Presentation on theme: "§ 5.3 Applications of the Natural Logarithm Function to Economics."— Presentation transcript:
1§ 5.3Applications of the Natural Logarithm Function to Economics
2Section OutlineRelative Rates of ChangeElasticity of Demand
3Relative Rate of Change DefinitionExampleRelative Rate of Change: The quantity on either side of the equationis often called the relative rate of change of f (t) per unit change of t (a way of comparing rates of change for two different situations).An example will be given immediately hereafter.
4Relative Rate of Change EXAMPLE(Percentage Rate of Change) Suppose that the price of wheat per bushel at time t (in months) is approximated byWhat is the percentage rate of change of f (t) at t = 0? t = 1? t = 2?SOLUTIONSincewe see that
5Relative Rate of Change CONTINUEDSo at t = 0 months, the price of wheat per bushel contracts at a relative rate of 0.22% per month; 1 month later, the price of wheat per bushel is still contracting, but more so, at a relative rate of 0.65%. One month after that (t = 2), the price of wheat per bushel is contracting, but much less so, at a relative rate of %.
7Elasticity of DemandEXAMPLE(Elasticity of Demand) A subway charges 65 cents per person and has 10,000 riders each day. The demand function for the subway is(a) Is demand elastic or inelastic at p = 65?(b) Should the price of a ride be raised or lowered in order to increase the amount of money taken in by the subway?SOLUTION(a) We must first determine E(p).
8Elasticity of DemandCONTINUEDNow we will determine for what value of p E(p) = 1.Set E(p) = 1.Multiply by 180 – 2p.Add 2p to both sides.Divide both sides by 3.So, p = 60 is the point at which E(p) changes from elastic to inelastic, or visa versa.
9Elasticity of DemandCONTINUEDThrough simple inspection, which we could have done in the first place, we can determine whether the value of the function E(p) is greater than 1 (elastic) or less than 1 (inelastic) at p = 65.So, demand is elastic at p = 65.(b) Since demand is elastic when p = 65, this means that for revenue to increase, price should decrease.