Question

Show that a straight line supply curve passing through the origin has \(e_{s}=1\)

Short answer

A straight line supply curve passing through the origin can be represented by the equation \(Q_s = mP\). The price elasticity of supply (e_s) formula is \(e_s = \frac{dQ_s}{dP} \cdot \frac{P}{Q_s}\). The derivative of the supply curve equation is \(m\). Substituting the derivative and the supply curve equation into the price elasticity of supply formula results in \(e_s = m \cdot \frac{P}{mP}\), which simplifies to \(e_s = 1\). Therefore, a straight line supply curve passing through the origin has price elasticity of supply equal to 1.

Step-by-step solution

Write the straight line supply curve equation

A linear supply curve passing through the origin can be represented by the equation: \[ Q_s = mP \] where \(Q_s\) is the quantity supplied, \(P\) is the price, and \(m\) is the slope of the supply curve.

Find the price elasticity of supply formula

The price elasticity of supply (e_s) is calculated as: \[ e_s = \frac{\% \Delta Q_s}{\% \Delta P} \] This formula can be rewritten using the derivatives of the supply curve equation with respect to the price: \[ e_s = \frac{dQ_s}{dP} \cdot \frac{P}{Q_s} \]

Calculate the derivative of the supply curve equation

To find the price elasticity of supply, we first need to calculate the derivative of the supply curve equation (\(Q_s = mP\)) with respect to the price (\(P\)): \[ \frac{dQ_s}{dP} = m \]

Substitute the derivative and the supply curve equation into the price elasticity of supply formula

Now, substitute the derivative (\(m\)) and the supply curve equation (\( Q_s = mP\)) into the price elasticity of supply formula: \[ e_s = m \cdot \frac{P}{mP} \]

Simplify the expression and find e_s

Now, we can simplify the expression: \[ e_s = m \cdot \frac{P}{mP} = \frac{mP}{mP} = 1 \] Thus, we have shown that a straight line supply curve passing through the origin has a price elasticity of supply (\(e_s\)) equal to 1.

Key concepts

Understanding the Supply Curve

When studying economics, the supply curve is a fundamental concept that depicts the relationship between the price of a good and the quantity of that good that suppliers are willing to offer for sale. In its simplest form, it's often shown as a straight line, indicating that as the price increases, suppliers are typically willing to supply more of the good.

A straight-line supply curve that passes through the origin is unique. It has a constant slope and suggests that the quantity supplied is directly proportional to the price; this means there's a constant rate of change. No matter what the price is, the quantity of goods supplied changes at a uniform rate. For example, if the slope (m) is 2, then for every one unit increase in price, the quantity supplied increases by two units. This linear and proportional relationship is crucial to understanding market dynamics.

Elasticity in Economic Analysis

Elasticity is a measure that economists use to understand how one variable responds to changes in another variable. In the context of a supply curve, price elasticity of supply describes how the quantity supplied of a good is affected by a change in price. It tells us how much the quantity supplied will rise or fall in response to a price increase or decrease.

What does it mean when the elasticity is equal to 1?

When we find out that a supply curve has an elasticity of 1, it's referred to as being unitary elastic. This means that the percentage change in quantity supplied is equal to the percentage change in price. On a practical level, if a product's price goes up by 10%, the amount of product supplied to the market also increases by 10%. It is a perfect one-to-one response and is crucial in understanding how suppliers react to price changes in the market.

The Importance of Economics Education

Economics education plays an imperative role in equipping students with the knowledge to analyze, understand, and make predictions about the economy, both on a small scale (individual businesses) and on a large scale (national economies). Key concepts, such as supply and demand curves, elasticity, and market equilibrium, are necessary for grasping how economies function.

Economics isn't just about money; it's about making choices and understanding the consequences of those choices. Introducing these concepts early in education not only prepares students for future studies in the field but also helps them become informed consumers and voters. Hands-on exercises, like the elasticity of a supply curve, are ideal examples of how quantitative analysis blends with economic theory to inform decision-making.

Quantitative Analysis in Economics

Quantitative analysis involves using mathematical models, statistical tools, and numerical data to understand and solve economic problems. By using quantitative techniques, economists can make sense of complex data, recognize patterns, and predict future trends.

In the exercise provided, quantitative analysis helps us define the elasticity of the supply curve mathematically. Through a calculation process which includes defining the supply curve equation, finding derivatives, and then applying those to the elasticity of supply formula, we achieve a clear, numerical understanding of how supply reacts to changes in price. It's the ability to apply these quantitative methods that allows us to move from theoretical concepts to practical, real-world applications in economics.