Question

Given the following market supply, \(Q^{S}=210-5 P\), find the producers' surplus when \(P=6\) and \(P=8\). Calculate a supply schedule and then draw a graph showing producer surplus.

Short answer

Producer surplus is 3240 at \(P=6\) and 2890 at \(P=8\).

Step-by-step solution

Understand the Concept of Producer Surplus

Producer surplus is the amount producers are willing to accept for a good versus what they actually receive. It's calculated as the area above the supply curve and below the price level.

Write Down the Supply Equation

The market supply equation is given as \(Q^{S} = 210 - 5P\). Here, \(Q^{S}\) represents quantity supplied, and \(P\) represents the price level.

Calculate Quantity Supplied at Given Prices

Substitute \(P = 6\) into the supply equation: \(Q^{S} = 210 - 5(6) = 210 - 30 = 180\). Substitute \(P = 8\) into the supply equation: \(Q^{S} = 210 - 5(8) = 210 - 40 = 170\).

Determine the Minimum Price Producers Will Accept (Price at Zero Quantity)

Set \(Q^{S} = 0\) to find the minimum price the producers will accept, which is the price at zero quantity: \(0 = 210 - 5P\). Solve for \(P\): \(5P = 210\) thus \(P = 42\).

Calculate Producer Surplus with P=6

Producer surplus is the area of the triangle above the supply curve and below the price \(P = 6\). The height of this triangle is \(6 - 42 = -36\) (we consider distance from zero, so take absolute value), and the base is \(180\). So, producer surplus = \(\frac{1}{2} \times 180 \times 36 = 3240\).

Calculate Producer Surplus with P=8

Similarly, producer surplus when \(P = 8\) is an area of a triangle. The height is \(8 - 42 = -34\), and the base is \(170\). Thus, producer surplus = \(\frac{1}{2} \times 170 \times 34 = 2890\).

Graphical Representation

On a graph, plot the supply curve starting at (0, 42). Mark the coordinates for \(P=6\) and \(P=8\) on the horizontal axis with their respective quantities 180 and 170. Shade the areas representing the producer surplus (triangles) for each price level.

Key concepts

Supply Curve

The supply curve is a vital concept in economics, representing the relationship between the price of a good and the quantity of that good that producers are willing to supply. In essence, as the price of a good increases, producers are generally willing to supply more, following the law of supply. The curve is usually upward sloping, indicating this direct relationship between price and quantity supplied.

• The higher the price, the more incentive there is to produce and supply more.
• A lower price generally results in a decrease in quantity supplied.

In graphical terms, the supply curve can shift depending on various factors such as changes in production technology, input costs, or government policies. These shifts represent changes in the quantities supplied at each price level. Understanding the supply curve helps illustrate the calculation of producer surplus, as the curve aids in identifying the minimum price that producers are willing to accept for providing a given quantity in the market.

Market Supply Equation

The market supply equation is a mathematical representation of the supply curve. It gives us a formula that shows how the quantity supplied in a market (\(Q^S\)) responds to changes in the price (\(P\)) of a good or service. For instance, our exercise gives us a specific supply equation:\[Q^{S} = 210 - 5P\]

• Here, \(Q^S\) represents the quantity supplied,
• \(P\) is the price of the good.
• The equation illustrates how quantity supplied decreases as price increases, indicating an inverse relationship.

In practical terms, such equations help businesses and economists predict how much of a product will be offered in a market at different price levels. By plugging different price values into the equation, you can derive the corresponding quantities, forming the basis of supply schedules and furthering analysis such as calculating producer surplus.

Graphical Analysis of Supply and Demand

Graphical analysis is a powerful visual tool that combines both supply and demand elements to better understand market dynamics. When we plot the supply curve on a graph, it typically starts from a point on the price axis where quantity supplied is zero and slopes upward to the right. In this context:

• The intersection of supply and demand curves determines the market equilibrium point, which includes the equilibrium price and quantity.
• Graphically, producer surplus appears as the area above the supply curve and below the market price.

In the exercise, we learn to visualize the producer surplus at specific price points by plotting the supply curve and identifying the triangular area between the curve and the horizontal price line.
This visual method allows students to see the real implications of producer surplus calculations instead of only relying on numerical data. Drawing graphs reinforces the understanding of how market mechanisms work and the impact of price changes on producer surplus.

Quantity Supplied Calculation

The calculation of quantity supplied is integral to understanding market dynamics and is straightforward when using a given market supply equation. As shown previously, the equation we dealt with is:\(Q^S = 210 - 5P\). To find the quantity supplied at various price points, it's essential to substitute different values of \(P\) into this equation.
Let's say:

• When \(P = 6\), substituting into the equation gives \(Q^{S} = 210 - 5(6) = 180\).
• When \(P = 8\), the equation results in \(Q^{S} = 210 - 5(8) = 170\).

By harnessing these calculations, one can chart the supply curve and better understand how producers respond to price variations.
This process exemplifies the quantity response of suppliers to price changes, an essential aspect when discussing economic supply and demand, and affords insights into market behavior and producer strategies.