Problem 23 - Calculus

Question

An item is initially sold at a price of \(p\text{dollars}\) per unit. Over time, market forces push the price toward the equilibrium price, \(p \text{dollars}^{*},\) at which supply balances demand. The Evans Price Adjustment model says that the rate of change in the market price, \(p\text{dollars},\) is proportional to the difference between the market price and the equilibrium price. (a) Write a differential equation for \(p\) as a function of \(t\) (b) Solve for \(p\) (c) Sketch solutions for various different initial prices, both above and below the equilibrium price. (d) What happens to \(p\) as \(t \rightarrow \infty ?\)

Step-by-Step Solution

Answer

(a) \( \frac{dp}{dt} = k(p^* - p) \); (b) \( p(t) = p^* - (p^* - p_0)e^{-kt} \); (d) \( p \to p^* \).

Step 1: Understanding the Differential Equation

The problem states that the rate of change of the market price is proportional to the difference between the current market price \( p \) and the equilibrium price \( p^* \). We can express this relationship as a differential equation: \[ \frac{dp}{dt} = k(p^* - p) \] where \( k \) is a constant of proportionality.

Step 2: Solving the Differential Equation

This is a separable differential equation. We can rewrite it as \( \frac{dp}{p^* - p} = k \, dt \). Integrating both sides gives \( \int \frac{1}{p^* - p} \, dp = \int k \, dt \). The left side integrates to \(-\ln|p^* - p|\) and the right side to \( kt + C \), where \( C \) is the integration constant. Therefore, we have \(-\ln|p^* - p| = kt + C\). Solving for \( p \) yields: \[ p = p^* - Ce^{-kt} \] where \( C \) is determined by initial conditions.

Step 3: Setting Up Initial Conditions

To find the constant \( C \), we consider the initial condition \( p(0) = p_0 \) (the initial price). Substituting it into our solution gives \( p_0 = p^* - C \), which simplifies to \( C = p^* - p_0 \). Substitute \( C \) back into the solution to get: \[ p(t) = p^* - (p^* - p_0)e^{-kt} \].

Step 4: Sketching the Solutions

The solution \( p(t) = p^* - (p^* - p_0)e^{-kt} \) shows exponential decay or growth towards the equilibrium \( p^* \). If the initial price \( p_0 > p^* \), \( p(t) \) decreases over time. If \( p_0 < p^* \), \( p(t) \) increases over time. Sketch curves for each scenario, showing \( p(t) \) approaching \( p^* \) as \( t \to \infty \).

Step 5: Analyzing Long-term Behavior

As \( t \to \infty \), \( e^{-kt} \) approaches zero, meaning \( p(t) = p^* \). In both cases (whether \( p_0 > p^* \) or \( p_0 < p^* \)), the market price \( p(t) \) will eventually converge to the equilibrium price \( p^* \).

Key Concepts

Equilibrium Price

The concept of equilibrium price is central to understanding market dynamics. It is the price at which the quantity of an item supplied matches the quantity demanded by consumers. This balance means that there are no surpluses or shortages.
This price is important because it signifies a stable market state. Market forces push prices towards equilibrium. When prices are above this level, excess supply leads to downward pressure on prices. Conversely, if the price is lower than this level, demand exceeds supply, causing prices to rise.

• This mechanism helps stabilize the market.
• Equilibrium is achieved when no further changes in price occur, assuming other conditions remain constant.

As a result, understanding the equilibrium price helps businesses and policymakers make informed decisions.

Market Forces

Market forces are the factors that influence the pricing and availability of goods and services. They include supply and demand, competition, and consumer preferences. These forces exert pressure on prices to adjust toward equilibrium over time.
In the context of the Evans Price Adjustment model, market forces dictate that the rate of price change is proportional to how far the current price is from the equilibrium price. This relationship can be expressed through a differential equation.
The key elements driving market forces include:

• Supply and Demand: Changes in supply or demand directly affect prices.
• Consumer Preferences: Shifts can prompt changes in demand.
• Competition: Competing firms can force prices to adjust.

Being aware of these forces helps in predicting how prices will change in response to market dynamics.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In differential equations, this concept is often used to model how prices adjust over time. This type of behavior is seen when prices move toward equilibrium.
In the given problem, solving the differential equation reveals the solution: \[ p(t) = p^* - (p^* - p_0)e^{-kt} \] This equation shows how the price (\( p(t) \)) approaches the equilibrium price (\( p^* \)) exponentially. The term \( e^{-kt} \) represents exponential decay, meaning the price difference decreases over time.

• If prices start above equilibrium, they decay downwards.
• Below equilibrium, they grow upwards, still exhibiting decay properties in reverse.

This process helps ensure that prices eventually settle at the equilibrium point.

Initial Conditions

Initial conditions in differential equations specify the starting values of the variables involved. They are crucial in solving these equations as they determine the specific solution to a problem.
For the Evans Price Adjustment model, the initial condition is the initial market price, \( p_0 \). It helps in finding the constant \( C \) in the solution of the differential equation: \[ p(t) = p^* - (p^* - p_0)e^{-kt} \] By plugging the initial condition into this equation, we find \( C \), ensuring the solution fits the real-world situation.

• The initial price affects how quickly the market price reaches equilibrium.
• Knowing \( p_0 \) allows us to predict how the price will evolve.

Accurate initial conditions aid in foreseeing the market's path towards price stability.