The Rise and Spread of Favoritism Practices

Appendix

Proof of Proposition 1. The proof follows from the analysis of the relative position of the members’ premium and the clique cost as a function of the clique size f. Focus on the cost function (Figure 2(b)); the intercept is pδ and the slope is given by c. This is the reason why the proposition fixes δ, the intercept, and then focuses on c, the slope.

The value of the premium function at f=1 is wp (Figure 2(a)). Then, since c cannot be negative, when δ≥w, the cost function is above the premium function for any f, and there is no favoritism at equilibrium.

The intercept of the premium function is p1−2p1−pw. For values of δ larger than 1−2p1−pw, the cost function is above the premium function for low values of f. If the link cost is also large enough (larger than cˉ), then the cost function does not intersect the premium function, and therefore the unique equilibrium is f∗=0. On the contrary, if the cost c is small, both functions intersect each other and there are two equilibria, f∗=0 and f∗=1.

Consider now values of δ smaller than 1−2p1−pw. In this case the cost function is below the premium function for low values of f. Hence, as the cost c increases, the cost function will first touch (at cˆ) the premium function with a tangency; and then they cross each other twice at the value of f that solves the quadratic equation pw1−ξ=pδ+cf (the smaller solution is denoted by fP). Note that the curvature of the premium function depends on p, and this is relevant because for some values of p, the tangency between the cost and the premium functions happens at values of f larger than 1. Once this is taken into account, the proof will be done.

Let us compute the slope of the premium function pw1−ξ at f=1.

Consider now the cost function with c equal to that value; then the corresponding value of δ is:

Note that δ˜ is positive for values of p smaller than 1/3 and negative otherwise. This explains the expression max0,1−3p1−2pw in (i) and (ii) of the proposition.

Consider δ smaller than δ˜ (this can only happen if p≤1/3). The tangency occurs for values of f outside the interval 0,1 and therefore for small values of c the equilibrium is f∗=1. The smaller solution of the quadratic equation fP is equilibrium of the economy when its value is equal or smaller than 1. This happens for values of c equal or larger than c‾. From that on, the unique equilibrium remains fP.

For δ larger than δ˜ (and smaller than 1−2p1−pw) the tangency occurs inside the interval 0,1. The value of c associated to the tangency is obtained by fixing A⋅=0 in the expression of fP.

Hence, if c is smaller than cˆ, the equilibrium is f∗=1. As c grows beyond cˆ, the equation pw1−ξ=pδ+cf has two solutions; the smaller value fP is always smaller than 1 and it is therefore always an equilibrium value of the economy. The second one is also smaller than 1 for low values of c (up to cˉ); however, this second value of f is unstable and sets the basin of attraction of fP (recall that f=1 is still an equilibrium value). For values of c larger than cˉ, the second value of f is larger than 1 so that f=1 is no longer an equilibrium. QED.