An economic model Consider the following economic model. Let...

An economic model Consider the following economic model. Let $P$ be the price of a single item on the market. Let $Q$ be the quantity of the item available on the market. Both $P$ and $Q$ are functions of time. If one considers price and quantity as two interacting species, the following model might be proposed: $$\begin{aligned} \frac{d P}{d t} &=a P\left(\frac{b}{Q}-P\right) \\ \frac{d Q}{d t} &=c Q(f P-Q) \end{aligned}$$ where $a, b, c,$ and $f$ are positive constants. Justify and discuss the adequacy of the model. a. If $a=1, b=20,000, c=1,$ and $f=30$ , find the equilibrium points of this system. If possible, classify each equilibrium point with respect to its stability. If a point cannot be readily classified, give some explanation. b. Perform a graphical stability analysis to determine what will happen to the levels of $P$ and $Q$ as time increases. c. Give an economic interpretation of the curves that determine the equilibrium points.

An economic model Consider the following economic model.
Let $P$ be the price of a single item on the market. Let $Q$ be the
quantity of the item available on the market. Both $P$ and $Q$ are
functions of time. If one considers price and quantity as two interacting species, the following model might be proposed:
$$\begin{aligned} \frac{d P}{d t} &=a P\left(\frac{b}{Q}-P\right) \\ \frac{d Q}{d t} &=c Q(f P-Q) \end{aligned}$$
where $a, b, c,$ and $f$ are positive constants. Justify and discuss the
adequacy of the model.
a. If $a=1, b=20,000, c=1,$ and $f=30$ , find the equilibrium points of this system. If possible, classify each equilibrium point with respect to its stability. If a point cannot be
readily classified, give some explanation.
b. Perform a graphical stability analysis to determine what will
happen to the levels of $P$ and $Q$ as time increases.
c. Give an economic interpretation of the curves that determine
the equilibrium points.

Key Concepts

Stability Analysis

Stability analysis involves determining whether small deviations from an equilibrium point will decay over time (leading the system back to the equilibrium) or grow (driving the system away from it). This analysis helps in understanding the resilience of economic equilibria and whether certain price or quantity levels are sustainable under perturbations.

Mathematical Modeling in Economics

This concept involves representing economic behavior and interactions using mathematical frameworks, such as systems of differential equations. It allows economists to formalize the dynamic relationships between economic variables and to analyze how changes over time may affect these variables within a theoretical structure.

Nonlinear Differential Equations

Nonlinear differential equations are equations involving derivatives where the dependent variables appear with exponents other than one or multiplied together. In the context of economic models, they capture complex interactions between variables, such as price and quantity, which may not increase or decrease in a straightforward, proportional manner.

Equilibrium Points

An equilibrium point in a dynamical system is a state where the system experiences no change over time, meaning the derivatives of the variables are zero. Finding equilibria is crucial because they represent steady states of the economic system where price and quantity remain constant, serving as a starting point for understanding the system's long?term behavior.

Phase Plane Analysis

Phase plane analysis is a graphical method used to study the trajectories of two-dimensional dynamical systems by plotting one variable against another. It enables visualization of the behavior of the system near equilibrium points, the directions of motion, and the overall dynamics of interacting economic variables like price and quantity.

Economic Interpretation of Dynamical Systems

This concept connects the mathematical analysis of dynamical systems with real-world economic implications. It involves interpreting curves, equilibria, and trajectories in economic terms, such as how different levels of supply and demand correspond to market stability or instability, thereby providing insights into market behavior and policy impacts.

Transcript

00:02 We are asked to consider an economic model.

00:04 We're going to let p equal the price of a single item on the market, and q is the quantity of peel available.

00:10 Both p and q are functions of time.

00:13 We're asked to consider p and q as two intersecting species.

00:19 And we've got the equation showing the model proposed, where a, b, and c are positive constants.

00:24 We're asked to justify and discuss adequacy.

00:28 Okay, so let's start that.

00:37 The quadratic terms, negative ap squared and negative cq squared, are both equations, limit the growth of both functions.

01:08 Limit growth of both functions.

01:22 The terms pq, p over q, in our first differential equation, differential equation, equation, equation every rate eq, and q in the second favorite growth of functions, the competition between the terms prevents the uncontrolled increase or decrease.

02:16 Competition prevents uncontrolled increase or decrease.

02:35 Okay, the first equation, p over q provides stability in the derivative for p, which is very much less than q because p over q approach is zero.

03:19 And i'm going to go to the next page, and then go back.

03:24 So, let's rewrite that.

03:27 The term qp in the second equation favors growth of p, growth of q, with an increase in both functions.

03:56 So if the price starts to grow rapidly, then negative p squared limits growth.

04:24 If the quantity on the market grows rapidly, negative q squared limits growth.

04:46 So with these, these factors give pretty reasonable adequacy, reasonably adequate.

05:13 On the other hand, the term that favors price growth, p over q favors price growth.

05:34 That's proportional, proportional to quantity, proportional to p over q is proportional to price, proportional to price and inversely proportional to quantity.

06:42 The reason will be adequate.

06:47 I'm going to erase this one.

06:55 P over q is proportional to price, proportional to price.

07:14 And that is reasonable.

07:19 And inversely, proportional.

07:23 To quantity of product.

07:34 When quantity of product grows, price growth is limited, which is reasonable.

07:58 And if quantity decreases, price begins to increase.

08:15 And that's reasonable.

08:19 Okay, so that is reasonable.

08:22 Okay, then a, i'm going to go back over here, asks us, says if we're given a equals 1, b equals 20 ,000 and c equals 1 and f equals 30 we're asked to find the equilibrium points.

08:39 Let's do this.

08:46 That means this would be true.

08:51 Assistant for parameters in a result in and that would be okay.

09:30 For dp or dt equals zero.

09:39 So p equals zero.

09:42 So pq equals b equals 20 ,000.

09:48 And for dq over dt equals zero, we'll have q equals zero.

09:58 And then we'll get fp equals q equals 30p equals q.

10:07 So it's possible to notice that the result, that the result will need two pairs satisfied and group of this take time.

10:17 So i got to go that next page, but the page after.

10:22 For p equals zero, and q equals 0.

10:28 And for q equals 30p.