Question

The market demand for a given good is \(Q^{D}=26-4 \mathrm{P}\), while the market supply is \(Q^{S}\) \(=2 \mathrm{P}-4\). Find the equilibrium price and quantity in the market. Now assume that the government introduces a specific tax \(t=3\) on the suppliers. Find the new equilibrium price and the new equilibrium quantity. Compare the pre-tax equilibrium with the after-tax equilibrium. What are the main differences?

Short answer

Pre-tax equilibrium: \(P = 5\), \(Q = 6\); Post-tax equilibrium: \(P = 6\), \(Q = 2\). Tax increases price and decreases quantity.

Step-by-step solution

Set Demand Equal to Supply

To find the equilibrium without tax, equate the demand function \(Q^D = 26 - 4P\) to the supply function \(Q^S = 2P - 4\) and solve for \(P\).

Solve for Equilibrium Price (\(P^*\))

Set the equations \(26 - 4P = 2P - 4\). Combine like terms to get: \(26 + 4 = 4P + 2P\) or \(30 = 6P\). Solve for \(P\):\[P = \frac{30}{6} = 5\]

Solve for Equilibrium Quantity (\(Q^*\))

Plug \(P = 5\) back into either the demand or supply equation. Using demand:\(Q^D = 26 - 4(5) = 26 - 20 = 6\).So, \(Q^* = 6\).

Introduction of Tax

The introduction of a tax \(t = 3\) affects the supply function. The new supply function becomes:\(Q^S = 2(P - 3) - 4 = 2P - 6 - 4 = 2P - 10\).

Set New Demand Equal to New Supply

Set the demand function \(Q^D = 26 - 4P\) equal to the new supply function \(Q^S = 2P - 10\) and solve for the new equilibrium price \(P_{new}\).

Solve for New Equilibrium Price (\(P_{new}^*\))

Set the equations \(26 - 4P = 2P - 10\). Combine like terms to get: \(26 + 10 = 4P + 2P\) or \(36 = 6P\). Solve for \(P\):\[P = \frac{36}{6} = 6\]

Solve for New Equilibrium Quantity (\(Q_{new}^*\))

Plug \(P = 6\) back into either the new demand or supply equation. Using demand:\(Q^D = 26 - 4(6) = 26 - 24 = 2\).So, \(Q_{new}^* = 2\).

Compare Pre-Tax and Post-Tax Equilibria

The pre-tax equilibrium price was 5, and the quantity was 6. The post-tax equilibrium price increased to 6, and the quantity decreased to 2. This shows that the tax led to a higher price for consumers and lower quantity sold, highlighting the burdens of taxes on market equilibrium.

Key concepts

Supply and Demand

In a marketplace, supply and demand are the core forces driving the economy. These are two basic but crucial concepts that determine the price and quantity of goods sold. The demand function, in general terms, shows the relationship between price and the quantity of a good consumers are willing and able to purchase. Typically, as the price decreases, the quantity demanded increases. This is called the Law of Demand, which is usually represented by a downward sloping curve.
On the flip side, the supply function illustrates how much producers are willing and able to sell at a given price. As prices rise, suppliers are generally more willing to supply more of the good, according to what's known as the Law of Supply, resulting in an upward sloping curve. Understanding these curves is essential because they form the basis for determining market equilibrium.
The intersection of the supply and demand curves shows the equilibrium point, where both the amount supplied and demanded is the same. In the exercise, this was calculated by setting the demand equation, \(Q^D = 26 - 4P\), exactly equal to the supply equation, \(Q^S = 2P - 4\). Solving this gives us the equilibrium price and quantity before taxes are applied.

Equilibrium Price

The equilibrium price is a key concept that occurs when the quantity demanded equals the quantity supplied in a market, meaning there is no excess demand or surplus. This is the price at which buyers and sellers trade the exact amount of a good.
In the given exercise, finding the equilibrium price involved setting the demand and supply functions equal. With the demand \(Q^D = 26 - 4P\) and the supply \(Q^S = 2P - 4\), by solving \(26 - 4P = 2P - 4\), the equilibrium price was determined to be \(P = 5\). This calculation is critical because it reflects the price at which the market clears—all goods that are produced are sold.
In a real-world context, understanding equilibrium price helps businesses and policymakers determine how to adjust production and pricing strategies in response to market conditions. It provides insights into resource allocation efficiency in the market, ensuring that supply meets demand.

Tax Impact on Markets

Taxes have a significant impact on market equilibrium, often altering both the equilibrium price and quantity. Specifically, a tax on suppliers typically shifts the supply curve upwards by the amount of the tax, creating a new market condition.
In the exercise provided, a tax of \(t = 3\) was introduced, adjusting the original supply equation to \(Q^S = 2(P - 3) - 4 = 2P - 10\). This reflects the added burden on producers who must now charge higher prices to maintain the same profit margin. Solving the new equations \(26 - 4P = 2P - 10\) showed that the new equilibrium price increased to \(P = 6\).
The quantity at this new equilibrium was also affected, reducing from 6 to 2. This demonstrates the real-world outcome of tax impacts: consumers face higher prices and lower quantities available on the market. Taxes, though necessary for public goods and services, can introduce inefficiencies by distorting the natural balance of supply and demand.