Answer to Question #188398 in Economics of Enterprise for faryal

Question #188398

Given the demand function. P=16π‘’βˆ’0.02𝑄

β€’ Determine the quantity and price at which total revenue will be maximized.

β€’ test the second-order condition.


1
Expert's answer
2021-05-05T13:37:16-0400

1.

P=16eβˆ’0.02QP=16e^{-0.02Q}


Total revenue=TR=PQ=16eβˆ’0.02Q.Q=TR=PQ=16e^{-0.02Q}.Q


Ξ΄TRΞ΄Q=16eβˆ’0.02Q+16Q(βˆ’0.02)eβˆ’0.02Q=0\frac{\delta TR}{\delta Q}=16e^{-0.02Q} +16Q(-0.02)e^{-0.02Q}=0


16eβˆ’0.02Q[1βˆ’0.02Q]=016e^{-0.02Q}[1-0.02Q]=0

[eβˆ’0.02Q[e^{-0.02Q } is always positive]


so, 1βˆ’0.02Q=01-0.02Q=0


Q=10.02=50Q=\frac{1}{0.02}=50


P=16eβˆ’0.02(50)P=16e^{-0.02(50)}


p=16e=162.72=5.88p=\frac{16}{e}=\frac{16}{2.72}=5.88



(b)

Ξ΄2TRΞ΄Q2=16eβˆ’0.02Q(βˆ’0.02)+(βˆ’0.032)eβˆ’0.02Q\frac{\delta ^2TR}{\delta Q^2}=16e^{-0.02Q}(-0.02)+(-0.032)e^{-0.02Q}


=βˆ’0.32Qeβˆ’0.02(0.02)=-0.32Qe^{-0.02}(0.02)


<0


=negative


second order is also satisfied.


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