Standard methods of variance component estimation used in the Fay-Herriot model for small areas can produce problems of inadmissible values (negative or zero) for these variances. This implies that the empirical best linear unbiased predictor of a small area mean does not take into account the variance of the random effect of the corresponding area, reducing it to a regression estimator. In this paper, we propose an approach based on the expectation-maximization (EM) algorithm to solve the problem of inadmissibility. As stated in the theory of variance component estimation, we confirm through Monte Carlo simulations that the EM algorithm always produces strictly positive variance component estimates. In addition, we compare the performance of the proposed approach with two recently proposed methods in terms of relative bias, mean square error and mean square predictor error. We illustrate our approach with official data related to food security and poverty collected in Mexico, showing their potential applications.