The typical consumer’s food basket in the base year 2022 is as follows:

        30 chickens at $2 each

        10 hams at $6 each

        10 steaks at $8 each

A chicken feed shortage causes the price of chickens to rise to $5.00 each in the year 2023. Hams rise to $7.00 each, and the price of steaks is unchanged.

Calculate the change in the “cost-of-eating” index between 2022 and 2023.
 

Year	Cost of the basket
2022	$ 
200
2023	$ 
300
Instructions: Enter your responses rounded to one decimal place.

The official cost-of-eating index has 
increased
 by 
50
 %.

Suppose that consumers are completely indifferent between two chickens and one ham in 2016. For this example, how large is the substitution bias in the official “cost-of-eating” index?
The change in the cost-of-eating index is 
70
 %. The 
overestimate
 of inflation in the cost of eating reflects substitution bias.



Let's go through the problem step by step to calculate the **cost-of-eating index** and **substitution bias**.

### Step 1: Calculate the cost of the basket in the base year 2022

For 2022, the given quantities and prices are:
- **30 chickens** at $2 each,
- **10 hams** at $6 each,
- **10 steaks** at $8 each.

The total cost for each item can be calculated as:
- Chickens: $ 30 \times 2 = 60 $
- Hams: $ 10 \times 6 = 60 $
- Steaks: $ 10 \times 8 = 80 $

So, the total **cost of the basket in 2022** is:
\[
60 + 60 + 80 = 200
\]

### Step 2: Calculate the cost of the basket in the year 2023

For 2023, the prices change:
- **Chickens** rise to $5 each,
- **Hams** rise to $7 each,
- **Steaks** remain at $8 each.

Now, let's calculate the cost for each item in 2023:
- Chickens: $ 30 \times 5 = 150 $
- Hams: $ 10 \times 7 = 70 $
- Steaks: $ 10 \times 8 = 80 $

So, the total **cost of the basket in 2023** is:
\[
150 + 70 + 80 = 300
\]

### Step 3: Calculate the official “cost-of-eating” index

The **cost-of-eating index** is based on the ratio of the cost in 2023 to the cost in 2022. The formula is:

\[
\text{Cost-of-eating index} = \left(\frac{\text{Cost in 2023}}{\text{Cost in 2022}}\right) \times 100
\]

Substituting the values we calculated:
\[
\text{Cost-of-eating index} = \left(\frac{300}{200}\right) \times 100 = 150
\]

So, the **cost-of-eating index** has increased by **50%** from 2022 to 2023.

### Step 4: Calculate the substitution bias

The problem states that consumers are indifferent between **two chickens** and **one ham**. This suggests that if chicken prices increase, consumers might substitute away from chickens towards hams.

#### Recalculating the cost of the basket considering substitution:
If consumers substitute **two chickens** for **one ham**, then the 30 chickens originally consumed can be replaced by:
\[
\text{New quantity of chickens} = 0 \quad \text{(substituted)}
\]
\[
\text{New quantity of hams} = 10 + \frac{30}{2} = 10 + 15 = 25
\]

The cost of this new basket in 2023 would be:
- **0 chickens** at $5 each = $0,
- **25 hams** at $7 each = $ 25 \times 7 = 175 $,
- **10 steaks** at $8 each = $ 10 \times 8 = 80 $.

So, the new total cost is:
\[
0 + 175 + 80 = 255
\]

### Step 5: Calculate the substitution bias

The official cost-of-eating index calculated without substitution was 150 (an increase of 50%). Now, using the substituted basket, the index would be based on $255 instead of $300. The new **cost-of-eating index** is:

\[
\text{New index} = \left(\frac{255}{200}\right) \times 100 = 127.5
\]

Thus, the increase is:
\[
127.5 - 100 = 27.5\%
\]

Finally, the **substitution bias** is the difference between the official index increase (50%) and the new index increase (27.5%):
\[
50 - 27.5 = 22.5\%
\]

Therefore, the **overestimate of inflation** due to substitution bias is **22.5%**.

### Summary:
- The official **cost-of-eating index** increased by **50%**.
- The substitution bias results in an **overestimate of inflation by 22.5%**.

Would you like further clarification, or details on any part of this explanation? 

Here are some related questions:
1. How would the cost-of-eating index change if the price of steaks increased as well?
2. Can the substitution bias be negative in some situations?
3. What factors influence a consumer's decision to substitute between goods?
4. How would the index change if the base year were shifted to 2023 instead?
5. What is the impact of income elasticity on consumer substitution patterns?

**Tip:** Substitution bias arises because price indices often assume a fixed consumption basket, which doesn't account for how consumers change their purchasing behavior when relative prices change.

The typical consumer’s food basket in the base year 2022 is as follows: 30 chickens at $2 each, 10 hams at $6 each, 10 steaks at $8 each. A chicken feed shortage causes the price of chickens to rise to $5.00 each in the year 2023. Hams rise to $7.00 each, and the price of steaks is unchanged. Calculate the change in the 'cost-of-eating' index between 2022 and 2023.
Year: Cost of the basket
2022: $200
2023: $300.
The official cost-of-eating index has increased by 50%.
Suppose that consumers are completely indifferent between two chickens and one ham in 2023. How large is the substitution bias in the official 'cost-of-eating' index? The change in the cost-of-eating index is 70%. The overestimate of inflation in the cost of eating reflects substitution bias.

Learn how to calculate the cost-of-eating index and account for substitution bias between 2022 and 2023. This problem involves inflation calculation, substitution bias, and changes in consumer behavior. Step-by-step solution included for Grade 10-12 students.

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