Given the following cost function
P1=210-5Q1
P2=125-1.25Q2
TC=4000+20Q where Q=Q1+Q2
With a clear solution compute Q and P
To find the values of Q and P, we need to equate the cost function (TC) with the sum of the individual cost functions for P1 and P2.
TC = P1(Q1) + P2(Q2)
4000 + 20Q = (210 - 5Q1)(Q1) + (125 - 1.25Q2)(Q2)
First, let's expand the individual cost functions:
4000 + 20Q = (210Q1 - 5Q1^2) + (125Q2 - 1.25Q2^2)
Simplifying further:
4000 + 20Q = 210Q1 - 5Q1^2 + 125Q2 - 1.25Q2^2
We also know that Q = Q1 + Q2, so we can substitute that into the equation:
4000 + 20(Q1 + Q2) = 210Q1 - 5Q1^2 + 125Q2 - 1.25Q2^2
Expanding further:
4000 + 20Q1 + 20Q2 = 210Q1 - 5Q1^2 + 125Q2 - 1.25Q2^2
Rearranging terms:
5Q1^2 + 1.25Q2^2 + 20Q1 - 125Q2 + 20Q2 - 210Q1 + 4000 - 4000 = 0
Simplifying:
5Q1^2 - 125Q2^2 + 20Q1 - 105Q2 = 0
This equation represents a system of two nonlinear equations. To find the values of Q1 and Q2, we can solve this system using numerical methods or software.
Once we find the values of Q1 and Q2, we can substitute them back into the cost function equations for P1 and P2 to find their respective prices (P).