Academia or the Private Sector? Sorting of Agents into Institutions and an Outside Sector

A AProof of Theorem 1

Since , , is defined as the closure of the set of types at which is strictly increasing, and . We first prove that if , , and , then and . Suppose to the contrary that . Then, there exists such that and . Since , agents of ability type strictly prefer A. This is a contradiction, since ability type is higher than the lowest ability type working for A and thus has the option of moving to A. Suppose now, to reach another contradiction, that . Then, there exists such that and . Since , agents of ability type strictly prefer A. This is again a contradiction, since type is higher than the lowest ability type working for A and thus has the option of moving to A.

We now show that if and , then and . To see this, note that implies that . Also, note that for , we have that , so .

Suppose now, in order to reach a contradiction, that . The support set is defined as the closure of the set of types at which is strictly increasing, so must be a limit point of some neighborhood that belongs to , and this neighborhood must necessarily be to the right of the minimum. Then, for , we have that

[2]

Since researchers always have the possibility of switching to the outside sector, it must hold that

[3]

because otherwise some type with close to would move to the outside sector. Inequality [3] implies that

[4]

Since for all and , it must be the case that

[5]

But inequalities [2], [4], and [5] imply that , which is equivalent to . Hence, a researcher in close to would be better off switching to B, which contradicts that . Thus, .

Suppose instead, to reach another contradiction, that . Then, it must hold that

[6]

since researchers of ability type have the option of moving to B, but are choosing C over B. Inequality [6] is equivalent to

[7]

Since is non-increasing in , and , it must be the case that

[8]

for all . Also, for all . This, together with inequalities [7] and [8] implies that

[9]

for all and for all . Because researchers can always switch to the outside sector, this implies that . Hence, assuming that has led to a contradiction, because it was assumed that . Thus, . An analogous argument shows that if and , then and .

We next show that , , and are intervals. To see this, suppose that is non-empty and not an interval. Then, there exist types and , with such that all ability types on choose B or C, while and . Thus, it must be that and either and/or . Furthermore, since is the closure of the set of types for which is strictly increasing, types in small neighborhoods below and above are in the support set of A.

If then, since types and have the option of switching between A and B, both ability types must be indifferent between A and B:

[10]

and

[11]

Since , eqs [10] and [11] imply that , which contradicts that .

If instead then, since all ability types have the option of switching between A and C, it must be the case that

[12]

for all and for all . Since , ability types in a small neighborhood below are also in . For these , because is non-increasing in , and . This together with inequality [12] implies that for all , so these ability types would prefer to switch to the outside sector, which contradicts that they belong to the support set of A.

Hence, assuming that there exist types and , with such that all types on choose B or C, while and has lead to a contradiction. It follows that is an interval. A similar argument proves that is an interval.

Suppose finally that is non-empty and not an interval. Then, there exist ability types and , with such that all ability types on choose A or B, while and . Suppose, without loss of generality, that .

Since ability types have the option of switching between B and it must be the case that

[13]

which gives

[14]

For in neighborhood above , , because is non-increasing in , and . With inequality [14], this implies that

[15]

This contradicts that so must be an interval.

Next, we establish that if and , then for heterogenous s and for a homogenous . A similar argument applies to show these inequalities with replacing when , , and .

Since researchers always have the option of switching to the outside sector, we must have that

[16]

This implies that for some

If some researchers of ability type prefer the outside sector to university B, that is, for some , then . Suppose instead, for a contradiction, that . Since researchers always have the option of switching to the outside sector, we must have that for all , which implies for all . This contradicts that for some . Therefore, if for some .

If , then inequality [16] holds for all and . To see this, suppose that . Then, we have . Under the supposition, we must have for some , which implies

[17]

Since is non-decreasing in and , we havethat This contradicts inequality [17], because and Therefore, if , then . Finally, by definition an equilibrium allocation is a feasible allocation and, thus, has for all , that is, in any equilibrium every researcher works in either A, B, or C. Therefore, and because , , and are intervals, . Then, again by feasibility of the equilibrium, it must also be the case that .

To establish that with homogenous , suppose that . A researcher of ability type is the lowest ability researcher who works for B and, therefore, has rank . Since it is already established that , it must, hence, be that

[18]

Researchers with must be indifferent between B and the outside option. Furthermore, for , since for all . We, thus, have that for ,

using the result in inequality [18]. This implies that , a contradiction with . Therefore, , which establishes that . Following the same argument as the case with heterogenous s, and . Hence, we have established that there exists such that , while work in the academic sector. ■

B Complete characterization for, and proof of, Theorem 2

Complete characterization for Theorem 2: Under Assumptions 1 and 2, there exists a unique stable sorting equilibrium.

Defining the following constants:

1. 

2. 

3. 

4. 

5. ,

and the following functions:

1. 

2. 

3. 

4. ,

the equilibrium is

1. UFS if , and

2. CFS if , , and

3. USU if , , and

4. UPO if , and

5. CPO if , , , and

6. UFO if , and

7. CFO if , , and

8. CSU if , and

9. CFO or CSU if , and

10. EAS if

Proof of Theorem 2

To prove Theorem 2, we first solve for the cut-offs v, z, y, and x, as well as for the slopes of the non-normalized type-distributions , , and across -space. We can, then, solve for the regions of -space in which the different equilibria exist, for the universities’ qualities, and for the influence of the outside option on the size of the academic sector in the constrained equilibria. We proceed to show that the equilibria described in Theorem 2 are stable and that there is a unique stable equilibrium across -space. Finally, we prove the comparative statics results.

Recall that according to Assumption 1, . Also, note that eq. [1] gives that

A researcher of type z is the highest ability researcher who works in the outside sector. Hence, is constant and equal to on . As established in Theorem 1, and, thus, , and it follows that . Since , it follows that On everyone works in the outside sector. We conclude that the non-normalized distribution of types in the outside sector is

Consider first full segregation. With full segregation, . The non-normalized distributions of types, , , and , under full segregation are illustrated in Figure 2. By Theorem 1, it is the top end of the ability distribution that will choose to work in academia. With full segregation, all of the highest ability researchers choose to work for the better university A, while the best of the remaining researchers choose to work for B. Only a researcher with is indifferent between A and B. It follows easily, in addition to the values for found above, that under full segregation

and

In particular, we have that and .

The peer effect at university A is then given by

while the peer effect at university B is given by

The full segregation equilibrium is unconstrained as long as , that is, as long as the peer effect in B when both universities’ demands are satisfied is as least as high as the wage difference between the outside and academic sectors for the threshold type . When this is satisfied, the lowest ranked researcher in B has higher utility from working at B than from working in the outside sector. Therefore, the boundary between UFS and CFS is given by

Full segregation also requires that the difference in peer effect is large enough that no researcher wishes to move from A to B. Under UFS, this condition is given by

[19]

As long as inequality [19] is satisfied, the lowest ranked researcher at A has at least as high utility from staying at A as from moving to B. Thus, the North boundary of UFS is given by

When instead , the full segregation equilibrium is constrained and the size of the academic sector is determined by the equation

[20]

That is, the size of the academic sector is pinned down by such that the lowest ranked researcher in B gets exactly the same utility from working at B as he would from working in the outside sector.

The lowest ranked researcher at B, , strictly prefers A to B in CFS, since his ranking is zero at either university and in CFS. Suppose, in order to reach a contradiction, that . Since , condition (ii) of Definition 2 implies that , which contradicts that strictly prefers A to B. We therefore have that in CFS. This together with eq. [20] implies that under CFS the size of university B, , is determined implicitly by the equation

[21]

In CFS, university B hires some researchers, . Setting in eq. [21] gives the CFS/USU boundary:

[22]

At the CFS/CPO boundary, the lowest type at university A is indifferent between university A and university B: . Rearranging, we have . Therefore, we need for

Substituting into eq. [21] gives the CFS/CPO boundary:

[23]

Totally differentiating eq. [23] gives that on the CFS/CPO boundary,

so the CFS/CPO boundary is downward sloping.

Consider now partial overlap. With partial overlap, . The non-normalized distributions of types, , , and , under partial overlap are illustrated in Figure 3. Again, by Theorem 1, it is the top end of the ability distribution that will choose to work in academia. When there is overlap, all researchers with ability in the interval are indifferent between the two universities, thus both and are increasing on as illustrated in Figure 3.

To find the non-normalized type-distributions and under partial overlap, we first find their values at the critical points z, y, and x. With , we see that , since researchers in are allocated exclusively to B. A researcher of type x is the highest ability researcher who works for B, hence and is constant thereafter. Since y is the lowest type who works for A, . Finally, researchers in are allocated exclusively to A, hence .

Since a researcher of type y is indifferent between A and B, we must have that

[24]

Since a researcher of type x is also indifferent between A and B, we have that

[25]

Eqs [24] and [25] imply that

[26]

Furthermore, researchers with are also indifferent between A and B. Hence, for ,

[27]

It follows from eq. [27] and the fact that on this interval , that and must be linear on , and if then is flatter than , and vice versa.

We can find the slopes of and on from the critical points above and conclude that under partial overlap the non-normalized distribution of types at university A is

while the non-normalized distribution of types at university B is

The peer effect at university B is then given by

[28]

while the peer effect at university A is

[29]

Using eqs [28] and [29], the difference in peer effect between A and B is given by

This and eq. [26] imply that

which together with eq. [25] can be used to find that

[30]

Plugging this back into [26] then gives that

[31]

Using eqs [28]–[31], we now have that

[32]

and

[33]

From eqs [30] and [31] it follows that is equivalent to

[34]

which is consistent with the North boundaries of UFS and CFS. From eq. [30] it also follows that if and only if

[35]

and if and only if inequality [35] holds with equality. When the latter is true it is also the case that , cf eq. [31], and the overlap is full rather than partial.

The partial overlap equilibrium is unconstrained, as long as the peer effect at university B, which is given in eq. [33], calculated with the universities actual demands and is at least as high as the wage difference between the outside and academic sectors. That is, the equilibrium is UPO as long as

[36]

Equality in inequality [36] defines a curve in -space, which is the boundary between UPO and CPO. Solving for in this equality, the boundary is givenby

[37]

It follows from inequalities [34] and [35] that the South boundary of UPO is given by

[38]

and the North boundary of UPO is given by

Totally differentiating eq. [37] gives that the UPO/CPO boundary is upward sloping,

because from eq. [38] in UPO.

When inequality [36] does not hold, the partial overlap equilibrium is constrained and the size of the academic sector is determined by the equation

[39]

That is, the size of the academic sector is pinned down by such that the lowest ranked researcher in B gets exactly the same utility from working at B as he would from working in the outside sector.

The lowest ranked researcher at B, , strictly prefers A to B in CPO, since his ranking is zero at either university and in CPO. Suppose, in order to reach a contradiction, that . Since , condition (ii) of Definition 2 implies that , which contradicts that strictly prefers A to B. We therefore have that in CPO. Thus, the size of university B in CPO is determined implicitly by the equation

[40]

We need in CPO. Setting in eq. [40] gives the CPO/USU boundary:

[41]

Totally differentiating eq. [41] gives that the boundary between CPO and USU is upward sloping,

because with and inequality [34] gives that

At the CPO/CFO boundary, university B hires researchers of the highest type, . Setting and in eq. [30] gives . Thus, if and only if at the CPO/CFO boundary. Substituting into eq. [40] gives CPO/CFO boundary:

[42]

Totally differentiating eq. [42] gives that the CPO/CFO boundary is downward sloping:

Consider now full overlap. With full overlap, and . In this case, A and B share all types in academia, that is, all types in academia are indifferent between A and B. Then

[43]

This scenario is illustrated in Figure 4.

The full overlap equilibrium is unconstrained as long as

[44]

If inequality [44] is not satisfied, the full overlap equilibrium is constrained, that is, CFO. Then the size of the academic sector is determined by

[45]

In CFO, all types in academia are indifferent between A and B. Thus, the size of each university is undetermined. However, the relative university sizes are irrelevant for the researchers’ utilities.

Setting in eq. [45] gives that the CFO/EAS boundary is

[46]

Setting and in eq. [45] gives that for the CFO equilibrium may in fact be a CSU equilibrium.

As we show later, CFO is only stable if . Plugging into eq. [45] gives that the South CFO boundary is

[47]

Consider finally single university. The non-normalized distributions of types, , , and , under single university are illustrated in Figure 5. By Theorem 1, the top end of the distribution will choose to work for the better university A, and the remaining researchers will work for the outside sector. It follow that under single university

and

In particular, we have that .

The peer effect at university A is then given by

and the peer effect at university B is .

Single university requires that no researcher would like to move from university A to university B. Under Assumption 2, a researcher will have a rank of zero if he moves to an empty university. The lowest type in A weakly prefers university A to university B if and only if . Under Assumption 2, this condition becomes . This condition always holds because , and the peer effect at university B is because there are no peers at the empty university B. Note that all other types in A also prefer to stay in A, since for those types .

The lowest type in A weakly prefers university A to the outside sector as long as . The size of university A is unconstrained if That is, if the peer effect in A is at least as high as the wage difference between the outside and academic sectors for the lowest type in A, . Therefore, the USU/CSU boundary is given by

If instead then the single university equilibrium is constrained and the size of university A is implicitly determined by

[48]

Setting gives the CSU/EAS boundary,

[49]

When this is satisfied, the highest type is indifferent between university A and the outside sector.

Note, for and , both CFO and CSU equilibria are possible.

In full overlap equilibria, since all types in academia are indifferent between A and B, eq. [10] is satisfied for all types in academia. Therefore, full overlap equilibria are in principle possible across -space. However, as we will show next, they are only stable when , resulting in the unique stable equilibrium being as depicted in Figures 6 and 7.

Figure 7

Sorting equilibrium with affine salary functions for

We show stability by applying Theorem 6.5 in Stokey and Lucas (1989). Let denote the mean type in academia. That is,

Note that

[50]

Let t denote the difference in average types between the universities, that is,

Under full overlap, , while under full segregation, . Under partial overlap,

Define also

By eq. [50],

[51]

A stationary point has . By Theorem 6.5 in Stokey and Lucas (1989), a stationary point is stable if the derivative is less than one in absolute value.

Taking the derivative of in eq. [51] gives

[52]

Taking the derivative w.r.t. t in eq. [29] yields

[53]

By eq. [24],

which gives that

[54]

Similarly, by eq. [25],

yielding that

[55]

Combining eqs [52]–[55], we now have that

[56]

Now, consider first full overlap allocations. For these, and . Hence, , so full overlap allocations are stationary points. Plugging and into eq. [56] gives that

Thus, if and only if . This implies that UFO allocations are stable for . Under CFO on the other hand, only North-East of the CPO/CFO boundary given by eq. [42]. Therefore, CFO allocations are stable for

[57]

For EAS, and from eq. [52]. It follows that . Therefore, is a stationary point. The EAS equilibrium is stable, because for all t.

Consider next full segregation allocations. For these and . By inserting this and the expression for into eq. [51], it followsthat . Therefore, is a stationary point. Under full segregations . Plugging this into eq. [56] shows that for all t. It follows that the full segregation equilibria are stable.

Next consider partial overlap allocations. By eqs [32] and [33], . Inserting eq. [29] and the expression for into eq. [51], it follows that . Therefore, is a stationary point. By eq. [24], , while by eq. [25], . Plugging these expressions into eq. [56] shows that . It follows that

Thus,

if and only if .

This implies that UPO allocations are stable for . Under CPO, on the other hand, only South-West of the CPO/CFO boundary given by eq. [42]. Therefore, CPO allocations are stable for

Finally, in SU equilibria, the peer effect at university B, , is zero. Therefore, does not depend on or the mean type in academia, . Also, by Assumption 2 the rank at B of any researcher who moves to the empty university B is zero. Therefore, any researcher in university A or the outside sector will prefer their location to university B.

Proof of base salary comparative statics:

(i) That is decreasing in for CFS and CPO equilibria followsfromtaking the total derivative in eqs [21] and [40] and solving for . In CFS, . In CPO, because . That is decreasing in for CFO follows from taking the total derivative of eq. [45] and finding .

1. Totally differentiating eq. [48] gives .

2. (iv)In CFS, , and it follows that since .

In CPO, . Taking the derivative gives that , which is negative since .

1. The result for CPO and CFS follows from (iv). For CFO, the quality difference is zero and does not change with . In the unconstrained equilibria, the academic sector is unaffected by changes in the outside option, and in particular the universities’ qualities do not depend on .

Proof of salary progression comparative statics:

1. That is increasing in for CFS and CPO equilibria follows from plugging into eqs [21] and [40], taking the total derivatives, and solving for . In CFS, . In CPO, That is increasing in for CFO follows from plugging into eq. [45], taking the total derivative, and solving for .

2. Following the proof of the base salary comparative statics, in CFS and CPO since .

3. The result for CPO and CFS follows from (iii). For CFO, the quality difference is zero and does not change with . In the unconstrained equilibria, the academic sector is unaffected by changes in the outside option, and in particular the universities’ qualities do not depend on .