Entrepreneurship and asymmetric information in input markets

Int Tax Public Finance (2011) 18: 166–192DOI 10.1007/s10797-010-9154-8

Entrepreneurship and asymmetric information in inputmarkets

Robin Boadway · Motohiro Sato

Published online: 3 September 2010© Springer Science+Business Media, LLC 2010

Abstract Entrepreneurs starting new firms face two sorts of asymmetric informationproblems. Information about the quality of new investments may be private, leadingto adverse selection in credit markets, and entrepreneurs may not observe the qualityof workers applying for jobs, resulting in adverse selection in labor markets. We con-struct a simple model to illustrate some consequences of new firms facing both sortsof asymmetric information. The market equilibrium can involve an excess supply ofworkers entering the entrepreneurial sector, as well as credit rationing. Equilibriumoutcomes mismatch workers to firms, and will generally result in an inefficient num-ber of both entrepreneurs and workers opting for the entrepreneurial sector. Taxes orsubsidies on new firms and on wages can improve efficiency, but a second-best opti-mum can only be achieved if it is optimal to induce an excess supply of workers toenter the entrepreneurial sector.

Keywords Entrepreneurship · Asymmetric information · Adverse selection

JEL Classification D82 · G14 · H25

1 Introduction

New firms and the entrepreneurs that initiate them are beset by problems of asym-metric information with respect to their prospects for success, as well as with respectto the quality of labor they are able to hire and their ability to obtain credit on good

R. Boadway (�)Queen’s University, Kingston, Canadae-mail: [email protected]

M. SatoHitotsubashi University, Tokyo, Japan

mailto:[email protected]

Entrepreneurship and asymmetric information in input markets 167

terms. The fact that new entrepreneurs are to a large extent indistinguishable fromone another means that creditors are unable to tailor financial terms to entrepreneurs’project qualities. As well, since they are hiring workers for the first time, they do nothave the experience to discern the quality of potential workers and whether they willbe a good match for the particular projects being initiated. This puts new firms ata disadvantage with respect to existing firms whose track records have been proven,who may have internal finance, and who have had a chance to sort out good or suitableworkers from bad or unsuitable ones. These problems naturally lead to the questionof whether public policies should actively encourage the entry of new entrepreneurialfirms. That is the focus of this paper.

The literature has recognized in a piecemeal way some of the problems that newfirms face due to asymmetric information, and has come to some surprising results.The seminal paper of Stiglitz and Weiss (1981) studied the adverse selection problemsthat arise when banks are unable to distinguish high- from low-quality projects andmust offer the same financial terms to all. In their case, the expected return couldbe observed, but not the riskiness of individual projects of a given expected return.In this setting, too few projects would be financed—those with the highest risk—suggesting a subsidy on the financing of new firms. Moreover, the possibility of creditrationing existed which exacerbated the underinvestment. Subsequently, de Meza andWebb (1987) considered the perhaps more realistic case in which banks knew projectreturns but not their probability of success. In this setting, the findings of Stiglitzand Weiss were reversed: there would be overinvestment in low-probability projectsand no possibility of credit rationing, leading to a presumption of taxing new firms.These results have been generalized by Boadway and Keen (2006) to allow for moregeneral patterns of project characteristics in the pool, and to allow for alternativeforms of finance. What emerges is a general presumption of overinvestment as low-risk investments opt for debt finance and high-risk ones for equity finance.

Asymmetric information has also been the focus of the venture capital (VC) liter-ature where the financing of new entrepreneurs is combined with managerial advice.Here, the emphasis has been more on moral hazard problems associated with the ef-fort of both the VC and the entrepreneur. Keuschnigg and Neilsen (2003) have arguedthat these moral hazard problems can be addressed by a tax on new firms combinedwith a preferential capital gains tax. Dietz (2002) has added adverse selection to theVC problem and allowed entrepreneurs to choose between VC financing (with man-agerial advice) and bank financing. He finds that high-risk projects choose the formerand low-risk projects the latter, but that too many low-risk projects end up beingfinanced by VCs.

There has been a limited amount of attention paid to the consequences for newfirms of imperfect information on other markets. Weiss (1980) considers the case ofadverse selection in labor markets when firms cannot observe the quality of workersthey hire. He shows that workers will tend to be drawn from the bottom of the skilldistribution—since a common wage is paid regardless of quality—and too few willbe hired. Moreover, there is a possibility of excess labor supply, or job queues, inequilibrium. A subsidy on employment would be welfare-improving in this context.The Weiss model focuses entirely on adverse selection in labor markets: there is noheterogeneity of entrepreneurs and no uncertainty of project success. Presumably

168 R. Boadway, M. Sato

other forms of asymmetric information in labor markets would particularly affectnew firms as well, such as unobservable effort (Shapiro and Stiglitz 1984) or searchproblems (Diamond 1982).

There are potentially many other ways in which the entry of new firms is renderedinefficient because of asymmetric information or externalities. Some of these includesignaling problems, knowledge externalities, and strategic barriers to entry. The po-tential consequences of various sources of inefficiency for tax policy toward newfirms are surveyed in broad terms in Boadway and Tremblay (2005). Rosen (2005)reviews the empirical effects of existing policies on entrepreneurship in the UnitedStates.

Our purpose is to study how adverse selection in both credit markets and labormarkets affects equilibrium and efficiency in the formation of new firms, and to con-sider the consequences for policy. To do so, we develop a simple but rather specificmodel designed to capture the main features of information asymmetry facing newfirms, while at the same time avoiding needless complications. This asymmetry istwo-sided: potential entrepreneurs do not know the quality of individual workers,and workers do not know the quality, or ability, of new entrepreneurs. The banks andthe government know neither.

The model we use builds on that of de Meza and Webb (1987) to study adverseselection in credit markets, but with the addition of an employment dimension. Inour basic model, entrepreneurs of varying ability must decide whether to undertakea risky project. Those that do each hire a fixed number of workers who are of vary-ing quality. If successful, an entrepreneur’s firm produces a given output, where theprobability of success depends on the ability of the entrepreneur and the size of out-put depends on the quality of workers hired. Entrepreneurs have no initial wealth,so must rely on credit to finance their operations. Equilibrium involves the highest-ability entrepreneurs choosing to undertake projects and hiring workers randomlyfrom the set of lowest-quality workers. There are several sorts of inefficiencies inthis context. Workers of different qualities will be mismatched with entrepreneurs ofdifferent ability. Labor markets or credit markets might not clear. In the basic model,there may be an excess number of workers entering the entrepreneurial sector, whilein some extensions there may be credit rationing. There will generally be an ineffi-cient number of entrepreneurs and workers entering the entrepreneurial sector. Thiswill lead to efficiency-enhancing policy intervention.

The basic model is outlined in the following section, and the full-information equi-librium in Sect. 3. This is followed by the analysis of equilibrium when the qualityof workers and ability of entrepreneurs are private. In Sect. 5, we investigate the ef-ficiency of market outcomes under asymmetric information and the implications forpolicy. The following section discusses two extensions to the basic model, one inwhich the number of workers hired by each firm is variable and the other in whichthe quality of workers affects the probability of success. Each of these cases increasesthe policy demands on the government. A concluding section discusses some exten-sions.


2 Elements of the basic model

The model we use has several specific features. They are chosen to highlight thekinds of issues that can arise when there is asymmetric information in labor andcredit markets, while at the same time avoiding complications that lead to excessivelycomplex analysis and obscure the phenomena we are trying to illustrate. Many of oursimplifications parallel those found in the literature on adverse selection in creditmarkets, such as the simple structure of project returns and the limitation on thenumber of dimensions of decision-making. While the results are model-specific, theyare intuitive and suggestive.

The model is partial equilibrium in the sense that it focuses on the sector of theeconomy consisting of new entrepreneurial firms, denoted as sector E. There is acontinuum of potential entrepreneurs, as well as a (separate) continuum of potentialworkers.1 Entrepreneurs differ in a single dimension called ability, denoted a, whileworkers differ by quality, denoted q , and both a and q are distributed uniformlyover [0,1]. The total population of entrepreneurs is normalized to unity, while thatof workers is normalized to n, so there are n workers per entrepreneur. These as-sumptions about the distributions of entrepreneurs and workers are useful because,as we shall see, they lead to perfect matching of workers and entrepreneurs in thefull-information outcome, thereby avoiding the complications that arise when match-ing is imperfect. Only a portion of both potential entrepreneurs and workers end upchoosing sector E, and those who do not have a fall-back option as discussed below.

Every potential entrepreneur has a project that may either be successful or un-successful. To undertake a project, each entrepreneur hires n workers, taken to befixed for simplicity in the basic case. Allowing firms to choose their labor demandis a straightforward extension taken up later. Success occurs with probability p andyields a return R(q), where q = (q1, . . . , qn) are the qualities of the n workers hiredby the firm. More specifically, we assume the probability of success equals the entre-preneur’s ability (p = a) and that the return if successful is given by

Rq = Rqα (1)

where q is the average (or expected) quality of workers hired and 0 < α < 1. If theproject fails, zero revenue is obtained (R = 0).2 Both a and q are private informationto the entrepreneur or the worker, respectively. Given that, the quality of workers anentrepreneur hires is simply a random draw from those who choose sector E.

When a project is undertaken, each worker is paid a wage up front, and for sim-plicity wage costs are the only costs to the entrepreneurs.3 Let c denote total wage

1An alternative approach would be to assume that entrepreneurs and workers come from the same pop-ulation, as in Kanbur (1981), and discussed in de Meza (2002). In our approach, differences in attributesof entrepreneurs and workers play a key role, and it simplifies matters to abstract from the possibility thatindividuals may possess various amounts of each attribute. This would add occupational choice to ouranalysis and would complicate things considerably.2These assumptions are equivalent to the de Meza and Webb (1987) case, as opposed to the Stiglitz andWeiss (1981) case where expected returns on projects are the same, or Boadway and Keen (2006) whereprojects are distributed over both R and p.3In principle, payments to workers could include an ex post bonus contingent on the return if successful,and this might be a way of separating workers by quality. However, an ex post bonus will not be useful


costs. Entrepreneurs are assumed to have no wealth so the amount c must be obtainedfrom the credit market.4 We assume that credit takes the form of a loan extended by abank at a gross interest rate of r (i.e., one plus the market interest rate). If the projectis successful, the entrepreneur repays rc to the bank. Otherwise, the firm goes bank-rupt and the bank receives no payment. We assume that banks can costlessly observewhether the project succeeds or fails. Since it would be in the interest of entrepre-neurs to declare bankruptcy even if the project is successful, it may be more realisticto require that banks monitor project returns ex post in the event of such a declaration.Adding an ex post monitoring cost would not change the results, so we leave it outfor simplicity.

All agents—entrepreneurs, workers, and banks—are assumed to be risk-neutral.Potential entrepreneurs choose whether or not to enter sector E and undertake aproject. Those who do not undertake their project have a risk-free alternative income,denoted π0. The expected profit of the project to an active entrepreneur, using (1), isthen:

π = p(

R(q) − rc) = a

(Rqα − rc

) ≥ π0 (2)

where π0 is assumed to be the same for all entrepreneurs. A marginal entrepreneurwill be one with ability a = a such that (2) holds with equality.

In the credit market, banks are perfectly competitive. Let ρ be the risk-free grossrate of return that banks must pay to their depositors. If banks knew the probabilityof success p of a given project—that is, the ability of the entrepreneur—as in thefull-information case, they would in equilibrium charge a gross interest rate r that,by their zero-expected profit condition, satisfies:

r(p) = ρ

a(3)

However, if a for individual entrepreneurs is not known, banks will charge a commongross interest rate r that satisfies:

r = ρ

a(4)

where a is the expected ability of entrepreneurs who obtain bank finance.Workers can seek employment either in sector E or in a ‘traditional’ sector, de-

noted sector T, where there is full information. To rule out the possibility of unem-ployment, we assume that there is ex post mobility between the two sectors: a workerwho chooses to seek employment in sector E can move to sector T in the same periodif a job offer is not received. Following Weiss (1980), those who opt for sector T

in our context given that R is determined by the average quality of workers and, assuming that n is largeenough, it is difficult to attribute the contribution of individual workers. If there were few enough workersat a firm such that each one recognized the effect of their own quality on R, it might be possible to use abonus to separate workers by quality. In a background paper, we studied this possibility for the extremecase in which each firm hires only one worker.4Adding a fixed capital cost that must be financed by credit, as in Stiglitz and Weiss (1981) and de Mezaand Webb (1987), adds nothing of substance in our context since credit is already required to finance wagecosts.


produce an output equal to their quality q and are paid a wage equal to q . There isno need for loan intermediation in sector T since wage payments are perfectly certainand there is no bankruptcy. However, in sector E, there may be asymmetric informa-tion in the sense that entrepreneurs cannot observe the quality of workers they hire(and they cannot observe workers’ productivity in sector T). In this case, all thosewho are employed in sector E earn the same wage w whatever their quality, and thecost of workers per firm is c = nw. Note that if q is private information, workerscannot observe the quality of other employees of the same entrepreneur. We assumethat n is large enough that each worker in a firm takes as given q and, therefore, theexpected return of the successful firm.

As we shall see, when q cannot be observed there may be an excess supply ofworkers opting for sector E, in which case available jobs are filled randomly. Let e bethe proportion of workers going to sector E who become employed there. Then, givenrisk neutrality, workers will seek work in sector T only if the following reservationconstraint is satisfied:

ew + (1 − e)q ≥ q or w ≥ q (5)

where e ≤ 1. Let q be the quality of workers who are just indifferent between sectorsE and T, so q = w. All workers with q ≤ q enter sector E, so the number of workersin E, given the uniform distribution assumed, is nq . This result that the lowest-qualityworkers enter sector E only applies when worker quality cannot be observed. In con-trast, higher-quality workers will generally be attracted to sector E when q is observ-able by entrepreneurs. This constitutes one source of inefficiency due to asymmetricinformation.

Let m denote the number of entrepreneurs who enter sector E. Then the endoge-nous variables are m, a, w, q and e. These are determined sequentially in three stages:

• Stage 1: Entrepreneurs decide whether to enter sector E, determining m and a.• Stage 2: The wage rate w in sector E is determined.• Stage 3: Workers decide whether to enter sector E, determining e and q . Those

who are not employed revert to T.

The outcome is assumed to be a subgame perfect Nash equilibrium, which we solvein reverse order. It is useful first to outline the features of an equilibrium under fullinformation. This will allow us to identify the sources of inefficiency induced byasymmetric information and to develop the intuition for policy intervention in thelatter case.

3 Equilibrium and optimality with full information

With full information, both the ability of each entrepreneur and the quality of eachworker are known to all agents, including the government. Equilibrium is character-ized first by the set of potential entrepreneurs and workers who choose to enter sectorE, and second by the assignment of workers to entrepreneurs. The nature of the equi-librium outcome depends on the parameters of the problem as well as the returnR(q). For illustration, we focus on a particular case: that in which the highest-quality


workers opt for sector E. The lowest-quality workers will go to sector T since theircontribution to expected revenue will be too low in sector E. The highest-ability en-trepreneurs will always enter sector E. In the case where the highest-quality workersenter, the nature of the equilibrium outcome is intuitive.

Consider first the optimal matching of workers by quality with entrepreneurs byability. The expected revenue of a type-a entrepreneur employing a set of workersq with expected quality q is given by aRqα , where a and each element of q arepublic information and, therefore, qα and a are known to the workers and the banks.For a given distribution of worker qualities and entrepreneurial abilities in sector E,aggregate expected revenue will be highest if higher-quality workers are matchedwith higher-ability entrepreneurs. To see this, consider two entrepreneurs of abilitya2 > a1 and two sets of workers, q1 and q2, whose qualities are such that q2 > q1.Thus, workers q2 are of higher expected quality than q1. Expected output will behighest if q2 is matched with a2, and q1 with a1, since

a1Rqα1 + a2Rqα

2 > a1Rqα2 + a2Rqα

1 or a2(qα

2 − qα1

)> a1

(qα

2 − qα1

)

Extending this logic to many types of entrepreneurs, expected output will be max-imized by matching the highest-quality workers with the highest-ability entrepre-neurs. In our context where the distributions of entrepreneurs and workers are uni-form, matching will be perfect. Each entrepreneur will hire n workers of identicalquality, and workers of higher quality will be matched with entrepreneurs of higherability.

Next, consider how the market might generate such an outcome. With full infor-mation, wage rates will be specific to workers’ abilities, so there will be no adverseselection and no excess supply of labor in E (unlike in the asymmetric-informationcase as we shall see in the next section). Therefore, there will be an equal numberof workers and jobs in sector E (n per entrepreneur). Given that the densities of q

and a are identical by assumption, we might expect that the full-information equi-librium will entail perfect matching of workers to entrepreneurs with q increasingin a. Moreover, if the highest-ability entrepreneurs and the highest-quality workersare the ones that opt for sector E, which is the case we shall assume, the matchingoutcome will imply q = a since the upper support of both distributions is the same.In Appendix 1, we show that q = a is in fact an equilibrium in the full-informationcase when both entrepreneurs and workers are drawn from the top of their respectivedistributions. The characteristics of the equilibrium are as follows. There will be anequilibrium wage function in sector E, w(q), such that w(q) ≥ q to ensure participa-tion, w′(q) > 1, w′′(q) ≥ 0 and w(q) = q for the marginal (lowest-quality) workersin E. The wage function w(q) will be such that an entrepreneur of ability a in E willprefer to hire n workers of quality q = a. The cutoff ability of active entrepreneurswill be determined by a zero expected-profit condition: π0 = aRaα − ρna. Sinceeach entrepreneur hires n workers, that will determine the cutoff quality of workersin sector E.

The full-information equilibrium will be efficient. To see that, we only have toshow that a is optimal. This determines the number of active entrepreneurs and work-ers. We already know that output is maximized when matching is perfect, which willbe the case when q = a. Entrepreneurs and workers in sector E obtain a surplus,


while those in T do not. Social surplus is then given by the following, where q = a inequilibrium:

S =∫ 1

a

[aRaα − ρnw(a) − π0

]da + ρ

∫ 1

a

n[w(a) − a

]da

where the first term is the surplus obtained by entrepreneurs and the second term thesurplus of workers, both measured in terms of second-period income.5 DifferentiatingS by a, we obtain:

dS

da= −[

aRaα − ρnw(a) − π0] − ρn

[w(a) − a

]

= −[aRaα − ρna − π0

] = 0

where the last equality follows from the zero expected-profit condition of the mar-ginal entrepreneur. Therefore, the number of entrepreneurs is optimal, and the full-information equilibrium is efficient.

The following proposition summarizes the results for the full-information case.

Proposition 1 In the model with full information when the highest-quality workerschoose sector E, equilibrium has the following features:

(i) The highest-a entrepreneurs will enter sector E.(ii) There is perfect matching of workers by quality and entrepreneurs by ability.

(iii) The number of active entrepreneurs is socially optimal.

The equilibrium outcome we have described is only one of many that can occur.We have chosen it partly for simplicity, but partly because of the stark differencesthat will exist between it and equilibria under asymmetric information discussed be-low. As mentioned, the set of workers who opt for sector E in equilibrium could fallalong an interval of dimension 1 − a in the interior of the quality distribution. Ofcourse, higher-quality workers would be matched with higher-entrepreneurs, and theequilibrium outcome under full information would still be socially optimal.

This characterization of the full-information equilibrium allows us to readily iden-tify the several types of inefficiencies that arise with asymmetric information, someof which can be addressed by policy and other which cannot, although they havean indirect influence on policy prescriptions. First, with asymmetric information, thelowest-quality workers will opt for sector E. While government policy cannot changethat, nonetheless there will be a tendency to prefer more workers to enter sector E tomove their average quality closer to the full-information average. At the same time,there may be a tendency for too many entrepreneurs to enter sector E since bankscannot offer actuarially fair loans. That is, low-ability entrepreneurs obtain loans atterms that are more favorable that their probability of success would warrant sinceloan terms will be based on the expected probability of success of entrepreneurs.

5Alternatively, dividing through by ρ would yield the surplus in present value terms. Since workers getpaid in the first period, their surplus occurs then, while entrepreneurs’ surplus occurs in the second period.


Taken together, the desire to have more workers and fewer entrepreneurs in E maylead to an excess supply of workers entering sector E. Finally, there will be imperfectmatching of workers to firms. High-ability entrepreneurs will hire workers whosequality is lower than optimal, and vice versa. While this cannot be directly corrected,it may nonetheless lead to a desire to attract more entrepreneurs at the lower end ofthe ability distribution than would otherwise be the case.

4 Equilibrium with asymmetric information

In this case, both entrepreneur ability a and worker quality q are private information.All workers who are employed in sector E obtain a common wage rate w, while allactive entrepreneurs face a common interest rate r . We consider each of the threestages of events outlined earlier, starting with the last stage and moving backward.

Stage 3: Workers’ choice of sector

At this stage, the number of entrepreneurs m and their average ability a havebeen determined, as well as the wage rate w. Suppose workers believe—correctlyin equilibrium—that the probability of being employed in sector E is e. Given thecommon wage rate w, the expected income of a type-q worker who opts for sectorE is ew + (1 − e)q . Since a worker of quality q can receive a wage of q in sectorT, the cutoff quality of workers by (5) is q = w. All workers with q < q—those ofthe lowest quality—choose sector E regardless of the parameters of the problem. Thiscontrasts with the social optimum achieved with full information where a segment ofhigher-quality workers will generally be in sector E.

Since there are nq workers in sector E and since by assumption each of the m

entrepreneur hires n workers, the employment rate e is given by

e = min

{nm

nq,1

}= min

{m

q,1

}(6)

For the case in which m < q , there is an excess supply of workers entering E (e < 1)and we have by (5) and (6), q = w > m. This implies that the quality of the marginalworker with e = 1 and e < 1 may be written

q ={

w = m if e = 1

w > m if e < 1(7)

Recall from (1) that an entrepreneur’s project return depends on q . When an entre-preneur hires workers, those workers are selected randomly from the pool of availableworkers with q ∈ [0, q], so the expected value of q , q , is the same for all entrepre-neurs. Given the uniform distribution of workers, q is given by

q = E[q|q ≤ q

] = q

2= w

2(8)


Then, for an entrepreneur of ability a whose probability of success is a, expectedprofits can be written, using (2), (4), and (8), as

π = a(Rqα − rnw

) = a

(R

2αwα − ρnw

a

)(9)

This assumes that all entrepreneurs who choose to become active can receive a loan atthe rate r = ρ/a. We return later to the issue of whether there can be credit rationingin this context, which would complicate matters considerably.

Stage 2: Determination of the wage rate

At this stage, the number and average quality of entrepreneurs, m and a, have beendetermined in Stage 1, and entrepreneurs in sector E anticipate the expected qualityof workers and the employment rate in Stage 3. Consider first the wage rate preferredby any given entrepreneur. Using (9), the value of w that maximizes the profits of anentrepreneur of any ability is the solution to the following problem:

max{w} π = a

(R

2αwα − ρnw

a

)(10)

Using the first-order conditions, the solution, denoted wo, is given by

wo =(

αR

2α

a

ρn

) 11−α

(11)

Since wo is independent of the ability level of the entrepreneur, all entrepreneurswould prefer the same wage rate wo. The second-order conditions are satisfied givenour assumption that α < 1. Starting at w = 0, profits will initially rise with w andeventually reach a peak at wo, assumed to be at wo < 1 so that we have an interiorsolution. The intuition is that the rise in w attracts better quality workers, which in-creases the return, but also increases labor costs. Given that R is concave in q thelatter eventually offsets the former.

Denote by we the market clearing wage rate, that is, the wage rate such that e = 1.Given m, the market clearing wage will be we = m, where the number of workersjust equals the demand for workers by entrepreneurs, nq = nm. Whether there is anexcess supply of workers in E then depends on the relative size of we and wo. Ifwo > we = m, entrepreneurs will bid up the wage rate above the market clearinglevel, attracting excess workers into sector E such that e < 1.6 On the other hand, ifwo ≤ we = m, the wage rate will be bid up to we by competition for workers, so allworkers opting for sector E become employed there, leading to e = 1. Consider theconsequences for expected profits of each outcome in turn.

6We can confirm that e < 1 is an equilibrium. Suppose, starting at e < 1, workers who have not beenoffered a job in E agree to work for w′ < wo . Their quality must be less than w′ , since those with q >

w′ prefer to return to sector T. The expected quality of workers who offer w′ will be w′/2 < wo/2.Entrepreneurs’ profits will therefore be higher at the profit-maximizing wage wo , so they will have noincentive to offer the wage w′ .


Case 1: e = 1, wo ≤ we = m

Expected profits of an entrepreneur of ability a follow from (9):

πe(a, a,m) = a

(R

2αmα − ρnm

a

), for wo ≤ we (12)

This is increasing in a and a, but the effect of m is ambiguous.

Case 2: e < 1, wo > we = m

In this case, the market wage is wo given by (11). Using that and (9), expectedprofits may be written

πo(a, a) = a

(R

2α

(wo

)α − ρnwo

a

)= a

1 − α

α

(αR

2α

) 11−α

(a

ρn

) α1−α

,

for wo > we (13)

This function for expected profits is increasing in both arguments, a and a.

Stage 1: The number of active entrepreneurs

Given that expected profits are increasing in ability a in both cases, there is a cutoffability level a such that all entrepreneurs with a ≥ a become active and the remainderobtain their reservation profits π0 elsewhere. Then given the uniform distribution thatwe have assumed, the number of active entrepreneurs is m = 1 − a and their averageability is a = (1 + a)/2, so m = 2 − 2a. Using this, we can rewrite expected profits in(12) as

πe(a, a) = a

(R(1 − a)α − ρn(2 − 2a)

a

)for wo ≤ we (14)

Whether or not there is an excess supply of worker in E depends on whether wo �we = m = 2 − 2a. Using (11), we obtain that

wo � we = 2 − 2a as

(αR

2α

a

ρn

) 11−α

� 2 − 2a (15)

The left-hand side of the second inequality in (15) is increasing in a, while the right-hand side is decreasing. Moreover, at a = 0, the right-hand side exceeds the left-handside. Therefore, there will be a value of a, denoted a, such that

(αR

2α

a

ρn

) 11−α = 2 − 2a (16)

For a ≤ a, wo ≤ we so e = 1, and vice versa. It must be the case that 0 < a < 1 (sincethe right-hand side is less that the left-hand side at a = 1). Note that a ≥ 1/2 sinceif m = 0, a = 1/2. In what follows, we shall assume that a > 1/2 to allow for thepossibility that e = 1. If a < 1/2, a would always exceed a so there would always bean excess supply of workers entering E (e < 1).


Fig. 1

It remains to determine a, the average quality of entrepreneurs, which de-pends upon how many entrepreneurs become active. For the marginal entrepreneur,πi(a, a) = π0, where i ∈ {e, o} depending whether e = 1 or e < 1 in equilibrium.By (13) and (14), for a > a, we have πi(a, a) > π0 since πi(a, a) is increasing ina, confirming our presumption that active entrepreneurs are those such that a ≥ a.In characterizing the number of active entrepreneurs, we can focus on the expectedprofit function for the marginal entrepreneur, defined as π(a) ≡ πi(a, a), i ∈ {e, o}.Using (13) and (14) and the fact that a = 2a − 1, we obtain

π(a)

{(2a − 1)(R(1 − a)α − ρn(2−2a)

a), if a ≤ a, e = 1

(2a − 1) 1−αα

(αR2α )

11−α ( a

ρn)

α1−α , if a > a, e < 1

(17)

Differentiating the first row of (17) with respect to a and using (15), we obtain thatπ (a) is increasing in the range where a ≤ a (the range where e = 1). From the secondrow, π(a) is also increasing. Therefore, the π (a) function is increasing as shown inFig. 1, with π(a) = 0 at a = 1/2.

The equilibrium value for a is that for which π(a) = π0. Figure 1 depicts one suchequilibrium, denoted a∗. This is a case where a∗ < a, so the market clearing wageapplies. The higher the value of π0, the higher will be a∗, and the more likely willthere be an excess supply of labor in equilibrium.

We have implicitly assumed that, while there may be an excess supply of laborin E in equilibrium, credit markets clear. Stiglitz and Weiss (1981) found that creditrationing could arise when projects were pooled by their expected return (pR), whichwas exogenously given. In the de Meza and Webb (1987) case where projects werepooled by their return R and the distribution of the probability of success p acrossentrepreneurs was given, credit rationing could not arise. Our basic model is an exten-sion of the de Meza–Webb model where entrepreneurs differ according to their prob-ability of success p = a. As such, credit rationing will not occur. Intuitively, banks


will not be reluctant to increase their interest rate in a situation of excess demand forloans. Doing so will ration out low-a entrepreneurs, which will increase bank profits.As we shall discuss below, when we extend our basic model to allow worker qualityto affect the probability of success of projects, credit rationing becomes a possibility.

The following proposition summarizes the results of the equilibrium for the basicmodel.

Proposition 2 In the basic model, equilibrium has the following features:

(i) The highest-a entrepreneurs and the lowest-q workers will enter sector E.(ii) t Workers receive a common w, which may exceed the market-clearing wage we.

(iii) Entrepreneurs face a common interest rate, and there will be no credit rationing.(iv) The equilibrium may involve either e = 1 or e < 1 depending on the value of π0.

5 Efficiency and policy with asymmetric information

In this section, we study the optimality properties of equilibrium outcomes withasymmetric information in credit and labor markets. We begin by investigating theefficiency of market equilibria, and then look at the consequences for governmentpolicy.

Efficiency can be captured by the surplus accruing to society from a given allo-cation of resources. Inactive entrepreneurs (who obtain reservation profits π0) andworkers who end up in sector T earn no surplus. (If there were credit rationing, activeentrepreneurs who are unable to obtain a loan would earn no surplus if there is expost immobility between the two sectors.) Social surplus S is given by

S =∫ 1

a

(a(

R(q) − rnw) − π0

)da + ρ

∫ q

0ne(w − q)dq

= am(Rqα − rnw

) − mπ0 + ρ

(newq − neq2

2

)

Since eq = m by (6), q = q/2 = w/2 by (8) and r = ρ/a by (4), social surplus canbe written as

S(a, q) = (2 − 2a)(aRqα − π0 − ρnq

)(18)

This expression for the surplus implicitly takes into account the fact that the lowest−q

workers and the highest—a entrepreneurs will enter sector E. At most, governmentpolicies can affect a and q . They cannot affect the range of qualities of workersentering E or the matching of workers with entrepreneurs.

5.1 The inefficiency of equilibria

The effect of changes in the average ability of entrepreneurs a and the average qualityof workers q on surplus are given by

Sa(a, q) = (2 − 4a)Rqα + 2π0 + 2ρnq (19)


Sq(a, q) = (2 − 2a)(aRαqα−1 − ρn

)(20)

If the values of a and q could be affected independently, the expressions in (19) and(20) would be set equal to zero in the optimum. As we shall see, this will be thecase if e < 1 in the optimum since q can be increased above the market-clearinglevel by changing wo. However, q will be tied to a if e = 1, where there is a one-to-one relationship between the demand for workers determined by the number ofentrepreneurs and their supply needed to clear the market (nm = nq). In this case,q = m/2 = 1 − a. Note that it is not possible for e > 1. This would require that thereis an excess demand for workers in E, so the wage rate would be bid up. Thus, theconstraint e ≤ 1 applies to the maximization of S. If the optimal values of a and q

that set (19) and (20) equal to zero entail e > 1, the constraint is binding and theoptimum cannot be achieved.

Suppose now that we are in a no-intervention equilibrium. As we have seen, thiscan involve either e = 1 or e < 1. The relationships between the equilibrium valuesq∗ and a∗ in these two cases are as follows, using the expression for wo in (11):

q∗ = w

2=

{1 − a∗ if e = 112 (αR

2αa∗ρn

)1

1−α if e < 1(21)

Note that q∗ is decreasing in a∗ if e = 1. Increasing a reduces the number of en-trepreneurs and their demand for labor, thus reducing the wage rate and the averagequality of workers. On the other hand, q∗ is increasing in a∗ if e < 1. An increase ina increases the preferred wage wo by reducing the interest rate ρ/a. More workersare attracted to E, increasing their average quality, and this can be done without theentrepreneurs providing jobs.

Consider the effect of a on S starting in equilibrium and holding q unchanged.For the marginal entrepreneur, π0 = (2a − 1)(Rqα − 2ρnq/a), since a = 2a − 1.Substituting this expression for π0 into (19), the effect of a on surplus in equilibriumcan be written as

Sa(a∗, q∗) = ρnq∗

(2

a∗ − 3

)� 0 as

2

3� a∗ (22)

Thus, holding q constant, an increase in a starting at equilibrium will increase effi-ciency if and only if a∗ < 2/3. Equivalently, if the number of active entrepreneurs islarge enough in equilibrium (i.e., π0 low enough), it will be efficiency-improving toreduce them.

Consider now the effect of average worker quality on S. The expression for π0 canbe written as follows

π0 = (2a − 1)

(Rqα − 2

ρnq

a

)= (2a − 1)

q

αa

(aRαqα−1 − 2αρn

)

Therefore, aRαqα−1 > 2αρn ≥ ρn if α ≥ 1/2. This implies by (20) that

Sq(a∗, q∗) = (2 − 2a∗)(a∗Rαq∗α−1 − ρn

)> 0 if α ≥ 1

2(23)


As long as α is large enough, so the probability of success is not too concave in q ,an increase in the number of workers entering E starting in the no-intervention equi-librium will be welfare improving: the beneficial effect on the probability of successwill outweigh the increase in the wage rate.

In the case where e < 1, so w = wo = 2q (20) reduces to the following, using (11):

Sq

(a∗,wo/2

) = (2 − 2a∗)ρn > 0 (24)

Thus, if e < 1, there are too few workers entering into sector E in equilibrium.Finally, given the equilibrium relationships between q and a in (21), the overall

change in surplus from a change in a with the market wage rate adjusting can bewritten

dS(a, q)

da= Sa(a, q) + Sq(a, q)

∂q

∂a(25)

Evaluating this at the no-intervention equilibrium using (21), we obtain

dS(a∗,1 − a∗)da

< 0 if a > a∗ >2

3, α ≥ 1

2

dS(a∗,wo/2)

da> 0 if a < a∗ <

2

3

(26)

where the latter takes account of the change in a on wo using (11). In general, theremay be too many or too few entrepreneurs in equilibrium, whether e = 1 or e < 1.

These are all local inefficiency properties of the no-intervention equilibrium. Weturn now to properties of the full optimum and the policies that can support them.

5.2 Optimal policies

We can characterize the optimal values of the average ability of entrepreneurs andthe average quality of workers as the values of a and q that maximize S in (18), thatis, that set the expressions in (19) and (20) equal to zero. Denote these by as and qs .If a and q can be influenced independently, the optimum can be implemented. Therelevant instruments for this would be a subsidy on active entrepreneurs, denoted τ ,and a subsidy on wages in sector E, denoted σ . If τ or σ are negative, they would betaxes.

It is apparent that the optimum can be implemented as long as the following con-dition is satisfied in the optimum:

Condition 1 qs ≥ 1 − as = ms/2

This condition implies that the number of workers seeking employment in sectorE in the optimum, 2nqs , is at least as great as the demand for workers by activeentrepreneurs, nms . The government can satisfy this condition by subsidizing wagesto induce the optimal entry of workers into E while at the same time controlling thenumber of active entrepreneurs by subsidizing (or taxing) them. On the other hand, ifthe optimum violates Condition 1, so qs < 1 − as = ms/2, it cannot be achieved by


government policy. The reason is that this would require a supply of labor less thanits demand, which is incompatible with labor market equilibrium since firms wouldbid up the wage rate to extinguish the excess demand. Consider these two cases inturn.

Case 1: Condition 1 satisfied

Suppose qs > 1 − as , where as and qs satisfy Sa(as , qs) = 0 = Sq(as , qs) in (19)

and (20). The policies for achieving the optimum are as follows. A wage subsidy σ

is paid in sector E and a subsidy τ is paid to active entrepreneurs, so the profit of atype—a entrepreneur in (9) becomes7

π = a

(R

2αwα − ρn(1 − σ)w

a

)+ τ

The preferred wage (the same for all entrepreneurs) maximizes π , which yields theanalog to (11):

wo =(

αR

2α

a

(1 − σ)ρn

) 11−α = 2

(αRa

2(1 − σ)ρn

) 11−α

The socially optimal wage, denoted ws , is obtained by setting Sq = 0 in (20):

ws = 2qs = 2

(αRa

ρn

) 11−α

(27)

The social optimum is then achieved by choosing a wage subsidy σ such that wo =ws , or σ = 1/2.

Next, consider the subsidy τ on active entrepreneurs. For the marginal entrepre-neur, we have, using w = 2q ,

π0 = (2a − 1)

(Rqα − 2ρn(1 − σ)q

a

)+ τ

Using this expression for π0 in (19) with σ = 1/2, we obtain after simplification

1

2Sa(a, q) = ρnq

(1

a− 1

)+ τ

The optimal subsidy on active entrepreneurs is obtained by choosing τ to set thisexpression to zero. Therefore, subsidy policies that will take us to the information-constrained optimum are

σ s |es<1 = 1

2, τ s |es<1 = −ρnqs

(1

as− 1

)< 0

7We have modeled τ as applying to all active entrepreneurs for simplicity. However, the same analysis andresults apply if τ only applied to successful entrepreneurs. This might be relevant if τ < 0 so is a tax.


Thus, if the social optimum involves an excess supply of workers entering sector E(as a way of increasing their average quality), a subsidy on wages combined with atax on active entrepreneurs will achieve the optimum. The wage subsidy serves toattract higher-quality workers into sector E. The tax on entrepreneurs precludes whatotherwise would be excess entry of entrepreneurs due the fact that all entrepreneursincluding the lowest-ability ones face a common interest rate, where the excess entryis exacerbated by the wage subsidy.

Case 2: Condition 1 not satisfied

Suppose qs < 1 − as , so it would be optimal to have a excess demand for workersin sector E in the optimum. Since this is not possible, the government cannot optimizea and q separately: the two are related by q = 1 − a through labor market clearing.One of the two previous policy instruments is sufficient, and let us assume it is τ . Theeffect of a change in τ on social surplus can be written as

dS

dτ= dS

da

da

dτ

where da/dτ > 0, and dS/da is given by (25). Using (20) and (22) revised to takeaccount of the subsidy to entrepreneurs, this can be written after routine manipulationas follows

dS

dτ= 2

da

dτ

(τ − aRαqα + 2ρnq

(1

a− 1

))

The optimal subsidy sets this expression to zero and is given by

τ = aRαqα − 2ρnq

(1

a− 1

)� 0

The ambiguity of the subsidy reflects the fact that a compromise must be made be-tween attracting entrepreneurs to increase the average quality of the workers in E andrepelling entrepreneurs to increase their average ability. The balance can go in eitherdirection.

The following proposition summarizes these results.

Proposition 3 The social optimum satisfies Sa(as , qs) = 0 = Sq(as, qs).

(i) If as > 1 − qs , so es < 1, the social optimum can be achieved by σ > 0 andτ < 0.

(ii) If as > 1− qs , the social optimum cannot be achieved since es > 1 is not feasible;the τ that maximizes S(a, q) subject to e = 1 can be positive or negative.

In what follows, we shall denote constrained optimal policy outcomes by a∗∗, q∗∗and e∗∗. In the case where es < 1, these are equal to socially optimal values as , qs

and es . When the constraint e ≤ 1 is binding, a∗∗, q∗∗ and e∗∗ maximize S subject toe = 1.


5.3 Global comparisons

As we have seen, two regimes may occur in the no-intervention equilibrium, e∗ = 1or e∗ < 1. Similarly, the constrained optimum may involve one of two regimes e∗∗ =1 or e∗∗ < 1. In this subsection, we explore the relationship between e∗ and e∗∗,indicating when it might be optimal for policy to induce e∗∗ < 1 when e∗ = 1, or viceversa.

To study this, it is useful first to derive some global properties of the surplus func-tion S(a, q) in (18). Recall the socially optimal choice of q , given by qs in (27). Thisindicates the value of q that an optimizing government would choose, if it could doso separately from a. In a decentralized economy, it can only do so as long as noexcess demand for labor results, that is, as long as e ≤ 1. As in the no-interventioncase, we can define as to be the cutoff value of a such that e < 1 if a > as , and e = 1otherwise. This will be the value of a such that there is no excess supply of labor, soqs = m/2 = 1 − as . Using (27), as uniquely satisfies

(αRas

ρn

) 11−α = 1 − as (28)

Comparing this with (16), which determines a in the no-intervention equilibrium, wesee that as < a.

Next, consider how S(a, q) varies with a, using (25). In the range where e = 1, thechoice of q is restricted to satisfy q = 1 − a, so ∂q/∂a = −1. When e < 1, q = qs(a)

by (27). Using this along with (19) and (20), the following properties of the functionS(a, q) are proven in Appendix 2.

Proposition 4 If α < 2/3 and as < 2/(2 + α), then

(i) The function S(a, q(a)) is strictly concave, with dS(a, q(a))/da continuous.(ii) If dS(as,1 − as)/da < 0, then e∗∗ = 1 so a∗∗ < as and q∗∗ = 1 − a∗∗.

(iii) If dS(as,1 − as)/da ≥ 0, then e∗∗ < 1 so a∗∗ > as and q∗∗ = qs(a∗∗).

We assume in what follows that these conditions on α and as , which are onlysufficient ones, are satisfied. We also assume that 1/2 < a∗∗ < 1, so the optimalsolution is in the interior. Then S varies with a according to a smooth single-peakedcurve where as can be below or beyond the peak, corresponding to e∗∗ < 1 or e∗∗ = 1,respectively.

Now turn to the relation between no-intervention and socially optimal outcomes.To get unambiguous results, we assume that a < 2/3. Given that as < a as shownabove, this guarantees that as < 2/(2 + α) as required for Proposition 4, althoughthe condition now is slightly more stringent. The following proposition is provenin Appendix 3.

Proposition 5 Assume that α < 2/3 and as < a < 2/3.

(i) If a∗ ≤ 2/3, a∗∗ > a∗(ii) If a∗ > 2/3, a∗∗ > a∗ iff Sa(a

∗, qs(a∗)) > 0


The implication of this proposition is as follows. Suppose that in the no-intervention equilibrium, a∗ < 2/3, that is, that π0 is low enough such that thereare not too few entrepreneurs entering sector E. Then by (i), e∗ < 1 implies e∗∗ < 1(a∗ > a implies a∗∗ > as ), and e∗∗ = 1 implies e∗ = 1 (a∗∗ < as implies a∗ < a).If there is excess supply of labor in E in the no-intervention equilibrium, there willbe in the optimum as well. And, if the labor market in E clears in the optimum, itwill in the no-intervention case as well. However, a regime shift can occur. It is pos-sible that e∗ = 1 in the market equilibrium, while e∗∗ < 1 in the optimum, so thatpolicy induces an excess supply of labor. These relations between e∗ and e∗∗ can besummarized in the following corollary.

Corollary

(i) If e∗ < 1, then e∗∗ < 1.(ii) If e∗∗ = 1, then e∗ = 1.

(iii) e∗ = 1 and e∗∗ < 1 are possible.

6 Extensions to the basic model

In this section, we alter the basic model in two separate ways, each of which broad-ens the scope for policy intervention. In the first case, the output of entrepreneurs isallowed to vary with the number of workers hired, which entrepreneurs are allowedto choose. In the second case, the quality of workers is assumed to influence theprobability of success instead of the revenue if successful. For clarity, each case isconsidered separately instead of including both influences in the same model. Sincethe form of the analysis follows closely that of the basic model, we concentrate onthe ways in which the analysis changes the results and policy implications ratherthan presenting the full analysis in detail. The results are intuitive, so this should besufficient.

6.1 Variable labor demand

The main revision to the basic model is that in the entrepreneurs’ revenue function,R is replaced by f (l) with f ′(l) > 0 > f ′′(l), where l is the number of workers hiredper firm. Given a wage rate w, expected profits for type-a entrepreneurs become

π = a(f (l)qα − rwl

)(29)

Since with asymmetric information all entrepreneurs face the same interest rate r ,wage rate w and expected quality of workers q , all will hire the same number ofworkers l.8

8This parallels the approach of Weiss (1980) where firms vary only in a fixed additive cost and are other-wise equally productive so hire the same number of workers.


The distribution of workers is uniform as before, with a density of n, so q = q/2,where q = w. Given that each firm hires l workers, while there are n workers of eachquality level, the employment rate e changes from (6) to

e = min

{lm

nq,1

}

The key feature added to the model is that entrepreneurs can now choose both thewage rate and the quantity of labor they hire. As before, their preferred wage rateis chosen to maximize profits now given by (29) with q = w/2, which leads to theanalog of (11) for wo with f (l) replacing R. The preferred wage rate wo will prevailif it is above the market clearing wage, which is now we = lm/n.

The choice of l satisfies the first-order condition from maximizing (29):

qαf ′(l) = rw (30)

which will apply whether e = 1 (w = we) or e < 1 (w = wo > we). This condition,which says that the expected value of the marginal product of the last worker hiredequals the cost per worker (including the borrowing cost), reflects the fact that thefirm hires randomly from the pool of workers in sector E and takes q as the expectedquality of the marginal worker it hires. From a social point of view, marginal workershired in E will be of quality q > q = q/2. Thus, the value of the social marginalproduct of labor exceeds the value of the expected marginal product from the pointof view of the firm, with the result that too few workers are hired per firm. This isshown more formally in Appendix 4.

The fact that too few workers are hired by each active entrepreneur is the only ad-ditional consequence added to the basic model by this extension. The other results ofthe basic model continue to hold. The equilibrium will be as characterized in Fig. 1.With e < 1, the wage rate will tend to be set too high and there will be too may en-trepreneurs. The policy prescription in the case where there is variable labor demandis apparent. An employment subsidy should be used whatever the value of e. Withe < 1 in the optimum, there should be a wage subsidy combined with a tax on activeentrepreneurs. With e = 1, either a subsidy or a tax on entrepreneurs can be optimal.

6.2 Worker quality affects probability of success

We revert to the assumption that the number of workers per firm is fixed at n, butnow instead of the average quality of workers affecting the return of successful entre-preneurs, it affects the probability of success. Let the return if successful be simplyR. The probability of success depends both on the ability of the entrepreneur and onq according to the function aqα , where 0 < q < 1. The expected profit of a type-aentrepreneur becomes

π = p(a, q)(R − rwn) = aqα(R − rnw) (31)

The analysis proceeds in a way that is parallel to the basic model. The no-intervention equilibrium of Fig. 1 continues to apply, and can involve either e < 1


or e = 1 depending on whether wo � we. The fewer entrepreneurs there are in equi-librium (the higher is a∗), the more likely is excess supply of labor to occur (giventhat wo is increasing in a).

The efficiency and policy implications parallel those in the basis model, and arederived in Appendix 5. With e = 1, there can be too many or too few active en-trepreneurs, so either a tax or subsidy on entrepreneurs entering sector E would beappropriate. With e < 1, the wage rate wo will be too low from a social point ofview. Intuitively, when a firm increases the wage rate, it attracts some higher qualityworkers into sector E, thereby increasing the average quality of workers that all firmscan hire, This reduces the interest rate r that all firms pay, which is an externalitythat individual entrepreneurs do not take into account in their choice of wo. As well,the average ability of entrepreneurs will be too low indicating that there is excessentry for the same reason as in the basic model. Policy would call for a wage subsidycombined with a tax on active entrepreneurs.

A further additional effect is that there is now a possibility of credit rationing,which will preclude an equilibrium occurring. As we know from Stiglitz and Weiss(1981), a necessary condition for credit rationing to occur is that bank profits decreasewhen the interest rate rises. Then, it is possible that banks will not be willing toraise interest rates in the face of an excess demand for loans. In the basic model, anincrease in the interest rate causes the least able entrepreneurs not to enter sector E.This increases a and the expected probability of success p, and, therefore, banks’expected profits. In the model where the probability of success also depends on theaverage quality of workers, there is another effect at work. When r is increased,not only do low-ability entrepreneurs not enter, but also, since the market wage falls,some higher quality workers do not enter. This tends to reduce the average probabilityof success, counteracting the effect of the increase in a. It is possible that the effect ofthe reduced value of q outweighs the effect of an increase in a so that banks are notwilling to raise r . It can be shown that p is decreasing in a for 1/(1 + α) < a < a (sowe are in the range where e = 1). In these circumstances, credit rationing can occur.

7 Concluding comments

The results in this paper are obviously model-specific. Nonetheless, they are sugges-tive and do indicate that once one combines adverse selection in labor markets withthose in credit markets, matters become much more complicated and policy prescrip-tions less clear cut. The market equilibrium may involve an excess supply of workersentering the entrepreneurial sector, and in some circumstances may also involve creditrationing. Depending on the nature of the equilibrium, efficiency consequences and,therefore, policy prescriptions, may differ. If e < 1, a presumption exists that therewill be too many entrepreneurs, and the wage rate will be too high. In an equilibriumwith e = 1, no such presumption exists. In either case, entrepreneurs will hire too fewworkers when their labor demand is variable. Unlike the case with adverse selectionapplying only in credit markets, there may be too few or too many entrepreneurs. Ad-verse selection in credit markets tends to induce too many low-ability entrepreneursto enter since the interest rate they face is too generous given their ability. At the


same time, the entry of more entrepreneurs mitigates the adverse selection problemin labor markets which results in too-few high-quality workers.

There are various ways the model could be fruitfully enriched or revised. The in-formational assumptions could be relaxed. For example, if entrepreneurs’ ability isobservable but worker quality is not, things simplify in an intuitive way. Type—a

entrepreneurs would pay an interest rate given by (3), while the wage rate wouldremain common. The preferred wage rate wo will now be higher for more-able en-trepreneurs, so in an equilibrium with e < 1, wo would be the wage preferred by thehighest-ability entrepreneur (a = 1). As a result, wo would be higher and a lowerthan in the base case. The equilibria depicted in Fig. 1 would continue to apply. Thecase for a wage subsidy would be enhanced. However, now there would be too fewentrepreneurs whatever the equilibrium value of e (since there is no adverse selectionin the credit market now), calling for a subsidy in both cases.

Alternatively, worker quality might be observable rather than entrepreneurs’ abil-ity. In this case, adverse selection remains in the credit market so all entrepreneurs paythe interest rate r . However, worker-specific wages are offered, and the labor marketclears in equilibrium (e = 1). Given that workers cannot observe firm characteristics,the wage profile w(q) must be such that all entrepreneurs are indifferent about whatquality of worker they prefer. Workers of higher quality will now enter sector E, butthey will still be allocated inefficiently (i.e., randomly) across firms. Given the pre-dominance of adverse selection, there will be too many entrepreneurs as in de Mezaand Webb (1987).

Another possibility would be to have both old and new firms competing for work-ers and in output markets. Assuming that established firms have informational advan-tages over new ones, one would expect to find a case for differential tax treatment ofthe two sorts of firms, although to which type’s advantage may not be obvious. Also,we have assumed in our basic model pooling on both labor and credit markets. Onecould extend the model to allow for the possibility of separating firms either on thebasis of information acquired from ex ante monitoring by banks or signaling by en-trepreneurs, or on the basis of other firm characteristics or behavior, such as firm sizeor the ability to provide collateral. These extensions would complicate the analysisconsiderably.

Acknowledgements We thank the referees for very helpful comments, as well as Matthias Polborn andparticipants at the 15th Spanish Public Economics Conference and seminars at the Universities of Calgary,Colorado, Illinois, Umeå, and Uppsala for helpful suggestions. Financial support is acknowledged fromthe SSHRC of Canada, the Center of Excellence Project of the Ministry of Education of Japan, and theJapan Society for the Promotion of Science, Grant-in-Aid for Scientific Research.

Appendix 1: Perfect matching equilibrium with full information

Suppose the wage function in sector E is w(q), where w(q) ≥ q to ensure partic-ipation. Suppose an entrepreneur of type a hires n(q) workers of type q , where∫ 1

0 n(q)dq = n. Given full information, the banks charge a gross interest rate ofr(p) = ρ/a by (3). Given the wage function w(q), the entrepreneur’s expected profits


can be written, using (2), as

π = aR

[∫ 10 n(q)qdq

n

]α

− ρ

∫ 1

0w(q)n(q) dq (1.1)

Entrepreneurs choose their mix of workers to maximize π , given w(q).Suppose w(q) is convex in equilibrium, so w′′(q) ≥ 0. We confirm below that this

will be the case. Then the following lemma applies:9

Lemma 1 If w′′(q) ≥ 0, then entrepreneur a prefers to hire all n workers of the samequality q∗ such that n∗(q) = n for q = q∗ and n∗(q) = 0 for q �= q∗.

Then expected profits (1.1) can be rewritten

π(a, q) = aRqα − ρnw(q) (1.2)

Since there are n workers of each quality in equilibrium, all entrepreneurs choosea different q and we expect that q = a will be an equilibrium. This will be so if profitsπ(a, q) are maximized where q = a. The first-order condition for an entrepreneur a’schoice of q is aRαqα−1 − ρnw′(q) = 0. This will be satisfied at q = a if

w′(q) = αR

ρnqα (1.3)

The second-order condition, evaluated at q = a, is: aR(α−1)αqα−2 −ρnw′′(q) < 0.Since the first term is negative, this will be satisfied if w′′(q) ≥ 0, which by (1.3) isthe case. Thus, q = a is clearly a candidate for equilibrium.

To verify that q = a is an equilibrium, integrate (1.3) to obtain

w(q) = αR

ρn(1 + α)qα+1 + F (1.4)

where F is a constant whose value is determined below. Assume w(q) is defined over[0,1] and is continuous. And, assume R0 is sufficiently large that in (1.3), w′(q) > 1for all workers in sector E. This will be so if w′(q) > 1 for the marginal worker, sincew′′(q) > 0.

Competition for workers ensures that for the marginal workers w(q) = q .10 Sincew′(q) > 1, w(q) > q for all workers with q > q , implying that the highest-qualityworkers enter sector E. Since, as we confirm below, the highest-a entrepreneurs areactive, the marginal entrepreneur with lowest ability a will hire the marginal workers

9Proof Equation (1.1) may be written π = aRE[q]α − ρnE[w(q)]. Since E[w(q)] ≥ w(E[q]) forw′′(q) ≥ 0, we have π = aRE[q]α − ρnE[w(q)] < aRE[q]α − ρnw(E[q]). Starting in a situationin which workers of different qualities are hired, the entrepreneur can increase profits by replacingthem with n workers of a type equal to the average quality of existing workers, q = E[q], since thenπ = aRE[q]α − ρnw(E[q]). �10Proof If w(q) > q , a worker of slightly lower quality q − ε will work for a slightly lower w. Themarginal entrepreneur would then prefer to employ this worker than one with quality q . �


q . And since the densities of the two distributions are the same, this implies thatq(a) = a so there is perfect matching. Given w(q) = q = a, we can infer by applying(1.4) for q that F satisfies

F = a − αR

ρn(1 + α)aα+1 (1.5)

w(q) adjusts so this is satisfied. All workers with q > a earn w(q) > q , a surplus.Consider the entrepreneurs. For a type-a entrepreneur, π = aqαR − ρnw(q),

where w(q) satisfies (1.4) and (1.5). The unique solution to the entrepreneur’s first-order condition will be q = a, so expected profits may be written

π(a) = a1+αR − ρn

(αR

ρn(1 + α)aα+1 + F

)= R

(1 + α)aα+1 − ρnF

Since π(a) is increasing in a, the marginal entrepreneur will have the lowest abilityamong active entrepreneurs, where a is determined, using (1.5), by π0 = Raα+1 −ρna. All entrepreneurs with a ≥ a enter, and the number of active entrepreneurs ism = 1 − a.

Appendix 2: Properties of the surplus function

Suppose e = 1. Using (19) (20) and dq = 1 − da, (25) simplifies to (assumingdS/da > 0 at a = 1/2 so the optimum is interior)

dS(a,1 − a)

da= 2π0 + 2R(1 − a)α

(1 − (2 + α)a

) + 4ρn(1 − a)

where

d2S(a,1 − a)

da2= 2R(1 − a)α−1(1 + α)

(a(2 + α) − 2

) − 4ρn < 0 if a <2

2 + α

This applies in the range a < as , so the inequality will apply if as < 2/(2 + α).Suppose e < 1. Since q will be chosen such that Sq(a, qs(a)) = 0, (25) becomes

dS(·)/da = ∂S(·)/∂a, or, using (19):

dS(a, qs(a))

da= 2π0 + (2 − 4a)Rqsα + 2ρnqs

where qs(a) is given by (27). Differentiating this with respect to a:

d2S(a, qs(a))

da2= 2R

11−α

(α

ρn

) α1−α

(a

α1−α

3α − 2

1 − α+ α

1 − αa

2α−11−α (1 − 2a)

)< 0

if α <2

3


We also have that

lima→as

dS(a,1 − a)

da= ∂S(as,1 − as)

∂a= lim

a→as

dS(a, qs(a))

da

since ∂S/∂q = 0 at a = as .Therefore, if as < 2/(2 + α) and α < 2/3, S(a, q(a)) is strictly concave in a with

dS(·)/da continuous.

Appendix 3: Comparing market and social outcomes

Differentiate Sa(a, q) in (19) with respect to q:

Saq (a, q) = (2−4a)Rαqα−1 +2ρn � 0 as q �(

αR(2a − 1)

ρn

) 11−α ≡ Q(a) (3.1)

Using (21), we can infer that Q(a) < q for a < 2/3.Assuming a < 2/3, there are four cases.

• Case 1: as < a < a∗ ≤ 2/3In this range, e∗ < 1. Since a∗ ≤ 2/3, then Sa∗(a∗, qs(a∗)) ≥ 0 by (22), and

qs(a∗) > qo(a∗). Then

dS(a∗, qs(a∗))da

= Sa

(a∗, qs(a∗)

)> Sa

(a∗, qo(a∗)

)> 0

using (3.1) for the first inequality and (22) for the second.• Case 2: as < a∗ < a

In this case, e∗ = 1, while e∗∗ < 1, so qs(a∗) > 1 − a∗, and the inequality in(22) still applies. Therefore,

dS(a∗, qs(a∗))da

= Sa

(a∗, qs(a∗)

)> Sa(a

∗,1 − a∗) > 0

again using (3.1) for the first inequality and (22) for the second.• Case 3: a∗ < as < a

In this case, we have e∗ = 1 and e∗∗ = 1, so q∗ = 1 − a∗ > qs(a∗), and theinequality in (22) still applies. Therefore,

dS(a∗,1 − a∗)da

= Sa(a∗,1 − a∗) − Sq(a∗,1 − a∗) > 0

using ∂q/∂a = −1 in the equality, and (22) and (q∗ > qs ) for the inequality.• Case 4: a∗ > 2/3

In this case, it is immediate that if Sa(a∗, qs(a∗)) ≤ 0, then a∗∗ ≤ a∗; otherwise,

a∗∗ > a∗.


Appendix 4: Inefficient hiring with variable labor demand

Social surplus is given by

S =∫ 1

a

(a(f (l)qα − ρwl/a

) − π0)da + ρ

∫ q

0ne(w − q)dq

This leads to the analog of (18), using we = lm/n:

S(a) ={

(2 − 2a)(af (l)(1 − a)α − π0 − (1 − a)l2) if a ≤ a

(2 − 2a)(af (l)(wo

2 )α − π0 − ρlwo

2 ) if a > a(4.1)

Differentiating the first and second rows in (4.1) and using the first-order conditionon l from the entrepreneur’s profit-maximization problem and m = 2 − 2a, we obtain

dSe

dl= ρm(1 − a) > 0,

dSo

dl= ρmwo

2> 0

Therefore, employment should be subsidized whether e = 1 or e < 1. The results ontaxing or subsidizing entrepreneurs are the same as in the basic model.

Appendix 5: Inefficient wage rate when quality affects probability of success

Expected profits for entrepreneur a in (31) can be written

π = a

(w

2

)α

(R − rnw) (5.1)

The preferred wage rate for all entrepreneurs maximizes π yielding, using r = ρ/p:

wo = α

1 + α

Rp

ρnwhere p = a

(wo

2

)α

(5.2)

With e = 1, m = w, so p = a(m/2)α . With e < 1, wo can be substituted into p in(5.2) to give

p = a1

1−α

(α

1 + α

R

2ρn

) α1−α

(5.3)

Social surplus when e = 1 and e < 1 can now be written

Se = (2 − 2a)(a(1 − a)αR − π0 − ρn(1 − a)

)(5.4)

So = (2 − 2a)(a(q)αR − π0 − ρnq

), q = wo/2 (5.5)

Differentiate (5.4) with respect to a and use (5.2) and the expression for π0 to give

∂Se

∂a= 2ρn

(1 − a

awe − (1 + α)wo

)� 0


Differentiate (5.5) with respect to a and q and use (5.2) and the expression for π0 togive

∂So

∂a= 2ρnq

(1

a− 1

)> 0,

∂So

∂q= ρnm(1 + 2α) > 0

Thus, with e = 1 in equilibrium, there may be too many or too few entrepreneurs.With e < 1, there will be too many entrepreneurs and the wage rate will be too low.

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Entrepreneurship and asymmetric information in input markets