a,b,c,d # Comparing a Quantity Tax to an Income Tax### Utility Function and Tax Policy Considerations**Utility Function:** Charlie's utility over goods X and Y is given by the function $ U = XY $. The prices of each good are $1, and Charlie’s total income is $100. The government is considering using either an income tax or a quantity tax to collect revenue from Charlie.**Marginal Utilities:** - $ MU_X = Y $ - $ MU_Y = X $ The government wants to determine which policy Charlie would prefer for a given amount of tax revenue.### Discussion Questionsa. **Allocation and Demand:** - How much of his income will Charlie allocate for good X? - What is the demand for X? What is the demand for Y?b. **Quantity Tax Scenario:** - Assume the government imposes a quantity tax of $1 for each unit of X that Charlie purchases. - Determine how many units of X and Y he will consume.c. **Table Completion:** - The table below is to be filled with details on consumption and utility under different tax scenarios.| | X | Y | U ||-----|---|---|---|| **No tax** | | | || **Quantity tax of $1** | | | || **Income tax of $....**| | | |d. **Equivalent Income Tax:** - What is the income tax equivalent to the quantity tax of $1?e. **Affordability Comparison:** - Is the consumption bundle under the quantity tax affordable under the income tax?f. **Table Completion:** - Fill in the last row of the table with appropriate values.g. **Graphical Representation:** - Use the provided graph grid to represent the two policies. - Determine which policy the government should implement.### Graph DescriptionThe graph consists of a grid of squares intended for plotting consumption bundles under different scenarios. Students should illustrate changes in consumption of goods X and Y under no tax, quantity tax, and income tax scenarios, potentially using different colors or patterns for clarity.

Note that MUX-Y and MUX-X Comparing a Quantity tax to an Income tax Charlie's utility over X and Y is given by U=XY. The prices of each good is $1 and his income is $100. The government is contemplating whether to use an income tax or a quantity tax to collect some money from Charlie. The government wants to know which policy Charlie would prefer for a given amount of tax revenue. a. How much of his income will Charlie allocate for good X? What is the demand for X? the demand for Y?

Note that MUX-Y and MUX-X Comparing a Quantity tax to an Income tax Charlie's utility over X and Y is given by U=XY. The prices of each good is $1 and his income is $100. The government is contemplating whether to use an income tax or a quantity tax to collect some money from Charlie. The government wants to know which policy Charlie would prefer for a given amount of tax revenue. a. How much of his income will Charlie allocate for good X? What is the demand for X? the demand for Y?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

a,b,c,d

# Comparing a Quantity Tax to an Income Tax

### Utility Function and Tax Policy Considerations

**Utility Function:**
Charlie's utility over goods X and Y is given by the function $ U = XY $.
The prices of each good are $1, and Charlie’s total income is $100.
The government is considering using either an income tax or a quantity tax to collect revenue from Charlie.

**Marginal Utilities:**
- $ MU_X = Y $
- $ MU_Y = X $

The government wants to determine which policy Charlie would prefer for a given amount of tax revenue.

### Discussion Questions

a. **Allocation and Demand:**
- How much of his income will Charlie allocate for good X?
- What is the demand for X? What is the demand for Y?

b. **Quantity Tax Scenario:**
- Assume the government imposes a quantity tax of $1 for each unit of X that Charlie purchases.
- Determine how many units of X and Y he will consume.

c. **Table Completion:**
- The table below is to be filled with details on consumption and utility under different tax scenarios.

| | X | Y | U |
|-----|---|---|---|
| **No tax** | | | |
| **Quantity tax of $1** | | | |
| **Income tax of $....**| | | |

d. **Equivalent Income Tax:**
- What is the income tax equivalent to the quantity tax of $1?

e. **Affordability Comparison:**
- Is the consumption bundle under the quantity tax affordable under the income tax?

f. **Table Completion:**
- Fill in the last row of the table with appropriate values.

g. **Graphical Representation:**
- Use the provided graph grid to represent the two policies.
- Determine which policy the government should implement.

### Graph Description

The graph consists of a grid of squares intended for plotting consumption bundles under different scenarios. Students should illustrate changes in consumption of goods X and Y under no tax, quantity tax, and income tax scenarios, potentially using different colors or patterns for clarity.

Expert Solution

Step 1

Utility function : U = X*Y

Px = Py = 1

Therefore , Budget Constraint : x + y = 100

Optimal consumption demand for any good is achieved at the point where marginal rate of substitution is equal to the price ratio .

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