Question

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Currently, the demand equation for pet hamsters is Q = 40 – 3P. The current price is $10 per pet hamster. Is this the best price to charge in order to maximize revenues?

Solve for the best price to charge in order to maximize revenues. Show any steps or processes used to reach the answer above. Explain your process as though you are teaching the concept to a student who is a beginner in economics.

Answer 1

(Here we have assumed there is no cost involved in selling of hamster as nothing is mentioned in question.)

Given that the demand equation for pet hamsters is: Q = 40 - 3P

Total Revenue(TR) is Price times quantity

TR = Price * Quantity = P*Q = P * (40 - 3P) = 40P - 3P²

Now we need to find the price at which TR=40P - 3P² is maximum

First Order Condition:
In order to find maximum or minimum we need to differentiate TR with respect to P and the equate the equation to 0 and find the corresponding P
d(TR)/d(P) = 0
d(40P - 3P²)/dP = 40 - 6P = 0
Therefore P = 40/6 = 6.67

Second Order Condition:
In order to check whether the P which we found out in the previous step corresponds to maxima of TR and not minima of TR we will use second order condition
If second derivative of TR with respect to P is less than 0 at P found in First Order Condition(P=6.67), then P=6.67 corresponds to maxima of TR
d²(TR)/dP² = d²(40P - 3P²)/dP² = -6
-6 > 0 at P=6.67. Therefore P=6.67 corresponds to maximum to total revenue

Therefore Total Revenue is maximum when price charged is $6.67

It can be confirmed that $10 is not the best price to sell hamsters:
TR_P=10 = 40(10) - 3(10*10) = 400 - 300 = $100
TR_P=6.67 = 40*(6.67) - 3(6.67*6.67) = $133.33
Therefore more Revenue can be made is Price charged is $6.67. So, $10 is not the best price.