Answer the following elasticity-related questions. Each part is an independent problem.

(a) Show that any linear inverse supply that passes through the origin (i.e., an inverse supply with the functional form p = c Q with c > 0) has a price elasticity of supply equal to one. Show that any linear inverse supply curve with a positive intercept (i.e., having the functional form p = k + c Q with c, k > 0) must be elastic.

(b) Revnol, a manufacturer of cosmetics, prices its popular pink lipstick at $8. On the basis of test-marketing, Revnol believes that women between the ages of 18 and 20 have an own-price elasticity of –1.0 and that 60 percent of them are likely to purchase the product. In the age group from 21 to 25 years, the own-price elasticity is –1.2 and 50 percent of them are likely to buy.

(i) In a market with 25,000 women aged 18 to 20, and 15,000 aged 21 to 25, how many lipsticks can the firm expect to sell at a price of $8 per unit?

(ii) If Revnol were to cut prices by 10 percent, approximately how many more pink lipsticks would it expect to sell? (Hint:use the elasticity)

(c) On a certain product market, 500 units are demanded at a price of $15. The own-price elasticity at that point is –1.5. What is the equation of a linear demand that passes through the point (Q= 500, p=15)?

(d) What is the equation of a constant-elasticity demand function that has an own-price elasticity of –2 and passes through the point (Q= 500, p=10)?