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
2 Section OutlineRelative Rates of ChangeElasticity of Demand
3 Relative 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.
4 Relative 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
5 Relative 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 %.
7 Elasticity 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).
8 Elasticity 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.
9 Elasticity 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.