Contrary to widespread belief, we show that low-pay workers might not generally prefer that the minimum wage rate be increased to a level where the labor demand is unitary elastic. Rather, there exists a critical value of elasticity of labor demand such that increases in the minimum wage rate make low-pay workers better off for higher elasticities, but worse off for lower elasticities. We demonstrate that the critical value decreases with the workers’ income-equivalent wage rate and increases with their risk aversion. It is also shown that there may not exist an optimal minimum wage rate, and if it does exist, may not be unique.