Brian Galle has posted this draft on SSRN.  Here is the abstract:

This Article presents the first empirical examination of giving to § 501(c)(4) organizations, which have recently become central players in U.S. politics. Although donations to a 501(c)(4) are not legally deductible, the elasticity of c(4) giving to the top-bracket tax-price of charitable giving is – 1.24, very close to the elasticity for charities. 501c(4) donations also correlate with changes in the tax savings from in-kind gifts. These responses could be driven either by donor-side behavior, such as misunderstandings or intentional over-claiming, or by firm-side fundraising.

I find evidence consistent with both explanations. 501(c)(4) fundraising is also highly responsive to the value of the deduction, with an elasticity of -2.9, and is more effective when the value of the deduction rises. These results imply that the U.S. is currently granting much larger subsidies to c(4) firms than is generally understood, and that subsidies for charity cause previously unobserved pressures on competing c(4)s.