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§ 5.3 Applications of the Natural Logarithm Function to Economics

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Section Outline Relative Rates of Change Elasticity of Demand

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**Relative Rate of Change**

Definition Example Relative Rate of Change: The quantity on either side of the equation is 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.

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**Relative Rate of Change**

EXAMPLE (Percentage Rate of Change) Suppose that the price of wheat per bushel at time t (in months) is approximated by What is the percentage rate of change of f (t) at t = 0? t = 1? t = 2? SOLUTION Since we see that

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**Relative Rate of Change**

CONTINUED So 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 %.

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Elasticity of Demand

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Elasticity of Demand EXAMPLE (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).

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Elasticity of Demand CONTINUED Now 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.

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Elasticity of Demand CONTINUED Through 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.

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