Problem 36 - Applied Calculus

Question

A tax of $$\$ 8$$ per unit is imposed on the supplier of an item. The original supply curve is \(q=0.5 p-25\) and the demand curve is \(q=165-0.5 p\), where \(p\) is price in dollars. Find the equilibrium price and quantity before and after the tax is imposed.

Step-by-Step Solution

Answer

Pre-tax equilibrium: \( \$190 \) and 70 units; Post-tax equilibrium: \( \$194 \) and 68 units.

Step 1: Understand Equilibrium Without Tax

The equilibrium occurs where the supply equals the demand, i.e., \( q_s = q_d \). The original supply curve is \( q_s = 0.5p - 25 \). The demand curve is \( q_d = 165 - 0.5p \). Set these two equations equal to each other to find the equilibrium price before tax.

Step 2: Solve The Equilibrium Equation Before Tax

Set the demand and supply equations equal:\[0.5p - 25 = 165 - 0.5p\] Combine like terms:\[0.5p + 0.5p = 165 + 25\]\[ p = 190 \]Now substitute \( p = 190\) back into either the demand or supply equation to find the equilibrium quantity:\[q = 0.5(190) - 25 = 95 - 25 = 70\]Hence, the equilibrium price before tax is \( \$190 \), and the equilibrium quantity is 70 units.

Step 3: Modify Supply Curve For Tax

With a tax of \( \$8 \) per unit, the new supply curve shifts upward by 8 dollars per unit. The new supply curve becomes:\[q_s = 0.5(p - 8) - 25 = 0.5p - 4 - 25 = 0.5p - 29\]

Step 4: Find New Equilibrium With Tax

Set the new supply curve equal to the demand curve to find the new equilibrium price after tax:\[0.5p - 29 = 165 - 0.5p\]Combine terms:\[0.5p + 0.5p = 165 + 29\]\[ p = 194 \]Substitute \( p = 194 \) into either equation to find the new equilibrium quantity:\[q = 0.5(194) - 29 = 97 - 29 = 68\]With the tax, the new equilibrium price is \( \$194 \) and the equilibrium quantity is 68 units.

Key Concepts

Supply and Demand Curves

Supply and demand curves are fundamental in economics, helping us understand how the market operates. These curves graphically represent the relationship between the price of a good and the quantity supplied or demanded by the market. The supply curve generally slopes upwards, reflecting the idea that higher prices incentivize producers to supply more of a product. In our original exercise, the supply curve is represented as \(q = 0.5p - 25\), indicating how the quantity supplied, \(q\), increases with price, \(p\).
A demand curve, on the other hand, typically slopes downwards, signifying that consumers tend to buy more at lower prices. For the exercise, the demand curve is \(q = 165 - 0.5p\). Here, it shows that as the price increases, the quantity demanded decreases.

• Supply Curve: Quantity supplied increases with price.
• Demand Curve: Quantity demanded decreases as price increases.
• Intersection: Equilibrium is found where these curves intersect.

Tax Impact on Supply

Taxes directly impact the supply curve. When a tax is imposed on producers, it effectively increases the cost of production per unit. This upward shift can be illustrated by shifting the supply curve vertically. In the exercise, a tax of \(\$8\) per unit changes the supply curve from \(q = 0.5p - 25\) to \(q = 0.5p - 29\). This equation shows that for any given price, the quantity supplied decreases by the amount of the tax.

• Effect of Tax: The supply curve shifts up, decreasing the quantity supplied at every price point.
• Consumer Impact: Prices tend to increase when the supply curve shifts upwards, affecting consumers.
• Business Impact: Producers adjust supplied quantities due to added costs, balancing between pricing and quantity supplied.

Understanding this shift helps to see the broader economic implications of taxing suppliers, such as potential decreases in production or adjustments in pricing strategies.

Equilibrium Calculation

Calculating equilibrium is crucial in analyzing market trends and setting strategic pricing. The equilibrium price and quantity occur where the supply and demand curves intersect, meaning the quantity demanded equals the quantity supplied. In the exercise without tax, the supply \( q_s = 0.5p - 25 \) and demand \( q_d = 165 - 0.5p \) were set equal to find the initial equilibrium price of \( \\(190 \) and quantity of 70 units.
After a tax is introduced, the supply curve shifts, creating a new point of intersection with the unchanged demand curve. This new equilibrium is determined by solving \( q_s = 0.5p - 29 \) against \( q_d = 165 - 0.5p \), leading to a new equilibrium price of \( \\)194 \) and a reduced equilibrium quantity of 68 units.

• Initial Conditions: Equilibrium found by setting original supply and demand equations equal.
• With Tax: New equilibrium derived from updated supply curve.
• Resulting Impact: Price covers cost of tax, quantity slightly decreases.

Equilibrium calculations show how market forces adjust to new economic conditions, such as taxes, ensuring a balance is maintained.