Is the bicycle an inferior good? It depends!
Sales of bicycles have significantly increase since the lockdown. According the the Guardian, Halfords have observed a doubling of demand and Wiggle an online retailer in increase of 192%. At the same time this week's Economist reports a decrease in GDP of 7.7% for Q1 and 8.7% for the year. This would suggest that the bicycle is indeed an inferior good with very high income elasticity of around -15.
However, an exam question may be "to what extent is the rise in cycle sales due to the fall in incomes" and what other factors may explain it?
The bicycle is a very good way of travelling in a pandemic while remaining relatively isolated and it is a substitute good for both cars and public transport. The demand curve of both of these has shifted due to the pandemic.
Public transport is much less popular than before due to the risks of transmission of the covid virus. For those who wish to make reactively short individual trips the bike is a strong substitute particularly in good weather. The costs of cycling have reduced too. Fear of being run over or injured on a bike are economic costs and limiting factors of cycle use. The quiet roads during the lockdown and the road closures and cycle lanes installed by local authorities have reduced this cost as has Police action on speeding and dangerous driving. This represents movement along the demand curve for cycling and therefore demand for its complementary good- a bike.
The price of car use has reduced as demand for petrol and diesel has reduced causing price cuts in addition to the supply side shift bought about by the spat between the Arabs and the Russians. However the economic cost of driving a car has increased as the time taken for a journey has increased due to cycle lanes, rigorous enforcement of speed limits (preventing motorists amusing themselves or saving time by speeding and externalising the cost in fear and danger to everyone else) and restricted parking. This means drivers have an additional cost of increased time making a journey, finding a parking scarce legal space or they are prevented from externalising the cost by parking illegally or selfishly space by fines which presumably they seek to avoid. They may also have the additional cost of having to walk further to get to their final destination. The psychological utility of driving is also reduced as for a lot of car users chugging along at 20mph is frustrating and a disutility.
So overall is the bike an inferior good? I don't think so. Some low-end bikes may be but high end ones costing £10,000 or more are probably quite highly income elastic. It would be interesting to see which bikes are in demand. I think the changes in travel patterns and the substitution effect is a much more likely explanation of the cycle boom than income elasticity although it would be very difficult to unpick!