Problem 20 - Calculus Concepts

Question

A demand function and \(a\) supply function for the same commodity is given. a. Locate the shutdown point. Write a sentence of interpretation for this point. b. Locate the point of market equilibrium. Write a sentence of interpretation for this point. \(D(p)=-p^{2}+49\) billion units; \(S(p)=\left\\{\begin{array}{ll}0 & \text { for } p<4 \\ p^{2} & \text { for } p \geq 4\end{array}\right.\) billion units; \(p\) dollars per unit

Step-by-Step Solution

Answer

Shutdown point is at \( p = 4 \). Market equilibrium is \( p \approx 4.95 \), 24.5 billion units.

Step 1: Understand the functions

The demand function is given as \(D(p) = -p^2 + 49\), and the supply function is specified in two parts, \(S(p) = 0\) for \(p < 4\) and \(S(p) = p^2\) for \(p \geq 4\). The demand function represents how many units are demanded at price \(p\), and the supply function represents how many units are supplied at a given price.

Step 2: Locate the shutdown point

The shutdown point for the supplier occurs when the price \(p\) is just at the point where suppliers start supplying, that is, for \(p = 4\). This is because the supply function transitions from \(0\) to a positive value at \(p=4\). This implies the supplier is just willing to start supplying goods at this price. Interpretation: At \(p = 4\), suppliers begin to supply goods to the market.

Step 3: Locate the market equilibrium

The market equilibrium is where demand equals supply, \(D(p) = S(p)\). For \(p \geq 4\), set \(-p^2 + 49 = p^2\). So, \(-2p^2 + 49 = 0\). Solving gives \(p^2 = 24.5\), then \(p = \sqrt{24.5}\). Thus, equilibrating: \(p \approx 4.95\). Substitute \(p\) back into \(D(p)\) or \(S(p)\) to find the quantity, \(-4.95^2 + 49 \approx 24.5\) billion units. Interpretation: At \(p \approx 4.95\), demand meets supply with roughly 24.5 billion units, meaning the market is in balance.

Key Concepts

Demand and Supply Functions

Demand and supply functions are fundamental concepts in economics used to understand how markets operate. In our exercise, the demand function is given by the equation \(D(p) = -p^2 + 49\), which represents the quantity of goods consumers are willing to buy at different price levels \(p\). As the price increases, demand typically decreases, as shown by the negative coefficient of \(p^2\). This quadratic equation suggests that demand decreases more steeply as prices rise.

The supply function in our example is represented by a piecewise equation: \(S(p) = 0\) for \(p < 4\) and \(S(p) = p^2\) for \(p \geq 4\). This reflects the quantity of goods producers are willing to sell at different prices. The supply function indicates that suppliers are not willing to supply any goods if the price is less than \(4, highlighting that this is their minimum acceptable price to cover production costs. As the price crosses \)4, the supply starts to increase quadratically, suggesting a positive correlation between the price and quantity supplied.

Market Equilibrium

Market equilibrium occurs when the quantity demanded equals the quantity supplied. This is a crucial point because it indicates a balance where neither excess demand (shortage) nor surplus occurs. In our example, the equilibrium is located where the demand function \(D(p) = -p^2 + 49\) equals the supply function \(S(p) = p^2\). By setting these equal, \(-p^2 + 49 = p^2\), we can solve for \(p\) to find the equilibrium price.

Solving \(-2p^2 + 49 = 0\) gives us \(p^2 = 24.5\), leading to \(p = \sqrt{24.5}\), approximately \(p \approx 4.95\). At this price, both demand and supply are approximately 24.5 billion units, indicating that the market clears at this point with no leftover products or unmet demand. Therefore, the market is considered balanced, fostering stable conditions for both buyers and sellers.

Shutdown Point Analysis

The shutdown point for a producer is the price level at which they will cease production due to financial unsustainability. It is not profitable for suppliers to produce below this point because they cannot cover their variable costs. In our exercise, the shutdown point is identified by the supply function \(S(p) = 0\) for \(p < 4\). This means that at prices below \(4, suppliers will not supply any goods, marking \)4 as the critical price where suppliers are just willing to start production.

The shutdown point is pivotal for understanding producers' behavior in the market. At precisely \(p = 4\), suppliers are breaking even on their variable costs, beginning production at this minimum price. If the market price drops below this point, suppliers will halt production to avoid loss, while prices at or above this allow for continued operations and potential profit. Understanding this point helps in comprehending market entry and exit decisions by firms.