Matching pursuit, or the pure greedy algorithm, is a widely used method for approximating a target function by a sparse linear combination of elements from a dictionary. When the target function is contained in the variation space corresponding to the dictionary, many impressive works over the past few decades have obtained upper and lower bounds on the error of matching pursuit, but they do not match. Here we close this gap and obtain a sharp characterization of the decay rate of matching pursuit. It turns out that, unlike other greedy algorithm variants, the convergence rate is suboptimal and is determined by the solution to a certain non-linear equation.