II. Supply and Demand
If p is the price in dollars of a given commodity at time t, then we can think of price as a function of time. Similarly, the number of units demanded by consumers at any time, and the number of units supplied by producers at any time, may also be considered as functions of time as well as functions of price.
Both the quantity demanded and the quantity supplied depend not only on the price at the time, but also on the direction and rate of change that consumers and producers ascribe to prices. For example, even when prices are high, if consumers feel that prices are rising, the demand may rise. Similarly, if prices are low but producers feel they may go lower, the supply may rise.
If we assume that prices are determined in the marketplace by supply and demand, then the equilibrium price is the one we seek.
Suppose the supply and demand functions for a certain commodity in a competitive market are given, in hundreds of units, by
where denotes the rate of change of the price with respect to time. If, at , the market equilibrium price is $12, we can express the market equilibrium price as a function of time.
Our goals are as follows.
To express the market equilibrium price as a function of time.
To determine whether there is price stability in the marketplace for this item (that is, to determine whether the equilibrium price approaches a constant over time).
To achieve these goals, do the following.
Integrate both sides of this separated differential equation.
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