Bayer, R.- C. and F. A. Cowell - DARP

allows substituting C 0 from (7) into (8). Applying (9) and using (4) gives asimpli…ed …rst-order condition for the optimal output choice:nXj6=i@ i (d (q)) @d j (q)[@d j @q i i ] + 1 t + @ i (d) @i (q)[i @d i i ] = 0:i @q i(10)The set of conditions (10), one for each …rm, characterises the equilibrium.Using this characterisation of equilibrium we can now present the mainresults in three steps: sections 4 to 6.4 Compliance decisionsIn this section we compare the di¤erent e¤ects that …xed and relative auditrules have on the extent of tax evasion. If the declaration stage has aninterior solution then, under weak conditions, a relative audit rule leads toless tax evasion than a comparable …xed audit rule. 12The intuition for the result appears simple: in addition to the typicalincentives provided by a …xed detection probability and the corresponding…ne, a relative audit rule provides a further incentive to increase the declaration,as this decreases the detection probability. If one replaces the relativeaudit rule in (5) with a …xed audit rule the …rst-order condition becomest 0 i [f + t] = C 0 i( i (q) d i ): (11)To compare the rules (7) and (11) we need a criterion that makes the two regimescomparable. Suppose we require the equilibrium detection probabilityin the relative-rule scenario to be equal to the …xed-detection probability.This ensures that the audit costs incurred by the authority are equal underboth regimes. Setting i (d 0 ) = 0 i and keeping the pro…ts the same in bothcases it is clear that (7) and (11) di¤er only by the “shading”term@ i (d)@d i[ i i ]:This is just the additional incentive to increase the declaration in order toreduce the detection probability and leads to higher declarations under arelative audit rule.However, this intuitive argument neglects the dependence of …rm’s decisionupon the decisions of the others under a relative audit rule. More is12 The result is also valid if …rms compete in prices rather than quantities.11