Explain the law of consumer's surplus.

Law of Consumer's Surplus

Consumer's surplus is the difference between the maximum price a consumer is willing to pay for a commodity and the actual market price they actually pay. It represents the net benefit or gain to the consumer from participating in the market.

Definition

Consumer's surplus is defined as the area under the demand curve and above the market price line, up to the quantity purchased. Mathematically, it is expressed as:

$$CS={\int }_0^{Q}P(Q)dQ-P_{0}Q$$

Where:

• $P(Q)$ is the demand function (price as a function of quantity).
• $P_0$ is the equilibrium market price.
• $Q$ is the equilibrium quantity.

Graphical Representation

The figure below illustrates the consumer's surplus for a linear demand curve. The shaded triangular area represents the surplus.

Explanation

1. Willingness to Pay: The demand curve shows the maximum price consumers are willing to pay for each unit. The first unit is valued highest, and the marginal willingness to pay decreases as quantity increases.
2. Market Price: In a competitive market, all units are sold at a single uniform price, $P_0$.
3. Surplus Calculation:
   • For the first few units, the price consumers are willing to pay is higher than $P_0$. The difference is the surplus for those units.
   • At the equilibrium quantity $Q^{\ast }$, the willingness to pay equals the market price, so the surplus for the last unit is zero.
   • The total consumer's surplus is the sum of these differences for all units purchased, which geometrically forms the triangle bounded by the demand curve, the price axis, and the horizontal line at $P_0$.

Example Calculation

Assume a linear demand function: $P=10-Q$. If the market price is $P_0=5$:

1. Find equilibrium quantity: $5=10-Q\Rightarrow Q=5$.
2. Calculate the area of the triangle: $$CS=\frac{1}{2}\times \text{Base}\times \text{Height}$$ $$CS=\frac{1}{2}\times 5\times (10-5)=\frac{1}{2}\times 5\times 5=12.5$$

The consumer's surplus is 12.5.

Significance

• It measures the welfare gain to consumers.
• A higher consumer's surplus indicates greater consumer satisfaction.
• It is used in policy analysis to evaluate the impact of taxes, subsidies, and price controls on consumer welfare.