# DECS-S-0034: Demand Function & Correlation Coefficient Explained
## From the whiteboard: how these concepts connect to your assignments

---

## PART 1: The Demand Function

### Whiteboard Content: qD = f(p)

**Quantity Demanded is a function of Price.** This is the Law of Demand:

- Price goes **down** (p₁ → p₂) → Quantity demanded goes **up** (q₁ → q₂)
- Movement is ALONG the demand curve (not a shift of the curve)
- Downward-sloping D curve = people buy more when it's cheaper

### Graph: Supply & Demand

```
 $/unit
   ↑
p₁ |----·  (high price → low quantity demanded at q₁)
   |      \
p₂ |--------·  (low price → high quantity demanded at q₂)
   |          \
   |            D
   +----------------→ #units
              q₁    q₂
```

The **demand curve (D)** is drawn in blue, sloping downward. The **supply curve (S)** is in red, sloping upward. A price drop from p₁ to p₂ causes a movement ALONG the D curve from q₁ to q₂ — more units are demanded at the lower price.

### How This Connects to Your Homework

**DECS-S-0034 SIMs Regression:** Your regression IS estimating a demand function:
```
Mobile_subs = f(GDP, Internet, Landlines, Tourism, YoungDep)
```

The coefficient on each variable tells you: "If this changes, how much does Mobile change?"

**QMM-02 Hot Dog Pricing:** This is literally a demand function:
```
MSHARE (Dubuque market share) = f(PDUB, PMAY, PBPREG, PBPALL)
```

| Variable | What It Represents | QMM-02 Result |
|----------|-------------------|:---:|
| PDUB | Own-price effect | **Negative** (Law of Demand ✓) |
| PMAY | Cross-price (competitor) | **Positive** (substitutes) |
| PBPREG | Cross-price (Ball Park) | Not significant |
| PBPALL | Cross-price (Ball Park) | Not significant |

The **own-price effect** is negative — when Dubuque raises its price, demand falls. This is the exact same downward-sloping D curve from the whiteboard.

---

## PART 2: The Correlation Coefficient (ρ)

### Whiteboard Content: −1 ≤ ρ ≤ +1

ρ (rho) measures the **strength and direction** of a linear relationship between two variables.

### The Scale

| ρ Value | Interpretation | Visual |
|:-------:|----------------|--------|
| **+1.0** | Perfect positive correlation | Both variables move together exactly |
| **+0.7** | Strong positive correlation | When X ↑, Y ↑ (usually) |
| **0** | No linear relationship | X tells you nothing about Y |
| **−0.7** | Strong negative correlation | When X ↑, Y ↓ (usually) |
| **−1.0** | Perfect negative correlation | Both variables move opposite exactly |

### ~ Uncorrelated (ρ ≈ 0)

When ρ ≈ 0, the two variables are **uncorrelated** — knowing one gives you NO information about the other. They might still be related in a non-linear way (U-shaped, exponential, etc.), but the straight-line relationship is zero.

### How This Connects to Your SIMs Assignment

Your 201-country correlation matrix:

| Pair | ρ | Meaning |
|------|:---:|---------|
| Mobile ↔ Internet | **+0.41** | Moderate positive — countries with more internet have more SIMs |
| Mobile ↔ GDP | +0.17 | Weak positive — barely related |
| Mobile ↔ Landlines | +0.22 | Weak positive |
| Mobile ↔ Tourism | +0.07 | ≈ Uncorrelated — tourism doesn't predict mobile use |
| Mobile ↔ Young Dep | **−0.38** | Moderate negative — young countries have FEWER SIMs (surprising!) |

This is why your R² is only 0.18 — the correlations are mostly weak. The only strong relationship in the data is Internet ↔ Young Dep (−0.78, a very strong negative).

### How This Connects to QMM-02 Hot Dog

In the hot dog data, you'd expect:
- **MSHARE ↔ PDUB:** Negative ρ (higher own price → lower share)
- **MSHARE ↔ PMAY:** Positive ρ (competitor raises price → you gain share)
- **MSHARE ↔ PBPREG:** Small or zero ρ (not a significant competitor)

---

## PART 3: Key Takeaway for DECS-S-0034

| Concept | What It Tells You |
|---------|-------------------|
| **Demand function** q=f(p) | How one variable responds to another — this IS what regression estimates |
| **Correlation ρ** | Direction and strength BEFORE you run regression — tells you which variables are worth including |
| **R²** | After regression — how much of the variation your model explains |
| **p-value** | Whether each coefficient is "real" or just noise |

**The workflow:**
1. Look at correlation (ρ) → screen variables
2. Run regression → estimate the demand function
3. Check R² and p-values → evaluate the model
4. Interpret coefficients → answer the business question

---

## PART 4: Quick Reference

### Demand Function
> qD = f(p) — the relationship between price and quantity demanded. In regression: Y = f(X₁, X₂, ..., Xₙ)

### Correlation vs. Regression
| | Correlation (ρ) | Regression (β) |
|---|---|---|
| What it measures | Strength of linear relationship | Effect size: change in Y per change in X |
| Range | −1 to +1 | Any number |
| Unit? | Unitless | Units: Y-units per X-unit |
| Controls? | No — just two variables | Yes — controls for other variables |

### Significance Codes
| Code | p-value | Meaning |
|:----:|:-------:|---------|
| *** | < 0.001 | Extremely significant |
| ** | < 0.01 | Very significant |
| * | < 0.05 | Significant |
| . | < 0.10 | Marginally significant |
| (none) | ≥ 0.10 | Not significant |

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*Reference for DECS-S-0034 Statistics for Business Decision Making*
*Sasin School of Management | Chulalongkorn University*