This paper investigates whether OECD countries compete with each other over corporation taxes, and whether such competition can explain the fall in statutory tax rates in the 1980s and 1990s. We develop a model in which multinational firms choose their capital stock in response to an effective marginal tax rate (EMTR), and simultaneously choose the location of their profit in response to differences in statutory tax rates. Governments engage in two-dimensional tax competition: they simultaneously compete over EMTRs for capital and over statutory rates for profit. We estimate the parameters of their reaction functions using data from 21 countries between 1982 and 1999. We find evidence that countries compete over both measures, and moreover, that the estimated slopes of reaction functions are consistent with our theoretical predictions. We find that - consistent with our model, but not some other forms of competition - evidence of strategic interaction is present only between open economies (i.e. those without capital controls in place). The Nash equilibrium average statutory rates implied by the empirical model fall substantially over the period, in line with falls in actual statutory rates. The reductions in equilibrium tax rates can be explained almost entirely by more intense competition generated by the relaxation of capital controls.