Answer to Question #188398 in Economics of Enterprise for faryal

Question #188398

Given the demand function. P=16๐‘’โˆ’0.02๐‘„

โ€ข Determine the quantity and price at which total revenue will be maximized.

โ€ข test the second-order condition.


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

1.

"P=16e^{-0.02Q}"


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


"\\frac{\\delta TR}{\\delta Q}=16e^{-0.02Q} +16Q(-0.02)e^{-0.02Q}=0"


"16e^{-0.02Q}[1-0.02Q]=0"

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


so, "1-0.02Q=0"


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


"P=16e^{-0.02(50)}"


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



(b)

"\\frac{\\delta ^2TR}{\\delta Q^2}=16e^{-0.02Q}(-0.02)+(-0.032)e^{-0.02Q}"


"=-0.32Qe^{-0.02}(0.02)"


<0


=negative


second order is also satisfied.


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