I am struggling to solve this problem. I formulated my profit equation involving total revenues from both products and then subtracting the cost function from this, yet I do not get correct answers. I do not see how I am getting this wrong: A two-product firm faces the following demand and cost functions:
1 = 40 − 21 − 2
2 = 35 − 1 − 2
C = 12 + 222 + 10
a) Find the output levels that satisfy the first-order condition to maximize the profit (40 marks)
My method has been taking the inverse demand functions for both goods: P=... then multiplying by Q and combining for total revenue. Profit=TR-TC, and taking first partial derivatives to find the maximising Q and P inputs. Yet I get woeful answers. Any help would be appreciated :)
- Workings:
Take inverse demand equation, P1=..., multiply by relevant Q, combine for TR. Subtract C from TR for profit equation.
Total Revenue = P1Q1+P2Q2
Profit(Q1, Q2, P1, P2) = TR-C = 20Q1-1.5Q12-0.5P2Q1+35Q2-P1Q2-3Q22-10
Taking the first partial derivative w.r.t Q1 & Q2 and setting equal to 0, yields Q2=2.35
Solution:
After some thought I used the demand functions in terms of P and I substituted Q1 & Q2 equations into the cost function and got the correct answers. This left a profit equation in terms of P1 and P2 only. I still do not see how my previous method failed to work though.
It appears you've solved this on your own, but as to your question of how your prior method failed: the profit equation you defined as a function of both Q and P before taking the partials w/r/t Q likely failed as you ignored the fact that P and Q are functions of one another (P varies with Q as defined by the first two equations written). You captured this dynamic in your final approach by substituting in the Q1 and Q2 equations, but if you wished to take the partials w/r/t Q1 and Q2 as before, you would've needed to solve for P1 and P2 in terms of Q1 and Q2 from the first two equations, and substitute in for the P terms.