In the present paper lattice packings of open unit discs are considered in the Euclidean plane. Usually, efficiency of a packing
is measured by its density, which in case of lattice packings is the quotient of the area of the discs and the area of the
fundamental domain of the packing. In this paper another measure, the expandability radius is introduced and its relation
to the density is studied. The expandability radius is the radius of the largest disc which can be used to substitute a disc
of the packing without overlapping the rest of the packing. Lower and upper bounds are given for the density of a lattice
packing of given expandability radius for any feasible value. The bounds are sharp and the extremal configurations are also
presented. This packing problem is related to a covering problem studied by Bezdek and Kuperberg [BK97].