Economists use demand functions to describe how much of a commodity can be sold at varying prices. For example, the demand function D(p) = 500 - 10p says that at a price of p = 10, a quantity of D(10) = 400 units of the commodity can be sold. The elasticity  of the demand gives the approximate percent change in the demand for every 1% change in the price.

a. Compute the elasticity of the demand function D(p) = 500 - 10p.

b. If the price is $12 and increases by 4.5%, what is the approximate percent change in the demand?

c. Show that for the linear demand function D(p) = a - bp, where a and b are positive real numbers, the elasticity is a decreasing function, for p ≥ 0 and p ≠ a/b.

d. Show that the demand function D(p) = a/p^b, where a and b are positive real numbers, has a constant elasticity for all positive prices.

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dD P dp D