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 classificd, 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 classificd, 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

Model Adequacy and Justification

Justifying and discussing the adequacy of a model requires evaluating the realism of its assumptions and the relevance of its parameters. This analysis assesses whether the model captures essential economic dynamics, such as interactions between price and quantity, and whether it can reliably predict or explain observable market behaviors.

Economic Interpretation of Differential Equations

Interpreting differential equations in an economic context involves understanding how various terms correspond to economic forces like demand, supply, and market adjustments. Each term in the model can be linked to specific economic behaviors, allowing one to infer the potential consequences of different market conditions or interventions.

Nullclines

Nullclines are curves in the phase plane along which one of the derivatives is zero. They help in determining the structure of the phase portrait by indicating regions where the system’s dynamics change direction, and they are crucial for a qualitative analysis of the system's behavior.

Dynamical Systems

Dynamical systems involve mathematical formulations that describe how a system evolves over time. In the context of economic models, differential equations are employed to represent the evolution of variables, such as prices and quantities, and capture the inherent feedback mechanisms between them.

Stability Analysis

Stability analysis involves assessing whether small deviations from an equilibrium point will decay or amplify over time. This is typically done through linearization of the system near the equilibrium and examining the eigenvalues of the Jacobian matrix, which indicates whether the equilibrium is stable, unstable, or of saddle type.

Equilibrium Points

Equilibrium points are the states at which the system's variables remain constant over time, meaning all time derivatives are zero. Identifying these points in an economic model helps to understand the long-run behavior of the system and under what conditions the market stabilizes.

Phase Plane Analysis

Phase plane analysis is a graphical method used to study two-dimensional dynamical systems by plotting the trajectories and nullclines in the status space. This approach provides visual insights into the behavior of the system over time and facilitates understanding of how trajectories converge to or diverge from equilibrium points.

Transcript

00:03 For a fitz at price, as q increases, q increases, dp by dt gets smaller, dt gets smaller, and possibly becomes negative, so becomes negative.

00:44 This observation implies that as the quantity supply increases, the price will not rise as fast.

00:52 So if q gets high enough, then the price will decrease.

01:01 Next, consider dq by dt.

01:04 So for a fixed quantity as b increases, dq by dt gets larger.

01:22 So thus as the market price increases, the quantity supplied will increase at a phosphate.

01:29 So if p is too small, if p is too small, then dq by dt will be negative.

01:46 It will be negative.

01:49 And the quantity supply with decrease.

02:01 So this observation is the traditional explanation of the effect of market price level on the quantity supply.

02:21 Now let's draw the graph so if this is q and this is dp by d.

02:42 Then there is a n minus apis so the graph looks like this so the p will be holdstop now about p d k by d t in that case so say it's minus c q square the graph will be like this now section a the d p by d t equal to zero and and dq by dt is also equal to zero.

04:15 Use the equilibrium point, that would frame point pq, green point of pq, that will be 0 and 0 ,0 and 25 .8 7775.

04:55 Now, dp by dt 870775.

04:57 Now, d .p by dt 810, when pq will be less than 20 ,000 and p will be j than 0 yeah here p will be 0 now otherwise d p by d t less than 0 so d k by d t then then then when d k by d t here than zero when when p will be greater than q by 30 and q will be greater than 0.

06:15 So otherwise dq by d2 will be less than 0...

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