Problem 40 - Applied Calculus

Question

A supply curve has equation \(q=4 p-20,\) where \(p\) is price in dollars. A \(\$ 2\) tax is imposed on suppliers. Find the equation of the new supply curve. Sketch both curves.

Step-by-Step Solution

Answer

The new supply curve is \(q = 4p - 28\). Both curves are parallel, with the new curve shifted downward.

Step 1: Understand the Initial Supply Curve

The initial supply curve is given by the equation \(q = 4p - 20\), where \(q\) is the quantity supplied and \(p\) is the price in dollars. This equation shows how quantity supplied changes with price before any tax is imposed.

Step 2: Identify the Effect of the Tax on Suppliers

A \(\$2\) tax is imposed on suppliers. This means that for every unit sold, suppliers effectively receive \(p - 2\) dollars instead of \(p\) dollars.

Step 3: Adjust the Supply Curve Equation for the Tax

Substitute \(p - 2\) into the original supply equation to account for the tax: \(q = 4(p - 2) - 20\). Simplify this equation to find the new supply curve: \(q = 4p - 8 - 20 = 4p - 28\).

Step 4: Sketch Both Supply Curves

The original supply curve, \(q = 4p - 20\), and the new supply curve, \(q = 4p - 28\), are both linear and have the same slope of 4. The new supply curve is shifted downward by 8 units (since \(28 - 20 = 8\)) compared to the original curve due to the tax.

Key Concepts

Tax Impact on Supply

When a tax is imposed on suppliers, it directly affects the supply curve. In this context, the tax reduces the supplier's net income per unit sold.
For instance, with a \(2 tax, suppliers now earn \)2 less per unit, effectively changing their earnings from \(p\) to \(p-2\).
This impact can be understood through economic behavior—increased costs lead to a decrease in supply. The entire supply curve shifts downwards because suppliers need a higher price to cover the costs and maintain production levels.

• The original supply curve is represented as a linear function, such as \( q = 4p - 20 \), where \( q \) is the quantity supplied.
• After the $2 tax, the supply curve transforms to \( q = 4(p - 2) - 20 \), which simplifies to \( q = 4p - 28 \).
• This new equation shows that for the same price \( p \), the quantity supplied is now less, reflecting the tax's impact.

Understanding this helps in analyzing real-world economic policies and their effects on markets.

Linear Functions

Linear functions are foundational in economics and represent relationships that change at a constant rate. In a supply curve, such as \( q = 4p - 20 \), the relationship between price \( p \) and quantity \( q \) is linear.
The form of a linear function is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• The slope \( m \) indicates the rate of change. Here, \( m = 4 \), meaning for every dollar change in price \( p \), the quantity \( q \) changes by 4 units.
• The y-intercept \( b \) is where the line crosses the y-axis (when \( p = 0 \)). In this case, \( b = -20 \).
• A tax changes the y-intercept, leading to a parallel shift in the line.

Linear functions simplify the analysis of economic models, providing a clear visualization of how different variables interact.

Economics in Calculus

Calculus in economics is crucial for understanding changes and their rates in economic models. With a supply curve like \( q = 4p - 20 \), calculus can help determine how small changes in price impact the quantity supplied.
Although the original problem deals with linear functions, calculus comes in handy for more complex, non-linear situations.

• Derivative concepts gauge marginal changes; here, since \( q \) changes by 4 for every unit of \( p \), \( dq/dp = 4 \).
• Understanding integrals can help in computing total quantities over a range, key when integrating supply curves with taxes or subsidies.
• Although not required for linear equations, these concepts expand into evaluating efficiency, economic equilibrium, and consumer/producer surpluses.

By bridging calculus and economics, a deeper comprehension of market dynamics is achieved, paving the way for informed decision-making and policy formulation.