Problem 37 - Applied Calculus

Question

The demand and supply curves for a product are given in terms of price, \(p\), by $$ q=2500-20 p \quad \text { and } \quad q=10 p-500 $$ (a) Find the equilibrium price and quantity. Represent your answers on a graph. (b) A specific tax of $$\$ 6$$ per unit is imposed on suppliers. Find the new equilibrium price and quantity. Represent your answers on the graph. (c) How much of the $$\$ 6$$ tax is paid by consumers and how much by producers? (d) What is the total tax revenue received by the government?

Step-by-Step Solution

Answer

(a) Equilibrium: price $100, quantity $500. (b) With tax: price $102, quantity $460. (c) Consumers pay $2, producers $4. (d) Tax revenue is $2760.

Step 1: Equilibrium Price and Quantity

To find the equilibrium price and quantity, equate the demand and supply functions: \(2500 - 20p = 10p - 500\). Combine like terms: \(2500 + 500 = 10p + 20p\) Giving us: \(3000 = 30p\). Solve for \(p\): \(p = \frac{3000}{30} = 100\). Now, substitute \(p\) back into one of the original equations to find \(q\): \(q = 2500 - 20(100) = 500\). Thus, the equilibrium price is \(100\), and the equilibrium quantity is \(500\).

Step 2: Representing Graphically

The graph of the demand and supply curves intersects at the equilibrium point. Plot the demand curve using points like \((0, 2500)\) and the supply curve with points like \((100, 500)\); both curves should intersect at \((100, 500)\).

Step 3: Calculate New Equilibrium with Tax

With a \$6 tax per unit imposed on suppliers, the new supply equation becomes \(q = 10(p-6) - 500\), simplifying to \(q = 10p - 560\). Equate the new supply to the demand: \(2500 - 20p = 10p - 560\). Combine like terms: \(2500 + 560 = 30p\) Solve for \(p\): \(p = \frac{3060}{30} = 102\). Substitute \(p = 102\) into the new supply equation: \(q = 10(102) - 560 = 460\). The new equilibrium price is \(102\), and the new quantity is \(460\).

Step 4: Graph New Equilibrium

Plot the new supply curve, which is shifted vertically by the tax amount, and find the new intersection with the demand curve at \((102, 460)\).

Step 5: Divide Tax Burden

Before tax, the price was \(100\) and after tax, the price consumers pay is \(102\); thus, consumers pay \(\\(2\) of the tax. The remaining \(\\)4\) (from \$6) is paid by producers because they effectively receive \(102 - 6 = 96\).

Step 6: Calculate Tax Revenue

The total tax revenue is calculated as the tax per unit multiplied by the quantity sold: \(\\(6 \times 460 = \\)2760\).

Key Concepts

Supply and Demand

Understanding supply and demand is key to finding equilibrium in markets. Demand shows how much quantity of a product consumers are willing to buy at different price levels. The demand equation in this exercise is given as \( q=2500-20p \). It indicates that as the price \( p \) increases, the quantity demanded \( q \) decreases.
On the other hand, supply reflects the quantity that producers are willing to supply at different prices. The supply equation here is \( q=10p-500 \). This means that as the price increases, suppliers are more willing to produce more product.
Equilibrium occurs where the quantity demanded equals the quantity supplied. This balance point determines the equilibrium price and quantity in the market, allowing both consumers and suppliers to agree on the transaction terms without any surplus or shortage.

Tax Impact

Taxes can significantly alter the equilibrium in a market. In this exercise, a specific tax of \( \\(6 \) is levied per unit on the suppliers. This tax changes the cost structure for producers, leading to a new supply equation.
The original supply function \( q = 10p - 500 \) is adjusted by subtracting the tax from the price received by suppliers, thus becoming \( q = 10(p-6) - 500 = 10p - 560 \). This tax elevates the supply curve vertically upwards by \( \\)6 \), indicating that suppliers will now supply less at any price than they did before the tax.
The new equilibrium price \( p \) and quantity \( q \) can be found by setting the modified supply equation equal to the demand equation. This demonstrates how taxes redistribute the economic burden between consumers and producers, affecting both prices they pay and the quantities sold.

Equilibrium Quantity

Equilibrium quantity is the amount of product bought and sold at the equilibrium price. Initially, without a tax, the equilibrium quantity is determined by setting the supply and demand equations equal: \( 2500-20p = 10p-500 \). Solving this gives a quantity of \( 500 \). This signifies the balanced point where market forces align, ensuring all products supplied meet consumer demand.
When a \( \$6 \) tax is introduced, the equations must be adjusted, causing the new quantity equilibrium to fall to \( 460 \). This reduction happens because the tax makes the product effectively more expensive, decreasing both the willingness of consumers to buy and producers to supply.
Equilibrium quantity changes highlight how sensitive market balance can be to external changes such as taxation, revealing the need for careful economic policy considerations.

Graphical Representation

Visualizing supply and demand through graphs helps to better grasp the concept of equilibrium. The demand curve typically slopes downward, indicating that as price decreases, demand increases. Conversely, the supply curve slopes upward, signifying that higher prices incentivize more production.
In this exercise, the intersection of these two curves at point \((100, 500)\) initially shows the equilibrium price and quantity. After the tax is applied, the supply curve shifts up to reflect the new costs imposed on producers. The curves now intersect at a new equilibrium point, \((102, 460)\), depicting the rise in price paid by consumers and the reduced quantity due to the tax.
Graphs are not just for visual appeal; they are fundamental tools for illustrating shifts and changes in economic conditions, helping students understand how different factors like taxes can affect market outcomes.