Question

Consider the following supply and demand curves for batteries: Q S B = 10 + 8PB + 7PC Q D B = 100 − 4PB − 2PC where PB is the cost of batteries and PC is the cost of copper.

a) Determine the equilibrium price and quantity for batteries (P ∗ B and Q∗ B) in terms of the variable PC .

b) Using comparative statics, discuss how the equilibrium P ∗ B and Q∗ B changes as PC increases and decreases.

c) Determine the price elasticity of supply and price elasticity of demand as a function of PB and PC .

d) Let PC = 2, for what values of PB is the price elasticity of demand elastic, unitary elastic, and inelastic?

e) Determine the cross-price elasticity of demand for batteries with respect to the price for copper, εQD B ,PC , what can we say about copper and batteries (demand substitutes or complements)?

Answer 1

(a) In equilibrium, QDB = QSB

100 - 4PB - 2PC = 10 + 8PB + 7PC

12PB = 90 - 9PC

PB = (90 - 9PC)/12

PB = (30 - 3PC)/4

QB = 100 - [4 x (30 - 3PC)/4] - 2PC = 100 - (30 - 3PC) - 2PC = 100 - 30 + 3PC - 2PC = 70 + PC

(b) From above equilibrium solution,

PB = (30 - 3PC)/4 = 7.5 - 0.75PC

dPB/dPC = -0.75

This means that as PC increases (increases) by 1 unit, equilibrium price decreases (increases) by 0.75 unit.

QB = 70 + PC

dQB/dPC = 1

This means that as PC increases (increases) by 1 unit, equilibrium quantity increases (decreases) by 1 unit.

(c)

Elasticity of demand (Ed) = ($\dpi{80} \partial$QDB/$\dpi{80} \partial$PB) x (PB/QDB) = -4 x [PB / (100 - 4PB - 2PC)] = -4PB / (100 - 4PB - 2PC)

Elasticity of supply (Es) = ($\dpi{80} \partial$QSB/$\dpi{80} \partial$PB) x (PB/QSB) = 8 x [PB / (10 + 8PB + 7PC)] = 8PB / (10 + 8PB + 7PC)

(d)

When PC = 2,

Ed = -4PB / [100 - 4PB - (2 x 2)] = -4PB / (100 - 4PB - 4) = -4PB / (96 - 4PB) = -PB / (24 - PB) [Dividing by 4]

Demand is elastic (inelastic) if absolute value of elasticity is higher (lower) than 1. Demand is unit elastic if absolute value of elasticity equals 1.

(A) If demand is elastic,

PB / (24 - PB) > 1

PB > (24 - PB)

2PB > 24

PB > 12

(B) If demand is inelastic,

PB / (24 - PB) < 1

PB < (24 - PB)

2PB < 24

PB < 12

(C) If demand is unit elastic,

PB / (24 - PB) = 1

PB = (24 - PB)

2PB = 24

PB = 12

(e)

Cross price elasticity = ($\dpi{80} \partial$QDB/$\dpi{80} \partial$PC) x (PC/QDB) = -2 x [PC / (100 - 4PB - 2PC)] = -2PC / (100 - 4PB - 2PC)

Since PB and PC are non-negative, cross-price elasticity is zero, which means that the goods are complements.