Question

Consider a competitive market for which the quantities demanded and supplied (per year) at various prices are given as follows: $$\begin{array}{|ccc|} \hline \begin{array}{c} \text { PRICE } \\ \text { (DOLLARS) } \end{array} & \begin{array}{c} \text { DEMAND } \\ \text { (MILLIONS) } \end{array} & \begin{array}{c} \text { SUPPLY } \\ \text { (MILIONS) } \end{array} \\ \hline 60 & 22 & 14 \\ \hline 80 & 20 & 16 \\ \hline 100 & 18 & 18 \\ \hline 120 & 16 & 20 \\ \hline \end{array}$$ a. Calculate the price elasticity of demand when the price is \(\$ 80\) and when the price is \(\$ 100\). b. Calculate the price elasticity of supply when the price is \(\$ 80\) and when the price is \(\$ 100\). c. What are the equilibrium price and quantity? d. Suppose the government sets a price ceiling of \(\$ 80 .\) Will there be a shortage, and if so, how large will it be?

Short answer

a. The price elasticity of demand at $80 is 0.4, and at $100 is 0.55. b. The price elasticity of supply at $80 is -0.5, and at $100 is 0.5. c. The equilibrium price is $100 and the equilibrium quantity is 18 million units. d. If a price ceiling at $80 is introduced, there will be a shortage of 4 million.

Step-by-step solution

Calculate the price elasticity of demand at $80

To find the price elasticity of demand at $80, use the formula \(E_d = \frac{%ΔQ}{%ΔP}\). In this case, %ΔP is the percent change from $80 to $100, or \( \frac{100 - 80}{80} = 0.25 \). %ΔQ is the percent change in quantity demanded, or \( \frac{20 - 18}{20} = 0.10 \). Plugging these values in, \(E_d = \frac{0.10}{0.25} = 0.4 .\) So, the price elasticity of demand at $80 is 0.4.

Calculate the price elasticity of demand at $100

Repeating the process for the price of $100, %ΔP is the percent change from $100 to $120, or \( \frac{120 - 100}{100} = 0.20 \). %ΔQ is the percent change in quantity demanded, or \( \frac{18 - 16}{18} = 0.11 \). Plugging these in, \(E_d = \frac{0.11}{0.20}=0.55 .\) Therefore, the price elasticity of demand at $100 is 0.55.

Calculate the price elasticity of supply at $80 and $100

The calculation for supply elasticity is similar to demand. At $80, %ΔP is \( \frac{100 - 80}{80} = 0.25 \) and %ΔQ is \( \frac{16 - 18}{16} = -0.125 \). Thus, \(E_s = \frac{-0.125}{0.25} = -0.50 .\) At $100, %ΔP is \( \frac{120 - 100}{100} = 0.20 \) and %ΔQ is \( \frac{20 - 18}{20} = 0.10 \). Thus, \(E_s = \frac{0.10}{0.20} = 0.50 .\) The price elasticity of supply is -0.50 at $80 and 0.50 at $100.

Determine the equilibrium price and quantity

The equilibrium price and quantity is found where quantity demanded equals quantity supplied. From the table, it can be seen that this occurs at a price of $100, with a quantity of 18 million. Therefore, the equilibrium price is $100, and the equilibrium quantity is 18 million.

Analyze the impact of a price ceiling at $80

A price ceiling means that prices can't rise above a certain level. If a price ceiling is set at $80, the quantity demanded will exceed the quantity supplied. Checking the table, at $80, quantity demanded is 20 million, but quantity supplied is only 16 million. Therefore, there will be a shortage, and it will be 4 million units.

Key concepts

Price Elasticity of Demand

The price elasticity of demand measures how sensitive the quantity demanded of a good is to a change in its price. It's an important concept in understanding how demand reacts to pricing in a competitive market.

Calculating the price elasticity of demand involves using the formula:

• \(E_d = \frac{\%\Delta Q_d}{\%\Delta P}\)

Where \(\%\Delta Q_d\) is the percentage change in quantity demanded and \(\%\Delta P\) is the percentage change in price.

For example, when the price increases from \(80 to \)100, the quantity demanded decreases from 20 million to 18 million. Calculating the percentage changes and applying the formula gives us the elasticity values. A price elasticity of demand less than 1, as calculated, indicates that the demand is inelastic. This means consumers are not very responsive to price changes in this range.

It's important to understand elasticity to anticipate how changes in price can impact sales volumes.

Price Elasticity of Supply

Price elasticity of supply measures how responsive the quantity supplied is to a change in price. This can influence how suppliers react to market changes and price alterations.

The elasticity is calculated in a similar manner to demand elasticity:

• \(E_s = \frac{\%\Delta Q_s}{\%\Delta P}\)

Where \(\%\Delta Q_s\) is the percentage change in quantity supplied and \(\%\Delta P\) is the percentage change in price.

For the example provided, calculating at prices \(80 and \)100, we determine how supply changes with price variations. At \(80, the elasticity is negative, meaning supply doesn’t respond as expected to a price increase (likely due to constraints in production in the short term). At \)100, the elasticity of supply is 0.50, indicating some responsiveness to price changes.

Understanding this concept helps predict how suppliers might adjust the quantity they bring to market when prices fluctuate.

Equilibrium Price

The equilibrium price in a competitive market occurs where the quantity demanded by consumers equals the quantity supplied by producers. This balance prevents shortages and surpluses, allowing efficient resource allocation.

In the problem presented, the equilibrium price is identified by matching the quantities in the demand and supply columns of the table provided. At $100, both demand and supply are 18 million units, indicating that this is the equilibrium point. At this price, the market clears, meaning all that is supplied is consumed, and there is neither excess nor shortage.

The equilibrium price and quantity are crucial as they signal the point at which the market functions efficiently without external forces, allowing economic balance.

Price Ceiling

A price ceiling is a government-imposed limit on how high a price can be charged in the market. It's generally intended to protect consumers from prices that are perceived to be too high.

In the given scenario, a price ceiling of $80 is set. However, at this price, the quantity demanded (20 million) exceeds the quantity supplied (16 million), leading to a situation of excess demand or a shortage.

This shortage amounts to 4 million units (20 million demanded minus 16 million supplied).

Price ceilings can create inefficiencies in the market, where the lack of supply at the lower price may drive up demand, resulting in shortages. Understanding price ceilings helps in analyzing potential market distortions, and the need for measures to address the resultant excess demand.