Question

The equation for a supply curve is \(4 \mathrm{P}=\mathrm{Q}\). What is the elasticity of supply as price rises from 3 to \(4 ?\) What is the elasticity of supply as the price rises from 7 to 8 ? Would you expect these answers to be the same?

Short answer

The elasticity of supply is 1 when the price rises from 3 to 4 and from 7 to 8. We would expect these answers to be the same because the supply curve has a constant slope, making the price elasticity of supply constant, equal to 1 in this case.

Step-by-step solution

Find the initial and final quantities supplied for both cases

Using the given supply curve equation, \(4P = Q\), we will find the quantity supplied at the initial and final prices for both cases: Case 1 (Price rises from 3 to 4): - Initial price: \(P_1 = 3\) - Final price: \(P_2 = 4\) - Initial quantity supplied: \(Q_1 = 4P_1 = 4(3) = 12\) - Final quantity supplied: \(Q_2 = 4P_2 = 4(4) = 16\) Case 2 (Price rises from 7 to 8): - Initial price: \(P_1' = 7\) - Final price: \(P_2' = 8\) - Initial quantity supplied: \(Q_1' = 4P_1' = 4(7) = 28\) - Final quantity supplied: \(Q_2' = 4P_2' = 4(8) = 32\)

Calculate the percentage changes in price and quantity supplied

Next, we will calculate the percentage changes in price and quantity supplied for both cases: Case 1 (Price rises from 3 to 4): - \(\% \Delta P = \frac{P_2 - P_1}{P_1} \times 100\% = \frac{4 - 3}{3} \times 100\% = 33.33\% \) - \(\% \Delta Q = \frac{Q_2 - Q_1}{Q_1} \times 100\% = \frac{16 - 12}{12} \times 100\% = 33.33\% \) Case 2 (Price rises from 7 to 8): - \(\% \Delta P' = \frac{P_2' - P_1'}{P_1'} \times 100\% = \frac{8 - 7}{7} \times 100\% = 14.29\% \) - \(\% \Delta Q' = \frac{Q_2' - Q_1'}{Q_1'} \times 100\% = \frac{32 - 28}{28} \times 100\% = 14.29\% \)

Calculate the elasticity of supply for both cases

Now, we will use the percentage changes in price and quantity supplied to calculate the elasticity of supply for both cases: Case 1 (Price rises from 3 to 4): - \(ES = \frac{\% \Delta Q}{\% \Delta P} = \frac{33.33\%}{33.33\%} = 1\) Case 2 (Price rises from 7 to 8): - \(ES = \frac{\% \Delta Q'}{\% \Delta P'} = \frac{14.29\%}{14.29\%} = 1\) So, the elasticity of supply is 1 in both cases. Finally, we need to answer whether we would expect these answers to be the same or not. Since the supply curve has a constant slope, the percentage change in quantity supplied will always be the same as the percentage change in price, making the price elasticity of supply constant, equal to 1 in this case. Thus, we would expect the answers to be the same.

Key concepts

Supply Curve

The supply curve illustrates the relationship between the price of a good and the amount that suppliers are willing to produce and sell. For instance, in the supply curve equation given, which is \(4P = Q\), \(Q\) represents the quantity supplied, while \(P\) is the price of the good. This specific equation implies that with every unit increase in price, the quantity supplied increases by 4 units.
Imagine this as a straight line on a graph, where price is on the vertical axis and quantity on the horizontal. The slope of this line is crucial as it determines how the quantity supplied responds to changes in price. A steeper slope would mean a larger increase in quantity for every unit increase in price.

• The supply curve is typically upward-sloping, indicating that as prices increase, producers are willing to supply more of the goods, reflecting typical market behavior.
• An equation like \(4P = Q\) represents a linear supply curve, which behaves predictably, as seen when solving for varying prices.

Percentage Change

Percentage change is a useful way to understand the relative change, rather than the absolute change, in any given variable. It is calculated by taking the difference between the final and initial values, dividing by the initial value, and then multiplying by 100 to convert it to a percentage.
This method allows you to capture how significant a change is relative to the starting point. In the case of the problem, this concept is used both for the change in price and the change in quantity supplied. Let's explore this through examples:

• In Case 1, where the price increases from 3 to 4, the percentage change in price is calculated as: \(\frac{4 - 3}{3} \times 100\% = 33.33\%.\)
• Similarly, the percentage change in quantity from 12 to 16 is: \(\frac{16 - 12}{12} \times 100\% = 33.33\%.\)

These calculations show not just the direction of change, but also its relative size, providing a clearer picture of how the market responds to price changes. Understanding percentage change is key to calculating other important measures like elasticity.

Price Elasticity

Price elasticity of supply measures how responsive the quantity supplied is to a change in price. It is calculated using the formula: \[ES = \frac{\% \Delta Q}{\% \Delta P}\]where \(\% \Delta Q\) is the percentage change in quantity supplied and \(\% \Delta P\) is the percentage change in price.
This ratio indicates whether quantity changes more or less than price. In our problem, the elasticity of supply was calculated to be 1 for both price changes from 3 to 4 and 7 to 8. This is because the percentage change in quantity was exactly the same as the percentage change in price.
When elasticity equals 1, it means that the supply is unit elastic, implying that the quantity supplied changes by the same percentage as the price does. This kind of supply elasticity implies a proportional relationship between price and quantity.

• In a perfectly elastic supply, suppliers can change production instantly without a change in cost.
• With inelastic supply, production cannot be easily changed.
• Knowing elasticity helps businesses and economists predict how changes in price will affect supply levels.

Understanding elasticity is fundamental for tuning supply chain strategies and making pricing decisions. It shows how supply might change with new market conditions.