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Chapter 1: Problem 40

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.

Short Answer

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

Step by step solution

01

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.
02

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.
03

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: \(q = 4p - 8 - 20 = 4p - 28\).
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攊ncreased 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.

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