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

Let, cost function (TC) = 100+12Q^2 and demand function (P) = 200 8Q . Find output and… – Economics NEB 2081 Solution

Equilibrium where Demand (P = 200 − 8Q) intersects MC (MC = 24Q) at Q* = 5, P* = 160.

Solution to Find Equilibrium Output and Price

Step 1: Understand the Given Functions

Total Cost (TC) is given by: $T C = 100 + 12 Q^{2}$ where $Q$ is the quantity produced.
Demand Function (Price, P) is given by: $P = 200 - 8 Q$

Step 2: Find the Marginal Cost (MC)

The Marginal Cost (MC) is the derivative of the Total Cost (TC) with respect to $Q$ : $M C = \frac{d ( T C )}{d Q} = \frac{d ( 100 + 12 Q ^{2} )}{d Q} = 24 Q$

Marginal Cost curve (MC = 24Q) derived from TC = 100 + 12Q².

Step 3: Find the Marginal Revenue (MR)

The Marginal Revenue (MR) is derived from the Demand Function (P). Since $T R = P \times Q = (200 - 8 Q) \times Q = 200 Q - 8 Q^{2}$ , $M R = \frac{d ( T R )}{d Q} = 200 - 16 Q$

Step 4: Find the Equilibrium Condition (MC = MR)

At equilibrium, Marginal Cost (MC) equals Marginal Revenue (MR): $24 Q = 200 - 16 Q$ $24 Q + 16 Q = 200$ $40 Q = 200$ $Q = \frac{200}{40} = 5$

Equilibrium condition: MR = MC at Q* = 5 (MR = 120).

Step 5: Find the Equilibrium Price (P)

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

Final Answer

Equilibrium Output (Q) = 5 units
Equilibrium Price (P) = Rs. 160