Demand Function is a mathematical representation that describes the relationship between the quantity demanded of a good and its various determinants. It shows how the quantity demanded changes in response to changes in price and other factors such as consumer income, preferences, and the prices of related goods.
Basic Form of the Demand Function:
The general form of a linear demand function can be expressed as:
Qd = a−bP + cY + dPr + eT + fN + gE
Where:
- Qd = Quantity demanded
- P = Price of the good
- Y = Consumer income
- Pr = Price of related goods (substitutes or complements)
- T = Consumer tastes/preferences
- N = Number of buyers
- E = Expectations about future prices
- a, b, c, d, e, f, g = Coefficients that represent the sensitivity of quantity demanded to changes in the respective factors.
Example of a Demand Function:
Let’s consider the demand function for a hypothetical product, Product X. We can express the demand function as follows:
Qd = 100 − 2P + 0 .5Y − 3Pr + 2T + 5N
Where:
- Qd is the quantity demanded of Product X.
- P is the price of Product X.
- Y is the average consumer income.
- Pr is the price of a substitute product.
- T reflects consumer preferences (assumed to be constant at 1).
- N is the number of consumers in the market.
Assumptions:
Let’s assume:
- Average consumer income (Y) = $40,000
- Price of a substitute product (Pr) = $10
- Number of buyers (N) = 100
- Consumer preferences (T) = 1 (constant)
Calculating Quantity Demanded:
Now, we can calculate the quantity demanded (Qd) for different prices of Product X (ranging from $5 to $25). Here is a table summarizing the calculations:
| Price of Product X (P) | Quantity Demanded (Q_d) |
| $5 | Qd = 100 − 2(5) + 0.5(40000) − 3(10) + 2(1) +5(100) = 100 − 10 + 20000−30 + 2 + 500 = 20562 |
| $10 | Qd = 100−2(10)+0.5(40000)−3(10)+2(1)+5(100) = 100 − 20 + 20000 − 30 + 2 + 500 = 20552 |
| $15 | Qd = 100−2(15)+0.5(40000)−3(10)+2(1)+5(100) = 100−30+20000−30+2+500 = 20542 |
| $20 | Qd = 100−2(20)+0.5(40000)−3(10)+2(1)+5(100) = 100−40+20000−30+2+500 = 20532 |
| $25 | Qd = 100−2(25)+0.5(40000)−3(10)+2(1)+5(100) = 100−50+20000−30+2+500 = 20522 |
Results Summary
The results can be summarized in the following table:
| Price of Product X (P) | Quantity Demanded (Q_d) |
| $5 | 20562 |
| $10 | 20552 |
| $15 | 20542 |
| $20 | 20532 |
| $25 | 20522 |
Graphing the Demand Function:
To visually represent the demand function, we can plot the quantity demanded against the price of Product X.
- X-Axis: Price of Product X (P)
- Y-Axis: Quantity Demanded (Q_d)
Demand Curve:
Analysis of the Demand Function:
- The demand for Product X decreases as the price increases, which is consistent with the law of demand.
- The quantity demanded is quite high at lower price levels but diminishes as the price rises, indicating consumer sensitivity to price changes.
- The steepness of the demand curve reflects how responsive consumers are to price changes.
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