Question

(More difficult) The demand and supply curves for T-shirts in Touristtown, U.S.A., are given by the following equations: \(Q=24,000-500 P \quad Q=6,000+1,000 P\) where \(P\) is measured in dollars and \(Q\) is the number of T-shirts sold per year. a. Find the equilibrium price and quantity algebraically. b. If tourists decide they do not really like T-shirts that much, which of the following might be the new demand curve? $$ Q=21,000-500 P \quad Q=27,000-500 P $$ Find the equilibrium price and quantity after the shift of the demand curve. c. If, instead, two new stores that sell T-shirts open up in town, which of the following might be the new supply curve? \(Q=4,000-1,000 P \quad Q=9,000+1,000 P\) Find the equilibrium price and quantity after the shift of the supply curve

Short answer

Answer: A decrease in demand leads to a decrease in the equilibrium price from $12 to $10 and a decrease in the equilibrium quantity from 18,000 to 16,000 T-shirts. On the other hand, an increase in supply leads to a decrease in the equilibrium price from $12 to $9 and an increase in the equilibrium quantity from 18,000 to 19,500 T-shirts.

Step-by-step solution

Set up the equations

We are given the demand and supply equations for T-shirts: \(Q = 24000 - 500P\) (demand equation) and \(Q = 6000 + 1000P\) (supply equation). In equilibrium, the quantity demanded equals the quantity supplied, so we will set these two equations equal to each other to find the equilibrium price and quantity.

Solve for P

Set the two equations equal to each other: \(24000 - 500P = 6000 + 1000P\) Now, solve for \(P\): \(1500P = 18000\) \(P = 12\)

Solve for Q

Substitute the value of \(P\) back into either the demand or supply equation to find the equilibrium quantity: \(Q = 24000 - 500(12)\) \(Q = 24000 - 6000\) \(Q = 18000\) So, the equilibrium price is $12 and the equilibrium quantity is 18,000 T-shirts. #b. Determine the new demand curve and find the equilibrium price and quantity:#

Analyze the change in demand

If tourists decide they do not like T-shirts that much, the demand for T-shirts will decrease. In this case, the new demand curve may be: \(Q = 21000 - 500P\)

Find the new equilibrium price and quantity

Set the new demand equation equal to the supply equation: \(21000 - 500P = 6000 + 1000P\) Now, solve for \(P\): \(1500P = 15000\) \(P = 10\) Now, substitute the value of \(P\) back into the new demand equation to find the new equilibrium quantity: \(Q = 21000 - 500(10)\) \(Q = 21000 - 5000\) \(Q = 16000\) So, the new equilibrium price is $10 and the new equilibrium quantity is 16,000 T-shirts. #c. Determine the new supply curve and find the equilibrium price and quantity:#

Analyze the change in supply

If two new stores that sell T-shirts open up in town, the supply of T-shirts will increase. In this case, the new supply curve may be: \(Q = 6000 + 1500P\)

Find the new equilibrium price and quantity

Set the demand equation equal to the new supply equation: \(24000 - 500P = 6000 + 1500P\) Now, solve for \(P\): \(2000P = 18000\) \(P = 9\) Now, substitute the value of \(P\) back into the demand equation to find the new equilibrium quantity: \(Q = 24000 - 500(9)\) \(Q = 24000 - 4500\) \(Q = 19500\) So, the new equilibrium price is $9 and the new equilibrium quantity is 19,500 T-shirts.

Key concepts

Demand Curve

The demand curve is a graphical representation of the relationship between the price of a good and the quantity demanded by consumers. In essence, it shows how much of a product people are willing and able to purchase at any given price. For the demand curve of T-shirts in Touristtown, we have the equation \(Q = 24,000 - 500P\). Here, \(Q\) represents the quantity demanded, and \(P\) is the price level of each T-shirt.

This equation illustrates a basic law of demand: as the price \(P\) increases, the quantity demanded \(Q\) decreases, reflecting a negative slope.

• A decrease in the popularity of T-shirts can shift this curve. For example, a new demand curve \(Q = 21,000 - 500P\) suggests a lower willingness to purchase the same quantity at previous prices, indicating that less is demanded at each price point.

Understanding shifts in and movements along demand curves is crucial for determining how changes in preferences affect market outcomes.

Supply Curve

The supply curve reflects how much producers are willing to sell at different price levels. It is generally upward sloping, indicating that as the price increases, the quantity supplied also increases. For Touristtown, the supply curve equation is \(Q = 6,000 + 1,000P\), with \(Q\) as the quantity supplied, influenced by the price \(P\).

This upward slope reflects producers' motivations: higher prices cover more of the costs and potentially increase profits, encouraging more production.

• Factors like the opening of new stores can shift the supply curve rightward, implying a greater quantity supplied at each price level. For instance, shifting the supply curve to \(Q = 6,000 + 1,500P\) indicates that more is supplied due to increased production capacity or more competitive market forces.

The understanding and analysis of supply curves help in interpreting how market changes and production expansions influence supply.

Market Equilibrium

Market equilibrium occurs when the quantity demanded equals the quantity supplied, leading to a stable price and quantity in the market. In equations, it is where the demand curve meets the supply curve. For T-shirts in Touristtown, we solve for equilibrium by equating demand \(Q = 24,000 - 500P\) and supply \(Q = 6,000 + 1,000P\) equations:

\[24,000 - 500P = 6,000 + 1,000P\]

This results in the equilibrium price \(P = 12\), where 18,000 T-shirts are sold.

With demand shifts (e.g., due to changing tastes), we find new equilibriums, \(P = 10\) and \(Q = 16,000\), by solving \(21,000 - 500P = 6,000 + 1,000P\) after preference changes. Similarly, with enhanced supply due to new stores, solving \(24,000 - 500P = 6,000 + 1,500P\) provides \(P = 9\) and \(Q = 19,500\).

Understanding market equilibrium aids in predicting how shifts affect both price and quantity in real-world scenarios.

Price Adjustment

Price adjustment refers to the market's response to a disequilibrium condition—a mismatch between demand and supply. When a change occurs in either demand or supply, prices adjust to restore market equilibrium.

For example, if demand decreases (customers buy less), surplus results at the current price. Prices will fall until equilibrium is reached again, as seen when the equilibrium price decreased from \\(12 to \\)10 due to diminished demand. Similarly, supply increases (more T-shirts available) might result in excess supply, causing prices to reduce further to \$9.

• This self-correcting nature signifies how markets inherently seek balance.
• Thus, prices not only reflect costs and consumer preferences but are signals helping adjust quantities available and purchased.

Understanding price adjustments assists in grasping market dynamics and the role of prices in economic stability.