Demand Supply Equations

Demand and Supply Equations

In economics, the demand and supply equations are mathematical representations of the relationships between the price of a good or service and the quantity demanded or supplied. These equations allow us to model how changes in price affect the amount of a good that consumers are willing to buy and that producers are willing to sell.

1. Demand Equation

The demand equation represents the relationship between the price (P) of a good and the quantity demanded (Qd) by consumers. It is typically written in the form of a linear equation:

$$Q_d = a - bP$$

Where:

• $Q_d$ is the quantity demanded at a given price.
• $P$ is the price of the good.
• $a$ is the intercept, representing the quantity demanded when the price is zero (i.e., the theoretical demand when the good is free).
• $b$ is the slope of the demand curve, showing how the quantity demanded changes as the price changes. It reflects the price sensitivity or elasticity of demand. The value of $b$ is typically negative because, according to the law of demand, as the price rises, the quantity demanded falls, and vice versa.

Example of a Demand Equation:

$$Q_d = 100 - 2P$$

• In this equation, $a = 100$, meaning that if the price of the good were zero, 100 units would be demanded.
• $b = 2$, meaning that for every 1-unit increase in price, the quantity demanded decreases by 2 units.

2. Supply Equation

The supply equation represents the relationship between the price (P) of a good and the quantity supplied (Qs) by producers. It is also typically written as a linear equation:

$$Q_s = c + dP$$

Where:

• $Q_s$ is the quantity supplied at a given price.
• $P$ is the price of the good.
• $c$ is the intercept, representing the quantity supplied when the price is zero (i.e., the theoretical quantity supplied when there is no incentive to produce).
• $d$ is the slope of the supply curve, showing how the quantity supplied changes as the price changes. It reflects the price sensitivity or elasticity of supply. The value of $d$ is typically positive because, according to the law of supply, as the price rises, the quantity supplied increases, and vice versa.

Example of a Supply Equation:

$$Q_s = 20 + 3P$$

• In this equation, $c = 20$, meaning that even when the price is zero, producers are willing to supply 20 units due to fixed costs or other factors.
• $d = 3$, meaning that for every 1-unit increase in price, the quantity supplied increases by 3 units.

3. Equilibrium Condition

In a competitive market, market equilibrium is achieved when the quantity demanded equals the quantity supplied. This means that the price at which the quantity demanded by consumers equals the quantity supplied by producers is the equilibrium price ($P^*$).

At equilibrium, we set the demand equation equal to the supply equation:

$$Q_d = Q_s$$

Using the equations for demand and supply:

$$a - bP = c + dP$$

Now, we solve for the equilibrium price $P^*$ and equilibrium quantity $Q^*$:

Solving for $P^*$:

$$a - c = bP + dP$$ $$a - c = (b + d)P$$ $$P^* = \frac{a - c}{b + d}$$

Solving for $Q^*$:

Once the equilibrium price $P^*$ is found, we substitute it back into either the demand or supply equation to find the equilibrium quantity $Q^*$. For example, using the demand equation:

$$Q^* = a - bP^*$$

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Example: Finding the Equilibrium Price and Quantity

Let’s use the following demand and supply equations:

Demand equation:

$$Q_d = 100 - 2P$$

Supply equation:

$$Q_s = 20 + 3P$$

To find the equilibrium price and quantity, follow these steps:

Step 1: Set the demand equal to supply.

$$100 - 2P = 20 + 3P$$

Step 2: Solve for the equilibrium price $P^*$.

$$100 - 20 = 3P + 2P$$ $$80 = 5P$$ $$P^* = \frac{80}{5} = 16$$

The equilibrium price is P = 16.

Step 3: Substitute $P^* = 16$ back into either the demand or supply equation to find $Q^*$.

Using the demand equation:

$$Q_d = 100 - 2(16) = 100 - 32 = 68$$

So, the equilibrium quantity is Q = 68.

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Conclusion

• The demand equation and supply equation describe the relationship between price and quantity for consumers and producers.
• The equilibrium price is found by setting the demand equal to supply, and the equilibrium quantity is found by substituting the equilibrium price back into either the demand or supply equation.
• These equations are fundamental tools in economic analysis, helping us understand how changes in market conditions affect prices and quantities.