The demand relation for beef takes the form of

*QB = 210 − 4.2PB + 0.6PF + 1.9I*

where *PB* is the price of beef, and *PF* is the price of fish, and *I* is the income. At one equilibrium, *QB* = 590.4, *PB* = 49, and *PF* = 67.

**(a)** We can infer that the income elasticity of demand for beef is Answer for part 1

at the equilibrium.

**(b)** We can further infer cross-price elasticity of demand for beef with respect to price of fish to be Answer for part 2

at the equilibrium.

solve b
|

explain each step deeply and if theres another word for partial derative use it instead of that cuz it confuisese me a lto

Let’s go through solving part (b), which asks for the **cross-price elasticity of demand** for beef with respect to the price of fish. Cross-price elasticity measures how the quantity demanded of one good responds to a change in the price of another good. Here's how we calculate and understand it step by step.

### 1. **Cross-price Elasticity Formula**

Cross-price elasticity \(\varepsilon_{QB, PF}\) is defined as:

\[
\varepsilon_{QB, PF} = \frac{\text{percentage change in quantity demanded of beef}}{\text{percentage change in price of fish}} = \frac{\frac{\Delta Q_B}{Q_B}}{\frac{\Delta P_F}{P_F}}
\]

This simplifies to:

\[
\varepsilon_{QB, PF} = \frac{\Delta Q_B}{\Delta P_F} \times \frac{P_F}{Q_B}
\]

Now, \(\frac{\Delta Q_B}{\Delta P_F}\) is the rate at which quantity demanded of beef changes with respect to the price of fish. This is the **coefficient** of \(P_F\) in the demand equation, which represents the **sensitivity** of beef demand to changes in the price of fish.

In your demand equation:
\[
Q_B = 210 - 4.2P_B + 0.6P_F + 1.9I
\]
the coefficient of \(P_F\) is \(0.6\). This tells us that for every 1-unit increase in the price of fish, the quantity demanded of beef increases by 0.6 units.

### 2. **Substitute into Cross-price Elasticity Formula**

Now we use the values at the equilibrium point where:
- \(Q_B = 590.4\)
- \(P_F = 67\)

Substitute these values and the coefficient of \(P_F\) (which is \(0.6\)) into the elasticity formula:

\[
\varepsilon_{QB, PF} = 0.6 \times \frac{67}{590.4}
\]

### 3. **Calculate Cross-price Elasticity**

Now, compute the value:

\[
\varepsilon_{QB, PF} = 0.6 \times 0.1135 = 0.0681
\]

### 4. **Interpretation**

The cross-price elasticity of beef with respect to the price of fish is approximately \(0.068\). This positive value suggests that beef and fish are **substitutes**, but the elasticity is quite low, meaning that the demand for beef is not highly sensitive to changes in the price of fish.

### Summary

The cross-price elasticity of demand for beef with respect to the price of fish is:

\[
\varepsilon_{QB, PF} = 0.068
\]

This indicates that a 1% increase in the price of fish would lead to a **0.068% increase** in the quantity of beef demanded, suggesting a weak substitutive relationship between beef and fish.

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Would you like to clarify any part of this solution, or dive deeper into some steps?

### 5 Related Questions
1. How does the income elasticity of demand for beef affect demand when income increases?
2. What happens to the quantity of beef demanded if both fish prices and income rise simultaneously?
3. How would the cross-price elasticity change if fish were a stronger substitute for beef?
4. Can cross-price elasticity ever be negative? What does that imply about the relationship between goods?
5. How would the demand for beef change if its price increased, holding fish price and income constant?

#### Tip
In economics, you can often replace the phrase "partial derivative" with "rate of change with respect to one variable," which keeps the idea simple without needing advanced calculus.

The demand relation for beef takes the form of QB = 210 − 4.2PB + 0.6PF + 1.9I where PB is the price of beef, PF is the price of fish, and I is the income. At one equilibrium, QB = 590.4, PB = 49, and PF = 67. (a) We can infer that the income elasticity of demand for beef is Answer for part 1 at the equilibrium. (b) We can further infer cross-price elasticity of demand for beef with respect to price of fish to be Answer for part 2 at the equilibrium.

Calculate and understand the cross-price elasticity of demand for beef with respect to the price of fish. This detailed explanation covers the formula for elasticity, step-by-step calculations, and economic interpretations. Suitable for students in Grades 11-12 or introductory undergraduate economics courses.

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