3. Consider the 6x6 system of equations below, of 3 goods and their respective demand
and supply functions:
Qd1 = 23 - 5P1 + P2 + P3 Qs1 = -8 + 6P1
Qd2 = 15 + P1
-3P2 + 2P3 Qs2 = -11 + 3P2
Qd3 = 19 + P1 +2P2
- 4P3 Qs3 = -5 + 3P3
a. Are these 3 goods substitutes or complements? Explain in detail using jargon.
b. Solve the system for each good’s equilibrium values: (Q1*, P1*), (Q2*, P2*) and
(Q3*, P3*)

### (a) Determining Whether Goods are Substitutes or Complements:

In economics, goods can be classified as **substitutes** or **complements** based on how the demand for one good changes when the price of another good changes.

- **Substitutes**: If the demand for good \( i \) increases when the price of good \( j \) increases, then the two goods are considered substitutes. Mathematically, the cross-price elasticity of demand is positive.
  
- **Complements**: If the demand for good \( i \) decreases when the price of good \( j \) increases, the goods are complements. In this case, the cross-price elasticity of demand is negative.

Let's analyze the demand functions provided:

#### Demand Functions:
- \( Qd1 = 23 - 5P1 + P2 + P3 \)
- \( Qd2 = 15 + P1 - 3P2 + 2P3 \)
- \( Qd3 = 19 + P1 + 2P2 - 4P3 \)

##### Cross-Price Effects:
1. **Effect of \( P2 \) on \( Qd1 \):**  
   - \( \frac{\partial Qd1}{\partial P2} = 1 \), meaning when \( P2 \) increases, the demand for good 1 increases.
   - **Conclusion:** Good 1 and good 2 are **substitutes**.

2. **Effect of \( P3

Consider the 6x6 system of equations below, of 3 goods and their respective demand and supply functions: Qd1 = 23 - 5P1 + P2 + P3 Qs1 = -8 + 6P1 Qd2 = 15 + P1 -3P2 + 2P3 Qs2 = -11 + 3P2 Qd3 = 19 + P1 +2P2 - 4P3 Qs3 = -5 + 3P3 a. Are these 3 goods substitutes or complements? Explain in detail using jargon. b. Solve the system for each good’s equilibrium values: (Q1*, P1*), (Q2*, P2*) and (Q3*, P3*)

Learn how to analyze whether three goods are substitutes or complements using a 6x6 system of supply and demand equations. The problem covers cross-price elasticity and solving for equilibrium values. Ideal for students in undergraduate economics or mathematics courses.

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