Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Evans Price Adjustment Model If there is excess demand for a commodity, the price will rise rapidly at first and then more slowly, according to what economists call the "Evans price adjustment model." A typical function describing this behavior is given by $$ p(t)=12-4 e^{-0.5 t} $$ where \(p(t)\) is the price in dollars after \(t\) days. Calculate \(\lim _{t \rightarrow \infty} p(t)\) and give an interpretation of its value.

Step-by-Step Solution

Answer

The limit is 12; the price stabilizes at $12 over time.

Step 1: Identify the Model's Limit at Infinity

The problem asks us to determine \( \lim _{t \rightarrow \infty} p(t) \). The given function is \( p(t) = 12 - 4e^{-0.5t} \). We need to evaluate the behavior of this function as the variable \( t \) approaches infinity.

Step 2: Evaluate the Exponential Component

As \( t \to \infty \), the term \( e^{-0.5t} \) tends towards 0. This is because the exponential function \( e^{-x} \) approaches zero as \( x \) approaches infinity.

Step 3: Calculate the Limit of the Entire Function

Substitute the limit of the exponential component into the function: \( \lim_{t \to \infty} p(t) = 12 - 4 \cdot 0 = 12 \). Thus, \( \lim _{t \rightarrow \infty} p(t) = 12 \).

Step 4: Interpret the Result

The interpretation of this limit is that, in the long term, the price \( p(t) \) will stabilize at 12 dollars. This reflects that the rapid initial price changes will slow down and eventually cease as the market reaches equilibrium.

Key Concepts

Exponential Functions

Exponential functions are math functions that grow or decay at a constant relative rate. They play a significant role in predicting behaviors that exponentially increase or decrease over time.
A common form of an exponential function is \( f(t) = a \cdot e^{kt} \) where:

• \( a \) is a constant
• \( e \) is the base of the natural logarithm
• \( k \) is the rate

In the Evans price adjustment model, the formula \( p(t) = 12 - 4e^{-0.5t} \) includes an exponential decay term \( e^{-0.5t} \). As time \( t \) increases, the term decreases rapidly towards zero. This reflects economic behavior, like a new market beginning at high-demand prices that slow down as time progresses.

Limits in Calculus

Limits are essential in calculus for understanding behavior as variables approach certain values. They help predict trends and define functions' long-term outcomes. Calculating a limit involves finding what value a function approaches as the input (or variable) gets indefinitely close to a specific point.
In the Evans Price Adjustment Model, the limit we calculate is \( \lim_{t \to \infty} p(t) \). As \( t \to \infty \), \( e^{-0.5t} \) approaches zero, leading the whole function \( p(t) \) to approach a stable value, which we've calculated to be 12. This means no matter how prices fluctuate initially, they'll stabilize at 12 when forecasting over a long period.

Market Equilibrium

Market equilibrium occurs when a market's supply and demand balance, resulting in stable prices. It's a critical concept in economics that ensures that resources are distributed efficiently.
In the Evans Price Adjustment Model, market equilibrium is modeled through the behavior of the price function \( p(t) = 12 - 4e^{-0.5t} \). Initially, prices may rise rapidly due to excess demand, but over time, they adjust and stabilize at a specific equilibrium point, which we've calculated to be 12. This model illustrates how, despite initial volatility, markets tend to settle into a stable state.

Price Stabilization

Price stabilization is a process where prices settle and stop fluctuating. It's essential for creating predictable economic conditions. Many factors drive initial price volatility, including excess demand or supply shocks. Over time, these factors' effects diminish, leading to a stable price level.
In the context of the Evans Price Adjustment Model, price stabilization occurs as \( t \to \infty \) in the function \( p(t) = 12 - 4e^{-0.5t} \). At this point, \( e^{-0.5t} \) becomes insignificant, meaning fluctuations no longer impact the price. As a result, prices stabilize at 12, providing a reliable forecast for economic planning.