Give step-by-step solution with explanation and final answer:
Novo Nordisk has a patent on a new drug, which gives it a monopoly on selling that drug. Novo Nordisk has two factories available that it

Question

Give step-by-step solution with explanation and final answer:
Novo Nordisk has a patent on a new drug, which gives it a monopoly on selling that drug. Novo Nordisk has two factories available that it can use to manufacture the new drug. Both facilities are already constructed and ready to start producing, and neither facility can be used for any other purpose. Novo Nordisk's total output of the drug is the sum of the amount it produces in each factory: Q = Q1 + Q2. The marginal cost of producing the drug in factory one is MC1(Q1) and the marginal cost of producing the drug in factory two is MC2(Q2). Show the optimal values of Q, Q1, and Q2 both graphically and mathematically.

please graph it for me

VIPSolutions · Accepted Answer

# Monopoly with Two Factories: Optimal Output Allocation

Novo Nordisk is a monopolist with two factories. Let:

- $ Q_1 $: Output from Factory 1  
- $ Q_2 $: Output from Factory 2  
- $ Q = Q_1 + Q_2 $: Total output

Let:
- $ MC_1(Q_1) $: Marginal cost in Factory 1  
- $ MC_2(Q_2) $: Marginal cost in Factory 2  
- $ P(Q) $: Inverse demand function (price as a function of total output)

---

## 1. Firm's Profit Maximization Problem

**Maximize profit:**
$$
\max_{Q_1, Q_2} \ \pi = P(Q) \cdot Q - C_1(Q_1) - C_2(Q_2)
$$
where $ C_1 $ and $ C_2 $ are the total cost functions, and $ Q = Q_1 + Q_2 $.

---

## 2. First-Order Conditions

Let’s write the Lagrangian, introducing $ Q = Q_1 + Q_2 $:

$$
\mathcal{L} = P(Q) Q - C_1(Q_1) - C_2(Q_2) + \lambda (Q - Q_1 - Q_2)
$$

Take partial derivatives and set to zero:
- With respect to $ Q_1 $:

$$
  \frac{\partial \mathcal{L}}{\partial Q_1} = -C_1'(Q_1) - \lambda =  \implies C_1'(Q_1) = -\lambda
  $$
- With respect to $ Q_2 $:

$$
  \frac{\partial \mathcal{L}}{\partial Q_2} = -C_2'(Q_2) - \lambda =  \implies C_2'(Q_2) = -\lambda
  $$
- With respect to $ Q $:

$$
  \frac{\partial \mathcal{L}}{\partial Q} = P(Q) + P'(Q) Q + \lambda = 
  \implies P(Q) + Q P'(Q) = -\lambda
  $$

Now, substitute:
$$
C_1'(Q_1) = C_2'(Q_2) = -\lambda = P(Q) + QP'(Q) = MR
$$

**Summary:**
- Marginal revenue (MR) equals marginal cost in each factory:
  $$
  MR = MC_1(Q_1) = MC_2(Q_2)
  $$
- And $ Q = Q_1 + Q_2 $

---

## 3. Step-by-Step Solution

**Step 1: Set MR equal to each MC**
$$
MR = MC_1(Q_1)
$$
$$
MR = MC_2(Q_2)
$$

**Step 2: Use the constraint**
$$
Q = Q_1 + Q_2
$$

**Step 3: Solve system for $ Q_1^*, Q_2^*, Q^* $**
- Find $ Q_1 $ and $ Q_2 $ such that their marginal costs are equal and both equal to the marginal revenue at the total output.

---

## 4. Graphical Representation

Below is a **schematic** of the graphical solution (since I can't generate images, I'll describe and show an ASCII representation).

### Graph 1: Marginal Revenue and Marginal Cost Curves

**Axes:**
- x-axis: Total output $ Q $
- y-axis: Marginal cost / marginal revenue

**Curves:**
- MR: Downward sloping marginal revenue curve from demand
- $ MC_1 $: Upward sloping marginal cost for Factory 1
- $ MC_2 $: Upward sloping marginal cost for Factory 2
- $ MC_{aggr} $: Combined "horizontal sum" marginal cost curve

**Graphical Steps:**
1. **Find $ Q^* $:** Where MR intersects the *horizontal sum* of the two MC curves (the *aggregate MC*).
2. **Allocate $ Q^* $ between factories:** At $ Q^* $, find the split $ Q_1^*, Q_2^* $ such that $ MC_1(Q_1^*) = MC_2(Q_2^*) = MR(Q^*) $.

```
Marginal Cost / Revenue
│
│         /
│        / (MC_2)
│---\   /
│   \_/    (MC_aggr)
│    /
│   /    (MC_1)
│--/------------------
│
│
│                     (MR)
│-------------------------------
           Q^*
```

- The intersection of MR and aggregate MC gives $ Q^* $.
- Drop a vertical line down to find $ Q^* $.

### Allocation

On a separate graph or table:

- At the equilibrium MR, mark on each MC curve the corresponding $ Q_1^* $ and $ Q_2^* $.
- The sum is $ Q^* $.

---

## 5. **Final Answers (General Form)**

$$
\boxed{
\begin{align*}
MR(Q^*) &= MC_1(Q_1^*) = MC_2(Q_2^*) \
Q^* &= Q_1^* + Q_2^*
\end{align*}
}
$$

Where:
- $ MR(Q) = \frac{d}{dQ}[P(Q) Q] $
- Solve for $ Q^* $ where $ MR(Q^*) = MC_{aggr}(Q^*) $
- Allocate $ Q^* $ across factories so that $ MC_1(Q_1^*) = MC_2(Q_2^*) $

---

## 6. **Summary Table**

| Step            | Equation                                 | Explanation                          |
|-----------------|------------------------------------------|--------------------------------------|
| 1. MR=MCs       | $ MR = MC_1(Q_1^*) = MC_2(Q_2^*) $    | Equalize MR and each factory's MC    |
| 2. Add outputs  | $ Q^* = Q_1^* + Q_2^* $               | Total output is sum from both        |

---

## 7. **Conclusion**

- **Mathematically**: Set MR equal to each MC, allocate output so MCs are equal, sum to get total output.
- **Graphically**: Find intersection of MR and aggregate MC for total output, then allocate across factories by setting MCs equal.

---

**If you have specific forms for MC or demand, I can show a worked numerical example and draw a more precise graph.**

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