Let, cost function (TC) = 100+12Q^2 and demand function (P) = 200 8Q . Find output and price at equilibrium line. [5]

Solution to Find Equilibrium Output and Price

Step 1: Understand the Given Functions

• Total Cost (TC) Function: $TC=100+12Q^2$ This represents the total cost of producing $Q$ units of output.

• Demand (Price) Function: $P=200-8Q$ This represents the price ($P$) that consumers are willing to pay for $Q$ units.

Step 2: Find the Marginal Cost (MC) Function

To determine the equilibrium, we need the Marginal Cost (MC), which is the derivative of the Total Cost (TC) function.

$$MC=\frac{d(TC)}{dQ}=\frac{d(100+12Q^2)}{dQ}=24Q$$

Step 3: Find the Marginal Revenue (MR) Function

Since the demand function is linear, the Marginal Revenue (MR) is the derivative of the Total Revenue (TR) function.

First, express Total Revenue (TR) as: $$TR=P\times Q=(200-8Q)\times Q=200Q-8Q^2$$

Now, find Marginal Revenue (MR): $$MR=\frac{d(TR)}{dQ}=\frac{d(200Q-8Q^2)}{dQ}=200-16Q$$

Step 4: Find the Equilibrium Condition

At equilibrium, Marginal Cost (MC) = Marginal Revenue (MR): $$24Q=200-16Q$$

Solve for $Q$: $$24Q+16Q=200$$ $$40Q=200$$ $$Q=\frac{200}{40}=5$$

So, the equilibrium output (Q) is 5 units.

Step 5: Find the Equilibrium Price (P)

Substitute $Q=5$ into the demand function: $$P=200-8(5)=200-40=160$$

Thus, the equilibrium price (P) is 160.

Verification (Optional but Good Practice)

Check if MC = MR at $Q=5$: $$MC=24(5)=120$$ $$MR=200-16(5)=200-80=120$$ Since $MC=MR$, the equilibrium condition is satisfied.

---

Final Answer

• Equilibrium Output (Q): 5 units
• Equilibrium Price (P): 160