Question

Taxes in \(\mathrm{Oz}\) are calculated according to the formula \\[ \mathbf{r}=. \mathbf{0} \mathbf{i} /^{2} \\] where \(T\) represents thousands of dollars of tax liability and /represents income measured in thousands of dollars. Using this formula, answer the following questions: a. How much tax do individuals with incomes of \(\$ 10,000, \$ 30,000,\) and \(\$ 50,000\) pay? What are the average tax rates for these income levels? At what income level does tax liability equal total income? b. Graph the tax schedule for Oz. Use your graph to estimate marginal tax rates for the in come levels specified in part (a). Also show the average tax rates for these income levels on your graph. c. Marginal tax rates in \(\mathrm{Oz}\) can be estimated more precisely by calculating tax owed if per sons with the incomes in part (a) get one more dollar. Make this computation for these three income levels. Compare your results by calculating the marginal tax rate function using calculus.

Short answer

Answer: For individuals with incomes of $10,000, $30,000, and $50,000, the average tax rates are 100%, 300%, and 500%, respectively, and their marginal tax rates are 200%, 600%, and 1000%, respectively. The income level at which tax liability equals total income is $10,000.

Step-by-step solution

a. Calculating the taxes and the average tax rates

To calculate the tax and the average tax rate for individuals with incomes of \(10,000\), \(30,000\) and \(50,000\), we need to use the tax formula \(T = 0.1I^2\). 1. For a \(10,000\) income: \(I = \frac{10,000}{1,000} = 10\) So, \(T = 0.1 \times 10^2 = 10\). Since it's in thousands of dollars, the tax for this income is \(10,000\) dollars. The average tax rate is: \(\frac{10,000}{10,000} = 1\) or 100%. 2. For a \(30,000\) income: \(I = \frac{30,000}{1,000} = 30\) So, \(T = 0.1 \times 30^2 = 90\). Since it's in thousands of dollars, the tax for this income is \(90,000\) dollars. The average tax rate is: \(\frac{90,000}{30,000} = 3\) or 300%. 3. For a \(50,000\) income: \(I = \frac{50,000}{1,000} = 50\) So, \(T = 0.1 \times 50^2 = 250\). Since it's in thousands of dollars, the tax for this income is \(250,000\) dollars. The average tax rate is: \(\frac{250,000}{50,000} = 5\) or 500%. Now, we need to find at what income level does tax liability equal total income. To achieve this, we need to solve for \(I\) when \(T = I\). In other words, \(0.1I^2 = I\). Let's solve it.

a. Calculating the income level when tax liability equals total income

To find the income level when tax liability equals total income, we need to solve the equation \(I = 0.1I^2\): \(I-0.1I^2 = 0 \Rightarrow I(1-0.1I) = 0\) Now, either \(I = 0\) or \(1-0.1I = 0\). The first option is trivial and has no effect on our result, so let's focus on the second option: \(1-0.1I = 0 \Rightarrow I = 10\) We found the income level when tax liability equals total income to be \(10\) thousand, which means \(10,000\) dollars.

b. Graphing the tax schedule

To graph the tax schedule for Oz, you would plot the function \(T = 0.1I^2\) using a graphing calculator or a suitable software tool. You would then estimate the marginal tax rates for the income levels specified in part (a) and show the average tax rates for these income levels on your graph. Unfortunately, as a text-based AI, I am unable to generate graphical representations for you, but I encourage you to complete this step using the available tools and our calculated values.

c. Calculating the marginal tax rates using calculus

To find the marginal tax rates for the given income levels, first, compute the derivative of the tax function using calculus: \(\frac{dT}{dI} = \frac{d(0.1I^2)}{dI} = 0.2I\) Now, we can compute the marginal tax rates for income levels of \(10,000\), \(30,000\), and \(50,000\) using the incomes in thousands, that is, \(I = 10\), \(I = 30\) and \(I = 50\): 1. For a \(10,000\) income: \(\frac{dT}{dI}(10) = 0.2 \times 10 = 2\) The marginal tax rate is \(2\) or 200%. 2. For a \(30,000\) income: \(\frac{dT}{dI}(30) = 0.2 \times 30 = 6\) The marginal tax rate is \(6\) or 600%. 3. For a \(50,000\) income: \(\frac{dT}{dI}(50) = 0.2 \times 50 =10\) The marginal tax rate is \(10\) or 1000%. In conclusion, for individuals with incomes of \(10,000\), \(30,000\), and \(50,000\), the marginal tax rates are 200%, 600%, and 1000%, respectively.

Key concepts

Marginal Tax Rates

Understanding marginal tax rates is essential when navigating the complexities of taxation. Imagine you're climbing a staircase where each step represents an additional income bracket, and as you ascend, the amount of tax applied to your next dollar of income increases. This is what marginal tax rates signify - the tax rate on any additional income earned.

When using calculus to analyze marginal tax rates, which is often the case in microeconomic tax analysis, we calculate the derivative of the tax function with respect to income. Taking our exercise problem, the formula being used is \( T = 0.1I^2 \), where \( T \) is the tax liability, and \( I \) is the income. To find the marginal tax rate, we find the derivative, which is \( \frac{dT}{dI} = 0.2I \). Plugging in the respective incomes in thousands (such as \( I = 10 \), \( I = 30 \), and \( I = 50 \)), we can determine the additional tax rate applied to every extra dollar earned at these income levels, which turns out to be quite steep, increasing from 200% to as much as 1000%.

Real-World Application

It's important to note that in real-world scenarios, tax systems are typically progressive, meaning the marginal tax rates increase as income increases but generally do not exceed 100%. This exercise, however, highlights the theoretical importance of understanding how quickly tax liability can rise with increasing income under different tax rules.

Average Tax Rates

The average tax rate is like a summary of the tax story. It answers the simple yet profound question: 'On average, what percentage of my income goes to taxes?' It's calculated by dividing the total tax liability by the total income. In our exercise scenario, the average tax rates are strikingly high due to the quadratic nature of the tax function. For instance, an individual with a \( \$10,000 \) income would have an average tax rate of 100%, meaning all of their income goes to taxes.

For incomes of \( \$30,000 \) and \( \$50,000 \), the respective average tax rates are 300% and 500%. These scenarios are hypothetical and exaggerated to demonstrate the math behind the concepts. They are not reflective of actual tax systems which strive to balance equity and efficiency without entirely disincentivizing work or economic growth.

Error in Tax Function

An important exercise improvement advice is to note that if an individual's tax rate exceeded their income, it would imply a tax function error or an extraordinarily punitive tax system which would likely be unfeasible.

Tax Liability Calculus

Tax liability calculus is the method of integrating mathematical tools to figure out how much tax an individual or an entity owes. Using the formula from our exercise example, \( T = 0.1I^2 \), a direct application of algebraic manipulation provided the specific tax amounts owing at various income levels. The calculus aspect came into play when we computed the marginal tax rates by taking derivatives.

When tax liability equals total income -- a point we determined to be at an income of \( \$10,000 \) in our exercise -- the mathematical analysis can reveal the unreasonable tax burdens or breakpoints within tax systems. Understanding these points can inform policymakers and individuals about the economic implications of tax policies. For example, calculus helps to engage with concepts such as deadweight loss, which occurs when high taxation creates a loss of economic efficiency.

Furthermore, dissecting tax liability through calculus allows for precision in estimating the effects of policy changes, making it an invaluable tool in both theoretical and practical applications of economics.