Q.Suppose there are two consumers in the market for a good and their demand functions are as follows:
$d_1(p) = 20 - p$ for any price less than or equal to 20, and $d_1(p) = 0$ at any price greater than 20.
$d_2(p) = 30 - 2p$ for any price less than or equal to 15 and $d_2(p) = 0$ at any price greater than 15.
Find out the market demand function.

The market demand function is the horizontal sum of individual demand functions. This means we add the quantities demanded by both consumers at each price, but only for prices where each consumer is actually in the market.

Consumer 1 demands $d_1(p) = 20 - p$ for $p \le 20$, and zero for $p > 20$.

Consumer 2 demands $d_2(p) = 30 - 2p$ for $p \le 15$, and zero for $p > 15$.

We need to consider three price ranges:

• $p > 20$: Both consumers demand zero. Market demand $D(p) = 0$.
• $15 < p \le 20$: Consumer 1 is active ($20-p$), but Consumer 2 demands zero. So $D(p) = (20 - p) + 0 = 20 - p$.
• $p \le 15$: Both are active. So $D(p) = (20 - p) + (30 - 2p) = 50 - 3p$. …

The core idea here is market demand aggregation. When we have multiple consumers in a market, the total quantity demanded at any given price is simply the sum of the quantities each consumer demands at that price. This is called horizontal summation — we add the quantities (on the horizontal axis), not the prices.

Why does this work? Because each consumer’s demand function tells us how much they are willing and able to buy at each price. The market doesn’t care who buys the good; it only cares about the total. So at a price of ₹10, if consumer 1 wants 10 units and consumer 2 wants 10 units, the market demand is 20 units.

Now, the tricky part in this question is that the two consumers have different price ranges in which they actually participate in the market. Consumer 1 demands a positive quantity only when $p \le 20$. Consumer 2 demands a positive quantity only when $p \le 15$. This means the market demand function will be piecewise — it changes its formula depending on the price range.

Let’s break it down step by step.

Step 1: Identify the price segments.

There are three natural price ranges to consider:

• Range A: $0 \le p \le 15$ — Both consumers are active (since $p \le 15$ satisfies both conditions).
• Range B: $15 < p \le 20$ — Consumer 1 is still active ($p \le 20$), but consumer 2 drops out ($p > 15$ means $d_2(p) = 0$).
• Range C: $p > 20$ — Both consumers demand zero.

Step 2: Write the market demand in each range.

• For $0 \le p \le 15$:

  $d_1(p) = 20 - p$

  $d_2(p) = 30 - 2p$

  So, $d_M(p) = (20 - p) + (30 - 2p) = 50 - 3p$.

• For $15 < p \le 20$:

  $d_1(p) = 20 - p$

  $d_2(p) = 0$

  So, $d_M(p) = 20 - p$.

• For $p > 20$:

  $d_1(p) = 0$, $d_2(p) = 0$

  So, $d_M(p) = 0$. …