Question

Let's assume that both the demand and supply are linear functions with respect to price. More explicitly, let D(p)=4−p and S(p)=1+p. Find a general solution for the price as a function of time (it will involve k ). What is the long term behaviour of the price? Does it tend to a specific value regardless of the initial price? If the demand for a product is greater than the supply, then the price will go up. If the supply is greater than the demand then the price will go down. Said succinctly D(p)−S(p)>0⇒dtdp>0D(p)−S(p)<0⇒dtdp<0 More precisely, we might assume the rate of change of the price (with respect to t ) will be directly proportional to D(p)−S(p). Thus we arrive at the governing differential equation: dtdp=k(D(p)−S(p)),k>0. Throughout this project we assume that the price function, p(t) satisfies this differential equation. Now let's assume that supply is linear, but demand is quadratic with respect to price. This would mean demand drops more steeply as the price increases. More explicitly assume that k=3,D(p)=100−p2, and S(p)=100+2p. Assuming the initial price is p=$15, give an explicit formula for price as a function of time. What will happen to the price in the long run? Any solution must be fully justified using techniques outlined in class. Hint: How did we found a find a general solution to the logistic equation?

Answer 1

The price as a function of time can be found by solving a differential equation with given demand and supply functions. The long-term behavior of the price depends on the derivative of the equation, but an explicit formula is needed to determine the exact behavior.

Step-by-step explanation:

The price as a function of time, represented by p(t), can be found by solving the differential equation dtp = k(D(p) - S(p)), where k is a constant. Substituting the given values D(p) = 100 - p^2, S(p) = 100 + 2p, and the initial price p = $15, we can find an explicit formula for p(t). The long-term behavior of the price depends on the sign of the derivative dtp/dp. If it is positive, the price will increase over time; if it is negative, the price will decrease over time. However, in this case, the explicit formula for p(t) is needed to determine the exact behavior of the price over time.