Inverse demand function

In the inverse demand function, price is a function of the quantity demanded. That contrasts with the demand function, where the quantity demanded is a function of price.

Example of calculation of inverse demand function

If Q is the quantity demanded and P is the price of the goods, then we can write the demand function as follows:

Q = f(P)

Say, the gasoline demand function has the following formula:

Q = 12 – 0.5P

From this function, you can see, if the price of gasoline is 1 dollar, the quantity demanded is 11.5 liters. If the price increases to 2 dollar, the quantity demanded decreases to 11 liters. In other words, for every 1 dollar increase in price, the quantity demanded decreases by 0.5 liters. The negative sign indicates that price is inversely proportional to quantity, as is the law of demand. The higher the price, the lower the demand for gasoline.

Furthermore, the inverse demand function can be formulated as P = f^-1(Q). Therefore, to calculate it, we can simply reverse P of the demand function. In the case of gasoline demand above, we can write the inverse function as follows:

Q -12 = -0.5P -> P = (Q-12) / -0.5 = -2Q + 24 = 24 – 2Q

Why it is important

Three reasons are why we need to look for reverse demand functions. First, with this function, it’s easy to calculate the impact of change in the quantity demanded to the product’s price. Compare if we only use the demand function as analysis.

Second, calculating quantities that maximize profit also becomes easy. Maximum profit when marginal revenue (MR) and marginal cost (MC). Marginal means additional revenue or costs when the company sells / produces one more product. Mathematically, marginal revenue is the first differential of total income. Meanwhile, marginal costs are the first differential of total costs.

In the previous example, the total revenue (TR) of gasoline sales is:

TR = P x Q = (24 – 2Qd)Q = 24Q – 2Q²

From this equation, the first differential is as follows:

MR = 24 – 4Q

Next, let’s assume, the total cost (TC) is:

120 + 12Q + Q²

The first differential of total costs is:

MC = 12 + 2Q

Because, the profit will be maximum when MR = MC, then:

MC = MR → 12 + 2Q = 24 – 4Q → 6Q = 24 – 12 → Q = 2

So, the company’s profit will be at maximum if it produces/sells 2 units.

Third, as the inverse supply function, the inverse demand function, is useful when drawing demand curves and determining the slope of the curve. Economists usually place price (P) on the vertical axis and quantity (Q) on the horizontal axis. That means the curve represents the inverse demand function. And, the slope of the curve is the quantity coefficient of the inverse function. From the example above, the slope of the curve is -2.

Let’s simulate the equation P = 24 – 2Q into table and curve data.

Q	P
1	22
2	20
3	18
4	16
5	14
6	12
7	10
8	8
9	6
10	4
11	2
12	0

If you input the table data above into excel, and create it with a scatter plot, the curve results are as follows: