Consider a market in which the following supply and demand schedules prevail.

The market-clearing price is the price at which the quantity demanded is equal to the quantity supplied.

There necessarily exist for both supply and demand critical prices such that beyond those critical levels they drop to zero. For more on this topic see Critical Prices.

Now suppose a control price of p_C is imposed. The quantity demanded at p_C is greater than the quantity supplied. A shortage is perceived in the market. Those not getting the commodity at p_C are tempted to offer to buy it from those who are getting it at p_C. The price gets bid up to p_B.

Now consider the social welfare components of the market. Consumer Surplus is the money value of the benefit consumers get from buying the amount Q they want at price P compared to the maximum amount they would pay to get that amount Q. Graphically it is the area above the price line for P and below the demand schedule. ( Usually in Economics the term used is Consumers' Surplus but this grammatically incorrect. English adhered to the rule that if a noun is used as an adjective it must be in its singular, as in a three mile race track. For more on this see Adjectival Nouns.)

On the supply side there is the Producer Surplus. It is the benefit the producers get from selling a quantity Q of their product at price P compared to the minimum payment they would require to supply that amount Q. Graphically it is the area below the price line for P and above the supply schedule.

The consumer and producer surplus are shown in the following diagram.

But there is a third component to social welfare in such a market. It is the profit of the black marketeers, shown in green in the above diagram. Note that in the particular case shown the profits of the black marketeers is greater than the combined consumer and producer surpluses.

The overall social welfare can be expressed as a function of the control price, as below.

Social welfare drops off from the maximum by an amount proportional to the square of the deviation of the control price from the market-clearing price. This means that what ever social damage the control price does when it deviates from the market-clearing price by $3 is qiuadrupled when it deviates by $6. If it deviates by $30 the social damage is 100 times as much.

However social welfare drops off from the maximum by an amount proportional to the square of the deviation of the control price from the market-clearing price until the control price hits either of the two critical levels. Then beyond the critical levels the social welfare drops to zero because either demand or supply catastrophically collapses to zero.

Thus while a control price which is small deviation from the market-clearing price may not cause much harm in the market that does not mean that a larger deviation won't cause serious or even catastrphic harm in the market.

The drop of social welfare from the maximum is called the Dead Weight Loss. It is shown graphically below.

The maximum social welfare is achieved only if the control price happens to equal the market-clearing price. The social welfare components of that situation are shown below.

So the harm caused by a control price is proportional to the square of its deviation from the market-clearing price until it goes beyond a critical level and then its harm becomes catastrophic.

The above analysis dealt with a control price set below the market-clearing price. An analogous situation prevails as in the case of a minimum wage. If the minimum wage is above the market-clearing wage for a particular labor market then some workers may be helped but others will be hurt. Again the harm done in that market is proportional to the square of the deviation of the minimum wage from the market-clearing wage rate for that labor, until the minimum wage exceeds the critical that results in a catastrophic collapse of labor demand to zero.

The equivalent of the black market would be where those fortunate enough to keep their job hired some unemployed person to fulfill their duties at less than the minimum wage. This is generally not feasible, but one can think of the pay of those fortunate enough to keep their job as being made up of two parts. One part is the wage necessary to get them to supply their labor and the other part is pay for their privileged situation in being able to keep their job when the wage rate is set above the market-clearing level.