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Consider a first-price sealed-bid auction with interdependent valuations and private budget constraints. Focusing on the two-bidder case, we identify new sufficient conditions for the existence of a symmetric equilibrium in pure strategies. In equilibrium, agents may adopt discontinuous bidding strategies that result in a stratification of competition along the budget dimension. Private budgets can simultaneously lead to more aggressive bidding (a high-budget agent leverages his wealth to outbid rivals) and more subdued bidding (competition becomes less intense among bidders at distinct budget levels). The presence of budget constraints may lead to multiple symmetric equilibria in the first-price auction.