Posted by Patrick on Saturday, September 22, 2007 at 5:25pm.
Suppose Nick produces to goods @ cost
C(q1,q2)=1000=.5q1^2+.5q2^2-5q1q2
If nick produces only one of he two goods his cost is given this cost function with the other good set = to 0.
The inverse demand functions for the two goods are given by
p1=a-.25q1 and p2=A-.05q2 a=17 A=10
How much more of good one does nick produce to maximize profits when he optimally produces both goods than when he produces only one?
Need help setting up and solving Please!!!
To find out how much more of good one Nick produces to maximize profits when he optimally produces both goods than when he produces only one, we need to first find the profit function.
The profit function can be calculated by subtracting the cost function from the revenue function.
The revenue function is given by multiplying the price of each good by the quantity of each good sold:
R(q1,q2) = p1q1 + p2q2
Substituting the given inverse demand functions, we get:
R(q1,q2) = (a - 0.25q1)q1 + (A - 0.05q2)q2
Now, let's find the profit function by subtracting the cost function from the revenue function:
π(q1,q2) = R(q1,q2) - C(q1,q2)
Substituting the given revenue function and cost function, we get:
π(q1,q2) = [(a - 0.25q1)q1 + (A - 0.05q2)q2] - [0.5q1^2 + 0.5q2^2 - 5q1q2]
Now that we have the profit function, we can find the optimal quantities of q1 and q2 that maximize the profits.
To do this, we need to find the partial derivatives of the profit function with respect to q1 and q2, and set them equal to zero:
∂π/∂q1 = 0
∂π/∂q2 = 0
By solving these equations, we can find the values of q1 and q2 that maximize the profits.
Once we have the optimal quantities, we can compare the difference between the quantity of good one produced when both goods are optimally produced and when only one good is produced.
This will give us the answer to how much more of good one Nick produces to maximize profits when he optimally produces both goods compared to when he produces only one.