Many multinational firms (MNEs) pay low or no corporation tax in high-tax countries because they shift taxable income to tax havens. We incorporate nonconvex costs of profit shifting and unobserved heterogeneity in profit-shifting ability in the MNEs’ value maximization problem to study responses of firms to tax policies. We estimate our model using UK corporate tax returns data and quantify: (i) the elasticities of tax base and capital stock with respect to tax rates, (ii) the fixed and variable components of profit-shifting costs for different firm types, and (iii) the government’s trade-off between raising tax revenue by reducing profit shifting and attracting investment. Accounting for extensive margin profit-reporting decisions, we reconcile most of the discrepancies between previous micro- and macro-level estimates of tax base elasticities. We test the predictions of the model using a quasi-natural experiment that restricted profit-shifting by Italian MNEs that operated in the UK and evaluate two types of tax policies that can be analyzed using our approach.