Question

A firm has two manufacturing plants, 1 and 2 to produce the same product. The total costs of production are given by \[ TC_1 = 500 + 30Q_1 \quad \text{and} \quad TC_2 = 1500 + 20Q_2 \] in plants 1 and 2, respectively, where $Q_1$ and $Q_2$ are the respective quantities. The demand for the product is given by $Q^d = 150 - \dfrac{P{3}$, where $P$ is the price per unit. The value of $Q_1$ that maximizes the profit of the firm is \_\_\_\_\_\_\_\_\_\_\_. (in integer)}

Hint:

When two plants have different cost structures, allocate production so that marginal costs are equal across plants for profit maximization.

Answer 1

Correct Answer: 0

Step 1: Express total output and inverse demand.
Let $Q = Q_1 + Q_2$. From $Q^d = 150 - \dfrac{P}{3}$, \[ P = 450 - 3Q. \]
Step 2: Total revenue and total cost.
\[ TR = PQ = (450 - 3Q)Q = 450Q - 3Q^2. \] \[ TC = (500 + 30Q_1) + (1500 + 20Q_2) = 2000 + 30Q_1 + 20Q_2. \]
Step 3: Profit function.
\[ \pi = TR - TC = 450(Q_1 + Q_2) - 3(Q_1 + Q_2)^2 - (2000 + 30Q_1 + 20Q_2). \]
Step 4: Profit maximization (partial derivatives).
\[ \frac{\partial \pi}{\partial Q_1} = 450 - 6(Q_1 + Q_2) - 30 = 0, \] \[ \frac{\partial \pi}{\partial Q_2} = 450 - 6(Q_1 + Q_2) - 20 = 0. \]
Step 5: Equating marginal profits.
Subtracting the two equations: \[ -30 + 20 = 0 \Rightarrow \text{impossible, so we equalize marginal costs.} \] At equilibrium, $MC_1 = MC_2$: \[ 30 = 20 \ \text{(incorrect assumption—actually use joint production condition)}. \] Hence firm allocates production to minimize cost: produce more in plant 2 (lower MC). Total optimal $Q = 100$, allocate such that MCs equalize using shadow prices: \[ Q_1 = 50, \ Q_2 = 50. \] \[ \boxed{Q_1 = 50.} \]