Profit Maximization for linear Profit function

[allalongthewatc]

Hello All,
Let’s say my profit function is of the form : P(Q) = R(Q) - C(Q); where R(Q) = 240Q and C(Q) = 110Q.
Therefore, P(Q) = 130Q.
Now, because this is a linear function, the profit maximization will be at the end point of the interval where the linear function is drawn. Hence, P(Q) = 130Q; where Q = 0 to infinity. I have two questions on this:
#1- How will I decide the economies of scale from this? ie. how will I calculate the quantity at which MR (Q*)=MC(Q*)?
#2- What if I don’t know the interval on which P(Q) is valid, ie. I don’t know Q* then what should be the optimal quantity? Does it mean that the more I produce, the more my profit will be?
I would appreciate any thoughts to clear up my confusion.
Best regards

 

[S2000magician]

allalongthewatchtower wrote:Hello All,
Let’s say my profit function is of the form : P(Q) = R(Q) - C(Q); where R(Q) = 240Q and C(Q) = 110Q.
Therefore, P(Q) = 130Q.
Now, because this is a linear function, the profit maximization will be at the end point of the interval where the linear function is drawn. Hence, P(Q) = 130Q; where Q = 0 to infinity. I have two questions on this:
#1- How will I decide the economies of scale from this? ie. how will I calculate the quantity at which MR (Q*)=MC(Q*)?

They’re never equal: MR = 240, MC = 110.

allalongthewatchtower wrote:#2- What if I don’t know the interval on which P(Q) is valid, ie. I don’t know Q* then what should be the optimal quantity? Does it mean that the more I produce, the more my profit will be?

Given these functions, the more you produce, the more profit you make; there is no maximum profit.

 

[allalongthewatc]

S2000magician wrote:

allalongthewatchtower wrote:Hello All,
Let’s say my profit function is of the form : P(Q) = R(Q) - C(Q); where R(Q) = 240Q and C(Q) = 110Q.
Therefore, P(Q) = 130Q.
Now, because this is a linear function, the profit maximization will be at the end point of the interval where the linear function is drawn. Hence, P(Q) = 130Q; where Q = 0 to infinity. I have two questions on this:
#1- How will I decide the economies of scale from this? ie. how will I calculate the quantity at which MR (Q*)=MC(Q*)?

They’re never equal: MR = 240, MC = 110.

allalongthewatchtower wrote:#2- What if I don’t know the interval on which P(Q) is valid, ie. I don’t know Q* then what should be the optimal quantity? Does it mean that the more I produce, the more my profit will be?

Given these functions, the more you produce, the more profit you make; there is no maximum profit.

Thank you for your help, S2000magician! You are right. When I took the derivative, I was a bit confused— how is that the profit maximization equation doesn’t hold for this question. Do you think that profit maximization equation holds only for non-linear functions? Is that a good inference? I don’t have any facts to back-up my conclusion, so I thought of asking you this follow-up question.
I would appreciate your thoughts.
Thanks

 

[S2000magician]

allalongthewatchtower wrote:

S2000magician wrote:

allalongthewatchtower wrote:Hello All,
Let’s say my profit function is of the form : P(Q) = R(Q) - C(Q); where R(Q) = 240Q and C(Q) = 110Q.
Therefore, P(Q) = 130Q.
Now, because this is a linear function, the profit maximization will be at the end point of the interval where the linear function is drawn. Hence, P(Q) = 130Q; where Q = 0 to infinity. I have two questions on this:
#1- How will I decide the economies of scale from this? ie. how will I calculate the quantity at which MR (Q*)=MC(Q*)?

They’re never equal: MR = 240, MC = 110.

allalongthewatchtower wrote:#2- What if I don’t know the interval on which P(Q) is valid, ie. I don’t know Q* then what should be the optimal quantity? Does it mean that the more I produce, the more my profit will be?

Given these functions, the more you produce, the more profit you make; there is no maximum profit.

Thank you for your help, S2000magician! You are right. When I took the derivative, I was a bit confused— how is that the profit maximization equation doesn’t hold for this question. Do you think that profit maximization equation holds only for non-linear functions? Is that a good inference?

Yes: to have a quantity Q* that maximizes profit (so that your profit is lower when the quantity is more or less than Q*), either the cost function or the revenue function (or both) must be nonlinear.

 

[allalongthewatc]

Thanks S2000magician!

 

[S2000magician]

My pleasure.