Quantity demanded in a market is given by q^d=p^-2 and quantity supplied is given by q^s=8 p If

Quantity demanded in a market is given by q^d=p^-2 and...

Key Concepts

Nonlinear Differential Equations

Nonlinear differential equations are equations in which the variables and their derivatives appear with exponents or in products, leading to dynamics that are more complex than linear systems. Many economic models incorporate nonlinearities because economic relationships are often not proportional over all ranges, making the analysis and understanding of their qualitative behavior vital in capturing realistic market dynamics.

Dynamical Systems and Stability

Dynamical systems theory is used to study how a system evolves over time from a given initial state. In economic models, stability analysis determines whether the system will return to equilibrium after a small disturbance. By analyzing the behavior of the differential equation near the equilibrium point, one can assess if the equilibrium is stable (returns to equilibrium) or unstable (diverges away), which is critical for understanding real-world market adjustments.

Price Adjustment Mechanism

The price adjustment mechanism describes how prices change over time in response to imbalances in the market, typically represented by excess demand. This is often modeled using differential equations where the rate of change of the price is proportional to the difference between demand and supply. This formulation provides a structured way to capture the dynamics of price evolution.

Excess Demand

Excess demand refers to the difference between the quantity demanded and the quantity supplied at a given price. It drives the adjustment process in the market. When excess demand is positive, prices tend to rise, and when it is negative, prices tend to fall, thus serving as the signal for market corrections toward equilibrium.

Market Equilibrium

Market equilibrium is the state where the quantity demanded equals the quantity supplied. At this point, there is no inherent pressure for the price to adjust because the forces of supply and demand are balanced. In dynamic analysis, identifying the equilibrium is the first step, as it allows one to study how the system behaves when perturbed from this state and whether it will return or move further away.

Transcript

00:01 So for this question, it says to 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.

00:10 So both p and q are function of the time.

00:12 If one considers price and quantity as two interacting species, the following model might be proposed, as you see on the screen.

00:19 For a, b, c, and f are positive constants.

00:22 So first, it says to justify and discuss the adequacy of this model.

00:27 So as you can see for fixed price as q increases, p prime is going to get smaller dpdp and possibly becomes negative.

00:37 So this implies that as the quantity supplied increases, the price will not rise as fast.

00:44 So if q gets high enough, the price will decrease.

00:47 So we see on this graph that it's going to be doing something like that.

01:00 So again, if q gets high enough, then the price will decrease.

01:08 On the other hand, for dqdt for a fixed quantity, as p the price increases, the q prime is going to get larger, as you see in that equation.

01:21 So as the market price increases, the quantity supplied will increase at a faster rate.

01:26 So if p is too small, it dq to t would be negative, and the quantity supplied will decrease.

01:33 And so we have something that looks like this.

01:40 And so this observation is the traditional explanation of the effect of market price levels on the quantity supplied.

01:51 So now looking at part a, we are told that a is equal to 1, b is equal 20 ,000, c is equal to 1, f is equal to 30, and we want to find the equilibrium points of the system, and then if possible, classify each with respect to its stability, and if it cannot be classified, then to give some explanation.

02:16 So we're told that a is equal to 1, b is equal to 20 ,000, c is equal to 1, and f is equal to 30.

02:27 So that means my new equations are that p prime is p times 20 ,000 over q minus p.

02:37 P q prime is equal to q times 30 p minus q which is equal to just that and then we want to find the equilibrium points so we're setting both of these equal to zero so when i do that that means that this um inside the parentheses here is going to equal to zero so in other words 20 000 is equal to pq so add the p the other side multiply by q and then here if this is equal to zero then that implies that 30 p minus q is equal to zero so 30 p is equal to q and so then to find our equilibrium points looking to see at what values of p and q this is true one obvious one is zero is zero if both p and q are zero the whole entire thing is zero that works but there's also one more so i'm going to go ahead and use some of my system of equations tricks here if i plug in 30p for q into this equation i get 20 ,000 is equal to 30p squared and then when i saw for p there i get that p is equal to 25 .8 and then when i plug in 25 .8 for p here i get that q is equal to 775.

04:31 So those are my two equilibrium points.

04:34 And so now i want to see the behavior around it at the derivative.

04:43 So i see that p prime is greater than zero when.

04:50 So inside here that's really, this is 20 ,000 minus pq is equal zero.

04:58 So i want pq to be less than 20 ,000 to make that positive.

05:08 And p is great than zero because i do have this p out here.

05:15 And then i see that p prime is less than zero otherwise.

05:24 And looking at q, q prime is greater than zero when p is greater than q over 30.

05:36 So again, i just divide that 30 to the other side.

05:39 So if p is greater than that, then this is positive.

05:44 Q prime is up, and q is greater than zero...