Problem 61 - Calculus: Early Transcendental Functions

Question

Suppose that a business has an income stream of $$ P(t) .\( The present value at interest rate \)r\( of this income for the next \)T\( years is \)\int_{0}^{t} P(t) e^{-r t} d t .\( Compare the present values at \)5 \%\( for three people with the following salaries for 3 years: \)\begin{aligned} &\text { A: } \quad P(t)=60,000 ; \quad \text { B: } \quad P(t)=60,000+3000 t ; \quad \text { and }\\\ &\mathbf{C}: P(t)=60,000 e^{0.05 t} \end{aligned}$

Step-by-Step Solution

Answer

The present values of incomes for person A, B, and C are calculated using the provided integral. These values are then compared to determine who has the highest present value of salary over the next three years.

Step 1: Calculate the present value of income for person A

We know \(P(t) = 60000\) for person A. Substituting these values in the equation we get: \(\int_{0}^{3} 60000 e^{-0.05 t} dt \). By performing the integral, we obtain the present value of income for person A.

Step 2: Calculate the present value of income for person B

The income stream for person B is described by the function \(P(t) = 60000 + 3000t\). We substitute these values into the integral: \(\int_{0}^{3} (60000 + 3000t) e^{-0.05 t} dt \). Solving this integral gives us the present value for person B.

Step 3: Calculate the present value of income for person C

For person C, \(P(t) = 60000e^{0.05t}\). We put these values into the integral: \(\int_{0}^{3} 60000e^{0.05t} e^{-0.05 t} dt \). We simplify and solve to find the present value for person C.

Step 4: Comparison

After calculating the present values of incomes for all individuals, perform a comparison to see who has the highest present value of income.

Key Concepts

Economic Model of Present Value

When we talk about the economic model of present value, we are essentially considering the worth of future income if it were received today. Why do we do this? Simply put, a dollar today is more valuable than a dollar tomorrow due to the potential for earning interest. To calculate the present value (PV) of an income stream, we can use the formula \( PV = \int_{0}^{T} P(t) e^{-rt} dt \), where \( P(t) \) is the income at time \( t \), \( r \) is the interest rate, and \( T \) is the total time period considered. This formula helps us understand the current value of expected cash flows over a period of time.

In the original exercise, the present value of different income scenarios over 3 years at a 5% interest rate is calculated. By integrating the given income functions, which represent different growth models, we can compare the present value of the incomes.

Exponential Growth in Income

Exponential growth in income means that income increases at a rate proportional to its current value. The mathematical representation of this concept is an exponential function, often written as \( P(t) = P_0e^{rt} \), where \( P_0 \) is the initial amount, \( r \) is the growth rate, and \( t \) is time. Such growth is common in industries with rapid technological innovation or where compound interest comes into play.

For instance, Person C in our exercise has a salary that grows exponentially, as modeled by the function \( P(t) = 60000e^{0.05t} \). The presence of the base \( e \) in the income function represents continuous growth, which can lead to significant increases in income over time.

Time Value of Money

The time value of money is a financial concept that reflects the idea that money available now is worth more than the same amount in the future due to its earnings potential. To put it simply, it's better to receive money today than the same amount at a future date because you can invest it and earn interest. The time value of money is the foundation for discounting cash flows, which is a technique for determining present value.

Time value is incorporated through the discount factor \( e^{-rt} \) in our model, which adjusts future income to reflect its value in present terms. When we apply this factor to the given income streams in our exercise, we get a concrete measure of the present worth of the income expected over 3 years, discounted at a rate of 5%.

Income Stream Evaluation

Income stream evaluation involves assessing the flow of income over a period to determine its current worth. In the exercise, we evaluate three different income streams over 3 years at a 5% interest rate. For Person A, the income stream is constant, whereas Person B has a linear increase in income, and Person C experiences exponential income growth. To evaluate these income streams, we use the integral which incorporates the decay function due to the time value of money.

By comparing the present values of these income streams, we can determine which person ultimately has the most valuable income scenario from today's economic standpoint. This assessment is crucial for financial planning and making investment decisions based on future income prospects.

Calculus in Finance

Calculus, particularly integral calculus, plays a critical role in finance, especially when evaluating income over time or assessing investment returns. The integration process helps in finding the present value of an income stream by summing up all infinitesimally small portions of the income over a continuous time frame.

In our original exercise, integration is used to calculate the present value of different salary structures over time. Each salary model, whether it be constant, linearly increasing, or exponentially growing, necessitates the use of integral calculus to properly evaluate its worth in present terms. This underscores the importance of calculus in financial modeling and analysis, as it provides a robust tool for understanding complex financial scenarios.