From Boring to Brilliant: How Math Makes Supply-Demand Curves Fascinating

In this blog, we delve into the concepts of supply and demand curves and polynomials, which are fundamental tools used in economics to understand market dynamics and model real-world scenarios. Most importantly you will see a real-world application of mathematical concepts that you learned in high school. Lets begin by understanding the basic concepts.

What is Supply Demand Curve

The supply and demand curve is a graphical representation of the relationship between the quantity of a good or service that producers are willing to sell and the quantity of that good or service that consumers are willing to buy, at different prices. The curve shows the equilibrium price and quantity, where the supply and demand for a product are balanced and there is no excess demand or excess supply.

The demand curve represents the willingness of consumers to pay for a product at different prices. It shows that as the price of a product increases, the quantity demanded by consumers decreases, and vice versa. This relationship is often referred to as the law of demand. The demand curve is downward sloping, indicating that as the price of the product decreases, the quantity demanded increases.

On the other hand, the supply curve represents the willingness of producers to supply a product at different prices. It shows that as the price of a product increases, the quantity supplied by producers increases, and vice versa. This relationship is often referred to as the law of supply. The supply curve is upward sloping, indicating that as the price of the product increases, the quantity supplied also increases.

The point where the supply and demand curves intersect is called the equilibrium point. This is where the market price and quantity of a product are determined. At this point, the quantity demanded by consumers is equal to the quantity supplied by producers, so there is no excess demand or excess supply. This is the most efficient allocation of resources and represents the market clearing price.

Modeling Supply Demand Curve using Polynomials

Let’s take a case of a startup that manufactures top-end Bluetooth headphone product. When they release their product, pricing is a crucial factor in determining the product’s success in the market. To determine the right price for the product, the startup can use a polynomial model to capture the relationship between the price of the product and the demand for it. The model will also take into account the fixed and variable costs associated with the production and distribution of the product.

The startup can use a polynomial equation to model the demand curve for its Bluetooth headphone product as follows:

D = a — bP + cP²

where D is the demand for the product, P is the price of the product, and a, b, and c are constants that represent the intercept, slope, and curvature of the equation, respectively.

The startup can use a similar equation to model the supply curve for its Bluetooth headphone product as follows:

S = d + eP

where S is the supply of the product, d is the fixed cost of producing and distributing the product, e is the variable cost of producing and distributing each unit of the product, and P is the price of the product.

To find the optimal price for the product, the startup needs to find the point where the supply and demand curves intersect. This is the point where the quantity of the product demanded is equal to the quantity of the product supplied, and the price at that point is the equilibrium price.

To find the equilibrium price, the startup needs to solve the system of equations:

D = S, i.e:

a — bP + cP² = d + eP

This can be done by substituting the equation for S into the equation for D and solving for P:

a — bP + cP² = d + eP cP² + (e — b)P + (d — a) = 0

Using the quadratic formula to solve for P, we get:

P = (-b + √(b² — 4ac))/2c

This equation gives us the equilibrium price for the Bluetooth headphone product. The startup can use this price to maximize its revenue while also accounting for the fixed and variable costs of producing and distributing the product.

Putting sample numbers in the quadratic equation

For example, we might assume that the maximum quantity demanded for top-end Bluetooth headphones in India is around 10 million units per year, which would give us a value for “a” of 10,000,000. We might also assume that the price elasticity of demand for mid-range smartphones in India is around -0.5, which would give us a value for “b” of 0.5.

Recall that the demand curve equation for the top-end Bluetooth headphones product is:

D = 10000000–0.5P + 0.0001P²

where D is the demand for the product and P is the price of the product in INR.

Assuming a fixed input cost of INR 5,00,00,000 (5 crore) and variable costs of producing and distributing each unit of the product at INR 6500, the total cost to produce and sell q units of the product is given by:

Total cost = Fixed input cost + Variable cost per unit x Quantity Total cost = 50000000 + 6500q

The revenue earned from selling q units of the product at a price of P per unit is given by:

Revenue = Price per unit x Quantity Revenue = Pq

To find the equilibrium quantity and price, we need to set the demand equal to the quantity supplied and solve for P and q:

D = Quantity supplied 10000000–0.5P + 0.0001P² = 50000000 + 6500q

Simplifying this equation, we get:

0.0001P² — 0.5P — 40000000 = 0

Solving for P using the quadratic formula, we get:

P = (0.5 ± √(0.5² — 4(0.0001)(-40000000))) / (2(0.0001)) P = -4456.5 or P = 9043.5

The solution P = -4456.5 is extraneous since the price cannot be negative. Therefore, the equilibrium price is INR 9043.5 per unit.

Substituting this price into the demand curve equation, we get:

D = 10000000–0.5(9043.5) + 0.0001(9043.5)² D = 3898900

Therefore, the equilibrium quantity is approximately 3.9 million units.

Using the equilibrium quantity and price, we can calculate the revenue and cost for one year:

Revenue = Price per unit x Quantity Revenue = 9043.5 x 3,900,000 Revenue = INR 35,257,650,000

Total cost = Fixed input cost + Variable cost per unit x Quantity Total cost = 50000000 + 6500 x 3,900,000 Total cost = INR 27,025,000,000

Therefore, the profit in the first year can be calculated as:

Profit = Revenue — Total cost Profit = INR 35,257,650,000 — INR 27,025,000,000 Profit = INR 8,232,650,000

In this blog we explored the concepts of supply and demand curves and polynomials and their applications in real-world scenarios. We discussed how the demand curve can be modeled using quadratic equations, and used real-world values to create a near-real-world model for pricing a product. We also used the quadratic equation to calculate the equilibrium price and quantity for a mid-range smartphone product, given the demand curve and other parameters. Additionally, we used polynomials to solve for equilibrium and calculate the profit for a hypothetical product, given fixed input costs and variable costs of production and distribution.