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In economics, demand and supply functions help us understand the behavior of consumer demand and seller supply in a market. They show how the quantity demanded or supplied changes with price. Let's break down what these functions represent in our pillow market problem.

The demand function given is \(Q^D = 100 - P\), where \(Q^D\) is the quantity demanded and \(P\) is the price. This equation tells us that as the price \(P\) increases, the quantity demanded decreases, following a typical downward-sloping demand curve.

On the other hand, the supply function is \(Q^S = -20 + 2P\), which indicates that as the price \(P\) goes up, the quantity supplied increases. This should come as no surprise, as suppliers are generally willing to supply more products when they stand to make more money from higher prices. This positive relationship is depicted by an upward-sloping supply curve.

Understanding these functions helps us find out where they intersect, known as the market equilibrium.

Price elasticity of demand measures how sensitive the quantity demanded of a good is to a change in its price. It's a crucial concept to grasp because it helps predict consumer behavior and optimize pricing strategies.

In our exercise, the price elasticity of demand \(E_d\) is calculated using:

• \(E_d = \frac{dQ^D/dP}{Q^D/P}\)

To find \(E_d\), we first need the derivative \(\frac{dQ^D}{dP}\), which represents change in quantity demanded per unit change in price. For \(Q^D = 100 - P\), this derivative is \(-1\).

At equilibrium, where \(Q^D = 60\) and \(P = 40\), \(E_d\) simplifies to \(-\frac{2}{3}\). This negative value signifies an inverse relationship between price and quantity demanded, which is typical for most goods. A value less than 1 in magnitude indicates inelastic demand, meaning consumers are not very responsive to price changes in this context.

Price elasticity of supply shows us how the quantity supplied responds to a change in price. Looking at our pillow example, this concept helps sellers predict how much more they might supply if prices rise.

The elasticity of supply \(E_s\) is determined by the formula:

• \(E_s = \frac{dQ^S/dP}{Q^S/P}\)

We find the derivative \(\frac{dQ^S}{dP}\), representing the rate of change in quantity supplied per unit change in price. For \(Q^S = -20 + 2P\), this derivative is 2.

At the equilibrium point of \(Q^S = 60\) and \(P = 40\), \(E_s\) turns out to be \(\frac{4}{3}\), showing that supply is elastic in this market. An elastic supply suggests suppliers are quite responsive to price changes, providing more pillows as prices increase.

The equilibrium price and quantity of a good are the point where the demand and supply curves intersect, leading to market stability. Both sides of the market are content, as demand matches supply perfectly.

In our pillow market scenario, the equilibrium price is found by setting the demand function equal to the supply function:

• \(100 - P = -20 + 2P\)

Solving this, we find \(P = 40\).

To find the equilibrium quantity, we substitute \(P = 40\) back into either the demand or supply equation. Doing this with \(Q^D = 100 - 40\) or \(Q^S = -20 + 80\), we find the equilibrium quantity is 60 units.

This equilibrium reflects a balanced market where the price ensures all pillows produced are purchased, maintaining a state where no surplus or shortage exists.