Labor Market and Search through Personal Contacts

Appendix

Proof of Proposition 1

Using the first and second conditions, let us define the following functions:

Existence:

First, we should prove that and implicitly define continuous relationship of u as a function of v. Note that function is continuous in all three arguments and has following properties:

Thus, for any fixed , there exists and it is unique. We still need to prove that is continuous.

Let us prove by contradiction; assume that is not continuous and there exists a point of discontinuity and . Let us choose any sequence of points in the domain of , such that this sequence has limit . Then, there exists a sequence of such that for , . Note that by properties of , function is bounded and hence is bounded. By the Bolzano–Weierstrass theorem, any bounded sequence has a convergent subsequence. Let us choose subsequences such that has a limit . Note that has the same limit as the original sequence, since any subsequence of convergent sequence converge to the same limit. By our assumption . By joint continuity of , we know that

thus we have and since and satisfy condition then both of them belong to . However, this is possible only if is not uniquely determined. We obtained a contradiction, since given the properties of , we know that is uniquely determined. Thus, we can conclude that is jointly continuous in v and s.

The second function is continuous in all three arguments. Recall that . We know that and it is easy to see that and , which is positive for any s and v as long as . Following the same lines of proof, we can argue that is unique and jointly continuous.

Thus, a solution obviously exists if at , the Beveridge curve is situated to the left of the labor-demand curve. We obtain and . So condition is .

Now let us prove that are continuous functions in s. Let us define following function:

by joint continuity of and we know that is jointly continuous in v and s. Additionally, we imposed restrictions that insure that and for any s. Using the same line of arguments as before, we can conclude that is continuous.

As we know, at , so . Taking one of them, e.g. we know that is continuous and as well, which implies that is continuous too.

Uniqueness:

To fix the socialization level of workers s, we first prove that the Beveridge curve, u is strictly decreasing in v. We already have seen that . Using (A1), it is easy to show that . Thus, by implicit function theorem , and the Beveridge curve is strictly decreasing in u in the plane.

Using (A1) and (A2), we can show that the derivative of with respect to v is negative:

Using (A1) and (A3), we can show that derivative of with respect to u is positive:

Thus by implicit function theorem . This implies that labor demand is monotonically increasing in unemployment u, which combined with the fact that the Beveridge curve is strictly decreasing, implies uniqueness of the solution for each given s.

Proof of Proposition 2

An increase in the cost of posting vacancy shifts the labor demand curve leaving the Beveridge curve unaffected. Thus, the effect of increase in on and depends on the sign of . Deriving with respect to we find that:

Using the fact that by the implicit function theorem we get that and the labor demand curve shifts downward. This in turn implies that increase in increases and at the same time decreases .

Proof of Proposition 3

is decreasing in s on the interval

Let us choose two values of search intensity and s s.t. , both belonging to the interval and compare position of curves on the plane. Observe that for a given value u we have , thus to equate the right-hand side v should decrease. Recall that . This in turn implies that the Beveridge curve shifts downward on the plane.

In Proposition 1, we have shown that . Taking derivative of with respect to s and rearranging we obtain:

Using the implicit function theorem, we can conclude that . This implies that the labor demand curve shifts upward. Combining both results, it is easy to see that .

By analogy if the matching function decreases on the interval then , and thus when we move from s to the Beveridge curve shifts upward on the plane. The derivative changes sign and becomes negative and thus the labor demand curve shifts downward. Combining both results, one can show that .

decreases in s onifand increases otherwise

Substituting matching function into the second condition, we can rewrite it in the following way:

The second condition does not depend on s and thus is unaffected by the change in s. This implies that it should hold for all socialization levels s. Expressing the inverse of v from the second equation, we get:

Deriving the right-hand side of the expression with respect to u, we obtain:

The derivative is negative if , where and positive otherwise. This in turn implies that the second curve is increasing on the interval and decreases afterward. Despite this non-monotonic shape of the second curve Proposition 1 implies uniqueness of the equilibrium. There are two points of intersection, but one of them and is not an equilibrium because by choosing we ensure that the vacancy rate is not zero when .

If a matching function increases on the interval , then we have seen that an increase in s shifts the Beveridge curve downward on the plane, leaving the second curve unaffected. This implies that the equilibrium unemployment decreases in s, while vacancy rate is decreasing in s if and increasing otherwise.

If a matching function decreases on the interval , then an increase in s shifts the Beveridge curve upward on the plane and the equilibrium unemployment increases in s. The vacancy rate increases in s, if and decreases otherwise.

Proof of Proposition 4

A necessary condition

A necessary condition is that is greater than the minimal level of unemployment, which is . Equalizing both expressions, we get . Let us assume that the denominator of is greater than zero, which is true when . Thus, we get the following inequality:

[7]

Solving the quadratic equation, we get that if , then the expression is positive if . If then should belong to interval , which contradicts the assumption that the denominator of is greater than zero. Thus, we get:

[8]

Assume now that the denominator of is less than zero and thus . In this case, inequality [7] is reversed. Thus, if , then the expression is negative if or . Taking into account the assumption that the denominator of is negative the only interval is . If , then we get or . Taking into account assumption, we get . Thus, we get:

[9]

Combining eqs [8] and [9], we get that if then condition is , while if then should be greater than , which in case equals to . Finally, we get that should be greater than .

A necessary condition

Assumption (A4) implies that equilibrium unemployment is a decreasing function in s. Thus to have non-monotonic behavior of the vacancy rate, should be greater than . Using (A4), we substitute into eqs [2] and [6], which gives the following system of equations:

Combining, we obtain:

Substituting , we get something negative and thus one root is positive and one is negative. Note that the expression is decreasing in . Substituting , we get that the expression is negative if the following holds:

Thus, a necessary condition for non-monotonic behavior of vacancy rate requires to be higher than .

A sufficient condition

Assumption (A4) and Proposition 3 imply that unemployment is monotonically decreasing in s. Thus, the lowest possible unemployment in the labor market is and . The last condition that we should check is that . Depending on the assumed mechanism of vacancies transmission, the minimal unemployment level may be different for different setups. However, we know that for any setup , when . By Proposition 2, we know that both and are increasing in , which combined with the fact that implies that for higher, but arbitrary close to we will have , which by Proposition 3 guaranties non-monotonicity of vacancy rate.

Proof of Proposition 5

To identify the effect of s on the market tightness, we express from the second equation:

This expression relates the inverse of the equilibrium market tightness to the equilibrium unemployment rate. The derivative of the right-hand side with respect to s is given by:

If a matching function is increasing on the interval , then by Proposition 3 the equilibrium unemployment level decreases in s, which implies that inline-formula> decreases in s too and thus market tightness increases in s. Consequently, equilibrium wage also increases in s.

If instead a matching function decreases on the interval , then the equilibrium unemployment level increases in s, and thus both market tightness and equilibrium wage are decreasing in s.

Proof of Lemma 1

(A1a)is increasing and concave in v

(A1b)is increasing and concave in u

Due to the exponential term, it is difficult to identify the sign of the derivative. Let us prove first that the function is concave:

This implies that minimum of the first derivative of the matching function with respect to u lays on boards. One can easily verify that the derivative is positive on both ends: , since and . Thus, we can conclude that for .

(A2), , and

By construction, the matching function is unemployment rate multiplied by the individual hiring probability and thus . By the model specification, the share u of vacancies goes to unemployed workers and results in immediate match. On the other hand, the share of vacancies is received by employed workers and not all of them result in match. This happens due to the miscoordination among workers when one worker can receive multiple offers or in the case when an employed worker does not have unemployed neighbors. This implies that by construction .

The last two properties can be easily verified by taking limits.

(A3)is decreasing in vacancies v andis decreasing in unemployment u

trivially follows from the fact that the matching function is concave in v.

Note that .

(A4)is increasing and concave in socialization level of workers s

Substituting to the matching function, we get .