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Recently, the gravitational instability (GI) model of giant planet and brown dwarf formation has been revisited and recast into what is often referred to as the ’tidal downsizing’ hypothesis. The fragmentation of self-gravitating protostellar discs into gravitationally bound embryos - with masses of a few to tens of Jupiter masses, at semimajor axes above 30-40 au - is followed by a combination of grain sedimentation inside the embryo, radial migration towards the central star and tidal disruption of the embryo’s upper layers. The properties of the resultant object depends sensitively on the time-scales upon which each process occurs. Therefore, GI followed by tidal downsizing can theoretically produce objects spanning a large mass range, from terrestrial planets to giant planets and brown dwarfs. Whether such objects can be formed in practice, and what proportions of the observed population they would represent, requires a more involved statistical analysis. We present a simple population synthesis model of star and planet formation via GI and tidal downsizing. We couple a semi-analytic model of protostellar disc evolution to analytic calculations of fragmentation, initial embryo mass, grain growth and sedimentation, embryo migration and tidal disruption. While there are key pieces of physics yet to be incorporated, it represents a first step towards a mature statistical model of GI and tidal downsizing as a mode of star and planet formation. We show results from four runs of the population synthesis model, varying the opacity law and the strength of migration, as well as investigating the effect of disc truncation during the fragmentation process. We find that a large fraction of disc fragments are completely destroyed by tidal disruption (typically 40 per cent of the initial population). The tidal downsizing process tends to prohibit low-mass embryos reaching small semimajor axis. The majority of surviving objects are brown dwarfs without solid cores of any kind. Around 40 per cent of surviving objects form solid cores of the order of 5-10 M-circle plus, and of this group a few do migrate to distances amenable to current exoplanet observations. Over a million disc fragments were simulated in this work, and only one resulted in the formation of a terrestrial planet (i.e. with a core mass of a few Earth masses and no gaseous envelope). These early results suggest that GI followed by tidal downsizing is not the principal mode of planet formation, but remains an excellent means of forming gas giant planets, brown dwarfs and low-mass stars at large semimajor axes.