Question

6. Consider the following supply and demand curves for bat teries: where PB is the cost of batteries and Pc is the cost of copper. a) Determine the equilibrium price and quantity for bat- teries (PR and QR) in terms of the variable Poc b) Using comparative statics, discuss how the equilib- rium Pß and QB changes as Pc increases and de creases c) Determine the price elasticity of supply and price elas- ticity of demand as a function of PB a d) Let Pe 2, for what values of PB is the price elastic- itv of demand elastic, unitarv elastic, and inelastic! e) Determine the cross-price elasticity of demand for bat- teries with respect to the price for copper. є08P what can we say about copper and batteries (demand substitutes or complements)?

Answer 1

a) The supply and demand equations are given as follows:

Q^S = 10 +8P_B +7P_C ------(i) &

Q^D =100-4P_B -2P_C ..........(ii)

Under market equilibrium

Q^S = Q^D

or, 10 +8P_B +7P_C = 100-4P_B -2P_C

or 12P_B +9P_C =90

or, 4P_B = 30-3P_C

or P_B = 3/4(10-P_C )

Putting the value of P_B in equation (i) or (ii) we get,

Q_B= Q^S =Q^D = 10+8.3/4(10-P_C ) + 7P_C = 70+P_C

b) From part a)

equilibrium price (P_B ) = 3/4(10-P_C )

& equilibrium quantity (Q_B ) = 70 +P_C

Here when value of P_C increases, the equilibrium price (P_B ) decreases and equilibrium quantity (Q_B ) increases.

But when value of P_C decreases, the equilibrium price (P_B ) increases and equilibrium quantity (Q_B ) decreases.

c) Price elasticity of demand (e_D ) = dQ^D /dP_B * P_B /Q^D

                                             = -4*P_B / (100-4P_B -2P_C )

                                         = 4P_B /(4P_B +2P_C -100)

Price elasticity of supply (e_s ) = dQ^S /dP_B *P_B /Q^S

                                         = 8P_B /(10+8P_B +7P_C )

d) If P_C = 2 then demand and supply equations is

Q^D = 100-4P_B -2*2= 96-4P_B &

Q^S = 10 +8P_B +7*2= 24+8P_B

Therefore, elasticity of demand (e_D ) = dQ^D /dP_C * P_B /Q^D = -2P_C /(96-4P_B ) = -4/(96-4P_B )

for unitary elastic demand, e_D = 1

or, -4/(96-4P_B ) = 1 or, 4P_B = 100 and P_B =25

For inelastic demand, e_D = 0 or, 96-4P_B =0 or, P_B =24

For elastic demand, e_D >0

Putting P_B =26 we get

e_D = -4/(96-4*26) = 4/8=1/2>0

e) Cross elasticity of demand for batteries with respect to price for copper is

e_CD = dQ^D /dP_C *P_C /Q^D = -2P_C /(100-4P_B -2P_C )