To find the output and price at equilibrium, we need to set the cost function (TC) equal to the demand function (P), as at equilibrium, the amount that producers are willing to supply at a given price is equal to the amount that consumers are willing to buy.
Given:
Cost function (TC) = 100 + 12Q
Demand function (P) = 200 - 8Q
Setting TC equal to P to find the equilibrium:
100 + 12Q = 200 - 8Q
Now, let's solve for Q to find the equilibrium quantity (Q_equilibrium):
1. Add 8Q to both sides of the equation to get all Q terms on one side:
100 + 12Q + 8Q = 200 - 8Q + 8Q
100 + 20Q = 200
2. Subtract 100 from both sides to isolate the Q terms:
100 + 20Q - 100 = 200 - 100
20Q = 100
3. Divide by 20 to solve for Q:
20Q / 20 = 100 / 20
Q = 5
We have found the equilibrium quantity (Q_equilibrium) to be 5. Next, we find the price at equilibrium (P_equilibrium) by substituting Q into the demand function:
P = 200 - 8Q
P = 200 - 8(5)
P = 200 - 40
P = 160
Therefore, the output at equilibrium (Q_equilibrium) is 5 units, and the price at equilibrium (P_equilibrium) is 160.currency units.
