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.

To find the income elasticity of demand for beef at the given equilibrium, we can use the formula for income elasticity of demand (E_I):

\[
E_I = \frac{\partial Q_B}{\partial I} \times \frac{I}{Q_B}
\]

Where:
- \(\frac{\partial Q_B}{\partial I}\) is the partial derivative of the demand function with respect to income \(I\).
- \(I\) is the income level.
- \(Q_B\) is the quantity demanded of beef.

### Step 1: Find the partial derivative with respect to income \(I\).
The demand function is given as:

\[
Q_B = 210 - 4.2P_B + 0.6P_F + 1.9I
\]

The partial derivative with respect to \(I\) is:

\[
\frac{\partial Q_B}{\partial I} = 1.9
\]

### Step 2: Determine the value of \(I\) at equilibrium.
We can rearrange the demand equation to solve for \(I\):

\[
Q_B = 210 - 4.2P_B + 0.6P_F + 1.9I
\]

Substitute the equilibrium values \(Q_B = 590.4\), \(P_B = 49\), and \(P_F = 67\):

\[
590.4 = 210 - 4.2(49) + 0.6(67) + 1.9I
\]

### Step 3: Simplify and solve for \(I\).
First, calculate the terms involving \(P_B\) and \(P_F\):

\[
590.4 = 210 - 205.8 + 40.2 + 1.9I
\]

\[
590.4 = 44.4 + 1.9I
\]

\[
1.9I = 590.4 - 44.4
\]

\[
1.9I = 546
\]

\[
I = \frac{546}{1.9} = 287.37
\]

### Step 4: Calculate the income elasticity \(E_I\).
Substitute the values into the elasticity formula:

\[
E_I = 1.9 \times \frac{287.37}{590.4}
\]

\[
E_I \approx 0.925
\]

Therefore, the income elasticity of demand for beef at the equilibrium is approximately **0.925**.

Would you like more details or have any other questions? Here are five related questions you might consider:

1. What does an income elasticity of 0.925 imply about beef?
2. How does the price of fish \(P_F\) influence the demand for beef?
3. What would happen to the demand for beef if the income \(I\) increases?
4. How could the demand function be adjusted for seasonal variations?
5. What is the cross-price elasticity of demand between beef and fish?

**Tip:** Income elasticity of demand indicates how sensitive the quantity demanded of a good is to changes in income. If it's positive and less than 1, the good is a normal good but a necessity rather than a luxury.

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.

This problem explores the income elasticity of demand for beef using a given demand equation. Learn how to calculate elasticity using partial derivatives and solve for equilibrium using algebraic methods. Example includes price of beef, fish, and income values with step-by-step guidance. Suitable for students studying economics at an advanced high school or undergraduate level.

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