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Singapore residents are not limited to one car. There's a lot of regional ultra rich in Singapore, and it's not unusual for them to have half a dozen or more cars, all of which count against the total cap.
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The total number of cars is fixed.

Singapore's population has exploded in the past 20 years, almost doubling due to immigration. Native birth rate has collapsed.

Many relatively wealthy foreign individuals have moved to Singapore, disproportionately middle aged and above. These folks also compete in buying car quotas. They can also buy more than one car. Their demand is more inelastic.

Younger folks, local or foreign born are priced out.

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That seems like a plausible hypothesis.

Btw, I don't think you need the assumption that the rich people's car demand is inelastic.

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