Q.Suppose there are 20 consumers for a good and they have identical demand functions:
$d(p) = 10 - 3p$ for any price less than or equal to $\frac{10}{3}$ and $d(p) = 0$ at any price greater than $\frac{10}{3}$.
What is the market demand function?

Market demand represents the total quantity of a good that all consumers in a market are willing and able to purchase at various prices during a given period. It is derived by horizontally summing the individual demand functions of all consumers.

When all consumers have identical demand functions, the market demand function is found by multiplying the individual demand function by the total number of consumers.

Given:

Number of consumers ($N$) = 20

Individual demand function $d(p)$:

$d(p) = 10 - 3p$ for $p \le \frac{10}{3}$

$d(p) = 0$ for $p > \frac{10}{3}$

To find the market demand function, we multiply the individual demand function by the number of consumers:

For prices $p \le \frac{10}{3}$: …

Understanding market demand begins with the concept of aggregation. Market demand represents the total quantity of a good that all consumers in a market are willing and able to purchase at various prices during a specific period. It is derived by summing up the individual demand curves of all consumers in the market. This process is often called "horizontal summation" because we sum the quantities (on the horizontal axis) demanded by each individual at every given price (on the vertical axis).

The economic intuition behind this is straightforward: at any given price, some consumers will demand a certain quantity. To find the total demand for the entire market at that price, we simply add up the quantities demanded by each individual consumer. If all consumers are identical, meaning they have the exact same demand function, this aggregation becomes even simpler: we just multiply the individual demand by the total number of consumers.

Let's derive the market demand function step-by-step:

1. Identify the individual demand function:

   The demand function for a single consumer is given as:

   $d(p) = 10 - 3p$ for $p \le \frac{10}{3}$

   $d(p) = 0$ for $p > \frac{10}{3}$

   Note

   The condition $p \le \frac{10}{3}$ is crucial. It indicates the maximum price at which a consumer would demand a positive quantity. If $p = \frac{10}{3}$, then $d(p) = 10 - 3(\frac{10}{3}) = 10 - 10 = 0$. For any price above $\frac{10}{3}$, the individual demand is zero, meaning consumers are not willing to buy the good at such high prices.

2. State the market demand aggregation principle:

   Market demand, $D(p)$, is the sum of the individual demands of all consumers in the market.

   Market Demand $D(p) = \sum_{i=1}^{N} d_i(p)$

   Since all $N$ consumers are identical, their individual demand functions are the same, $d_i(p) = d(p)$ for all $i$. Therefore, the market demand function simplifies to: …