Markets for Influence

As before, we assume that the success probabilities are given by

e»

n

∑ ^eJ

j=i

where we assume, for simplicity, that R = 1 and the prize is normalized to one
so that i’s payoff is given by π⅛ — e». One can immediately see that such context
is essentially isomorphic to a oligopsony game as described below.

Proposition 2 A standard Tullock contest characterised by payoff function
—ff--e», i = 1, ...,n, is strategically equivalent to a oligopsony game where:

∑≡j

J=l

(i) the strategic variable for firm i is the quantity purchased of an input
x» > 0;

(ii) output is given by the production function f (x») = X»;

(iii) the (constant) output priceis p = A — 1

(iv) A is sufficiently large that the input supply price w = A ^¹— is always
Σ ^i
i=l

positive.

Proof: Each firm i chooses X» to maximise profits, which can be written as:

Then replace X» with e». □

Proposition 2 above suggests that a Tullock contest will deliver outcomes
that are as competitive as those where the strategic variable is the quantity of
influenced purchased. This result is the oligopsony equivalent of Proposition 1.
This raises the following question: Would outcomes be more competitive if the
strategic variable were total expenditure?

The analogy with oligopoly can help us to answer this question. Grant
and Quiggin (1994) show that the equilibrium outcome with revenue as the
strategic variable is less competitive (higher price, lower aggregate quantity,
higher profit) than the Cournot-Nash equilibrium. This is because (loosely
speaking) if one player chooses to deviate by increasing revenue, this entails an
increase their own output and a reduction in the market price, and the Nash
assumption that other players will hold revenue constant implies that they must
increase quantity. Converse reasoning for the oligopsony case suggests that the
outcome of a standard Tullock contest with expenditure as a strategic variable
will be more competitive (lower price, higher aggregate quantity, more rent
dissipation) than the Cournot-Nash equilibrium. This is because an increase

<<   <   4   5   6   7   8   9   10   >   >>