay wage to their associates, since the time they spend working in the PA must satisfy a participation constraint given by the unit market wage. The full cost of the investment imputed to each member is w + [f.sub.PA]([L.sub.PA]), larger than w since each PA member holds an equal option to delay entry. This option is not owned by employees of F since it is in the hands of the entrepreneur, managing the enterprise on behalf of shareholders. Would-be associates are workers endowed with an option to build a partnership. This ability to give rise to a new venture is embodied in the option and makes for an individual income larger than w. By the decreasing marginal product of labor, PA will have a smaller size at entry than its twin mate F (i.e., [L.sub.PA] < [L.sub.F]). This is consistent with the empirical finding that PA is on average smaller than corresponding F. This is also the case of labor-managed enterprises, which are "... smaller than their capitalist counterparts in the short-run when profits are positive" (Bonin and Putterman 1987, p. 15). The same applies to the long run if profits are strictly positive (p. 57).

The conclusion that PA and F have different dimensions opens the question concerning entry price reactions as size changes.

6. The Effects of Uncertainty on Entry

PA and F enter when the market price rises above the average total cost A C(L) [equivalent to] [(wL/[rho]) + I]/Q(L) multiplied by a coefficient [[beta]/([beta] - 1)][rho]. However, we do not know the reactions to uncertainty of the two enterprises. We fill this gap by going through some comparative statics. First, we see whether Proposition 1 holds when uncertainty disappears. We can show that

PROPOSITION 2. If [sigma] = 0, the optimal size at entry for both F and PA is the minimum efficient scale; that is,

[L.sub.PA = [??] = [L.sub.F],

with coincident entry strategies

[p.sub.F]([??]) = [p.sub.PA]([??]).

PROOF. Straightforward.

From Equations 6 and 9, we realize that certainty leads to zero profit. As with competition, all rents are dissipated, and the option value to delay goes to zero (i.e., [f.sub.PA] = 0). Both enterprises would like to enter at the minimum of the U-shaped average cost curve, where the equilibrium individual income in PA is equal to the competitive wage paid by F.

Uncertainty destroys this symmetry. Both enterprises require positive expected profits before committing to an irreversible investment. If, at the time of entry, V(p; L) - I is positive, the discounted value of expected net individual dividend y(p; L) exceeds w because members pocket the rents. Because the dimension of the project is fixed, PA will be more "capital-intensive" than F (i.e., [L.sub.PA] < [L.sub.F]), whose cost of labor is w (Bonin and Putterman 1987; Delbono and Rossini 1992).

Consider now the effect of an increase in uncertainty:

PROPOSITION 3. AS market price volatility grows, the entry price goes up for both enterprises:

[partial derivative][p.sub.PA] / [partial derivative][sigma] > 0 and [partial derivative][p.sub.F] / [partial derivative][sigma] > 0,

and the size difference widens; that is,

[partial derivative] ([L.sub.F] - [L.sub.PA]) / [partial derivative][sigma] > 0.

PROOF. See the Appendix.

As the Real Option Theory predicts, increasing risk puts off investment timing; that is, the entry price increases with uncertainty because of the "bad news principle of irreversible investment" (Bernanke 1983). Higher market risk drives up volatility of the investment return, with positive effects on the option to invest. However, the net marginal benefit of waiting, arising from shunning investment in the bad state, increases with uncertainty. This induces an entry delay.

As uncertainty soars, F gets larger and PA gets smaller. The higher entry price makes F react by increasing the optimal size so as to keep the nominal marginal product in line with the market wage. On the contrary, for PA, the "supplemented wage" imputed to each member goes up with [sigma] (down with [beta]), and the enterprise downsizes to adjust the nominal marginal product.

According to Proposition 3, uncertainty makes the two enterprises delay entry.

Quite interesting is the investigation of entry prices, even though we are able to provide ranking in terms of entry prices only locally. When [sigma] [right arrow] [infinity], both [p.sub.PA] and [p.sub.F] tend to infinity: PA and F look alike because it is optimal to delay investment indefinitely. However, by looking at entry prices for low volatility, we see that an enterprise invests before the other and shows different "riskiness," since the PA set of entry prices is "less convex" than that of F. That is,

PROPOSITION 4. Starting from low price volatility, the entry trigger of PA is lower than the entry trigger of F. Therefore, we can say that PA is in general, less affected by risk than F.

PROOF. See the Appendix.

The entry boundary increases in different manners for PA and F. Because the members of PA equally share the option to invest, they might demand a higher reward and require a smaller price to compensate for the increased risk, This lowers the net marginal benefit of waiting of each individual associate, reducing the incentive to delay entry.

7. A Numerical Example

A numerical example might better illustrate the relationship between entry trigger prices, optimal dimension, and market volatility.

We adopt a standard Cobb-Douglas technology: Q(L) = [lambda][L.sup.[alpha]] with [alpha] [member of] (0, 1] and [lambda] [member of] (0, [infinity]).

Our choice of parameter values follows numerical examples provided by field literature (Dixit and Pindyck 1994; Pastor and Veronesi 2004; Smit and Trigeorgis 2004): p = 0.08, [lambda] = 1, [alpha] = 0.5, w = 0.2. In two cases selected according to the level of the investment (I = 50, I = 100), we investigate how trigger price and size (11) of the two enterprises change as uncertainty unfolds. For each case, we deal with three levels of uncertainty.

First Case: I = 50

(a) Low uncertainty ([sigma] = 0.01, [beta] = 40.50): PA enters at price 1.78 and size 20, Fat price 1.78 and size 22.

(b) Medium uncertainty ([sigma] = 0.08, [beta] = 5.52): PA enters at price 2.17 and size 13, F at price 2.18 and size 33.

(c) High uncertainty ([sigma] = 0.25, [beta] = 2.18): PA enters at price 5.60 and size 2, F at price 6.68 and size 326.

Second Case: I = 100

(a) Low uncertainty ([sigma] = 0.01, [beta] = 40.50): PA enters at price 2.50 and size 40, F at price 2.50 and size 44.

(b) Medium uncertainty ([sigma] = 0.08, [beta] = 5.52): PA enters at price 3.05 and size 27, F at price 3.06 and size 66.

(c) High uncertainty ([sigma] = 0.25, [beta] = 2.18): PA enters at price 7.87 and size 4, F at price 9.39 and size 652.

From the numerical example, it appears that the PA enters always at a trigger that is weakly smaller than that of F (i.e., at less favorable market conditions and smaller size).

8. Conclusions

We have gone through the comparative investigation of size and entry of partnerships and conventional firms. The analysis has been stimulated by the observation of a larger number of net entries of nonemployer enterprises--mainly partnerships--vis a vis conventional firms (employer) during a period of high financial volatility in the United States. The empirical observation is quite reliable because the United States is the country with the lowest amount of institutional (fiscal, financial, administrative, etc.) asymmetries between the two types of enterprise.

Four propositions show that the partnership is a more suitable entrepreneurial organization in times of high volatility, such as the 1997-2001 period. It enters at a lower market price and a smaller size. This is consistent with the statement that volatility boosts the value of an enterprise even if there is no bubble, as shown in Pastor and Veronesi (2004, 2005), who explain the stock exchange growth between 1997 and 2001 with increasing financial and real uncertainty brought about by the information technology revolution.

Our results might explain (i) why so many partnerships entered during a period of high volatility, such as the years between 1997 and 2001 in the United States; and (ii) the smaller operation scale of partnerships.

The divergence between the two entry policies is due to the irreversible commitment under uncertainty and the different internal organization of the two types of enterprise.

Partnership members hold an option to enter on the basis of their ability to set up a fresh venture. The option value increases with market volatility and the size of the required irreversible commitment. The value of this option adds to the market wage, making the total income received by members higher with respect to the conventional firm, even in the long run.

The associates of the partnership equally share the option to invest. By demanding a higher reward and requiring a smaller size to compensate for the increased risk, they lower the net marginal benefit of waiting, reducing the incentive to delay entry. Then, the partnership turns out to be more suitable than the conventional firm for periods of high volatility. This is due to a lower convexity of the entry price set of the partnership or, in simpler words, to a less fearful attitude to risk.

Possible avenues for future research should consider the opportunity to vary the size of the investment in a two-factor technology and the possibility of exit.

Appendix

PROOF OF LEMMA 1. Although the optimal triggers (Eqn. 10) look alike, they are not because at entry, the two enterprises have different sizes. As a proof, consider first the PA. Substituting Equation 10 into Equation 6 and rearranging, we write the Lth member's value of the project prior to investing

[f.sub.PA](P; L) = A(L)[p.sup.[beta]] for p < [p.sub.PA](L), (A1)

where the constant A(L) is given by

A(L) [equivalent to] [([beta] - 1).sup.[beta] - 1] / [([rho][beta]).sup.[beta]] [AC(L).sup.-[beta]] [(w/[rho])L + I] / L > 0 (A2)

By Equation A1, the optimal dimension of PA requires choosing L for which A(L) is the largest. This is equivalent to maximizing

a(L) [equivalent to] [AC(L).sup.-[beta]][(w/[rho])L + I] / L.

The first order condition (Eqn. 11) reported in the text follows by deriving In a(L) with respect to L:

Q'(L) / Q'(L) - ([beta] - 1) / [beta] (w/[rho]) / [(w/[rho])L + I] - 1 / [beta]L = 0.

Multiplying both sides by L and recalling that [(w/[rho])L]/[(w/[rho])L + I] [equivalent to] 1 - [[eta].sub.QL], where [[eta].sub.QL] [equivalent to] I/[(w/[rho])L + 1], we get

LQ'(L) / Q(L) - [1 - ([beta] - 1) / [beta] [[eta].sub.QL]] = 0. (A3)

Because ([beta] - 1)/[beta] < 1 and [[eta].sub.QL] < 1, a necessary condition for an interior solution requires that [[epsilon].sub.QL] [equivalent to] LQ'(L)/Q(L) < 1; that is, the average productivity Q(L)IL must be a decreasing function of labor, as from Assumption 3. The second-order condition for an interior solution is

Q"(L)Q(L) - Q'[(L).sup.2] / Q[(L).sup.2] + ([beta] - 1) / [beta] [(w/[rho]).sup.2] / [[(w/[rho])L + I].sup.2] + 1 / [beta][L.sup.2] < 0.

Rearranging and making use of Equation A3, we get the local condition

Q"(L)[L.sup.2] / Q(L) + (1 - [[epsilon].sub.QL])([[epsilon].sub.QL] + [[eta].sub.QL]) [equivalent to] [[epsilon].sub.QL]([d[epsilon].sub.QL] / dL L / [[epsilon].sub.QL] - [[eta].sub.QL]) + [[eta].sub.QL] < 0,

where

[d[epsilon].sub.QL] / dL = 1 / L [Q"(L)[L.sup.2] / Q(L) + (1 - [[epsilon].sub.QL])[[epsilon].sub.QL]].

A necessary, yet nonsufficient, requirement for the second-order condition entails the output elasticity to decrease in L. If this is not the case, the optimum comes from a binary comparison between the smallest dimension [L.bar] and the largest one [bar.L].

For the second part of Lemma 1, substituting Equation 10 into Equation 9 and rearranging, we get the shareholders value as

[F.sub.F](p; L) = B(L)[p.sup.[beta]] for p < [p.sub.F](L), (A4)

where the constant B(L) = LA(L). By Equation A4, optimality requires finding L that maximizes B(L), which, by Equation A2, is equivalent to maximizing La(L). In particular, the first order condition (Eqn. 12) in the text follows by deriving In La(L) with respect to L:

Q'(L) / Q(L) - ([beta] - 1) / [beta] w/[rho] / [(w/[rho])L + I] = 0.

Multiplying both sides by L and recalling [[eta].sub.QL] we get

LQ'(L) / Q(L) - ([beta] - 1) / [beta] [[eta].sub.QL]= 0. (A5)

Again, a necessary condition for an interior solution requires that [[epsilon].sub.QL] [equivalent to] LQ'(L)/Q(L) < 1, and the second-order condition is

Q"(L)Q(L) - Q'[(L).sup.2] / Q[(L).sup.2] + ([beta] - 1) / [beta] [(w/[rho]).sup.2] / [[(w/[rho])L + I)].sup.2] < 0.

Rearranging and making use of Equation A5, we get the local condition

[d[epsilon].sub.QL] / dL L + [[epsilon].sub.QL] (2[beta] - 1 / [beta] - 1 [[epsilon].sub.QL] - 3) + ([beta] - 1) / [beta] < 0.

As we have seen for PA, because the right-hand side of Equation 12 is <1, a necessary condition is a production elasticity [[epsilon].sub.QL] < 1; whereas, d[[epsilon].sub.QL]/dL < 0 is necessary but not sufficient to secure that the optimal dimension lies within the range [[L.bar], [bar.L]]. If entry costs are null, the condition in Equation 12 reduces to LQ'(L)/Q(L) = ([[beta].sub.1] - 1)/[[beta].sub.1] < 1, equivalent to the condition proposed by Dixit (1993) for an F choosing among investment projects of different dimensions. In Moretto (2003), there is an analogous condition for an F that incrementally reduces capacity.

PROOF OF PROPOSITION 1. First part of the proposition, define b(L) [equivalent to] La(L). We know from Lemma 1 that the F optimal size is given by

b'(L) = a(L) + La'(L) = 0,

and the second-order condition is

b"(L) = 2a'(L) + La"(L) < 0.

In general, a"(L) < 0 does not imply that b"(L) < 0: The two regions, where the second-order condition holds, overlap only partially over the range where the second-order condition holds a'([L.sub.PA]) = 0. Therefore, b'([L.sub.PA]) = a([L.sub.PA)] > 0. If an LF exists such that b'([L.sub.F]) = 0, this will necessarily be

[L.sub.PA] < [L.sub.F].

For the second part of the proposition, define the average cost function AC(L) [equivalent to] [(w/[rho])L + I]/Q(L). By the concavity of Q(L), it is easy to show that [lim.sub.L [right arrow] 0]AC(L) = +[infinity] and [lim.sub.L [right arrow] +[infinity] AC(L) = +[infinity]. By taking the derivative with respect to L, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A6)

Then, a value [??] > 0 exists such that [partial derivative]AC/[partial derivative]L = 0, and it is given by

[??]Q'([??]) / Q([??]) = (1 - I / [(w/[rho])[??] + I]). (A7)

The second order condition confirms that AC(L) is a convex function with a minimum represented by [??].

Because ([beta] - 1)/[beta] < 1, by comparing Equations A7 and 12, we have

([beta] - 1) / [beta] (1 - I / [(w/[rho])L + I]) < 1 - I / [(w/[rho])L + I],

which, in the range where the second-order condition holds, implies that [??] < [L.sub.F]. On the contrary, by comparing Equations A7 and 11, we notice that PA operates only in the descending branch of the average cost curve to the left of the minimum. That is

1 - ([beta] - 1) / [beta] I / [(w/[rho])L + I] > 1 - I / [(w/[rho])L + I], or 1 / [beta] I / [(w/[rho])L + I] > 0,

which implies that [??] > [L.sub.PA].

PROOF OF PROPOSITION 3. Applying the implicit function theorem to Equations 12 and 11, it can be shown that [partial derivative][L.sub.F]/[partial derivative][beta] [less than or equal to] 0 [less than or equal to] [partial derivative][L.sub.PA]/[partial derivative][beta]. Then, since [partial derivative][beta]/[partial derivative][sigma] < 0, ([beta] - 1)/[beta] decreases, and the opposite effect on optimal dimension follows. Moreover, totally differentiating Equation 10 for the two enterprises yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A9)

By the above result and Equation A6, it is easy to ascertain the positivity of both. In particular, if [sigma] [right arrow] [infinity], we have [beta] [right arrow] 1 and ([beta] - 1)/[beta] [right arrow] 0: Neither type of enterprise enters.

PROOF OF PROPOSITION 4. The slope of the entry price at [sigma] = 0 can be found by evaluating Equations A8 and A9 at [L.sub.F] = [L.sub.PA] = [??] Because AC'([??]) = 0, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then, both enterprises have the same slope of the entry price at [sigma] = 0. Differentiating Equations A8 and A9 once more with respect to [sigma] and evaluating the result at zero yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because [partial derivative][L.sub.F]/[partial derivative][[sigma].sub.|[sigma]=0] > 0 and [([partial derivative][L.sub.PA]/[partial derivative][sigma]).sub.|[sigma]=0] < 0, we conclude that [([[partial derivative].sup.2][p.sub.F]/[partial derivative][[sigma].sup.2]).sub.|[sigma]=0] > [([[partial derivative].sup.2][p.sub.PA]/[partial derivative][[sigma].sup.2]).sub.|[sigma]=0.

We acknowledge the financial support of the Universities of Brescia, Bologna, and Padova under the 60% scheme for the academic years 2004-2006. The paper has been presented at the 2005 International Industrial Organization Conference (IIOC) in Atlanta, Georgia, and at the 2005 European Association for Research in Industrial Economics (EARIE) Conference in Oporto, Portugal. We thank the editor and two anonymous referees for their comments and suggestions while keeping any responsibility for the content of the paper.

Received January 2005; accepted July 2007.

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(1) Proprietorships are very small and quite close to the self-employed category of the European nomenclature. See for instance Steingold (1999) and Parker, Barmby, and Belghitar (2005). Here are the U.S. Bureau of Census definitions: "Individual proprietorship ... is an unincorporated business owned by an individual." Self-employed persons are included in this category. "Partnership ... is an unincorporated business owned by two or more persons having a shared financial interest in the business" (i.e., sharing profits and losses and responsibilities having a general or limited liability). "A Nonemployer Corporation is a legally incorporated business under state laws," without employees. We follow the U.S. Bureau of Census nomenclature, whose definitions are available at http://www.census.gov/econ/ census02/text/sector00/00aidesc.htm (accessed February 5, 2008).

(2) Evidence presented is confined to the United States, where better data is coupled with the mildest asymmetries between the two kinds of enterprise with respect to fiscal, financial, and administrative entry barriers and other institutional aspects (OECD 2000, 2006).

(3) Data accessed July 9, 2007. Available http://www.census.gov/epcd/www/smallbus.html#RcptSize.

(4) Over the period 1997-2001, establishments of partnerships increased by 26%. This is the largest rate of growth among all categories belonging to the nonemployer group. Average receipt of partnerships in 2001 was $123.000, more than the overall figure for nonemployers, but still less than that of employers. Receipts of partnerships increased over the same time span by 39%.

(5) There are long-run affinities and short-run beterogeneities between a competitive labor-managed enterprise and the corresponding conventional firm. Quite remarkable is the "perverse" response of the short-run supply function of the labor-managed enterprise (Ward 1958, Vanek 1970, Pestieau and Thisse 1979, Delbono and Rossini 1992, Pencavel and Craig 1994), which vanishes with tradeability of memberships (i.e., in workers' enterprises; Sertel 1993, 1997).

(6) Excellent surveys of mainstream theory are in Pindyck (1991), Dixit (1992), Dixit and Pindyck (1994), Trigeorgis (1996), and Smit and Trigeorgis (2004). A recent contribution on an affine topic using real option theory is in Sodal (2006).

(7) The intense financial volatility (i.e., the high degree of variability of stock prices), of this period in the U.S. economy is widely documented in Pastor and Veronesi [2004, 2005]. The high financial variability is mirrored by the high variability of the relative prices of goods and services sold by innovative firms.

(8) This avoids the analysis of operating options differing across the two kinds of enterprise. The most relevant option corresponds to the ability to reduce output and to shut down. Operating options increase the value of the enterprise. See McDonald and Siegel (1986) and, for a thorough discussion, Dixit and Pindyck (1994, chs. 6, 7).

(9) Introducing risk aversion does not change the results because the analysis can be developed under a risk-neutral probability measure (Cox and Ross 1976; Harrison and Kreps 1979).

(10) The solution to [E.sub.0][[e.sup.-[rho]T]] can be obtained via the usual dynamic programming decomposition (Dixit, Pindyck, and Sodal 1999, p. 184). Because the process p' is continuous, the expected discount factor is increasing in p and decreasing in [p.sub.PA]; then, it can be defined by a function D(p; [p.sub.PA]). Over the infinitesimal time interval dt, p will change by the small value dp; hence, we get the following Bellman equation: [rho]D(p; [p.sub.PA]) dt = E[dD(p; [p.sub.PA])]. By applying Ito's Lemma to dD we obtain the following differential equation: (1/2)[[sigma].sup.2][p.sup.2]D" - [rho]D = 0. We solve it, subject to the two boundary conditions [lim.sub.p [right arrow] [infinity]] D(p; [p.sub.PA]) = 0, [lim.sub.p [right arrow] p] D(p; [p.sub.PA]) = 1, and we get D(p; [p.sub.PA]) = [(p/[p.sub.PA]).sup.[beta]], where 1 < [beta] < [infinity] is the positive root of the auxiliary quadratic equation: [PSI]([beta]) = (1/2)[[sigma].sup.2][beta]([beta] - 1) - [rho] = 0.

(11) The values used in the example are natural numbers, since size refers, respectively, to the number of partners of PA and employees of F. Numbers are approximations because we do not use integer programming.

Michele Moretto * and Gianpaolo Rossini ([dagger])

* Dipartimento di Scienze Economicbe, University of Padova, via del Santo, 33, Padova, Italy, E-mail michele.moretto@unipd.it.

([dagger]) Dipartimento di Scienze Economicbe, University of Bologna, Strada Maggiore, 45, Bologna, Italy; E-mail gianpaolo.rossini@unibo.it; corresponding author.