Demand Function: Example, Linear vs. Nonlinear

What’s it: A demand function is a mathematical equation representing the relationship between demand and its determinants. The function shows us how the demand for a product is affected by factors such as its own price, buyer income, and the related products’ price (substitutes and complements). From the equation, we know how they affect – whether positive or negative – and how significantly they affect the demand for a product.

A simple demand function uses its own price to describe the quantity demanded. Economists assume the two have a linear relationship. Thus, price changes will have a constant effect on changes in quantity demanded at different price points.

Economists describe this function as the demand curve after conversion into an inverse demand function. We have to invert the demand function to get the demand curve because it uses price as the Y axis and quantity as the X axis – showing price as a function of the quantity demanded.

What is an example of a demand function, and how to read it?

If we don’t know the impact and significance of each demand determinant, we can write the demand function as follows:

• Qd = f(P, INC, PR)

Qd represents the quantity, P is the product’s price, INC is the consumer’s income, and PR is the price of the related product. Qd is the dependent variable, which we can explain through the dependent variables P, INC, and PR. We read the function above: “the quantity demanded for a product is a function of its own price, consumer income, and the price of the related product.”

For example, let’s translate the above function into the following mathematical equation:

• Qd = α – β1 * P + β2 * INC + β3 * PR

The above equation can only be obtained if we know the relationship between price, consumer income, and product price related to demand. But, we don’t know how significantly they affect us.

β represents the coefficient. It shows us how much influence each variable has on the quantity demanded. Meanwhile, the sign on the coefficient reflects how the variable affects the quantity demanded, whether negative or positive.

Take β1 as an example. The negative sign on the coefficient shows the price has a negative effect on the quantity demanded. If the price increases by $1, the quantity demanded will decrease by β1. On the other hand, if the price decreases by $1, the quantity demanded will increase by β1.

As mentioned earlier, to illustrate the above mathematical equation, we must convert it to an inverse demand function. And to do so, we rearrange the above formula into an inverse demand function as follows:

• P = (α/β1) – (Qd/β1) + (β2/β1) * INC + (β3/β1) * PR

Since we use 3 variables to describe demand, we will find it difficult to describe it in a two-dimensional graph. For this reason, economists only use the price variable to simplify the model. First, we must find the quantity demanded for each price point to draw a demand curve. Then, we plot the combination of price and quantity demanded onto a curve.

Example: gasoline demand function

For example, the relationship between the demand for gasoline (Q) and factors such as its price (P), consumer income (I), and the price of cars (PC) is defined as follows:

• Qd = 9.3 – 0.7P + 0.2I – 0.03 PC

Qd is in liters. Meanwhile, P, I, and PC are in US dollars. Let’s read the equation above.

The constant 9.3 represents the quantity of gasoline demanded even when there is no change in price, consumer income, or the price of cars.

The quantity demanded (Qd), and price (P) have a negative relationship with a coefficient of 0.7. It shows that price and quantity are inversely related. And if the price increases by $1, then the quantity demanded will decrease by 0.7 and vice versa.

Consumer income (I) positively correlates with the quantity demanded (Qd). Thus, a $1 increase in consumer income leads to an increase in gasoline demand by 0.2 liters and vice versa.

The sign on the price of the car (PC) is negative, indicating the car price is inversely related to the demand for gasoline (Qd). Thus, if the car price increases by $1, the demand for gasoline decreases by 0.03 liters and vice versa. Specifically, the negative relationship shows cars are complementary goods to gasoline.

Drawing the demand function into a graph

Let’s use the gasoline demand function above. And we only use price and quantity as variables. So the equation we use is:

• Qd = 9.3 – 0.7P

First, we must obtain the inverse demand function to graph the function. The trick is moving P from the equal sign’s right to the left. On the other hand, we move Qd from the left of the equal sign to the right. So we get:

• 0.7P = 9.3 – Qd
• P = (9.3/0.7) – (1/0.7)*Qd
• P = 13.3 – 1.4Qd

1.4 represents the slope of the demand curve. Meanwhile, the negative sign indicates a downward sloping curve. To illustrate the above equation, we have to get the gasoline price data, then calculate the quantity using the above equation. Say the result is as follows:

The table above shows us the quantity demanded for each price point. If we plot the data in the table into a graph, it will produce a demand curve like the one on the right. The orange dots show the quantity demanded for each price point. For example, if the price equals $4, the quantity demanded equals 6.5 liters.

What are the types of demand function?

Two types of demand function:

1. Individual demand function
2. Market demand function

The individual demand function focuses on one person. It relates to the quantity demanded at various prices, given his income and tastes. It may be identical between individuals, although often it is not. For example, some people may be sensitive to price changes for reasons such as income. Meanwhile, whereas others may not. Thus, the impact on the quantity demanded will also differ if the price changes.

The market demand function focuses on everyone who buys the product. Suppose each individual has an identical demand function. In that case, we multiply the total consumers in the market by the individual demand function to obtain it. For example, there are 100 buyers in the gasoline market with the following individual demand functions:

• Qd = 9.3 – 0.7P

The market demand function for the above case is:

• Qd = 100 (9.3 – 0.7P) = 930 – 70P

Thus, if the gasoline price increases by $1, the demand for gasoline in the market will decrease by 860 liters (930 – 70). On the other hand, every $1 decrease in the gasoline price will increase the demand for 860 liters.

The demand function can also be distinguished based on its mathematical equation. It can:

1. Linear
2. Nonlinear

A linear function means the quantity demanded at each price level forms a straight line. You can see the function and demand curve above as an example.

Economists usually use linear functions as examples because they are easier to explain and understand for us. As we know, economic models try to describe reality more simply.

Nonlinear function means the relationship between the quantity demanded and the price forms other than a straight line if we plot it. The mathematical equation may be quadratic, exponential, or otherwise. Here is an example of a nonlinear function.

The curve above shows that demand has different responsiveness in response to price changes. For example, at the price of $2 gasoline, a $1 increase in price (to $3) would decrease the demand for 0.9 liters (5.8 – 4.9). However, when the price was already high and reached $12, the $1 increase (to $13) had little effect on the quantity demanded, which remained at around 2.8 liters.