Building upon a continuous-time model of search with Nash bargaining in a stationary environment, we analyze the effect of changes in minimum wages on labor market outcomes and welfare. While minimum wage increases invariably lead to employment losses in our model, they may be welfare-improving to labor market participants using any one of a number of welfare criteria. A key determinant of the welfare impact of a minimum wage increase is the Nash bargaining power parameter. We discuss identification of this model using Current Population Survey data on accepted wages and unemployment durations, and demonstrate that key parameters are not identified when the distribution of match values belongs to a location-scale family. By incorporating a limited amount of information from the demand side of the market, we are able to obtain credible and precise estimates of all primitive parameters, including bargaining power. Direct estimates of the welfare impact of the minimum wage increase from $4.25 to $4.75 in 1996 provide limited evidence of a small improvement. Using estimates of the primitive parameters we show that more substantial welfare gains for labor market participants could have been obtained by doubling the minimum wage rate in 1996, though at the cost of a perhaps unacceptably high unemployment rate.