In this paper we introduce inverse demand systems that include quadratic scale terms. These systems are similar to regular quadratic demand systems introduced by Howe, Pollack, and Wales. A unique feature of these specifications is that they maintain linear scale curves as a special case. For illustrative purposes we estimate the Normalized Quadratic Inverse Demand-Quadratic Scale System using monthly South Atlantic fish landings and valuation data, 1980-1996. In estimation concavity is maintained locally, and the rank reduction procedures advocated by Diewert and Wales (1988b) are employed. The estimated model is then used to obtain welfare estimates associated with catch restrictions.