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The elastic moduli of tissues are connected to their states of health and function. The epithelial monolayer is a simple, minimal, tissue model that is often used to gain understanding of mechanical behavior at the cellular or multi-cellular scale. Here we investigate how the elastic modulus of Madin Darby Canine Kidney (MDCK) cells depends on their packing density. Rather than measuring elasticity at the sub-cellular scale with local probes, we characterize the monolayer at the multicellular scale, as one would a thin slab of elastic material. We use a micro-indentation system to apply gentle forces to the apical side of MDCK monolayers, applying a normal force to approximately 100 cells in each experiment. In low-density confluent monolayers, we find that the elastic modulus decreases with increasing cell density. At high densities, the modulus appears to plateau. This finding will help guide our understanding of known collective behaviors in epithelial monolayers and other tissues where variations in cell packing density are correlated with cell motion.