Question 6.40: (a) Given the demand function P = 60 − Q², calculate the coe......
(a) Given the demand function P = 60 − Q², calculate the coefficient for point elasticity of demand at P = 44.
(b) Given the demand function Q = 45e^{−0.04P} , calculate the coefficient for point elasticity of demand at P = 10.
(a) P = 60 − Q²
Step 1: Find dQ/dP. Since the demand function is written as P = 60 − Q², find dP/dQ.
\frac {dP}{dQ} = −2Q, then \frac {dQ}{dP} = \frac {1}{\frac {dP}{dQ}} = \frac {1}{−2Q}
Step 2: Derive an expression for ε_d in terms of P
ε_d = \frac {dQ}{dP}.\frac {P}{Q} = -\frac {1}{2Q} \frac {P}{Q} = -\frac {P}{2Q²}= -\frac {P}{2(60 − P)}
Step 3: Evaluate ε_d at P = 44:
ε_d = -\frac {P}{2(60 − P)} = -\frac {44}{2(60 − 44)}= −1.375
Since ε_d < −1, demand is elastic.
(b) Q = 45e^{−0.04P}
Step 1: Find dQ/dP. Since the demand function is written as Q = 45e^{−0.04P},
\frac {dQ}{dP} = 45(−0.04)e^{−0.04P }= −1.8e^{−0.04P}Step 2: Derive an expression for ε_d in terms of P
ε_d = \frac {dQ}{dP}. \frac {P}{Q} = \frac {−1.8e^{−0.04P}}{1}.\frac {P}{45e^{−0.04P }}ε_d = −0.04P
Step 3: Evaluate ε_d at P = 10:
This one is easy!
ε_d = −0.04P = −0.04(10) = −0.4
Since ε_d > −1, demand is inelastic.