Problem 2 - Active Calculus

Question

The table below gives the percentage, \(P\), of households with a VCR, as a function of year. \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline Year & 1978 & 1979 & 1980 & 1981 & 1982 & 1983 & 1984 \\ \hline P & 0.3 & 0.5 & 1.1 & 1.8 & 3.1 & 5.5 & 10.6 \\ \hline Year & 1985 & 1986 & 1987 & 1988 & 1989 & 1990 & 1991 \\ \hline P & 20.8 & 36.0 & 48.7 & 58.0 & 64.6 & 71.9 & 71.9 \\ \hline \end{tabular} (a) A logistic model is a good one to use for these data. Explain why this might be the case: logically, how large would the growth in VCR ownership be when they are first introduced? How large can the ownership ever be? We can also investigate this by estimating the growth rate of \(P\) for the given data. Do this at the beginning, middle, and near the end of the data: \(P^{\prime}(1980) \approx\) \(P^{\prime}(1985) \approx\) \(P^{\prime}(1990) \approx\) Be sure you can explain why this suggests that a logistic model is appropriate. (b) Use the data to estimate the year when the point of inflection of \(P\) occurs. The inflection point occurs approximately at (Give the year in which it occurs.) What percent of households had VCRs then? \(P=\) What limiting value \(L\) does this point of inflection predict (note that if the logistic model is reasonable, this prediction should agree with the data for 1990 and 1991 )? \(L=\) (c) The best logistic equation (solution to the logistic differential equation) for these data turns out to be the following. $$ P=\frac{75}{1+316.75 e^{-0.699 t}} . $$ What limiting value does this predict? $$ L= $$

Step-by-Step Solution

Answer

The logistic model is appropriate because of initial rapid growth that slows as saturation occurs. The inflection point is around 1987 with 48.7% ownership. The logistic function predicts a limiting value of 75%.

Step 1: Understanding the Logistic Model

A logistic model is often used to describe growth that starts exponentially but slows down as it approaches a limiting value. This is appropriate here because initially, VCR adoption grows quickly, but the growth rate decreases as more households get VCRs, due to market saturation.

Step 2: Initial Growth Explanation

Initially, the growth in VCR ownership would be very rapid as few households had VCRs and many were interested in adopting the new technology. However, the ownership can never exceed 100% because not every household will adopt VCRs.

Step 3: Calculating the Growth Rate at 1980

Use the formula for rate of change: \[ P'(t) = \frac{P(t+1) - P(t-1)}{2} \]At 1980: \[ P'(1980) = \frac{P(1981) - P(1979)}{2} = \frac{1.8 - 0.5}{2} = 0.65 \text{ percent per year} \]

Step 4: Calculating the Growth Rate at 1985

Using the same formula: \[ P'(1985) = \frac{P(1986) - P(1984)}{2} = \frac{36.0 - 10.6}{2} = 12.7 \text{ percent per year} \]

Step 5: Calculating the Growth Rate at 1990

At the end, the growth rate starts to decrease: \[ P'(1990) = \frac{P(1991) - P(1989)}{2} = \frac{71.9 - 64.6}{2} = 3.65 \text{ percent per year} \]

Step 6: Analyzing the Growth Rate

Initially, the growth rate is small, then it increases significantly, and finally decreases again. This change in growth rate suggests the logistic model: rapid initial growth, a period of steep increase, and then a leveling off as the market becomes saturated.

Step 7: Estimating the Point of Inflection

The inflection point is where the growth rate transitions from increasing to decreasing. Reviewing the data, it seems to occur around 1987 as this is where the rate of change in growth starts to decline notably. At this year, the percentage of households with VCRs is 48.7%.

Step 8: Determining the Limiting Value from Inflection Point

For a logistic model, the inflection point occurs at half the limiting value, so: \[ 2 \times 48.7 = 97.4 \text{ percent } \] However, 100% is the actual terminal limit realistically. The logistic model predicts a limiting value, but not exceeding this realistic expectation.

Step 9: Using the Given Logistic Equation

Examine the given logistic function:\[ P = \frac{75}{1 + 316.75 e^{-0.699t}} \]As time goes to infinity (t \rightarrow t \rightarrow \[\frac{75}{1 + 316.75 \times 0} = 75 \text{ percent} \]This indicates that the predicted limiting value of households with VCRs is 75%.

Step 10: Summary

Using the logistic model, we observe that VCR ownership starts quickly and then levels off. The limiting value indicated by our data and the logistic function converges closely towards 75%.

Key Concepts

Logistic Model

A logistic model is vital in understanding technology adoption patterns, such as the growth of VCR ownership in households. This model is characterized by an initial exponential growth phase followed by a slowdown as it approaches a limiting value, reflecting the concept of market saturation. Initially, when VCRs were introduced, the growth rate was rapid because only a few households had VCRs, and many were eager to adopt this new technology. However, as more people acquired VCRs, the rate of new adopters slowed down, eventually stabilizing when the market reached near saturation. This model helps predict the long-term adoption trends and the point at which growth rates will decrease.

Growth Rate Calculation

Calculating the growth rate at different points in time provides insight into the technology adoption process. The formula used is: \[ P'(t) = \frac{P(t+1) - P(t-1)}{2} \]Applying this to the given data:

• At 1980: \[ P'(1980) = \frac{1.8 - 0.5}{2} = 0.65 \% \text{ per year} \]
• At 1985: \[ P'(1985) = \frac{36.0 - 10.6}{2} = 12.7 \% \text{ per year} \]
• At 1990: \[ P'(1990) = \frac{71.9 - 64.6}{2} = 3.65 \% \text{ per year} \]

As observed, the growth rate starts small, increases significantly during the middle years, and decreases again nearing the end. This behavior aligns perfectly with the logistic model's rapid initial growth, a period of steep increase, and then leveling off.

Inflection Point Analysis

The inflection point in a logistic growth curve is crucial as it indicates the transition from increasing to decreasing growth rates. For VCR adoption, this point of inflection can be estimated by evaluating when the rate of change starts to decline notably. Based on the given data, this occurs around 1987, with 48.7% of households owning VCRs at that time. The inflection point is where the curve changes concavity, symbolizing a shift in consumer behavior from rapid adoption to a more saturated market. Understanding this point helps in predicting when the market will start to experience diminishing growth rates.

Limiting Value Prediction

In the context of logistic growth, the limiting value, often denoted as \( L \), represents the maximum potential adoption level. This value reflects the upper ceiling of the market saturation point. For instance, in the VCR data scenario, the given logistic model equation is:\[ P = \frac{75}{1 + 316.75 e^{-0.699t}} \]As time approaches infinity:\[ t \rightarrow \infty, e^{-0.699t} \rightarrow 0 \]Hence,\[ P = \frac{75}{1 + 316.75 \times 0} = 75 \% \]This model predicts that the limiting value of VCR ownership will be 75%. Though, realistically speaking, not every household will adopt a VCR, suggesting a logical terminal limit near 100% market penetration is improbable.

Market Saturation

Market saturation occurs when the growth of a product's adoption slows down significantly, as most of the potential users have already adopted the product. In the case of VCRs, market saturation is evident in the data towards the late years (1990 and 1991), where the percentage of households owning VCRs stabilizes at around 71.9%. This stage represents a mature market where new adoptions are minimal, and growth rates are flat. Recognizing market saturation helps in making strategic decisions about product lifecycle management, future investments, and marketing strategies, based on the understanding that further growth in adoption will be limited.