I was trying to derive for A from the function: profit=PQ(P,A)-C(Q(A))-A
these are my steps
profit=Pdq/da-Cqdq/da-1
dprofit/dA=P(dq/da)-MC(dq/da)-1=0
(p-MC)change in q/change in A=1
p-mc/p[A/Q *change in q/change in A]=A
A=TR Ea/Ed
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this is what is is suppose to equal after deriving i just wanted to know if my steps were correct, thanks!
To derive the function for A from the given profit function, let's go through your steps to see if they are correct:
Step 1: profit = P*Q(P,A) - C(Q(A)) - A
It seems like you're assuming that P, Q, and C are functions of A. However, you haven't explicitly stated this in your steps, so I'll proceed with this assumption.
Step 2: differentiate the profit function with respect to A
Taking the partial derivative of profit with respect to A, using the chain rule:
dprofit/dA = P * (dQ/dA) - (dC/dQ) * (dQ/dA) - 1
Step 3: Set the derivative equal to zero to find the critical points
Setting dprofit/dA = 0:
0 = P * (dQ/dA) - (dC/dQ) * (dQ/dA) - 1
Step 4: Solve for (dQ/dA)
Rearranging the equation:
(dC/dQ) * (dQ/dA) = P * (dQ/dA) - 1
(P * (dQ/dA) - (dC/dQ) * (dQ/dA)) = 1
((P - (dC/dQ)) * (dQ/dA)) = 1
(dQ/dA) = 1 / (P - (dC/dQ))
Step 5: Solve for A
Now that you have (dQ/dA), you can solve for A using the following equation:
A = (dTR/dA) * (Ea/Ed)
It looks like you have correctly derived the equation for A, which is A = (dTR/dA) * (Ea/Ed). This equation represents the change in total revenue with respect to A, divided by the elasticity of demand.
So, overall, your steps seem to be correct!