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Question 1200875: How do I write a demand and a supply curve expressed in the form of an exponential and
logarithmic functions respectively?
Then how would I Demonstrate graphically or otherwise, so that the
functions reflect all the characteristics of a demand and supply curve?
Answer by asinus(45)   (Show Source):
You can put this solution on YOUR website! **1. Demand Curve (Exponential)**
* **Function:**
* **Q_d = a * e^(-bP)**
* Where:
* Q_d is the quantity demanded
* P is the price
* a and b are positive constants
* **Characteristics:**
* **Negative Slope:** As price (P) increases, the quantity demanded (Q_d) exponentially decreases. This captures the inverse relationship between price and quantity demanded.
* **Asymptotic to the Price Axis:** As price increases indefinitely, the quantity demanded approaches zero but never reaches it.
**2. Supply Curve (Logarithmic)**
* **Function:**
* **Q_s = c * ln(P) + d**
* Where:
* Q_s is the quantity supplied
* P is the price
* c and d are constants (c should be positive to ensure a positive slope)
* **Characteristics:**
* **Positive Slope:** As price (P) increases, the quantity supplied (Q_s) increases logarithmically. This reflects the direct relationship between price and quantity supplied.
* **Increasing at a Decreasing Rate:** The rate of increase in quantity supplied slows down as price increases.
**Demonstrating Graphically**
1. **Choose Values for Constants:** Select appropriate values for the constants (a, b, c, and d) in the demand and supply functions to create realistic curves. For example:
* **Demand:** Q_d = 100 * e^(-0.05P)
* **Supply:** Q_s = 20 * ln(P) + 50
2. **Plot the Curves:**
* Use graphing software (like Excel, Desmos, or a graphing calculator) to plot the demand and supply curves on the same graph.
* **X-axis:** Price (P)
* **Y-axis:** Quantity (Q_d or Q_s)
3. **Observe Characteristics:**
* **Demand Curve:** The curve should slope downward, showing a decrease in quantity demanded as price increases. It should approach but never touch the price axis.
* **Supply Curve:** The curve should slope upward, showing an increase in quantity supplied as price increases. The slope should gradually decrease as price increases.
4. **Equilibrium:**
* Identify the point where the demand and supply curves intersect. This point represents the equilibrium price and quantity.
**Note:**
* These are simplified examples. Real-world demand and supply curves can be more complex and may involve other factors like consumer preferences, input costs, technology, and government regulations.
* The specific values of the constants (a, b, c, and d) will significantly impact the shape and position of the curves.
By following these steps, you can effectively demonstrate the characteristics of demand and supply curves using exponential and logarithmic functions.
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