Problem 17 - 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. \(D(p)=35-7 \ln p\) million units; \(S(p)=\left\\{\begin{array}{ll}0 & \text { for } p<9 \\\ 3\left(1.081^{p}\right) & \text { for } p \geq 9\end{array}\right.\) million units: \(p\) dollars per unit

Step-by-Step Solution

Answer

Shutdown point is at \( p = 9 \). Market equilibrium is approximately at \( p = 12.24 \) with 18.51 million units.

Step 1: Determine the Shutdown Point

The shutdown point in the context of a supply curve is where the supplier can start providing the commodity. For this problem, this occurs at the smallest price where supply is non-zero. For this supply function, it is given as a piecewise function. For prices less than 9, the supply is zero, and for prices greater than or equal to 9, the supply becomes positive. Thus, the shutdown point occurs at \( p = 9 \). Interpretation: The shutdown point, \( p = 9 \), is the minimum price required for suppliers to enter the market and begin supplying the commodity.

Step 2: Calculate Demand and Supply at Equilibrium

The market equilibrium occurs where demand equals supply. For \( p \geq 9 \), set \( D(p) = S(p) \). The demand function is \( D(p) = 35 - 7 \ln p \) and the supply function for \( p \geq 9 \) is \( S(p) = 3(1.081^{p}) \). Equating them gives: \[ 35 - 7 \ln p = 3(1.081^{p}) \]

Step 3: Solve the Equation for Equilibrium Price

Solving \( 35 - 7 \ln p = 3(1.081^{p}) \) indicates a numerical or graphical solution may be necessary due to the complexity of this transcendental equation. Approximating solutions yields \( p \approx 12.24 \) (using a numerical solver or graphing calculator).

Step 4: Calculate Equilibrium Quantity

Substitute the equilibrium price back into either the demand or supply function to find the equilibrium quantity. Using the demand function:\[ D(12.24) = 35 - 7 \ln(12.24) \approx 18.51 \] million units. This value should match or approximately match the supply at this price.

Step 5: Interpret the Market Equilibrium

Interpretation: At the market equilibrium, when the price is approximately \( p = 12.24 \), both the demanded and supplied quantity equal approximately 18.51 million units. This price and quantity reflect a balance where consumers’ willingness to purchase matches suppliers’ willingness to sell.

Key Concepts

Shutdown Point

The shutdown point is a critical concept in understanding supply behavior in economics. In simple terms, the shutdown point is the lowest price at which a supplier is willing to supply a good or service. Below this price, the supplier would rather not produce at all due to the costs outweighing potential revenue. This is particularly pertinent in a situation where suppliers face a stepped or piecewise supply function.

For example, consider the supply function given in the exercise:

• For prices less than $9, the supply is 0.
• For prices greater than or equal to $9, the supply becomes positive and is described by a specific function.

The point where the supply curve switches from zero to positive is our shutdown point, thus at price $p=9$. This point serves as a boundary, ensuring that above it, suppliers can operate without losses. Therefore, if the market price falls below this shutdown threshold, suppliers halt production to avoid incurring losses.

Reaching the shutdown point marks the entry point into the market for suppliers, indicating they can cover their variable costs at the very least.

Demand Function

A demand function is a mathematical expression that illustrates the relationship between the price of a good and the amount consumers are willing and able to purchase. Generally, as prices rise, demand tends to fall, and vice versa; this relationship is depicted in the demand curve.

In the exercise, the demand function is expressed as:

• \(D(p)=35-7 \ln p\)

This specific function suggests that the quantity demanded (\(D(p)\)) depends on the natural logarithm of the price \(p\). Here are some important aspects of this function:

• The negative sign in front of the \(\ln p\) indicates that as the price \(p\) increases, the logarithmic term grows, reducing the overall demand.
• The constant 35 sets a base level of maximum potential demand when \(p\) is at its lowest practical level.

Understanding demand functions allows businesses and economists to predict how changes in price, whether due to market conditions, taxes, or other factors, can influence consumer purchasing behavior.

Supply Function

The supply function is another mathematical tool used to express how much of a good or service suppliers are willing to sell at different price levels. The supply curve usually slopes upwards, reflecting that higher prices incentivize suppliers to produce more.

In our exercise:

• The supply is a piecewise function where for \( p < 9 \), supply \(S(p)=0\), and for \( p \geq 9 \), supply becomes \( S(p)=3(1.081^{p})\).

This means:

• Below \(9, the supply is nonexistent as producers find it unviable to enter the market.
• Once prices are \)9 or above, supply increases exponentially with price \(p\), indicated by the base of the exponent greater than 1.

Such a supply function is reflective of markets where production or supply begins only when a certain minimum price level is reached, ensuring that suppliers can cover their fixed and variable costs. Beyond the shutdown point, the exponential nature of the function highlights the rapid increase in supply as price incentives grow.

For suppliers, understanding their supply function helps in setting production targets and making strategic pricing decisions to maximize profitability.