Question

The demand for Widgets (QX) is a function of the price of widgets (PX), the price of woozles (PY), and per capita income (I): QX = 1950 - 10 PX + 5 PY - 0.1I Currently, PX = 25, PY = 10, and I = 15,000. (a) Calculate the elasticity of demand for widgets with respect to its own price, the price of woozles, and income. (b) Over what range of prices is the demand for widgets elastic? (c) If the cost per widget is 10 and the manufacturer behaves as a monopolist, how many widgets will be sold and at what price. (d) By how much must the price of widgets change if there is a 1% decrease in per capita income and the goal is to keep QX constant. (e) What happens to the elasticity of demand for widgets if the price of woozles doubles?

Answer 1

QX = 1950 - 10PX + 5PY - 0.1I

Plugging in given values,

QX = 1950 - 10PX + 5 x 10 - 0.1 x 15,000 = 1950 - 10PX + 50 - 1,500 = 500 - 10PX

When PX = 25, QX = 500 - 10 x 25 = 500 - 250 = 250

(a)

(i) Own Price Elasticity of demand (Ed) = ($\dpi{80} \partial$QX/$\dpi{80} \partial$PX) x (PX/QX) = - 10 x (25/250) = - 1

(ii) Cross price Elasticity of demand (Ey) = ($\dpi{80} \partial$QX/$\dpi{80} \partial$PY) x (PY/QX) = 5 x (10/250) = 0.2

(iii) Income Elasticity of demand (Ei) = ($\dpi{80} \partial$QX/$\dpi{80} \partial$I) x (I/QX) = - 0.1 x (15,000/250) = - 6

(b)

When PX = 25, |Ed| = 1, so demand is unit elastic. When Demand is elastic, price will be higher.

When QX = 0, PX = 500/10 = 50 (vertical intercept of demand curve). So,

Demand is elastic when 50 >= PX > 25.

(c)

QX = 500 - 10PX

10PX = 500 - QX

PX = 50 - 0.1QX

Total revenue (TR) = PX.QX = 50QX - 0.1QX²

Marginal revenue (MR) = dTR/dQX = 50 - 0.2QX

Monopolist will equate MR and MC:

50 - 0.2QX = 10

0.2QX = 40

QX = 200

PX = 50 - 0.1 x 200 = 50 - 20 = 30

(d)

When income decreases by 1%, new value of I = 15,000 x 0.99 = 14,850

QX = 1950 - 10PX + 5 x 10 - 0.1 x 14,850 = 1950 - 10PX + 50 - 1,485 = 515 - 10PX

Since QX = 250:

515 - 10PX = 250

10PX = 265

PX = 26.5

NOTE: As HOMEWORKLIB Answering Policy, 1st 4 parts have been answered.