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This reasoning is problematic because it assumes perfect rationing. But rationing under price controls is never perfect. Under rent control, for example, apartments do not automatically go to those who value the apartments the most. The misallocation due to imperfect rationing makes the actual welfare cost of price controls much higher than the standard deadweight loss triangle.
In many cases, economists are deeply skeptical of price controls. If the costs of price controls were similar to those of taxes, I suspect that this skepticism would be substantially less. By applying off-the-shelf welfare analysis to price controls without thinking through the inefficiency of most rationing systems, teachers of introductory economics mislead their students about the effects of these policies.
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