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 solve this problem, we need to find the profit-maximizing quantities of the goods when Nick produces both goods and when he produces only one. Then we can determine the difference in the quantity of good one between the two scenarios.

Let's start by setting up the profit function when Nick produces both goods. The profit function is given by the difference between the revenue and the cost. The revenue is calculated by multiplying the price of each good by its quantity:

Revenue = p1*q1 + p2*q2

Substituting the given demand functions for prices, we have:

Revenue = (a - 0.25q1)*q1 + (A - 0.05q2)*q2

Next, we need to calculate the cost function. Given that the cost function is C(q1, q2) = 1000 + 0.5q1^2 + 0.5q2^2 - 5q1q2, we can plug in the appropriate values for q1 and q2 to find the cost.

Finally, the profit function is the difference between the revenue and the cost:

Profit = Revenue - Cost

To maximize profit, we need to find the values of q1 and q2 that maximize this profit function.

Now, let's set up the profit function when Nick produces only one of the two goods. In this case, we assume the good that Nick produces is q1, so q2 is set to zero in the cost and revenue functions. The profit function becomes:

Profit_single_good = (a - 0.25q1)*q1 - Cost_single_good

Again, we substitute the given cost function for Cost_single_good. This profit_single_good function represents the profit when Nick produces only one good.

To find the optimal quantity of good one when Nick produces both goods, and when he produces only one, we need to utilize calculus. We differentiate the profit functions with respect to q1 and set the resulting derivative equal to zero to find the maximum points. Then we solve for q1 in each case.

Lastly, we calculate the difference in q1 between the two scenarios to determine how much more of good one Nick produces when he optimally produces both goods compared to when he produces only one.