Question

Suppose that the demand for broccoli is given by $$Q=1,000-5 P$$ where \(Q\) is quantity per year measured in hundreds of bushels and \(P\) is price in dollars per hundred bushels. The long-run supply curve for broccoli is given by $$Q=4 P-80$$ a. Show that the equilibrium quantity here is \(Q=400 .\) At this output, what is the equilibrium price? How much in total is spent on broccoli? What is consumer surplus at this equilibrium? What is producer surplus at this equilibrium? b. How much in total consumer and producer surplus would be lost if \(Q=300\) instead of \(Q=400 ?\) c. Show how the allocation between suppliers and demanders of the loss of total consumer and producer surplus described in part (b) depends on the price at which broccoli is sold. How would the loss be shared if \(P=140 ?\) How about if \(P=95 ?\) d. What would be the total loss of consumer and producer surplus if \(Q=450\) rather than \(Q=400 ?\) Show that the size of this total loss also is independent of the price at which the broccoli is sold.

Short answer

Based on the given demand and supply equations, the equilibrium quantity was found to be 400, and the equilibrium price was $120. Consumer and producer surpluses at this equilibrium are both $16,000. When the quantity is decreased to 300, the total loss in consumer and producer surplus was determined to be $12,500, with a loss of $7,000 in consumer surplus and $5,500 in producer surplus. The allocation of loss between consumer and producer surplus depends on the price at which broccoli is sold, as shown with different prices used in part c. When the quantity is increased to 450, the total loss in consumer and producer surplus was found to be $500, with a loss of -$2,000 in consumer surplus (an increase) and $2,500 in producer surplus (a decrease). This total loss is independent of the price at which broccoli is sold.

Step-by-step solution

Set the demand equation equal to the supply equation

In order to find the equilibrium quantity and price, we need to set the demand equation equal to the supply equation: $$1000 - 5P = 4P - 80$$

Solve for P

Add 5P and 80 to both sides of the equation and solve for P: $$5P + 4P = 1000 + 80$$ $$9P = 1080$$ $$P = 120$$ So the equilibrium price is $120.

Solve for Q

Now that we have the equilibrium price, we can find the equilibrium quantity by substituting P into either the demand or supply equation. Let's use the demand equation: $$Q = 1000 - 5(120)$$ $$Q = 1000 - 600$$ $$Q = 400$$ So the equilibrium quantity is 400. #a. Calculate total spending on broccoli#

Multiply the equilibrium quantity by the equilibrium price

To find the total spending on broccoli, multiply the equilibrium quantity (400) by the equilibrium price (120): Total spending = Q * P Total spending = 400 * 120 Total spending = $48,000 #a. Calculate consumer surplus and producer surplus#

Find consumer surplus

To calculate the consumer surplus, we need to find the area of the triangle above the equilibrium price on the demand curve. The base of the triangle is the difference between the maximum price and the equilibrium price (1000/5 - 120 = 80), and the height is the equilibrium quantity (400). The formula for the area of a triangle is 0.5 * base * height: Consumer surplus = 0.5 * 80 * 400 Consumer surplus = $16,000

Find producer surplus

To calculate the producer surplus, we need to find the area of the triangle below the equilibrium price on the supply curve. The base of the triangle is the difference between the equilibrium price and the minimum price (120 - 80/4 = 80), and the height is the equilibrium quantity (400). The formula for the area of a triangle is 0.5 * base * height: Producer surplus = 0.5 * 80 * 400 Producer surplus = $16,000 #b. Calculate the total loss in consumer and producer surplus when Q = 300#

Find the new price at Q = 300

Substitute Q = 300 into the demand equation to find the new price: $$300 = 1000 - 5P$$ $$5P = 1000 - 300$$ $$P = \frac{700}{5}$$ $$P = 140$$

Calculate the new consumer and producer surplus at Q = 300

At the new price and quantity, the consumer surplus is now: Consumer surplus = 0.5 * (1000/5 - 140) * 300 Consumer surplus = $9,000 Producer surplus is now: Producer surplus = 0.5 * (140 - 80/4) * 300 Producer surplus = $10,500

Calculate the total loss in consumer and producer surplus

Subtract the new values from the original consumer and producer surplus to find the loss in each: Loss in consumer surplus = \(16,000 - \)9,000 Loss in consumer surplus = $7,000 Loss in producer surplus = \(16,000 - \)10,500 Loss in producer surplus = $5,500 Total loss in consumer and producer surplus = \(7,000 + \)5,500 Total loss = $12,500 #c. Show how the allocation between suppliers and demanders of the loss of total consumer and producer surplus depends on the price at which broccoli is sold# Price = \(140: From Step 9, the loss in consumer surplus is \)7,000 and the loss in producer surplus is $5,500. Price = $95: Let's first find the quantity at this price. $$Q = 1000 - 5(95)$$ $$Q = 1000 - 475$$ $$Q = 525$$ The consumer surplus is now: 0.5 * (1000/5 - 95) * 525 = $11,362.50 The producer surplus is now: 0.5 * (95 - 80/4) * 525 = $12,637.50 Loss in consumer surplus = 16,000 - 11,362.50 = $4,637.50 Loss in producer surplus = 16,000 - 12,637.50 = $3,362.50 Total loss for P = \(95: \)4,637.50 + \(3,362.50 = \)8,000 As we can see, the allocation of loss between consumer and producer surplus depends on the price at which broccoli is sold. #d. Calculate the total loss in consumer and producer surplus when Q = 450#

Find the new price at Q = 450

Substitute Q = 450 into the demand equation to find the new price: $$450 = 1000 - 5P$$ $$5P = 1000 - 450$$ $$P = \frac{550}{5}$$ $$P = 110$$

Calculate the new consumer and producer surplus at Q = 450

At the new price and quantity, the consumer surplus is now: Consumer surplus = 0.5 * (1000/5 - 110) * 450 Consumer surplus = $18,000 Producer surplus is now: Producer surplus = 0.5 * (110 - 80/4) * 450 Producer surplus = $13,500

Calculate the total loss in consumer and producer surplus

Subtract the new values from the original consumer and producer surplus to find the loss in each: Loss in consumer surplus = \(16,000 - \)18,000 Loss in consumer surplus = -$2,000 Loss in producer surplus = \(16,000 - \)13,500 Loss in producer surplus = $2,500 Total loss in consumer and producer surplus = -\(2,000 + \)2,500 Total loss = $500 Notice that the total loss in consumer and producer surplus when the quantity is 450 is independent of the price at which broccoli is sold (i.e., $500).

Key concepts

Demand and Supply Equations

Understanding demand and supply equations is crucial in microeconomics. These equations represent the relationship between price and the quantity of a product that consumers are willing to purchase (demand) or producers are willing to sell (supply).

For demand, the equation typically has a negative slope, indicating that as the price increases, the quantity demanded decreases. Conversely, the supply equation usually has a positive slope, showing that as the price rises, the quantity supplied also increases.

In our broccoli market example, the demand equation is expressed as \(Q = 1000 - 5P\), and the supply equation as \(Q = 4P - 80\). To find the equilibrium, these two equations are set equal to each other, and solved for the price (\(P\)) and quantity (\(Q\)). The point where the two curves intersect is crucial for market analysis as it dictates the equilibrium conditions.

Equilibrium Price and Quantity

The equilibrium price and quantity are where the market for a good or service is stable. This equilibrium occurs where the demand and supply curves intersect, representing a balance between what consumers are willing to pay and what producers are willing to accept.

In our calculation, setting the demand and supply equations equal to one another and solving for \(P\) gave us the equilibrium price of \(120 per hundred bushels of broccoli. Substituting this price back into either equation, we found the equilibrium quantity to be 400 hundred bushels.

Total Spending on Broccoli

Multiplying the equilibrium quantity by the equilibrium price yields the total spending on broccoli at equilibrium, which in this case is \)48,000. This scenario assumes the market functions perfectly without any external influences or government interventions.

Consumer and Producer Surplus

Consumer and producer surplus are measures of the economic welfare that consumers and producers derive from participating in the market. The consumer surplus is the difference between what consumers are willing to pay and what they actually pay, while the producer surplus is the difference between the market price and the lowest price at which producers are willing to sell.

In our broccoli example, the consumer surplus was calculated as \(16,000 and the producer surplus was also \)16,000 at equilibrium. These values represent the areas of the respective triangles above and below the equilibrium price on the demand and supply curves.

Changes in Surplus

When the quantity supplied and demanded deviates from equilibrium, as in the scenarios where \(Q=300\) or \(Q=450\), there's a loss in total surplus, representing a reduction in economic welfare. The calculations show how this loss is divided between consumers and producers depending on the selling price, underlining the importance of equilibrium for optimal market outcomes.