Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Find the consumer surplus and producer surplus for the pair of demand and supply equations. Supply \(p=15+0.00004 x\) Demand $$p=60-0.00001 x$$

Step-by-Step Solution

Answer

The consumer surplus is 4,050,000 and the producer surplus is 16,200,000.

Step 1: Determine the equilibrium price and quantity.

First, find the equilibrium by setting the supply equation equal to the demand equation and solving for \(x\), which gives \[15 + 0.00004x = 60 - 0.00001x\]This can be simplified to \[0.00005x = 45\]Finally, solving for \(x\) gives\[x = 900,000\]Then, substitute \(x = 900,000\) into either the demand or supply equation to find the equilibrium price:\[p = 60 - 0.00001 \times 900,000 = 51\]

Step 2: Calculate the Consumer Surplus

The consumer surplus (CS) is the area of the triangle formed by the demand curve, the price axis, and the line \(p = p_0\). The formula for consumer surplus is\[CS = 0.5 \times (p_{max} - p_{0}) \times x_0\]Substituting the given values gives:\[CS = 0.5 \times (60 - 51) \times 900,000 = 4,050,000\]

Step 3: Calculate the Producer Surplus

The producer surplus (PS) is the area of the triangle formed by the supply curve, the price axis, and the line \(p = p_0\). The formula for producer surplus is\[PS = 0.5 \times (p_{0} - p_{min}) \times x_0\]Substituting our values gives:\[PS = 0.5 \times (51 - 15) \times 900,000 = 16,200,000\]

Key Concepts

Consumer Surplus

Consumer surplus represents the difference between what consumers are willing to pay and what they actually pay. Imagine you're at the store, ready to pay up to \(60 for a new gadget, but you find it's priced at \)51. That \(9 is consumer surplus, your saving at this price.
In a market graph, consumer surplus is the area above the equilibrium price and below the demand curve. It's a triangle under the curve where the top point represents maximum willingness to pay.

• The height of the triangle is the difference between the highest price consumers will pay and the actual market price.
• The base is the quantity sold in the market.

To find consumer surplus, use the formula: \[CS = 0.5 \times (p_{\text{max}} - p_0) \times x_0\]In our example with prices from \)60 down to \(51 and selling 900,000 units, the consumer surplus is a hearty \)4,050,000. This represents the total extra value consumers receive, beyond the amount they pay.

Producer Surplus

Producer surplus is a measure of producer welfare. It calculates the extra amount producers receive from selling a good over what they were willing to accept. Imagine you expect to sell your homemade cake for at least \(15, but the market price is \)51. That means you're making \(36 more per cake.
The area below the market price and above the supply curve on a graph represents the producer surplus. It forms a triangular section where:

• The height is the difference between the equilibrium price and the minimum price producers would accept.
• The base is again the quantity sold.

To calculate producer surplus, the formula is:\[PS = 0.5 \times (p_0 - p_{\text{min}}) \times x_0\]Substituting in our scenario of prices from \)15 up to \(51 with 900,000 units sold, we find the producer surplus is an impressive \)16,200,000. This highlights the additional benefit producers gain by selling at the market price.

Demand and Supply

Demand and supply are the heart of economic analysis, shaping prices and quantities in the market. Demand refers to how much of a product consumers are willing and able to purchase at various prices. Conversely, supply represents how much producers are willing to create and sell.

• Demand curve slopes downward, indicating that higher prices lead to less quantity demanded.
• Supply curve slopes upward, meaning higher prices incentivize producers to supply more.

Understanding demand and supply helps determine market dynamics. In our example, the demand equation is given as:
\(p = 60 - 0.00001 x\)
The supply equation is:\(p = 15 + 0.00004 x\)
These equations tell us how price (\(p\)) varies based on different quantities (\(x\)). The intersection where these curves meet represents the market equilibrium.

Equilibrium Price and Quantity

Equilibrium occurs where the quantity demanded equals the quantity supplied, balancing market forces. At this point, the market is most efficient, with no inherent tendency to change.

• The equilibrium price is where the demand and supply curves intersect.
• The equilibrium quantity is the amount bought and sold at this price.

To find equilibrium, set the demand and supply equations equal and solve for quantity:\[15 + 0.00004x = 60 - 0.00001x\]Simplifying:\[0.00005x = 45\]So,\[x = 900,000\]At equilibrium quantity, plug back into either equation to find the equilibrium price, which in this case is:\[p = 51\]Hence, our equilibrium is at 900,000 units sold at $51 each. This establishes a state where market supply matches demand, avoiding surpluses or shortages.