D(x) is the price, in dollars per unit, that consumers are willing to pay for x units of an item, and S(x) is the price, in dollars per unit, that producers are willing to accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. D(x) = (x - 9)?, S(x) = x² + 2x + 21

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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### Supply and Demand Analysis

**Problem Statement:**

Let \( D(x) \) represent the price, in dollars per unit, that consumers are willing to pay for \( x \) units of an item. Similarly, \( S(x) \) signifies the price, in dollars per unit, that producers are willing to accept for \( x \) units. Your tasks are to determine:
- (a) The equilibrium point.
- (b) The consumer surplus at the equilibrium point.
- (c) The producer surplus at the equilibrium point.

**Given Functions:**
- Demand function: \( D(x) = (x - 9)^2 \)
- Supply function: \( S(x) = x^2 + 2x + 21 \)

**Steps to Solve:**

1. **Equilibrium Point:**
   - Set \( D(x) = S(x) \) and solve for \( x \).

2. **Consumer Surplus:**
   - Calculate the area between the demand curve and the equilibrium price up to the equilibrium quantity.

3. **Producer Surplus:**
   - Calculate the area between the supply curve and the equilibrium price up to the equilibrium quantity.

By solving these equations and calculations, you will understand how consumer and producer behaviors balance in a market to reach an equilibrium point, and how surpluses are quantified at that point.
Transcribed Image Text:### Supply and Demand Analysis **Problem Statement:** Let \( D(x) \) represent the price, in dollars per unit, that consumers are willing to pay for \( x \) units of an item. Similarly, \( S(x) \) signifies the price, in dollars per unit, that producers are willing to accept for \( x \) units. Your tasks are to determine: - (a) The equilibrium point. - (b) The consumer surplus at the equilibrium point. - (c) The producer surplus at the equilibrium point. **Given Functions:** - Demand function: \( D(x) = (x - 9)^2 \) - Supply function: \( S(x) = x^2 + 2x + 21 \) **Steps to Solve:** 1. **Equilibrium Point:** - Set \( D(x) = S(x) \) and solve for \( x \). 2. **Consumer Surplus:** - Calculate the area between the demand curve and the equilibrium price up to the equilibrium quantity. 3. **Producer Surplus:** - Calculate the area between the supply curve and the equilibrium price up to the equilibrium quantity. By solving these equations and calculations, you will understand how consumer and producer behaviors balance in a market to reach an equilibrium point, and how surpluses are quantified at that point.
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